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of End Chapter Solutions Corporate Finance: Core Principles and Applications 3rd edition Ross, Westerfield, Jaffe, and Jordan Updated 09-28-2010 Prepared by Joe Smolira Belmont University CHAPTER 1 INTRODUCTION TO CORPORATE FINANCE Answers to Concept Questions 1. The three basic forms are sole proprietorships, partnerships, and corporations. Some disadvantages of sole proprietorships and partnerships are: unlimited liability, limited life, difficulty in transferring ownership, and hard to raise capital funds. Some advantages are: simpler, less regulation, the owners are also the managers, and sometimes personal tax rates are better than corporate tax rates. The primary disadvantage of the corporate form is the double taxation to shareholders on distributed earnings and dividends. Some advantages include: limited liability, ease of transferability, ability to raise capital, and unlimited life. When a business is started, most take the form of a sole proprietorship or partnership because of the relative simplicity of starting these forms of businesses. To maximize the current market value (share price) of the equity of the firm (whether its publicly traded or not). In the corporate form of ownership, the shareholders are the owners of the firm. The shareholders elect the directors of the corporation, who in turn appoint the firms management. This separation of ownership from control in the corporate form of organization is what causes agency problems to exist. Management may act in its own or someone elses best interests, rather than those of the shareholders. If such events occur, they may contradict the goal of maximizing the share price of the equity of the firm. Such organizations frequently pursue social or political missions, so many different goals are conceivable. One goal that is often cited is revenue minimization; i.e., provide whatever goods and services are offered at the lowest possible cost to society. A better approach might be to observe that even a not-for-profit business has equity. Thus, one answer is that the appropriate goal is to maximize the value of the equity. Presumably, the current stock value reflects the risk, timing, and magnitude of all future cash flows, both short-term and long-term. If this is correct, then the statement is false. An argument can be made either way. At the one extreme, we could argue that in a market economy, all of these things are priced. There is thus an optimal level of, for example, unethical and/or illegal behavior, and the framework of stock valuation explicitly includes these. At the other extreme, we could argue that these are non-economic phenomena and are best handled through the political process. A classic (and highly relevant) thought question that illustrates this debate goes something like this: A firm has estimated that the cost of improving the safety of one of its products is $30 million. However, the firm believes that improving the safety of the product will only save $20 million in product liability claims. What should the firm do? The goal will be the same, but the best course of action toward that goal may be different because of differing social, political, and economic institutions. 2. 3. 4. 5. 6. 7. CHAPTER 1 B - 3 8. The goal of management should be to maximize the share price for the current shareholders. If management believes that it can improve the profitability of the firm so that the share price will exceed $35, then they should fight the offer from the outside company. If management believes that this bidder or other unidentified bidders will actually pay more than $35 per share to acquire the company, then they should still fight the offer. However, if the current management cannot increase the value of the firm beyond the bid price, and no other higher bids come in, then management is not acting in the interests of the shareholders by fighting the offer. Since current managers often lose their jobs when the corporation is acquired, poorly monitored managers have an incentive to fight corporate takeovers in situations such as this. We would expect agency problems to be less severe in other countries, primarily due to the relatively small percentage of individual ownership. Fewer individual owners should reduce the number of diverse opinions concerning corporate goals. The high percentage of institutional ownership might lead to a higher degree of agreement between owners and managers on decisions concerning risky projects. In addition, institutions may be better able to implement effective monitoring mechanisms on managers than can individual owners, based on the institutions deeper resources and experiences with their own management. The increase in institutional ownership of stock in the United States and the growing activism of these large shareholder groups may lead to a reduction in agency problems for U.S. corporations and a more efficient market for corporate control. 9. 10. How much is too much? Who is worth more, Ray Irani or Tiger Woods? The simplest answer is that there is a market for executives just as there is for all types of labor. Executive compensation is the price that clears the market. The same is true for athletes and performers. Having said that, one aspect of executive compensation deserves comment. A primary reason that executive compensation has grown so dramatically is that companies have increasingly moved to stock-based compensation. Such movement is obviously consistent with the attempt to better align stockholder and management interests. When stock prices soar, management cleans up. It is sometimes argued that much of this reward is simply due to rising stock prices in general, not managerial performance. Perhaps in the future, executive compensation will be designed to reward only differential performance, i.e., stock price increases in excess of general market increases. CHAPTER 2 FINANCIAL STATEMENTS AND CASH FLOW Answers to Concept Questions 1. Liquidity measures how quickly and easily an asset can be converted to cash without significant loss in value. Its desirable for firms to have high liquidity so that they have a large factor of safety in meeting short-term creditor demands. However, since liquidity also has an opportunity cost associated with it - namely that higher returns can generally be found by investing the cash into productive assets - low liquidity levels are also desirable to the firm. Its up to the firms financial management staff to find a reasonable compromise between these opposing needs The recognition and matching principles in financial accounting call for revenues, and the costs associated with producing those revenues, to be booked when the revenue process is essentially complete, not necessarily when the cash is collected or bills are paid. Note that this way is not necessarily correct; its the way accountants have chosen to do it. The bottom line number shows the change in the cash balance on the balance sheet. As such, it is not a useful number for analyzing a company. The major difference is the treatment of interest expense. The accounting statement of cash flows treats interest as an operating cash flow, while the financial cash flows treat interest as a financing cash flow. The logic of the accounting statement of cash flows is that since interest appears on the income statement, which shows the operations for the period, it is an operating cash flow. In reality, interest is a financing expense, which results from the companys choice of debt/equity. We will have more to say about this in a later chapter. When comparing the two cash flow statements, the financial statement of cash flows is a more appropriate measure of the companys operating performance because of its treatment of interest. Market values can never be negative. Imagine a share of stock selling for $20. This would mean that if you placed an order for 100 shares, you would get the stock along with a check for $2,000. How many shares do you want to buy? More generally, because of corporate and individual bankruptcy laws, net worth for a person or a corporation cannot be negative, implying that liabilities cannot exceed assets in market value. For a successful company that is rapidly expanding, for example, capital outlays will be large, possibly leading to negative cash flow from assets. In general, what matters is whether the money is spent productively, not whether cash flow from assets is positive or negative. Its probably not a good sign for an established company, but it would be fairly ordinary for a startup, so it depends. 2. 3. 4. 5. 6. 7. CHAPTER 2 B - 5 8. For example, if a company were to become more efficient in inventory management, the amount of inventory needed would decline. The same might be true if it becomes better at collecting its receivables. In general, anything that leads to a decline in ending NWC relative to beginning would have this effect. Negative net capital spending would mean more long-lived assets were liquidated than purchased. If a company raises more money from selling stock than it pays in dividends in a particular period, its cash flow to stockholders will be negative. If a company borrows more than it pays in interest and principal, its cash flow to creditors will be negative. 9. 10. The adjustments discussed were purely accounting changes; they had no cash flow or market value consequences unless the new accounting information caused stockholders to revalue the derivatives. Solutions to Questions and Problems NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. To find owners equity, we must construct a balance sheet as follows: CA NFA TA Balance Sheet CL LTD OE $36,400 TL & OE $7,500 28,900 $5,900 18,700 ?? $36,400 We know that total liabilities and owners equity (TL & OE) must equal total assets of $36,400. We also know that TL & OE is equal to current liabilities plus long-term debt plus owners equity, so owners equity is: OE = $11,800 NWC = $1,600 2. The income statement for the company is: Income Statement Sales Costs Depreciation EBIT Interest EBT Taxes (35%) Net income $753,000 308,000 46,000 $399,000 21,500 $377,500 132,125 $245,375 CHAPTER 2 B-6 One equation for net income is: Net income = Dividends + Addition to retained earnings Rearranging, we get: Addition to retained earnings = Net income Dividends Addition to retained earnings = $245,375 67,000 Addition to retained earnings = $178,375 3. To find the book value of current assets, we use the NWC equation, that is: NWC = CA CL Rearranging to solve for current assets, we get: CA = NWC + CL CA = $750,000 + 1,800,000 CA = $2,550,000 So, the book value balance sheet will be: Book Value Balance Sheet Current assets $2,550,000 Fixed assets 4,500,000 Total assets $7,050,000 The market value of current assets is given, so the market value balance sheet is: Market Value Balance Sheet Current assets $2,700,000 Fixed assets 5,200,000 Total assets $7,900,000 4. Taxes = 0.15($50,000) + 0.25($25,000) + 0.34($25,000) + 0.39($285,000 100,000) Taxes = $94,400 The average tax rate is the total tax paid divided by net income, so: Average tax rate = .3312 or 33.12% The marginal tax rate is the tax rate on the next $1 of earnings, so the marginal tax rate is 39 percent. CHAPTER 2 B - 7 5. To calculate OCF, we first need the income statement: Income Statement Sales Costs Depreciation expense EBIT Interest expense EBT Taxes (40%) Net income $25,300 9,100 1,700 $14,500 950 $13,550 5,420 $ 8,130 Using the equation for OCF, we get: OCF = $10,780 6. The net capital spending is the increase in fixed assets, plus depreciation, so: Net capital spending = $1,360,000 7. The long-term debt account will increase by $6 million, the amount of the new long-term debt issue. Since the company sold 8 million new shares of stock with a $1 par value, the common stock account will increase by $8 million. The capital surplus account will increase by $21 million, the value of the new stock sold above its par value. Since the company had a net income of $7 million, and paid $2.5 million in dividends, the addition to retained earnings was $4.5 million, which will increase the accumulated retained earnings account. So, the new long-term debt and stockholders equity portion of the balance sheet will be: Long-term debt Total long-term debt Shareholders equity Preferred stock Common stock ($1 par value) Capital surplus Accumulated retained earnings Total equity Total Liabilities & Equity $ 41,000,000 $ 41,000,000 $ 4,000,000 19,000,000 47,000,000 79,500,000 $ 149,500,000 $ 190,500,000 CHAPTER 2 B-8 8. The cash flow to creditors is the interest paid minus the change in long-term debt, so: Cash flow to creditors = $95,000 9. The cash flow to stockholders is the dividends paid minus any new equity purchased by shareholders, so: Cash flow to stockholders = $345,000 Note: APIS is the additional paid-in surplus. 10. We know that the cash flow from assets must be equal to the cash flow to creditors plus the cash flow to stockholders, so: Cash flow from assets = $250,000 Now, we can use the relationship between the cash flow from assets and the operating cash flow, change in net working capital, and capital spending to find the operating cash flow. Doing so, we find: Operating cash flow Intermediate 11. a. The accounting statement of cash flows explains the change in cash during the year. The accounting statement of cash flows will be: Statement of cash flows Operations Net income Depreciation Changes in other current assets Change in accounts payable Total cash flow from operations Investing activities Acquisition of fixed assets Total cash flow from investing activities Financing activities Proceeds of long-term debt Dividends Total cash flow from financing activities Change in cash (on balance sheet) $71 50 3 3 $127 = $475,000 $(138) $(138) $40 (22) $18 $7 CHAPTER 2 B - 9 b. The change in net working capital is the ending net working capital minus the beginning net working capital, so: Change in NWC = $1 c. To find the cash flow generated by the firms assets, we need the operating cash flow, and the capital spending. Since there are no interest payments, EBIT is the same as EBT. Calculating each of these, we find: Operating cash flow EBT Depreciation Taxes Operating cash flow $110 50 39 $121 Next, we will calculate the capital spending, which is: Capital spending Ending fixed assets Beginning fixed assets Depreciation Capital spending $408 320 50 $138 Now we can calculate the cash flow generated by the firms assets, which is: Cash flow from assets Operating cash flow Capital spending Change in NWC Cash flow from assets $121 (138) (1) $18 Notice that the accounting statement of cash flows shows a small positive cash flow, but the financial cash flows show a negative cash flow. The financial cash flow is a better number for analyzing the firms performance. 12. To construct the cash flow identity, we will begin cash flow from assets. Cash flow from assets is: OCF = $118,305 Next, we will calculate the change in net working capital which is: Change in NWC = $9,085 Now, we can calculate the capital spending. The capital spending is: Net capital spending = $116,461 CHAPTER 2 B-10 Cash flow from assets = $7,241 The company spent $7,241 on its assets. The cash flow from operations was $118,305, and the company spent $9,085 on net working capital and $116,461 in fixed assets. The cash flow to creditors is: Cash flow to creditors = $4,096 The cash flow to stockholders is a little trickier in this problem. First, we need to calculate the new equity sold. The equity balance increased during the year. The only way to increase the equity balance is to add addition to retained earnings or sell equity. To calculate the new equity sold, we can use the following equation: New equity = $20,937 What happened was the equity account increased by $55,770. Of this increase, $34,833 came from addition to retained earnings, so the remainder must have been the sale of new equity. Now we can calculate the cash flow to stockholders as: Cash flow to stockholders = $11,337 The company paid $4,096 to creditors and raised $11,337 from stockholders. Finally, the cash flow identity is: Cash flow from assets = Cash flow to creditors + Cash flow to stockholders $7,241 = $4,096 + $11,337 The cash flow identity balances, which is what we expect. 13. With the information provided, the cash flows from the firm are the capital spending and the change in net working capital, so: Cash flows from the firm Capital spending Additions to NWC Cash flows from the firm $(15,000) (2,100) $(17,100) And the cash flows to the investors of the firm are: Cash flows to investors of the firm Sale of long-term debt Sale of common stock Dividends paid Cash flows to investors of the firm (12,000) (3,000) 6,000 $(9,000) CHAPTER 2 B - 11 14. a. The interest expense for the company is the amount of debt times the interest rate on the debt. So, the income statement for the company is: Income Statement Sales Cost of goods sold Selling expenses Depreciation expense EBIT Interest expense EBT Taxes Net income $870,000 280,000 155,000 86,000 $349,000 39,000 $310,000 108,500 $201,500 b. And the operating cash flow is: OCF = $326,500 15. To find the OCF, we first calculate net income. Income Statement Sales Costs Other expenses Depreciation expense EBIT Interest expense EBT Taxes Net income Dividends Addition to retained earnings a. The operating cash flow was: OCF = $67,880 b. The cash flow to creditors is the interest paid minus any net new long-term debt, so: CFC = $17,900 Note that the net new long-term debt is negative because the company repaid part of its long-term debt. c. The cash flow to stockholders is the dividends paid minus and net new equity, or: $193,000 96,500 5,100 13,800 $77,600 10,400 $67,200 23,520 $43,680 $12,500 $31,180 CHAPTER 2 B-12 CFS = $6,500 d. We know that CFA = CFC + CFS, so: CFA = $17,900 + 6,500 = $24,400 Net capital spending = $41,800 Solving for the change in NWC gives $1,680, meaning the company increased its NWC by $1,680. 16. The solution to this question works the income statement backwards. Starting at the bottom: Net income = $6,900 EBT = $10,615 Now we can calculate: EBIT = $12,515 The last step is to use: Depreciation = $4,485 17. The balance sheet for the company looks like this: Cash Accounts receivable Inventory Current assets Tangible net fixed assets Intangible net fixed assets Total assets Common stock = $485,000 18. The market value of shareholders equity cannot be negative. A negative market value in this case would imply that the company would pay you to own the stock. The market value of shareholders equity can be stated as: Shareholders equity = Max [(TA TL), 0]. So, if TA is $15,100, equity is equal to $2,600, and if TA is $10,200, equity is equal to $0. We should note here that the book value of shareholders equity can be negative. 19. a. Taxes Growth = $17,490 Taxes Income = $2,924,000 Balance Sheet $175,000 Accounts payable 240,000 Notes payable 405,000 Current liabilities $820,000 Long-term debt Total liabilities 3,650,000 730,000 Common stock Accumulated ret. earnings $5,200,000 Total liab. & owners equity $435,000 160,000 $595,000 2,140,000 $2,735,000 ?? 1,980,000 $5,200,000 CHAPTER 2 B - 13 b. 20. a. Each firm has a marginal tax rate of 34% on the next $10,000 of taxable income, despite their different average tax rates, so both firms will pay an additional $3,400 in taxes. The income statement for the company is: Income Statement Sales Costs Administrative and selling expenses Depreciation expense EBIT Interest expense EBT Taxes Net income OCF = $95,000 $835,000 620,000 120,000 85,000 10,000 $68,000 $58,000 0 $58,000 b. c. Net income was negative because of the tax deductibility of depreciation and interest expense. However, the actual cash flow from operations was positive because depreciation is a non-cash expense and interest is a financing expense, not an operating expense. 21. A firm can still pay out dividends if net income is negative; it just has to be sure there is sufficient cash flow to make the dividend payments. Cash flow from assets = $95,000 0 0 = $95,000 Cash flow to stockholders = $45,000 0 = $45,000 Cash flow to creditors = $50,000 Net new LTD = $18,000 22. a. The income statement is: Income Statement Sales $25,700 Cost of good sold 18,400 Depreciation 3,450 EBIT $ 3,850 Interest 790 Taxable income $ 3,060 Taxes (40%) 1,224 Net income $ 1,836 b. c. OCF = $6,076 Change in NWC = $550 CHAPTER 2 B-14 Net capital spending = $7,820 CFA = $2,294 The cash flow from assets can be positive or negative, since it represents whether the firm raised funds or distributed funds on a net basis. In this problem, even though net income and OCF are positive, the firm invested heavily in both fixed assets and net working capital; it had to raise a net $2,294 in funds from its stockholders and creditors to make these investments. d. Cash flow to creditors = $790 Cash flow to stockholders = $3,084 We can also calculate the cash flow to stockholders as: Cash flow to stockholders = Dividends Net new equity Solving for net new equity, we get: Net new equity = $4,184 The firm had positive earnings in an accounting sense (NI > 0) and had positive cash flow from operations. The firm invested $550 in new net working capital and $7,820 in new fixed assets. The firm had to raise $2,294 from its stakeholders to support this new investment. It accomplished this by raising $4,184 in the form of new equity. After paying out $1,100 of this in the form of dividends to shareholders and $790 in the form of interest to creditors, $2,294 was left to meet the firms cash flow needs for investment. 23. a. Total assets 2009 = $4,340 Total liabilities 2009 = $2,330 Owners equity 2009 = $2,010 Total assets 2010 = $4,595 Total liabilities 2010 = $2,510 Owners equity 2010 = $2,085 b. NWC 2009 NWC 2010 Change in NWC = $410 = $435 = $25 c. We can calculate net capital spending as: Net capital spending = $1,130 So, the company had a net capital spending cash flow of $1,130. We also know that net capital spending is: Fixed assets sold Fixed assets sold = $1,900 1,130 = $770 CHAPTER 2 B - 15 To calculate the cash flow from assets, we must first calculate the operating cash flow. The operating cash flow is calculated as follows (you can also prepare a traditional income statement): EBIT = $5,290 EBT = $4,900 Taxes = $1,715 OCF = $4,505 Cash flow from assets = $3,350 d. Net new borrowing = $150 Debt retired = $290 Cash flow to creditors = $240 24. Cash Accounts receivable Inventory Current assets Net fixed assets Total assets Balance sheet as of Dec. 31, 2009 $13,695 Accounts payable 18,130 Notes payable 32,235 Current liabilities $64,060 Long-term debt $114,850 Owners' equity $178,910 Total liab. & equity Balance sheet as of Dec. 31, 2010 $14,010 Accounts payable 20,425 Notes payable 33,125 Current liabilities $67,560 Long-term debt $117,590 Owners' equity $185,150 Total liab. & equity $14,885 2,645 $17,530 $45,865 115,515 $178,910 Cash Accounts receivable Inventory Current assets Net fixed assets Total assets $13,950 2,485 $16,435 $53,510 115,205 $185,150 CHAPTER 2 B-16 2009 Income Statement Sales $26,115.00 COGS 8,985.00 Other expenses 2,130.00 Depreciation 3,750.00 EBIT $11,250.00 Interest 1,345.00 EBT $9,905.00 Taxes (35%) 3,466.75 Net income $6,438.25 Dividends Additions to RE 25. OCF = $12,450.25 Change in NWC = $4,595 Net capital spending = $6,495 Cash flow from assets = $1,360.25 Cash flow to creditors = $5,635 Net new equity = $3,490.25 Cash flow to stockholders = $6,995.25 Cash flow from assets = $1,360.25 Challenge $3,184.00 $3,254.25 2010 Income Statement Sales $28,030.00 COGS 10,200.00 Other expenses 1,780.00 Depreciation 3,755.00 EBIT $12,295.00 Interest 2,010.00 EBT $10,285.00 Taxes (35%) 3,599.75 Net income $6,685.25 Dividends Additions to RE $3,505.00 3,180.25 26. We will begin by calculating the operating cash flow. First, we need the EBIT, which can be calculated as: EBIT = $795 Now we can calculate the operating cash flow as: Operating cash flow Earnings before interest and taxes Depreciation Current taxes Operating cash flow $795 221 (231) $785 The cash flow from assets is found in the investing activities portion of the accounting statement of cash flows, so: CHAPTER 2 B - 17 Cash flow from assets Acquisition of fixed assets Sale of fixed assets Capital spending $415 (53) $362 The net working capital cash flows are all found in the operations cash flow section of the accounting statement of cash flows. However, instead of calculating the net working capital cash flows as the change in net working capital, we must calculate each item individually. Doing so, we find: Net working capital cash flow Cash Accounts receivable Inventories Accounts payable Accrued expenses Notes payable Other NWC cash flow $49 65 (51) (41) 21 (12) (5) $26 Except for the interest expense and notes payable, the cash flow to creditors is found in the financing activities of the accounting statement of cash flows. The interest expense from the income statement is given, so: Cash flow to creditors Interest Retirement of debt Debt service Proceeds from sale of long-term debt Total $120 240 $360 (131) $229 And we can find the cash flow to stockholders in the financing section of the accounting statement of cash flows. The cash flow to stockholders was: Cash flow to stockholders Dividends Repurchase of stock Cash to stockholders Proceeds from new stock issue Total $198 32 $230 (62) $168 27. Net capital spending = NFAend NFAbeg + Depreciation = (NFAend NFAbeg) + (Depreciation + ADbeg) ADbeg = (NFAend NFAbeg)+ ADend ADbeg CHAPTER 2 B-18 = (NFAend + ADend) (NFAbeg + ADbeg) = FAend FAbeg 28. a. b. The tax bubble causes average tax rates to catch up to marginal tax rates, thus eliminating the tax advantage of low marginal rates for high income corporations. Assuming a taxable income of $335,001, the taxes will be: Taxes = $113,900 Average tax rate = 0.34 or 34% The marginal tax rate on the next dollar of income is 34 percent. For corporate taxable income levels greater than $18,333,334, average tax rates are equal to marginal tax rates. Taxes = $6,416,667 Average tax rate = 0.35 or 35% The marginal tax rate on the next dollar of income is 35 percent. For corporate taxable income levels over $18,333,334, average tax rates are again equal to marginal tax rates. c. X = 0.4575 or 45.75% CHAPTER 3 FINANCIAL STATEMENTS ANALYSIS AND LONG-TERM PLANNING Answers to Concept Questions 1. Time trend analysis gives a picture of changes in the companys financial situation over time. Comparing a firm to itself over time allows the financial manager to evaluate whether some aspects of the firms operations, finances, or investment activities have changed. Peer group analysis involves comparing the financial ratios and operating performance of a particular firm to a set of peer group firms in the same industry or line of business. Comparing a firm to its peers allows the financial manager to evaluate whether some aspects of the firms operations, finances, or investment activities are out of line with the norm, thereby providing some guidance on appropriate actions to take to adjust these ratios if necessary. Both allow an investigation into what is different about a company from a financial perspective, but neither method gives an indication of whether the difference is positive or negative. For example, suppose a companys current ratio is increasing over time. It could mean that the company had been facing liquidity problems in the past and is rectifying those problems, or it could mean the company has become less efficient in managing its current accounts. Similar arguments could be made for a peer group comparison. A company with a current ratio lower than its peers could be more efficient at managing its current accounts, or it could be facing liquidity problems. Neither analysis method tells us whether a ratio is good or bad, both simply show that something is different, and tells us where to look. If a company is growing by opening new stores, then presumably total revenues would be rising. Comparing total sales at two different points in time might be misleading. Same-store sales control for this by only looking at revenues of stores open within a specific period. The reason is that, ultimately, sales are the driving force behind a business. A firms assets, employees, and, in fact, just about every aspect of its operations and financing exist to directly or indirectly support sales. Put differently, a firms future need for things like capital assets, employees, inventory, and financing are determined by its future sales level. Two assumptions of the sustainable growth formula are that the company does not want to sell new equity, and that financial policy is fixed. If the company raises outside equity, or increases its debtequity ratio, it can grow at a higher rate than the sustainable growth rate. Of course, the company could also grow faster than its profit margin increases, if it changes its dividend policy by increasing the retention ratio, or its total asset turnover increases. 2. 3. 4. CHAPTER 3 B-20 5. The sustainable growth rate is greater than 20 percent, because at a 20 percent growth rate the negative EFN indicates that there is excess financing still available. If the firm is 100 percent equity financed, then the sustainable and internal growth rates are equal and the internal growth rate would be greater than 20 percent. However, when the firm has some debt, the internal growth rate is always less than the sustainable growth rate, so it is ambiguous whether the internal growth rate would be greater than or less than 20 percent. If the retention ratio is increased, the firm will have more internal funding sources available, and it will have to take on more debt to keep the debt/equity ratio constant, so the EFN will decline. Conversely, if the retention ratio is decreased, the EFN will rise. If the retention rate is zero, both the internal and sustainable growth rates are zero, and the EFN will rise to the change in total assets. 6. Common-size financial statements provide the financial manager with a ratio analysis of the company. The common-size income statement can show, for example, that cost of goods sold as a percentage of sales is increasing. The common-size balance sheet can show a firms increasing reliance on debt as a form of financing. Common-size statements of cash flows are not calculated for a simple reason: There is no possible denominator. It would reduce the external funds needed. If the company is not operating at full capacity, it would be able to increase sales without a commensurate increase in fixed assets. Presumably not, but, of course, if the product had been much less popular, then a similar fate would have awaited due to lack of sales. Since customers did not pay until shipment, receivables rose. The firms NWC, but not its cash, increased. At the same time, costs were rising faster than cash revenues, so operating cash flow declined. The firms capital spending was also rising. Thus, all three components of cash flow from assets were negatively impacted. 7. 8. 9. 10. Financing possibly could have been arranged if the company had taken quick enough action. Sometimes it becomes apparent that help is needed only when it is too late, again emphasizing the need for planning. 11. All three were important, but the lack of cash or, more generally, financial resources ultimately spelled doom. An inadequate cash resource is usually cited as the most common cause of small business failure. 12. Demanding cash upfront, increasing prices, subcontracting production, and improving financial resources via new owners or new sources of credit are some of the options. When orders exceed capacity, price increases may be especially beneficial. 13. ROE is a better measure of the companys performance. ROE shows the percentage return for the year earned on shareholder investment. Since the goal of a company is to maximize shareholder wealth, this ratio shows the companys performance in achieving this goal over the period. 14. The EBITDA/Assets ratio shows the companys operating performance before interest, depreciation, and taxes. This ratio would show how a company has controlled costs. While taxes are a cost, and interest and depreciation can be considered costs, they are not as easily controlled by company management. Conversely, depreciation and amortization can be altered by accounting choices. This ratio only uses revenue minus costs that are directly related to operations in the numerator. As such, it gives a better metric to measure management performance over a period than does ROA. CHAPTER 3 B - 21 Solutions to Questions and Problems NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. Using the Du Pont identity, the ROE is: ROE = (PM)(TAT)(EM) ROE = (.064)(1.15)(2.50) ROE = .1840 or 18.40% 2. The equity multiplier is: EM = 1 + D/E EM = 1 + 0.75 EM = 1.75 One formula to calculate return on equity is: ROE = (ROA)(EM) ROE = .104(1.75) ROE = .1820 or 18.20% ROE can also be calculated as: ROE = NI / TE So, net income is: NI = ROE(TE) NI = (.1820)($900,000) NI = $163,800 3. This is a multi-step problem involving several ratios. The ratios given are all part of the Du Pont identity. The only Du Pont identity ratio not given is the profit margin. If we know the profit margin, we can find the net income since sales are given. So, we begin with the Du Pont identity: ROE = 0.15 = (PM)(TAT)(EM) = (PM)(S / TA)(1 + D/E) Solving the Du Pont identity for profit margin, we get: PM = [(ROE)(TA)] / [(1 + D/E)(S)] PM = [(0.15)($3,218)] / [(1 + 0.65)( $4,350)] PM = .0673 or 6.73% CHAPTER 3 B-22 Now that we have the profit margin, we can use this number and the given sales figure to solve for net income: PM = .0673 = NI / S NI = .0673($4,350) NI = $292.55 4. An increase of sales to $38,420 is an increase of: Sales increase = ($38,420 34,000) / $34,000 Sales increase = .13 or 13% Assuming costs and assets increase proportionally, the pro forma financial statements will look like this: Pro forma income statement Sales $38,420.00 Costs 29,154.00 EBIT 9,266.00 Taxes (34%) 3,150.44 Net income $ 6,115.56 Assets Total Pro forma balance sheet $ $ 113,339 Debt Equity 113,339 Total $ 26,500.00 78,080.89 $104,580.89 The payout ratio is constant, so the dividends paid this year is the payout ratio from last year times net income, or: Dividends = ($1,623.60 / $5,412)($6,115.56) Dividends = $1,834.67 The addition to retained earnings is: Addition to retained earnings = $6,115.56 1,834.67 Addition to retained earnings = $4,280.89 And the new equity balance is: Equity = $73,800 + 4,280.89 Equity = $78,080.89 So the EFN is: EFN = Total assets Total liabilities and equity EFN = $113,339 104,580.89 EFN = $8,758.11 5. The maximum percentage sales increase is the sustainable growth rate. To calculate the sustainable growth rate, we first need to calculate the ROE, which is: ROE = NI / TE ROE = $14,916 / $105,000 ROE = .1421 or 14.21% CHAPTER 3 B - 23 The plowback ratio, b, is one minus the payout ratio, so: b = 1 .30 b = .70 Now we can use the sustainable growth rate equation to get: Sustainable growth rate = (ROE b) / [1 (ROE b)] Sustainable growth rate = [.1421(.70)] / [1 .1421(.70)] Sustainable growth rate = .1104 or 11.04% So, the maximum dollar increase in sales is: Maximum increase in sales = $59,000(.1104) Maximum increase in sales = $6,514.79 6. We need to calculate the retention ratio to calculate the sustainable growth rate. The retention ratio is: b = 1 .25 b = .75 Now we can use the sustainable growth rate equation to get: Sustainable growth rate = (ROE b) / [1 (ROE b)] Sustainable growth rate = [.13(.75)] / [1 .13(.75)] Sustainable growth rate = .1080 or 10.80% 7. We must first calculate the ROE using the Du Pont ratio to calculate the sustainable growth rate. The ROE is: ROE = (PM)(TAT)(EM) ROE = (.065)(2.50)(1.10) ROE = .1788 or 17.88% The plowback ratio is one minus the dividend payout ratio, so: b = 1 .60 b = .40 Now, we can use the sustainable growth rate equation to get: Sustainable growth rate = (ROE b) / [1 (ROE b)] Sustainable growth rate = [.1788(.40)] / [1 .1788(.40)] Sustainable growth rate = .0770 or 7.70% 8. An increase of sales to $9,660 is an increase of: Sales increase = ($9,660 8,400) / $8,400 Sales increase = .15 or 15% CHAPTER 3 B-24 Assuming costs and assets increase proportionally, the pro forma financial statements will look like this: Pro forma income statement Sales Costs Net income $ 9,660.00 7,118.50 $ 2,541.50 Assets Total Pro forma balance sheet $24,725.00 $24,725.00 Debt Equity Total $ 4,200.00 19,841.50 $24.041.50 If no dividends are paid, the equity account will increase by the net income, so: Equity = $17,300 + 2,541.50 Equity = $19,841.50 So the EFN is: EFN = Total assets Total liabilities and equity EFN = $24,725 24,041.50 EFN = $683.50 9. a. First, we need to calculate the current sales and change in sales. The current sales are next years sales divided by one plus the growth rate, so: Current sales = Next years sales / (1 + g) Current sales = $317,400,000 / (1 + .15) Current sales = $276,000,000 And the change in sales is: Change in sales = $317,400,000 276,000,000 Change in sales = $41,400,000 We can now complete the current balance sheet. The current assets, fixed assets, and short-term debt are calculated as a percentage of current sales. The long-term debt and par value of stock are given. The plug variable is the additions to retained earnings. So: Assets Current assets Liabilities and equity Short-term debt Long-term debt Common stock Accumulated RE Total equity Total liabilities and equity $55,200,000 $41,400,000 $40,000,000 $20,000,000 202,200,000 $222,200,000 $303,600,000 Fixed assets 248,400,000 Total assets $303,600,000 CHAPTER 3 B - 25 b. We can use the equation from the text to answer this question. The assets/sales and debt/sales are the percentages given in the problem, so: Spontaneous debt Assets Sales Sales (PM Projected sales) (1 d) EFN = Sales Sales EFN = (.20 + .90) $41,400,000 (.15 $41,400,000) (.10 $317,400,000) (1 .40) EFN = $20,286,000 c. The current assets, fixed assets, and short-term debt will all increase at the same percentage as sales. The long-term debt and common stock will remain constant. The accumulated retained earnings will increase by the addition to retained earnings for the year. We can calculate the addition to retained earnings for the year as: Net income = Profit margin Sales Net income = .10($317,400,000) Net income = $31,740,000 The addition to retained earnings for the year will be the net income times one minus the dividend payout ratio, which is: Addition to retained earnings = Net income(1 d) Addition to retained earnings = $31,740,000(1 .40) Addition to retained earnings = $19,044,000 So, the new accumulated retained earnings will be: Accumulated retained earnings = $202,200,000 + 19,044,000 Accumulated retained earnings = $221,244,000 The pro forma balance sheet will be: Assets Current assets Liabilities and equity Short-term debt Long-term debt Common stock Accumulated RE Total equity Total liabilities and equity $63,480,000 $47,610,000 $40,000,000 $20,000,000 221,244,000 $241,244,000 $328,854,000 Fixed assets 285,660,000 Total assets The EFN is: $349,140,000 EFN = Total assets Total liabilities and equity EFN = $349,140,000 328,854,000 EFN = $20,286,000 CHAPTER 3 B-26 10. a. The sustainable growth rate is: Sustainable growth rate = (ROE b) / [1 (ROE b)] where: b = Retention ratio = 1 Payout ratio = .70 So: Sustainable growth rate = [.0845(.70)] / [1 .0845(.70)] Sustainable growth rate = .0629 or 6.29% b. It is possible for the sustainable growth rate and the actual growth rate to differ. If any of the actual parameters in the sustainable growth rate equation differs from those used to compute the sustainable growth rate, the actual growth rate will differ from the sustainable growth rate. Since the sustainable growth rate includes ROE in the calculation, this also implies that changes in the profit margin, total asset turnover, or equity multiplier will affect the sustainable growth rate. The company can increase its growth rate by doing any of the following: Intermediate 11. The solution requires substituting two ratios into a third ratio. Rearranging D/TA: Firm A D / TA = .35 (TA E) / TA = .35 (TA / TA) (E / TA) = .35 1 (E / TA) = .35 E / TA = .65 E = .65(TA) Rearranging ROA, we find: NI / TA = .10 NI = .10(TA) NI / TA = .12 NI = .12(TA) Firm B D / TA = .30 (TA E) / TA = .30 (TA / TA) (E / TA) = .30 1 (E / TA) = .30 E / TA = .70 E = .70(TA) Increase the debt-to-equity ratio by selling more debt or repurchasing stock Increase the profit margin, most likely by better controlling costs. Decrease its total assets/sales ratio; in other words, utilize its assets more efficiently. Reduce the dividend payout ratio. c. Since ROE = NI / E, we can substitute the above equations into the ROE formula, which yields: ROE = .10(TA) / .65(TA) = .1538 or 15.38% ROE = .12(TA) / .70(TA) = .1714 or 17.14% CHAPTER 3 B - 27 12. PM = NI / S = 18,351 / 163,184 = .1125 or 11.25% As long as both net income and sales are measured in the same currency, there is no problem; in fact, except for some market value ratios like EPS and BVPS, none of the financial ratios discussed in the text are measured in terms of currency. This is one reason why financial ratio analysis is widely used in international finance to compare the business operations of firms and/or divisions across national economic borders. The net income in dollars is: NI = PM Sales NI = 0.1125($261,070) NI = $29,358.86 13. a. The equation for external funds needed is: Spontaneous debt Assets Sales (PM Projected sales) (1 d) Sales EFN = Sales Sales where: Assets/Sales = $40,500,000/$37,000,000,000 = 1.0946 Sales = Current sales Sales growth rate = $37,000,000(.18) = $6,660,000 Spontaneous debt/Sales = $6,500,000/$37,000,000 = .1757 Profit margin = Net income/Sales = $5,265,000/$37,000,000 = .1423 Projected sales = Current sales (1 + Sales growth rate) = $37,000,000(1 + .18) = $43,660,000 d = Dividends/Net income = $1,579,500/$5,265,000 = .30 so: EFN = (1.0946 $6,660,000) (.1757 $6,660,000) (.1423 $43,660,000) (1 .30) EFN = $1,771,110 b. The current assets, fixed assets, and short-term debt will all increase at the same percentage as sales. The long-term debt and common stock will remain constant. The accumulated retained earnings will increase by the addition to retained earnings for the year. We can calculate the addition to retained earnings for the year as: Net income = Profit margin Sales Net income = .1423($43,660,000) Net income = $6,212,700 The addition to retained earnings for the year will be the net income times one minus the dividend payout ratio, which is: Addition to retained earnings = Net income(1 d) Addition to retained earnings = $6,212,700(1 .30) Addition to retained earnings = $4,348,890 CHAPTER 3 B-28 So, the new accumulated retained earnings will be: Accumulated retained earnings = $24,000,000 + 4,438,890 Accumulated retained earnings = $28,348,890 The pro forma balance sheet will be: Assets Current assets Liabilities and equity Short-term debt Long-term debt Common stock Accumulated RE Total equity Total liabilities and equity $12,390,000 $7,670,000 $7,000,000 $3,000,000 28,348,890 $31,348,890 $46,018,890 Fixed assets 35,400,000 Total assets The EFN is: $47,790,000 EFN = Total assets Total liabilities and equity EFN = $47,790,000 46,018,890 EFN = $1,771,110 c. The sustainable growth is: Sustainable growth rate = (ROE b) / [1 (ROE b)] where: ROE = Net income/Total equity = $5,265,000/$27,000,000 = .1950 b = Retention ratio = Retained earnings/Net income = $3,685,500/$5,265,000 = .70 So: Sustainable growth rate = [.1950(.70)] / [1 .1950(.70)] Sustainable growth rate = .1581 or 15.81% CHAPTER 3 B - 29 d. With a lower dividend, the company can meet its goals. With a zero dividend, EFN is: Assets Current assets Liabilities and equity Short-term debt Long-term debt Common stock Accumulated retained earnings Total equity Total liabilities and equity $12,390,000 $7,670,000 $7,000,000 $3,000,000 30,212,700 $33,212,700 $47,882,700 Fixed assets 35,400,000 Total assets The EFN is: $47,790,000 EFN = Total assets Total liabilities and equity EFN = $47,790,000 47,882,700 EFN = $92,700 If the dividend decrease is not sufficient, the company does have several other alternatives. It can increase its asset utilization and/or its profit margin. The company could also increase the debt in its capital structure. This will decrease the equity account, thereby increasing ROE. 14. This is a multi-step problem involving several ratios. It is often easier to look backward to determine where to start. We need receivables turnover to find days sales in receivables. To calculate receivables turnover, we need credit sales, and to find credit sales, we need total sales. Since we are given the profit margin and net income, we can use these to calculate total sales as: PM = 0.065 = NI / Sales 0.065 = $187,000 / Sales Sales = $2,876,923 Credit sales are 80 percent of total sales, so: Credit sales = $2,876,923(0.80) Credit sales = $2,301,538 Now we can find receivables turnover by: Receivables turnover = Credit sales / Accounts receivable Receivables turnover = $2,301,538 / $145,900 Receivables turnover = 15.77 times Days sales in receivables = 365 days / Receivables turnover Days sales in receivables = 365 / 15.77 Days sales in receivables = 23.14 days CHAPTER 3 B-30 15. The solution to this problem requires a number of steps. First, remember that current assets plus fixed assets equal total assets. So, if we find the current assets and the total assets, we can solve for net fixed assets. Using the numbers given for the current ratio and the current liabilities, we solve for current assets: Current ratio = Current assets / Current liabilities 1.25 = Current assets / $1,075 Current assets = $1,343.75 To find the total assets, we must first find the total debt and equity from the information given. So, we find the sales using the profit margin: Profit margin = Net income / Sales .0850 = Net income / $6,180 Net income = $525.30 We now use the sales figure as an input into ROE to find the total equity: ROE = Net income / Total equity .1625 = $525.30 / Total equity Total equity = $3.232.62 Next, we need to find the long-term debt. The long-term debt ratio is: Long-term debt ratio = 0.40 = LTD / (LTD + TE) Inverting both sides gives: 1 / 0.40 = (LTD + TE) / LTD = 1 + (TE / LTD) Substituting the total equity into the equation and solving for long-term debt gives the following: 1 + $3,232.62 / LTD = 2.5 LTD = $3,232.62 / 1.5 LTD = $2,155.08 Now, we can find the total debt of the company: Total debt = Current liabilities + long-term debt Total debt = $1,075 + 2,155.08 Total debt = $3,230.08 And, with the total debt, we can find the total debt and equity, which is equal to total assets: Total assets = Total debt + Total equity Total assets = $3,230.08 + 3,232.62 Total assets = $6,462.69 CHAPTER 3 B - 31 And finally, we are ready to solve the balance sheet identity as: Net fixed assets = Total assets Current assets Net fixed assets = $6,462.69 1,343.75 Net fixed assets = $5,118.94 16. This problem requires you to work backward through the income statement. First, recognize that: Net income = (1 tC)EBT Plugging in the numbers given and solving for EBT, we get: Net income = (1 tC)EBT $17,590 = (1 0.34)EBT EBT = $26,651.52 Now, we can add interest to EBT to get EBIT as follows: EBIT = EBT + Interest paid EBIT = $26,651.52 + 4,150 EBIT = $30,801.52 To get EBITD (earnings before interest, taxes, and depreciation), the numerator in the cash coverage ratio, add depreciation to EBIT: EBITD = EBIT + Depreciation EBITD = $30,801.52 + 5,820 EBITD = $36,621.52 Now, simply plug the numbers into the cash coverage ratio and calculate: Cash coverage ratio = EBITD / Interest Cash coverage ratio = $36,621.52 / $4,150 Cash coverage ratio = 8.82 times 17. The only ratio given which includes cost of goods sold is the inventory turnover ratio, so it is the last ratio used. Since current liabilities are given, we start with the current ratio: Current ratio = Current assets / Current liabilities 1.25 = CA / $325,000 Current assets = $406,250 Using the quick ratio, we solve for inventory: Quick ratio = (Current assets Inventory) / Current liabilities 0.85 = ($406,250 Inventory) / $325,000 Inventory = $406,250 (0.85 $325,000) Inventory = $130,000 CHAPTER 3 B-32 Now, we can solve for the cost of goods sold using the inventory turnover, which gives us: Inventory turnover = COGS / Inventory 9.50 = COGS / $130,000 COGS = $1,235,000 18. The common-size balance sheet answers are found by dividing each category by total assets. For example, the cash percentage for 2009 is: $13,582 / $346,628 = .0392 or 3.92% This means that cash is 3.92% of total assets. The common-base year answers are found by dividing each category value for 2010 by the same category value for 2009. For example, the cash common-base year number is found by: $15,675 / $13,582 = 1.1541 The common-size and common-base year balance sheets for the company are: Common size Assets 3.92% 6.24% 10.62% 20.78% 79.22% 100% Common size Common base year 2009 Current assets Cash Accounts receivable Inventory Total Fixed assets Net plant and equipment Total assets 2010 $13,582 21,640 36,823 72,045 274,583 $346,628 $15,675 22,340 39,703 $77,718 $290,586 $368,304 4.26% 6.07% 10.78% 21.10% 78.90% 100.00% 1.1541 1.0323 1.0782 1.0787 1.0583 1.0625 Liabilities and Owners' Equity Current liabilities Accounts payable Notes payable Total Long-term debt Owners' equity Common stock Accumulated RE Total Total L&E $19,085 24,530 $43,615 $35,000 $45,000 223,013 $268,013 $346,628 5.51% 7.08% 12.58% 10.10% 12.98% 64.34% 77.32% 100% $20,640 25,305 $45,945 $50,000 $45,000 227,359 $272,359 $368,304 5.60% 6.87% 12.47% 13.58% 12.22% 61.73% 73.95% 100.00% 1.0815 1.0316 1.0534 1.4286 1.0000 1.0195 1.0162 1.0625 CHAPTER 3 B - 33 19. To determine full capacity sales, we divide the current sales by the capacity the company is currently using, so: Full capacity sales = $725,000 / .92 Full capacity sales = $788,043 The maximum sales growth is the full capacity sales divided by the current sales, so: Maximum sales growth = ($788,043 / $725,000) 1 Maximum sales growth = .0870 or 8.70% 20. To find the new level of fixed assets, we need to find the full capacity ratio. Doing so, we find: Full capacity ratio = Fixed assets / Full capacity sales Full capacity ratio = $645,000 / $788,043 Full capacity ratio = 0.8185 Next, we calculate the total dollar amount of fixed assets needed at the new sales figure. Total fixed assets = 0.8185($850,000) Total fixed assets = $695,710 The new fixed assets necessary is the total fixed assets at the new sales figure minus the current level of fixed assets. New fixed assets = $695,710 645,000 New fixed assets = $50,710 21. Assuming costs vary with sales and a 20 percent increase in sales, the pro forma income statement will look like this: MOOSE TOURS INC. Pro Forma Income Statement Sales $ 1,114,800 Costs 867,600 Other expenses 22,800 EBIT $ 224,400 Interest 14,000 Taxable income $ 210,400 Taxes(35%) 73,640 Net income $ 136,760 The payout ratio is constant, so the dividends paid this year is the payout ratio from last year times net income, or: Dividends = ($33,735/$112,450)($136,760) Dividends = $41,028 CHAPTER 3 B-34 And the addition to retained earnings will be: Addition to retained earnings = $136,760 41,028 Addition to retained earnings = $95,732 The new retained earnings on the pro forma balance sheet will be: New retained earnings = $182,900 + 95,732 New retained earnings = $278,632 The pro forma balance sheet will look like this: MOOSE TOURS INC. Pro Forma Balance Sheet Assets Current assets Cash Accounts receivable Inventory Total Fixed assets Net plant and equipment $ 30,360 48,840 104,280 183,480 Liabilities and Owners Equity Current liabilities Accounts payable Notes payable Total Long-term debt Owners equity Common stock and paid-in surplus Retained earnings Total Total liabilities and owners equity $ $ 81,600 17,000 98,600 158,000 $ 495,600 $ $ $ 140,000 278,632 418,632 675,232 Total assets So the EFN is: $ 679,080 EFN = Total assets Total liabilities and equity EFN = $679,080 675,232 EFN = $3,848 22. First, we need to calculate full capacity sales, which is: Full capacity sales = $929,000 / .80 Full capacity sales = $1,161,250 The full capacity ratio at full capacity sales is: Full capacity ratio = Fixed assets / Full capacity sales Full capacity ratio = $413,000 / $1,161,250 Full capacity ratio = .35565 CHAPTER 3 B - 35 The fixed assets required at full capacity sales is the full capacity ratio times the projected sales level: Total fixed assets = .35565($1,114,800) = $396,480 So, EFN is: EFN = ($183,480 + 396,480) $675,232 = $95,272 Note that this solution assumes that fixed assets are decreased (sold) so the company has a 100 percent fixed asset utilization. If we assume fixed assets are not sold, the answer becomes: EFN = ($183,480 + 413,000) $675,232 = $78,752 23. The D/E ratio of the company is: D/E = ($85,000 + 158,000) / $322,900 D/E = .7526 So the new total debt amount will be: New total debt = .7526($418,632) New total debt = $315,044 This is the new total debt for the company. Given that our calculation for EFN is the amount that must be raised externally and does not increase spontaneously with sales, we need to subtract the spontaneous increase in accounts payable. The new level of accounts payable will be, which is the current accounts payable times the sales growth, or: Spontaneous increase in accounts payable = $68,000(.20) Spontaneous increase in accounts payable = $13,600 This means that $13,600 of the new total debt is not raised externally. So, the debt raised externally, which will be the EFN is: EFN = New total debt (Beginning LTD + Beginning notes payable + Spontaneous increase in AP) EFN = $315,044 ($158,000 + 68,000 + 17,000 + 13,600) = $58,444 CHAPTER 3 B-36 The pro forma balance sheet with the new long-term debt will be: MOOSE TOURS INC. Pro Forma Balance Sheet Assets Current assets Cash Accounts receivable Inventory Total Fixed assets Net plant and equipment $ 30,360 48,840 104,280 183,480 Liabilities and Owners Equity Current liabilities Accounts payable Notes payable Total Long-term debt Owners equity Common stock and paid-in surplus Retained earnings Total Total liabilities and owners equity $ $ 81,600 17,000 98,600 216,444 $ 495,600 $ $ $ 140,000 278,632 418,632 733,676 Total assets $ 679,080 The funds raised by the debt issue can be put into an excess cash account to make the balance sheet balance. The excess debt will be: Excess debt = $733,676 679,080 = $54,596 To make the balance sheet balance, the company will have to increase its assets. We will put this amount in an account called excess cash, which will give us the following balance sheet: CHAPTER 3 B - 37 MOOSE TOURS INC. Pro Forma Balance Sheet Assets Current assets Cash Excess cash Accounts receivable Inventory Total Fixed assets Net plant and equipment Liabilities and Owners Equity Current liabilities Accounts payable $ Notes payable Total Long-term debt Owners equity Common stock and paid-in surplus Retained earnings Total Total liabilities and owners equity $ $ 30,360 54,596 48,840 104,280 238,076 81,600 17,000 98,600 216,444 $ 495,600 $ $ $ 140,000 278,632 418,632 733,676 Total assets $ 733,676 The excess cash has an opportunity cost that we discussed earlier. Increasing fixed assets would also not be a good idea since the company already has enough fixed assets. A likely scenario would be the repurchase of debt and equity in its current capital structure weights. The companys debt/assets and equity/assets are: Debt/assets = .7526 / (1 + .7526) = .43 Equity/assets = 1 / (1 + .7526) = .57 So, the amount of debt and equity needed will be: Total debt needed = .43($679,080) = $291,600 Equity needed = .57($679,080) = $387,480 So, the repurchases of debt and equity will be: Debt repurchase = ($98,600 + 216,444) 291,600 = $23,444 Equity repurchase = $418,632 387,480 = $31,152 Assuming all of the debt repurchase is from long-term debt, and the equity repurchase is entirely from the retained earnings, the final pro forma balance sheet will be: CHAPTER 3 B-38 MOOSE TOURS INC. Pro Forma Balance Sheet Assets Current assets Cash Accounts receivable Inventory Total Fixed assets Net plant and equipment $ 30,360 48,840 104,280 183,480 Liabilities and Owners Equity Current liabilities Accounts payable Notes payable Total Long-term debt Owners equity Common stock and paid-in surplus Retained earnings Total Total liabilities and owners equity $ $ 81,600 17,000 98,600 193,000 $ 495,600 $ $ $ 140,000 247,480 387,480 679,080 Total assets Challenge $ 679,080 24. The pro forma income statements for all three growth rates will be: MOOSE TOURS INC. Pro Forma Income Statement 15 % Sales 20% Sales Growth Growth $1,068,350 $1,114,800 831,450 867,600 21,850 22,800 $215,050 $224,400 14,000 14,000 $201,050 $210,400 70,368 73,640 $130,683 $136,760 $39,205 91,478 $41,028 95,732 Sales Costs Other expenses EBIT Interest Taxable income Taxes (35%) Net income Dividends Add to RE 25% Sales Growth $1,161,250 903,750 23,750 $233,750 14,000 $219,750 76,913 $142,838 $42,851 99,986 We will calculate the EFN for the 15 percent growth rate first. Assuming the payout ratio is constant, the dividends paid will be: Dividends = ($33,735/$112,450)($130,683) Dividends = $39,205 And the addition to retained earnings will be: Addition to retained earnings = $130,683 39,205 CHAPTER 3 B - 39 Addition to retained earnings = $91,478 The new retained earnings on the pro forma balance sheet will be: New retained earnings = $182,900 + 91,478 New retained earnings = $274,378 The pro forma balance sheet will look like this: 15% Sales Growth: MOOSE TOURS INC. Pro Forma Balance Sheet Assets Current assets Cash Accounts receivable Inventory Total Fixed assets Net plant and equipment $ 29,095 46,805 99,935 175,835 Liabilities and Owners Equity Current liabilities Accounts payable Notes payable Total Long-term debt Owners equity Common stock and paid-in surplus Retained earnings Total Total liabilities and owners equity $ $ $ 78,200 17,000 95,200 158,000 $ 474,950 $ $ $ 140,000 274,378 414,378 667,578 Total assets So the EFN is: $ 650,785 EFN = Total assets Total liabilities and equity EFN = $650,785 667,578 EFN = $16,793 At a 20 percent growth rate, and assuming the payout ratio is constant, the dividends paid will be: Dividends = ($33,735/$112,450)($136,760) Dividends = $41,028 And the addition to retained earnings will be: Addition to retained earnings = $136,760 41,028 Addition to retained earnings = $95,732 The new retained earnings on the pro forma balance sheet will be: New retained earnings = $182,900 + 95,732 New retained earnings = $278,632 CHAPTER 3 B-40 The pro forma balance sheet will look like this: 20% Sales Growth: MOOSE TOURS INC. Pro Forma Balance Sheet Assets Current assets Cash Accounts receivable Inventory Total Fixed assets Net plant and equipment $ 30,360 48,840 104,280 183,480 Liabilities and Owners Equity Current liabilities Accounts payable Notes payable Total Long-term debt Owners equity Common stock and paid-in surplus Retained earnings Total Total liabilities and owners equity $ $ $ 81,600 17,000 98,600 158,000 $ 495,600 $ $ $ 140,000 278,632 418,632 675,232 Total assets So the EFN is: $ 679,080 EFN = Total assets Total liabilities and equity EFN = $679,080 675,232 EFN = $3,848 At a 25 percent growth rate, and assuming the payout ratio is constant, the dividends paid will be: Dividends = ($33,735/$112,450)($142,838) Dividends = $42,851 And the addition to retained earnings will be: Addition to retained earnings = $142,838 42,851 Addition to retained earnings = $99,986 The new retained earnings on the pro forma balance sheet will be: New retained earnings = $182,900 + 99,986 New retained earnings = $282,886 The pro forma balance sheet will look like this: CHAPTER 3 B - 41 25% Sales Growth: MOOSE TOURS INC. Pro Forma Balance Sheet Assets Current assets Cash Accounts receivable Inventory Total Fixed assets Net plant and equipment $ 31,625 50,875 108,625 191,125 Liabilities and Owners Equity Current liabilities Accounts payable Notes payable Total Long-term debt Owners equity Common stock and paid-in surplus Retained earnings Total Total liabilities and owners equity $ $ $ 85,000 17,000 102,000 158,000 $ 516,250 $ $ $ 140,000 282,886 422,886 682,886 Total assets So the EFN is: $ 707,375 EFN = Total assets Total liabilities and equity EFN = $707,375 682,886 EFN = $24,889 25. The pro forma income statements for all three growth rates will be: MOOSE TOURS INC. Pro Forma Income Statement 20% Sales 30% Sales Growth Growth $1,114,800 $1,207,700 867,600 939,900 22,800 24,700 $224,400 $243,100 14,000 14,000 $210,400 $229,100 73,640 80,185 $136,760 $148,915 $41,028 95,732 $44,675 104,241 Sales Costs Other expenses EBIT Interest Taxable income Taxes (35%) Net income Dividends Add to RE 35% Sales Growth $1,254,150 976,050 25,650 $252,450 14,000 $238,450 83,458 $154,993 $46,498 108,495 At a 30 percent growth rate, and assuming the payout ratio is constant, the dividends paid will be: Dividends = ($33,735/$112,450)($148,915) Dividends = $44,675 CHAPTER 3 B-42 And the addition to retained earnings will be: Addition to retained earnings = $148,915 44,675 Addition to retained earnings = $104,241 The new addition to retained earnings on the pro forma balance sheet will be: New addition to retained earnings = $182,900 + 104,241 New addition to retained earnings = $287,141 The new total debt will be: New total debt = .7556($427,141) New total debt = $321,447 So, the new long-term debt will be the new total debt minus the new short-term debt, or: New long-term debt = $321,447 105,400 New long-term debt = $58,047 The pro forma balance sheet will look like this: Sales growth rate = 30% and debt/equity ratio = .7526: MOOSE TOURS INC. Pro Forma Balance Sheet Assets Current assets Cash Accounts receivable Inventory Total Fixed assets Net plant and equipment $ 32,890 52,910 112,970 198,770 Liabilities and Owners Equity Current liabilities Accounts payable Notes payable Total Long-term debt Owners equity Common stock and paid-in surplus Retained earnings Total Total liabilities and owners equity $ $ 88,400 17,000 105,400 216,047 $ 536,900 $ $ $ 140,000 287,141 427,141 748,587 Total assets So the excess debt raised is: $ 735,670 Excess debt = $748,587 735,670 Excess debt = $12,917 At a 35 percent growth rate, and assuming the payout ratio is constant, the dividends paid will be: Dividends = ($30,810/$102,700)($154,993) CHAPTER 3 B - 43 Dividends = $46,498 And the addition to retained earnings will be: Addition to retained earnings = $154,993 46,498 Addition to retained earnings = $108,495 The new retained earnings on the pro forma balance sheet will be: New retained earnings = $182,900 + 108,495 New retained earnings = $291,395 The new total debt will be: New total debt = .75255($431,395) New total debt = $324,648 So, the new long-term debt will be the new total debt minus the new short-term debt, or: New long-term debt = $324,648 108,800 New long-term debt = $215,848 CHAPTER 3 B-44 Sales growth rate = 35% and debt/equity ratio = .75255: MOOSE TOURS INC. Pro Forma Balance Sheet Assets Current assets Cash Accounts receivable Inventory Total Fixed assets Net plant and equipment $ 34,155 54,945 117,315 206,415 Liabilities and Owners Equity Current liabilities Accounts payable Notes payable Total Long-term debt Owners equity Common stock and paid-in surplus Retained earnings Total Total liabilities and owners equity $ $ $ 91,800 17,000 108,800 215,848 $ 557,550 $ $ $ 140,000 291,395 431,395 756,043 Total assets So the excess debt raised is: $ 763,965 Excess debt = $756,043 763,965 Excess debt = $7,922 At a 35 percent growth rate, the firm will need funds in the amount of $7,922 in addition to the external debt already raised. So, the EFN will be: EFN = $57,848 + 7,922 EFN = $65,770 26. We must need the ROE to calculate the sustainable growth rate. The ROE is: ROE = (PM)(TAT)(EM) ROE = (.062)(1 / 1.90)(1 + 0.55) ROE = .0506 or 5.06% Now, we can use the sustainable growth rate equation to find the retention ratio as: Sustainable growth rate = (ROE b) / [1 (ROE b)] Sustainable growth rate = .09 = (.0506b) / (1 .0506b) b = 1.63 This implies the payout ratio is: Payout ratio = 1 b Payout ratio = 1 1.63 Payout ratio = 0.63 CHAPTER 3 B - 45 This is a negative dividend payout ratio of 63 percent, which is impossible. The growth rate is not consistent with the other constraints. The lowest possible payout rate is 0, which corresponds to retention ratio of 1, or total earnings retention. The maximum sustainable growth rate for this company is: Maximum sustainable growth rate = (ROE b) / [1 (ROE b)] Maximum sustainable growth rate = [.0506(1)] / [1 .0506(1)] Maximum sustainable growth rate = .0533 or 5.33% 27. We know that EFN is: EFN = Increase in assets Addition to retained earnings The increase in assets is the beginning assets times the growth rate, so: Increase in assets = A g The addition to retained earnings next year is the current net income times the retention ratio, times one plus the growth rate, so: Addition to retained earnings = (NI b)(1 + g) And rearranging the profit margin to solve for net income, we get: NI = PM(S) Substituting the last three equations into the EFN equation we started with and rearranging, we get: EFN = A(g) PM(S)b(1 + g) EFN = A(g) PM(S)b [PM(S)b]g EFN = PM(S)b + [A PM(S)b]g 28. We start with the EFN equation we derived in Problem 27 and set it equal to zero: EFN = 0 = PM(S)b + [A PM(S)b]g Substituting the rearranged profit margin equation into the internal growth rate equation, we have: Internal growth rate = [PM(S)b ] / [A PM(S)b] Since: ROA = NI / A ROA = PM(S) / A We can substitute this into the internal growth rate equation and divide both the numerator and denominator by A. This gives: Internal growth rate = {[PM(S)b] / A} / {[A PM(S)b] / A} CHAPTER 3 B-46 Internal growth rate = b(ROA) / [1 b(ROA)] To derive the sustainable growth rate, we must realize that to maintain a constant D/E ratio with no external equity financing, EFN must equal the addition to retained earnings times the D/E ratio: EFN = (D/E)[PM(S)b(1 + g)] EFN = A(g) PM(S)b(1 + g) Solving for g and then dividing numerator and denominator by A: Sustainable growth rate = PM(S)b(1 + D/E) / [A PM(S)b(1 + D/E )] Sustainable growth rate = [ROA(1 + D/E )b] / [1 ROA(1 + D/E )b] Sustainable growth rate = b(ROE) / [1 b(ROE)] 29. In the following derivations, the subscript E refers to end of period numbers, and the subscript B refers to beginning of period numbers. TE is total equity and TA is total assets. For the sustainable growth rate: Sustainable growth rate = (ROEE b) / (1 ROEE b) Sustainable growth rate = (NI/TEE b) / (1 NI/TEE b) We multiply this equation by: (TEE / TEE) Sustainable growth rate = (NI / TEE b) / (1 NI / TEE b) (TEE / TEE) Sustainable growth rate = (NI b) / (TEE NI b) Recognize that the denominator is equal to beginning of period equity, that is: (TEE NI b) = TEB Substituting this into the previous equation, we get: Sustainable rate = (NI b) / TEB Which is equivalent to: Sustainable rate = (NI / TEB) b Since ROEB = NI / TEB The sustainable growth rate equation is: Sustainable growth rate = ROEB b For the internal growth rate: Internal growth rate = (ROAE b) / (1 ROAE b) CHAPTER 3 B - 47 Internal growth rate = (NI / TAE b) / (1 NI / TAE b) We multiply this equation by: (TAE / TAE) Internal growth rate = (NI / TAE b) / (1 NI / TAE b) (TAE / TAE) Internal growth rate = (NI b) / (TAE NI b) Recognize that the denominator is equal to beginning of period assets, that is: (TAE NI b) = TAB Substituting this into the previous equation, we get: Internal growth rate = (NI b) / TAB Which is equivalent to: Internal growth rate = (NI / TAB) b Since ROAB = NI / TAB The internal growth rate equation is: Internal growth rate = ROAB b 30. Since the company issued no new equity, shareholders equity increased by retained earnings. Retained earnings for the year were: Retained earnings = NI Dividends Retained earnings = $95,000 43,000 Retained earnings = $52,000 So, the equity at the end of the year was: Ending equity = $230,000 + 52,000 Ending equity = $282,000 The ROE based on the end of period equity is: ROE = $95,000 / $282,000 ROE = .3369 or 33.69% The plowback ratio is: Plowback ratio = Addition to retained earnings/NI Plowback ratio = $52,000 / $95,000 Plowback ratio = .5474 or 54.74% CHAPTER 3 B-48 Using the equation presented in the text for the sustainable growth rate, we get: Sustainable growth rate = (ROE b) / [1 (ROE b)] Sustainable growth rate = [.3369(.5474)] / [1 .3369(.5474)] Sustainable growth rate = .2261 or 22.61% The ROE based on the beginning of period equity is ROE = $95,000 / $230,000 ROE = .4130 or 41.30% Using the shortened equation for the sustainable growth rate and the beginning of period ROE, we get: Sustainable growth rate = ROE b Sustainable growth rate = .4130 .5474 Sustainable growth rate = .2261 or 22.61% Using the shortened equation for the sustainable growth rate and the end of period ROE, we get: Sustainable growth rate = ROE b Sustainable growth rate = .3369 .5474 Sustainable growth rate = .1844 or 18.44% Using the end of period ROE in the shortened sustainable growth rate results in a growth rate that is too low. This will always occur whenever the equity increases. If equity increases, the ROE based on end of period equity is lower than the ROE based on the beginning of period equity. The ROE (and sustainable growth rate) in the abbreviated equation is based on equity that did not exist when the net income was earned. CHAPTER 4 DISCOUNTED CASH FLOW VALUATION Answers to Concept Questions 1. 2. 3. 4. 5. 6. 7. Assuming positive cash flows and interest rates, the future value increases and the present value decreases. Assuming positive cash flows and interest rates, the present value will fall and the future value will rise. The better deal is the one with equal installments. Yes, they should. APRs generally dont provide the relevant rate. The only advantage is that they are easier to compute, but, with modern computing equipment, that advantage is not very important. A freshman does. The reason is that the freshman gets to use the money for much longer before interest starts to accrue. Its a reflection of the time value of money. TMCC gets to use the $24,099 immediately. If TMCC uses it wisely, it will be worth more than $100,000 in thirty years. Oddly enough, it actually makes it more desirable since TMCC only has the right to pay the full $100,000 before it is due. This is an example of a call feature. Such features are discussed at length in a later chapter. The key considerations would be: (1) Is the rate of return implicit in the offer attractive relative to other, similar risk investments? and (2) How risky is the investment; i.e., how certain are we that we will actually get the $100,000? Thus, our answer does depend on who is making the promise to repay. The Treasury security would have a somewhat higher price because the Treasury is the strongest of all borrowers. 8. 9. 10. The price would be higher because, as time passes, the price of the security will tend to rise toward $100,000. This rise is just a reflection of the time value of money. As time passes, the time until receipt of the $100,000 grows shorter, and the present value rises. In 2019, the price will probably be higher for the same reason. We cannot be sure, however, because interest rates could be much higher, or TMCCs financial position could deteriorate. Either event would tend to depress the securitys price. CHAPTER 4 B-50 Solutions to Questions and Problems NOTE: All-end-of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. The simple interest per year is: $6,000 .07 = $420 The total balance will be $6,000 + 4,200 = $10,200 With compound interest, we use the future value formula: FV = $11,802.91 The difference is: = $1,602.91 2. To find the FV of a lump sum, we use: a. b. c. d. FV = $4,477.12 FV = $5,397.31 FV = $8,017.84 Because interest compounds on the interest already earned, the interest earned in part c is more than twice the interest earned in part a. With compound interest, future values grow exponentially. 3. To find the PV of a lump sum, we use: PV = PV = PV = PV = = $8,404.32 = $30,741.75 = $56,554.56 = $10,002.91 4. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: r = 8.10% r = 8.26% r = 12.64% r= 9.01% 5. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: CHAPTER 4 B-51 t = 10.64 years t = 21.81 years t = 25.25 years t = 14.07 years 6. To find the length of time for money to double, triple, etc., the present value and future value are irrelevant as long as the future value is twice the present value for doubling, three times as large for tripling, etc. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: t = 9.01 years The length of time to quadruple your money is: t = 18.01 years Notice that the length of time to quadruple your money is twice as long as the time needed to double your money (the difference in these answers is due to rounding). This is an important concept of time value of money. 7. To find the PV of a lump sum, we use: PV = $223,091,225.37 8. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: r = 4.46% Notice that the interest rate is negative. This occurs when the FV is less than the PV. 9. A consol is a perpetuity. To find the PV of a perpetuity, we use the equation: PV = $3,555.56 10. To find the future value with continuous compounding, we use the equation: a. b. c. d. FV = $3,624.75 FV = $2,154.99 FV = $3,624.75 FV = $3,697.98 11. To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a lump sum, we use: PV@5% = $3,500.05 PV@13% = $2,890.61 CHAPTER 4 B-52 PV@18% = $2,590.89 12. To find the PVA, we use the equation: At a 9 percent interest rate: X@9%: = $35,971.48 Y@9%: = $33,062.04 And at a 21 percent interest rate: X@21%: = $23,432.61 Y@21%: = $24,870.87 Notice that the PV of Cash flow X has a greater PV at a 9 percent interest rate, but a lower PV at a 21 percent interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the total cash flow is more important since the cost of waiting (the interest rate) is not as great. At a higher interest rate, Y is more valuable since it has larger cash flows. At a higher interest rate, these bigger cash flows early are more important since the cost of waiting (the interest rate) is so much greater. 13. To find the PVA, we use the equation: PVA@15 yrs: PVA@40 yrs: PVA@75 yrs: = $59,916.35 = $83,472.29 = $87,227.59 To find the PV of a perpetuity, we use the equation: PV = $87,500.00 Notice that as the length of the annuity payments increases, the present value of the annuity approaches the present value of the perpetuity. The present value of the 75-year annuity and the present value of the perpetuity imply that the value today of all perpetuity payments beyond 75 years is only $272.41 . 14. This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation: PV = $416,666.67 r = .0575 or 5.75% 15. For discrete compounding, to find the EAR, we use the equation: EAR =.1587 or 15.87% CHAPTER 4 B-53 EAR = .1268 or 12.68% EAR = .0942 or 9.42% To find the EAR with continuous compounding, we use the equation: EAR = .1388 or 13.88% 16. Here, we are given the EAR and need to find the APR. Using the equation for discrete compounding: EAR = .102 EAR = .084 EAR = .159 APR =.0995 or 9.95% APR = .0809 or 8.09% APR = .1478 or 14.78% Solving the continuous compounding EAR equation: APR = .1714 or 17.14% 17. For discrete compounding, to find the EAR, we use the equation: First National: EAR =.1619 or 16.19% First United: EAR= .1510 or 16.10% Notice that the higher APR does not necessarily mean the higher EAR. The number of compounding periods within a year will also affect the EAR. 18. The cost of a case of wine is 10 percent less than the cost of 12 individual bottles, so the cost of a case will be: Cost of case = $108 r = .0198 or 1.98% per week So, the APR of this investment is: APR = 1.0277 or 102.77% And the EAR is: EAR = 1.7668 or 176.68% The analysis appears to be correct. He really can earn about 177 percent buying wine by the case. The only question left is this: Can you really find a fine bottle of Bordeaux for $10? CHAPTER 4 B-54 19. Here, we need to find the length of an annuity. We know the interest rate, the PV, and the payments. t = 42.52 months 20. Here, we are trying to find the interest rate when we know the PV and FV. Using the FV equation: r = 33.33% per week The interest rate is 33.33% per week. To find the APR, we multiply this rate by the number of weeks in a year, so: APR = 1,733.33% And using the equation to find the EAR: EAR = 313,916,515.69% Intermediate 21. To find the FV of a lump sum with discrete compounding, we use: a. b. c. FV= $2,395.80 FV= $2,412.17 FV= $2,426.73 To find the future value with continuous compounding, we use the equation: FV = PVeRt d. FV = $2,429.75 e. The future value increases when the compounding period is shorter because interest is earned on previously accrued interest. The shorter the compounding period, the more frequently interest is earned, and the greater the future value, assuming the same stated interest rate. 22. The total interest paid by First Simple Bank is the interest rate per period times the number of periods. In other words, the interest by First Simple Bank paid over 10 years will be: r = .0545 or 5.45% 23. We need to find the annuity payment in retirement. Our retirement savings ends and the retirement withdrawals begin, so the PV of the retirement withdrawals will be the FV of the retirement savings. So, we find the FV of the stock account and the FV of the bond account and add the two FVs. C = $17,477.92 24. Since we are looking to quintuple our money, the PV and FV are irrelevant as long as the FV is five times as large as the PV. The number of periods is four, the number of quarters per year. So: r = .4953 or 49.53% CHAPTER 4 B-55 25. Here, we need to find the interest rate for two possible investments. Each investment is a lump sum, so: G: H: 6. r = .1076 or 10.76% r = .1136 or 11.36% This is a growing perpetuity. The present value of a growing perpetuity is: PV = $2,333,333.33 It is important to recognize that when dealing with annuities or perpetuities, the present value equation calculates the present value one period before the first payment. In this case, since the first payment is in three years, we have calculated the present value two years from now. To find the value today, we simply discount this value as a lump sum. Doing so, we find the value of the cash flow stream today is: PV = $1,860,119.05 27. The dividend payments are made quarterly, so we must use the quarterly interest rate. The quarterly interest rate is: Quarterly rate = .0225 Using the present value equation for a perpetuity, we find the value today of the dividends paid must be: PV = $133.33 28. We can use the PVA annuity equation to answer this question. The annuity has 19 payments, not 18 payments. Since there is a payment made in Year 4, the annuity actually begins in Year 3. So, the present value of the annuity is: PVA = $51,356.87 This is the value of the annuity one period before the first payment, or Year 3. So, the value of the cash flows today is: PV = $40,768.74 29. We need to find the present value of an annuity. Using the PVA equation, and the 12 percent interest rate, we get: PVA = $5,108.15 CHAPTER 4 B-56 This is the value of the annuity in Year 5, one period before the first payment. Finding the value of this amount today, we find: PV = $3,319.95 30. The amount borrowed is the value of the home times one minus the down payment, or: Amount borrowed = $562,500 The monthly payments with a balloon payment loan are calculated assuming a longer amortization schedule, in this case, 30 years. The payments based on a 30-year repayment schedule would be: C = $3,555.38 Now, at time = 8, we need to find the PV of the payments which have not been made. The balloon payment will be: PVA = $498,693.81 31. Here, we need to find the FV of a lump sum, with a changing interest rate. We must do this problem in two parts. After the first six months, the balance will be: Interest = $619.93 32. The company would be indifferent at the interest rate that makes the present value of the cash flows equal to the cost today. Since the cash flows are a perpetuity, we can use the PV of a perpetuity equation. Doing so, we find: r = .0697 or 6.97% 33. The company will accept the project if the present value of the increased cash flows is greater than the cost. The cash flows are a growing annuity, so the present value is: PV = $107,030.69 The company should accept the project since the cost less than the increased cash flows. 34. Since your salary grows at 4 percent per year, your salary next year will be: Next years salary = $78,000 This means your deposit next year will be: Next years deposit = $7,800 Since your salary grows at 4 percent, you deposit will also grow at 4 percent. We can use the present value of a growing perpetuity equation to find the value of your deposits today. Doing so, we find: PV = $125,844.67 CHAPTER 4 B-57 Now, we can find the future value of this lump sum in 35 years. We find: FV = $2,568,989.11 This is the value of your savings in 35 years. 35. The relationship between the PVA and the interest rate is: PVA falls as r increases, and PVA rises as r decreases FVA rises as r increases, and FVA falls as r decreases The present values of $7,000 per year for 15 years at the various interest rates given are: PVA@10% = $47,695.84 PVA@5% = $62,042.76 PVA@15% = $37,944.33 36. Here, we are given the FVA, the interest rate, and the amount of the annuity. We need to solve for the number of payments. Using the FVA equation: t = 118.19 payments 37. Here, we are given the PVA, number of periods, and the amount of the annuity. We need to solve for the interest rate. Using the PVA equation: r = 0.560% The APR is the periodic interest rate times the number of periods in the year, so: APR = 6.71% 38. The amount of principal paid on the loan is the PV of the monthly payments you make. So, the present value of the $1,150 monthly payments is: PVA = $189,770.61 The monthly payments of $1,150 will amount to a principal payment of $189,770.61. The amount of principal you will still owe is: Amount still owed = $70,229.39 This remaining principal amount will increase at the interest rate on the loan until the end of the loan period. So the balloon payment in 30 years, which is the FV of the remaining principal will be: Balloon payment = $435,777.30 CHAPTER 4 B-58 39. We are given the total PV of all four cash flows. If we find the PV of the three cash flows we know, and subtract them from the total PV, the amount left over must be the PV of the missing cash flow. So, the PV of the cash flows we know are: = $2,219.33 40. To solve this problem, we simply need to find the PV of each lump sum and add them together. It is important to note that the first cash flow of $1 million occurs today, so we do not need to discount that cash flow. The PV of the lottery winnings is: PV = $19,150,500.91 41. Here, we are finding the interest rate for an annuity cash flow. We are given the PVA, number of periods, and the amount of the annuity. We need to solve for the interest rate. We should also note that the PV of the annuity is not the amount borrowed since we are making a down payment on the warehouse. The amount borrowed is: APR = 0.0756 or 7.56% And the EAR is: EAR = 0.0782 or 7.82% 42. The profit the firm earns is just the PV of the sales price minus the cost to produce the asset. We find the PV of the sales price as the PV of a lump sum: PV = $97,027.02 And the firms profit is: Profit = $6,027.02 To find the interest rate at which the firm will break even, we need to find the interest rate using the PV (or FV) of a lump sum. Using the PV equation for a lump sum, we get: r = 0.1544 or 15.44% 43. We want to find the value of the cash flows today, so we will find the PV of the annuity, and then bring the lump sum PV back to today. The annuity has 24 payments, so the PV of the annuity is: PVA = $26,321.90 Since this is an ordinary annuity equation, this is the PV one period before the first payment, so it is the PV at t = 6. To find the value today, we find the PV of this lump sum. The value today is: PV = $16,587.26 44. This question is asking for the present value of an annuity, but the interest rate changes during the life of the annuity. We need to find the present value of the cash flows for the last eight years first. The PV of these cash flows is: CHAPTER 4 B-59 PVA2 = $116,039.35 Note that this is the PV of this annuity exactly seven years from today. Now, we can discount this lump sum to today. The value of this cash flow today is: PV = $50,304.85 Now, we need to find the PV of the annuity for the first seven years. The value of these cash flows today is: PVA1 = $96,302.37 The value of the cash flows today is the sum of these two cash flows, so: PV = $146,607.22 45. Here, we are trying to find the dollar amount invested today that will equal the FVA with a known interest rate, and payments. First, we need to determine how much we would have in the annuity account. Finding the FV of the annuity, we get: FVA = $480,979.15 Now, we need to find the PV of a lump sum that will give us the same FV. So, using the FV of a lump sum with continuous compounding, we get: PV = $144,868.14 46. To find the value of the perpetuity at t = 7, we first need to use the PV of a perpetuity equation. Using this equation we find: PV = $25,609.76 Remember that the PV of a perpetuity (and annuity) equations give the PV one period before the first payment, so, this is the value of the perpetuity at t = 14. To find the value at t = 7, we find the PV of this lump sum as: PV = $14,750.77 47. To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the interest rate quoted in the problem is only relevant to determine the total interest under the terms given. The interest rate for the cash flows of the loan is: PVA = $15,000 Again, we cannot solve this equation for r, so we need to solve this equation on a financial calculator, using a spreadsheet, or by trial and error. Using a spreadsheet, we find: r = 2.502% per month CHAPTER 4 B-60 So the APR is: APR = 30.03% And the EAR is: EAR = .3452 or 34.52% 48. The cash flows in this problem are semiannual, so we need the effective semiannual rate. The interest rate given is the APR, so the monthly interest rate is: Monthly rate = .00833 or 0.833% To get the semiannual interest rate, we can use the EAR equation, but instead of using 12 months as the exponent, we will use 6 months. The effective semiannual rate is: Semiannual rate = .0511 or 5.11% We can now use this rate to find the PV of the annuity. The PV of the annuity is: PVA @ t = 8: $76,823.89 Note, that this is the value one period (six months) before the first payment, so it is the value at t = 9. So, the value at the various times the questions asked for uses this value 9 years from now. PV @ t = 5: $51,582.02 Note, that you can also calculate this present value (as well as the remaining present values) using the number of years. To do this, you need the EAR. The EAR is: EAR = .1047 or 10.47% So, we can find the PV at t = 5 using the following method as well: PV @ t = 5: $51,582.02 The value of the annuity at the other times in the problem is: PV @ t = 3: PV @ t = 3: PV @ t = 0: PV @ t = 0: = $42,266.80 = $42,266.80 = $31,350.96 = $31,350.96 CHAPTER 4 B-61 49. a. If the payments are in the form of an ordinary annuity, the present value will be: PVA = $51,341.26 If the payments are an annuity due, the present value will be: PVAdue = $55,961.98 b. We can find the future value of the ordinary annuity as: FVA = $121,543.44 If the payments are an annuity due, the future value will be: FVAdue = $132,482.35 c. Assuming a positive interest rate, the present value of an annuity due will always be larger than the present value of an ordinary annuity. Each cash flow in an annuity due is received one period earlier, which means there is one period less to discount each cash flow. Assuming a positive interest rate, the future value of an annuity due will always be higher than the future value of an ordinary annuity. Since each cash flow is made one period sooner, each cash flow receives one extra period of compounding. 50. We need to use the PVA due equation, that is: C = $1,665.65 Notice, to find the payment for the PVA due we simply compound the payment for an ordinary annuity forward one period. 51. The payment for a loan repaid with equal payments is the annuity payment with the loan value as the PV of the annuity. So, the loan payment will be: C = $27,258.78 The interest payment is the beginning balance times the interest rate for the period, and the principal payment is the total payment minus the interest payment. The ending balance is the beginning balance minus the principal payment. The ending balance for a period is the beginning balance for the next period. The amortization table for an equal payment is: Year 1 2 3 Beginning Balance $69,000.00 47,951.22 25,008.05 Total Payment $27,258.78 27,258.78 27,258.78 Interest Payment $6,210.00 4,315.61 2,250.72 Principal Payment $21,048.78 22,943.17 25,008.05 Ending Balance $47,951.22 25,008.05 0 In the third year, $2,250.72 of interest is paid. Total interest over life of the loan = $12,776.33 CHAPTER 4 B-62 52. This amortization table calls for equal principal payments of $23,000 per year. The interest payment is the beginning balance times the interest rate for the period, and the total payment is the principal payment plus the interest payment. The ending balance for a period is the beginning balance for the next period. The amortization table for an equal principal reduction is: Year 1 2 3 Beginning Balance $69,000.00 46,000.00 23,000.00 Total Payment $29,210.00 27,140.00 25,070.00 Interest Payment $6,210.00 4,140.00 2,070.00 Principal Payment $23,000.00 23,000.00 23,000.00 Ending Balance $46,000.00 23,000.00 0 In the third year, $2,070 of interest is paid. Total interest over life of the loan = $12,420 Notice that the total payments for the equal principal reduction loan are lower. This is because more principal is repaid early in the loan, which reduces the total interest expense over the life of the loan. Challenge 53. The monthly interest rate is the annual interest rate divided by 12, or: Monthly interest rate = .00958 Now we can set the present value of the lease payments equal to the cost of the equipment, or $3,500. The lease payments are in the form of an annuity due, so: C = $162.38 54. First, we will calculate the present value of the college expenses for each child. The expenses are an annuity, so the present value of the college expenses is: PVA = $185,250.01 This is the cost of each childs college expenses one year before they enter college. So, the cost of the oldest childs college expenses today will be: PV = $69,533.81 And the cost of the youngest childs college expenses today will be: PV = $60,450.71 Therefore, the total cost today of your childrens college expenses is: Cost today = $129,984.52 This is the present value of your annual savings, which are an annuity. So, the amount you must save each year will be: CHAPTER 4 B-63 C = $14,497.78 55. The salary is a growing annuity, so using the equation for the present value of a growing annuity. The salary growth rate is 3.5 percent and the discount rate is 9 percent, so the value of the salary offer today is: PV = $791,062.31 The yearly bonuses are 10 percent of the annual salary. This means that next years bonus will be: Next years bonus = $5,200 Since the salary grows at 3.5 percent, the bonus will grow at 3.5 percent as well. Using the growing annuity equation, with a 3.5 percent growth rate and a 9 percent discount rate, the present value of the annual bonuses is: PV = $79,106.23 Notice the present value of the bonus is 10 percent of the present value of the salary. The present value of the bonus will always be the same percentage of the present value of the salary as the bonus percentage. So, the total value of the offer is: PV = $880,168.54 56. a. Here, we need to compare two options. In order to do so, we must get the value of the two cash flow streams to the same time, so we will find the value of each today. We must also make sure to use the aftertax cash flows, since it is more relevant. For Option A, the aftertax cash flows are: Aftertax cash flows = $260,000 The aftertax cash flows from Option A are in the form of an annuity due, so the present value of the cash flow today is: PVAdue = $2,710,997.76 b. For Option B, the aftertax cash flows are: Aftertax cash flows = $188,500 The aftertax cash flows from Option B are an ordinary annuity, plus the cash flow today, so the present value: PV = $2,676,973.38 You should choose Option A because it has a higher present value on an aftertax basis. CHAPTER 4 B-64 57. We need to find the first payment into the retirement account. The present value of the desired amount at retirement is: PV = $126,078.82 This is the value today. Since the savings are in the form of a growing annuity, we can use the growing annuity equation and solve for the payment. Doing so, we get: C = $10,251.54 This is the amount you need to save next year. So, the percentage of your salary is: Percentage of salary = .1281 or 12.81% Note that this is the percentage of your salary you must save each year. Since your salary is increasing at 3 percent, and the savings are increasing at 3 percent, the percentage of salary will remain constant. 58. Since she put $1,000 down, the amount borrowed will be: Amount borrowed = $29,000 So, the monthly payments will be: C = $585.24 The amount remaining on the loan is the present value of the remaining payments. Since the first payment was made on October 1, 2008, and she made a payment on October 1, 2010, there are 35 payments remaining, with the first payment due immediately. So, we can find the present value of the remaining 34 payments after November 1, 2010, and add the payment made on this date. So the remaining principal owed on the loan is: PV = $17,801.29 She must also pay a one percent prepayment penalty and owe the current month, so the total amount of the payment is: Total payment = $18,564.54 59. The cash flows for this problem occur monthly, and the interest rate given is the EAR. Since the cash flows occur monthly, we must get the effective monthly rate. One way to do this is to find the APR based on monthly compounding, and then divide by 12. So, the pre-retirement APR is: EAR = .10 APR = 9.57% And the post-retirement APR is: EAR = .08 APR = 12[(1.08)1/12 1] = 7.72% First, we will calculate how much he needs at retirement. The amount needed at retirement is the PV of the monthly spending plus the PV of the inheritance. The PV of these two cash flows is: CHAPTER 4 B-65 PVA = $1,831,165.95 PV = $214,548.21 So, at retirement, he needs: = $2,045,714.16 He will be saving $2,000 per month for the next 10 years until he purchases the cabin. The value of his savings after 10 years will be: FVA = $399,727.71 After he purchases the cabin, the amount he will have left is: = $99,727.71 He still has 20 years until retirement. When he is ready to retire, this amount will have grown to: FV = $670,918.19 So, when he is ready to retire, based on his current savings, he will be short: = $1,374,795.98 This amount is the FV of the monthly savings he must make between years 10 and 30. So, finding the annuity payment using the FVA equation, we find his monthly savings will need to be: C = $1,914.07 60. To answer this question, we should find the PV of both options, and compare them. Since we are purchasing the car, the lowest PV is the best option. The PV of the leasing is simply the PV of the lease payments, plus the $1,500. The interest rate we would use for the leasing option is the same as the interest rate of the loan. The PV of leasing is: PV = $15,860.31 The PV of purchasing the car is the current price of the car minus the PV of the resale price. The PV of the resale price is: PV = $14,957.84 The PV of the decision to purchase is: = $15,042.16 In this case, it is cheaper to buy the car than lease it since the PV of buying is lower. To find the breakeven resale price, we need to find the resale price that makes the PV of the two options the same. In other words, the PV of the decision to buy should be: CHAPTER 4 B-66 PV of resale price = $14,139.69 The resale price that would make the PV of the lease versus buy decision is the FV of this value, so: Breakeven resale price = $17,960.75 61. To find the quarterly salary for the player, we first need to find the PV of the current contract. The cash flows for the contract are annual, and we are given an APR with daily compounding. We need to find the EAR so the interest compounding is the same as the timing of the cash flows. The EAR is: EAR = 0.0513 or 5.13% The PV of the current contract offer is the sum of the PV of the cash flows. So, the PV is: PV = $33,748,414.59 The player wants the contract increased in value by $1,500,000, so the PV of the new contract will be: PV = $35,248,414.59 The player has also requested a signing bonus payable today in the amount of $8 million. We can simply subtract this amount from the PV of the new contract. The remaining amount will be the PV of the future quarterly paychecks. $35,248,414.59 8,000,000 = $27,248,414.59 To find the quarterly payments, first realize that the interest rate we need is the effective quarterly rate. Using the daily interest rate, we can find the quarterly interest rate using the EAR equation, with the number of days being 91.25, the number of days in a quarter (365 / 4). The effective quarterly rate is: Effective quarterly rate = 0.01258 or 1.258% Now, we have the interest rate, the length of the annuity, and the PV. Using the PVA equation and solving for the payment, we get: C = $1,322,389.91 62. To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the interest rate quoted in the problem is only relevant to determine the total interest under the terms given. The cash flows of the loan are the $20,000 you must repay in one year, and the $17,200 you borrow today. The interest rate of the loan is: r = 0.1628 or 16.28% Because of the discount, you only get the use of $17,200, and the interest you pay on that amount is 16.28%, not 14%. CHAPTER 4 B-67 63. Here, we have cash flows that would have occurred in the past and cash flows that would occur in the future. We need to bring both cash flows to today. Before we calculate the value of the cash flows today, we must adjust the interest rate, so we have the effective monthly interest rate. Finding the APR with monthly compounding and dividing by 12 will give us the effective monthly rate. The APR with monthly compounding is: APR = 0.0678 or 6.78% To find the value today of the back pay from two years ago, we will find the FV of the annuity, and then find the FV of the lump sum. Doing so gives us: FVA = $39,204.27 And the value of this years salary today is: FV = $41,948.57 Notice we found the FV of the annuity with the effective monthly rate, and then found the FV of the lump sum with the EAR. Alternatively, we could have found the FV of the lump sum with the effective monthly rate as long as we used 12 periods. The answer would be the same either way. Now, we need to find the value today of last years back pay: FVA = $41,267.66 Next, we find the value today of the five years future salary: PVA = $190,356.23 The value today of the jury award is the sum of salaries, plus the compensation for pain and suffering, and court costs. The award should be for the amount of: Award = $503,572.47 As the plaintiff, you would prefer a lower interest rate. In this problem, we are calculating both the PV and FV of annuities. A lower interest rate will decrease the FVA, but increase the PVA. So, by a lower interest rate, we are lowering the value of the back pay. But, we are also increasing the PV of the future salary. Since the future salary is larger and has a longer time, this is the more important cash flow to the plaintiff. 64. Again, to find the interest rate of a loan, we need to look at the cash flows of the loan. Since this loan is in the form of a lump sum, the amount you will repay is the FV of the principal amount, which will be: Loan repayment = $10,900 The amount you will receive today is the principal amount of the loan times one minus the points. Amount received = $9,800 Now, we simply find the interest rate for this PV and FV. CHAPTER 4 B-68 r = 0.1122 or 11.22% 65. This is the same question as before, with different values. Assuming a $10,000 face value loan, we get: Loan repayment amount = $11,300 Amount received = $9,700 r = 0.1649 or 16.49% The effective rate is not affected by the loan amount, since it drops out when solving for r. 66. First, we will find the APR and EAR for the loan with the refundable fee. Remember, we need to use the actual cash flows of the loan to find the interest rate. With the $2,500 application fee, you will need to borrow $227,500 to have $225,000 after deducting the fee. Solving for the payment under these circumstances, we get: C = $1,590.71 We can now use this amount in the PVA equation with the original amount we wished to borrow, $225,000. Solving for r, we find: PVA = $225,000 Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives: r = 0.6344% per month APR = 7.61% EAR = 0.0788 or 7.88% With the nonrefundable fee, the APR of the loan is simply the quoted APR since the fee is not considered part of the loan. So: APR = 7.50% EAR = 0.0776 or 7.76% 67. Be careful of interest rate quotations. The actual interest rate of a loan is determined by the cash flows. Here, we are told that the PV of the loan is $2,000, and the payments are $86.72 per month for three years, so the interest rate on the loan is: PVA = $2,000 Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives: r = 2.81% per month CHAPTER 4 B-69 APR = 33.68% EAR = 0.3939 or 39.39% Its called add-on interest because the interest amount of the loan is added to the principal amount of the loan before the loan payments are calculated. 68. Here, we are solving a two-step time value of money problem. Each question asks for a different possible cash flow to fund the same retirement plan. Each savings possibility has the same FV, that is, the PV of the retirement spending when your friend is ready to retire. The amount needed when your friend is ready to retire is: PVA = $1,483,161.99 This amount is the same for all three parts of this question. a. If your friend makes equal annual deposits into the account, this is an annuity with the FVA equal to the amount needed in retirement. The required savings each year will be: C = $15,701.35 b. Here we need to find a lump sum savings amount. Using the FV for a lump sum equation, we get: PV = $194,838.72 c. In this problem, we have a lump sum savings in addition to an annual deposit. Since we already know the value needed at retirement, we can subtract the value of the lump sum savings at retirement to find out how much your friend is short. Doing so gives us: FV of trust fund deposit = $98,357.57 So, the amount your friend still needs at retirement is: Amount short at retirement = $1,384,804.43 Using the FVA equation, and solving for the payment, we get: C = $14,660.10 This is the total annual contribution, but your friends employer will contribute $2,000 per year, so your friend must contribute: Friends contribution = $12,660.10 CHAPTER 4 B-70 69. We will calculate the number of periods necessary to repay the balance with no fee first. We simply need to use the PVA equation and solve for the number of payments. Without fee and annual rate = 19.20%: t = 178.49 months Without fee and annual rate = 9.20%: t = 78.51 months So, you will pay the card off: = 99.98 months sooner. We have already calculated the time to pay off the current card with no fee as 178.49 months. The time to repay the new card with a transfer fee is: With fee and annual rate = 9.20%: t = 81.78 months So, you will pay the card off: = 96.71 months sooner. 70. We need to find the FV of the premiums to compare with the cash payment promised at age 65. We have to find the value of the premiums at year 6 first since the interest rate changes at that time. So: FV1 = $1,127.36 FV2 = $1,024.87 FV3 = $1,064.80 FV4 = $968 FV5 = $990 Value at year six = $6,075.03 Finding the FV of this lump sum at the childs 65th birthday: FV = $569,573.51 The policy is not worth buying; the future value of the policy is $569,573.51, but the policy contract will pay off $500,000. The premiums are worth $69,573.51 more than the policy payoff. Note, we could also compare the PV of the two cash flows. The PV of the premiums is: CHAPTER 4 B-71 PV = $3,429.19 And the value today of the $500,000 at age 65 is: PV = $3,010.32 The premiums still have the higher cash flow. At time zero, the difference is $418.88. Whenever you are comparing two or more cash flow streams, the cash flow with the highest value at one time will have the highest value at any other time. Here is a question for you: Suppose you invest $418.88, the difference in the cash flows at time zero, for six years at a 10 percent interest rate, and then for 59 years at an 8 percent interest rate. How much will it be worth? Without doing calculations, you know it will be worth $69,573.51, the difference in the cash flows at time 65! 71. Since the payments occur at six month intervals, we need to get the effective six-month interest rate. We are assuming 365 per year, in other words, we are ignoring leap year. We can calculate the daily interest rate since we have an APR compounded daily, so the effective six-month interest rate is: Effective six-month rate = 00.0408 or 4.08% Now, we can use the PVA equation to find the present value of the semi-annual payments. Doing so, we find: PVA = $19,557,514.31 This is the value six months from today, which is one period (six months) prior to the first payment. So, the value today is: PV = $18,790,735.56 This means the total value of the lottery winnings today is: Value of winnings today = $22,790,735.56 You should not take the offer since the value of the offer is less than the present value of the payments. 72. Here, we need to find the interest rate that makes the PVA, the college costs, equal to the FVA, the savings. The PV of the college costs are: r = 7.31% 73. Here, we need to find the interest rate that makes us indifferent between an annuity and a perpetuity. To solve this problem, we need to find the PV of the two options and set them equal to each other. The PV of the perpetuity is: r = 0.0770 or 7.70% CHAPTER 4 B-72 74. The cash flows in this problem occur every two years, so we need to find the effective two year rate. One way to find the effective two year rate is to use an equation similar to the EAR, except use the number of days in two years as the exponent. (We use the number of days in two years since it is daily compounding; if monthly compounding was assumed, we would use the number of months in two years.) So, the effective two-year interest rate is: Effective 2-year rate = 0.2712 or 27.12% We can use this interest rate to find the PV of the perpetuity. Doing so, we find: PV = $62,684.59 This is an important point: Remember that the PV equation for a perpetuity (and an ordinary annuity) tells you the PV one period before the first cash flow. In this problem, since the cash flows are two years apart, we have found the value of the perpetuity one period (two years) before the first payment, which is one year ago. We need to compound this value for one year to find the value today. The value of the cash flows today is: PV = $70,675.29 The second part of the question assumes the perpetuity cash flows begin in four years. In this case, when we use the PV of a perpetuity equation, we find the value of the perpetuity two years from today. So, the value of these cash flows today is: PV = $49,311.39 75. To solve for the PVA due: C C C + + .... + 2 (1 + r ) (1 + r ) (1 + r ) t C C + .... + PVAdue = C + (1 + r ) (1 + r ) t - 1 PVA = C C C PVAdue = (1 + r ) (1 + r ) + (1 + r ) 2 + .... + (1 + r ) t PVAdue = (1 + r) PVA And the FVA due is: FVA = C + C(1 + r) + C(1 + r)2 + . + C(1 + r)t 1 FVAdue = C(1 + r) + C(1 + r)2 + . + C(1 + r)t FVAdue = (1 + r)[C + C(1 + r) + . + C(1 + r)t 1] FVAdue = (1 + r)FVA 76. We are given the value today of an annuity with the first payment occurring seven years from today. While we can solve as an annuity due, since the payments occur in the future, it is irrelevant if we calculate as an ordinary annuity, or an annuity due, as long as we are correct in the number of periods. We need to find the value of the lump sum six years from now (one period before the first payment). This will be the PVA. So, the value of the lump sum in six years is: CHAPTER 4 B-73 C = $22,212.70 77. a. The APR is the interest rate per week times 52 weeks in a year, so: APR = 364% EAR = 32.7253 or 3,272.53% b. In a discount loan, the amount you receive is lowered by the discount, and you repay the full principal. With a 7 percent discount, you would receive $9.30 for every $10 in principal, so the weekly interest rate would be: r = 0.0753 or 7.53% Note the dollar amount we use is irrelevant. In other words, we could use $0.93 and $1, $93 and $100, or any other combination and we would get the same interest rate. Now we can find the APR and the EAR: APR = 391.40% EAR = 42.5398 or 4,253.98% c. Using the cash flows from the loan, we have the PVA and the annuity payments and need to find the interest rate, so: r = 16.75% per week APR = 871.00% EAR = 3,142.1572 or 314,215.72% 78. To answer this, we can diagram the perpetuity cash flows, which are: (Note, the subscripts are only to differentiate when the cash flows begin. The cash flows are all the same amount.) C2 C1 C3 C2 C1 . C1 Thus, each of the increased cash flows is a perpetuity in itself. So, we can write the cash flows stream as: C1/R C2/R C3/R C4/R . So, we can write the cash flows as the present value of a perpetuity with a perpetuity payment of: CHAPTER 4 B-74 C2/R C3/R C4/R . The present value of this perpetuity is: PV = (C/R) / R = C/R2 So, the present value equation of a perpetuity that increases by C each period is: PV = C/R + C/R2 79. Since it is only an approximation, we know the Rule of 72 is exact for only one interest rate. Using the basic future value equation for an amount that doubles in value and solving for t, we find: It is not possible to solve this equation directly for R, but using Solver, we find the interest rate for which the Rule of 72 is exact is 7.846894 percent. 80. We are only concerned with the time it takes money to double, so the dollar amounts are irrelevant. So, we can write the future value of a lump sum with continuously compounded interest as: $2 = $1eRt 2 = eR t Rt = ln(2) Rt = .693147 t = .693147 / R Since we are using percentage interest rates while the equation uses decimal form, to make the equation correct with percentages, we can multiply by 100: t = 69.3147 / R Calculator Solutions 1. Enter Solve for $11,802.91 10,200 = $1,602.91 2. Enter 10 6% $2,500 10 N 7% I/Y $6,000 PV PMT FV $11,802.91 CHAPTER 4 B-75 N Solve for Enter Solve for Enter Solve for 3. Enter Solve for Enter Solve for Enter Solve for Enter Solve for 4. Enter Solve for 3 N 27 N 16% I/Y 21 N 14% I/Y 6 N 9% I/Y 9 N 7% I/Y 20 N 6% I/Y $2,500 PV 10 N 8% I/Y $2,500 PV I/Y PV PMT FV $4,477.12 PMT FV $5,397.31 PMT FV $8,017.84 $15,451 FV PV $8,404.32 PMT PV $30,741.75 PMT $51,557 FV PV $56,554.56 PMT $886,073 FV PV $10,002.91 $243 PV PMT $550,164 FV I/Y 8.10% PMT $307 FV CHAPTER 4 B-76 Enter Solve for Enter Solve for Enter Solve for 5. Enter Solve for Enter Solve for Enter Solve for Enter Solve for 6. Enter Solve for Enter Solve for 7. Enter Solve for 10 N I/Y 8.26% $405 PV PMT $896 FV 13 N I/Y 12.64% $34,500 PV PMT $162,181 FV 26 N I/Y 9.01% 7% I/Y $51,285 PV PMT $483,500 FV N 10.64 $625 PV PMT $1,284 FV N 21.81 8% I/Y $810 PV PMT $4,341 FV N 25.25 13% I/Y $18,400 PV PMT $402,662 FV N 14.07 16% I/Y $21,500 PV PMT $173,439 FV N 9.01 8% I/Y $1 PV PMT $2 FV N 18.01 20 N 8% I/Y $1 PV PMT $4 FV 6.25% I/Y PV $223,091,225.3 PMT $750,000,000 FV CHAPTER 4 B-77 7 CHAPTER 4 B-78 8. Enter Solve for 11. 4 N I/Y 4.46% $0 $850 1 $740 1 $1,090 1 $1,310 1 $12,377,50 0 PV $10,311,500 PMT FV CFo C01 F01 C02 F02 C03 F03 C04 F04 I=5 NPV CPT $3,500.05 12. Enter Solve for Enter Solve for Enter Solve for Enter Solve for 13. Enter Solve for Enter Solve for 40 N 15 N 5 N 9 N 5 N 9 N CFo C01 F01 C02 F02 C03 F03 C04 F04 I = 13 NPV CPT $2,890.61 9% I/Y $0 $850 1 $740 1 $1,090 1 $1,310 1 CFo C01 F01 C02 F02 C03 F03 C04 F04 I = 18 NPV CPT $2,590.89 $6,000 PMT $0 $850 1 $740 1 $1,090 1 $1,310 1 PV $35,971.48 FV 9% I/Y PV $33,062.04 $8,500 PMT FV 21% I/Y PV $23,432.61 $6,000 PMT FV 21% I/Y PV $24,870.87 $8,500 PMT FV 8% I/Y PV $59,916.35 $7,000 PMT FV 8% I/Y PV $83,472.29 $7,000 PMT FV CHAPTER 4 B-79 Enter Solve for 15. Enter Solve for Enter Solve for Enter Solve for 16. Enter Solve for Enter Solve for Enter Solve for 17. Enter Solve for Enter Solve for 18. Enter Solve for 75 N 8% I/Y PV $87,227.56 4 C/Y $7,000 PMT FV 15% NOM EFF 15.87% 12% NOM EFF 12.68% 12 C/Y 9% NOM EFF 9.42% 10.2% EFF 8.4% EFF 15.9% EFF 365 C/Y NOM 9.95% NOM 8.09% NOM 14.78% 15.1% NOM 2 C/Y 12 C/Y 52 C/Y EFF 16.19% 12 C/Y 15.5% NOM 2nd BGN 2nd SET 12 N EFF 16.10% 2 C/Y I/Y 1.98% $108 PV $10 PMT FV APR = 1.98% 52 = 102.77% CHAPTER 4 B-80 Enter Solve for 19. Enter Solve for 20. Enter Solve for 21. Enter Solve for Enter Solve for Enter Solve for 23. Enter Solve for 102.77% NOM EFF 176.68% 0.9% I/Y 52 C/Y N 42.52 1,733.33% NOM $13,200 PV $375 PMT FV EFF 313,916,515.69 % 10% I/Y 52 C/Y 3 N $1,800 PV PMT FV $2,395.80 32 N 10%/2 I/Y $1,800 PV PMT FV $2,412.17 3 12 N Stock account: 360 N 10%/12 I/Y $1,800 PV PMT FV $2,426.73 11% / 12 I/Y PV $700 PMT FV $1,963,163.8 2 Bond account: Enter Solve for Savings at retirement = $1,963,163.82 + 301,354.31 = $2,264,518.33 Enter 300 N 8% / 12 I/Y $2,2664,518. 33 PV 360 N 6% / 12 I/Y PV $300 PMT FV $301,354.31 PMT FV CHAPTER 4 B-81 Solve for $17,477.92 CHAPTER 4 B-82 24. Enter Solve for 25. Enter Solve for Enter Solve for 28. Enter Solve for Enter Solve for 29. Enter Solve for Enter Solve for 30. Enter Solve for Enter Solve for 31. Enter Solve for 12 / 3 N I/Y 49.53% $1 PV PMT $5 FV 5 N I/Y 10.76% $75,000 PV PMT $125,000 FV 11 N I/Y 11.36% 8% I/Y 75,000 PV PMT $245,000 FV 15 N PV $51,356.87 $6,000 PMT FV 3 N 8% I/Y PV $40,768.74 PMT $51,356.87 FV 15 N 12% I/Y PV $5,108.15 $750 PMT FV 5 N 9% I/Y PV $3,319.95 .75($750,000) PV PMT $5,108.15 FV 30 12 N 6.5%/12 I/Y PMT $3,555.38 $3,555.38 PMT FV 22 12 N 6.5%/12 I/Y PV $498,693.81 $6,000 PV FV 6 N 1.80% / 12 I/Y PMT FV $6,054.20 CHAPTER 4 B-83 Enter Solve for 6 N 18% / 12 I/Y $6,054.20 PV PMT FV $6,619.93 $6,619.93 6,000 = $619.93 35. Enter Solve for Enter Solve for Enter Solve for 36. Enter Solve for 37. Enter Solve for 0.560% 12 = 6.71% 38. Enter Solve for $260,000 189,770.61 = $70,229.39 Enter Solve for 360 N 6.1% / 12 I/Y $70,229.39 PV 360 N 6.1% / 12 I/Y $1,150 PMT N 118.19 60 N 10% / 12 I/Y 12 N 15% I/Y 12 N 5% I/Y 12 N 10% I/Y $7,000 PMT PV $47,695.84 FV PV $62,042.76 $7,000 PMT FV PV $37,944.33 $7,000 PMT FV PV $125 PMT $25,000 FV I/Y 0.560% $75,000 PV $1,475 PMT FV PV $189,770.61 FV PMT FV $435,777.30 CHAPTER 4 B-84 39. CFo C01 F01 C02 F02 C03 F03 C04 F04 I = 10% NPV CPT $4,150.84 $0 $1,750 1 $0 1 $1,380 1 $2,230 1 PV of missing CF = $5,985 4,150.84 = $1,834.16 Value of missing CF: Enter Solve for 40. CFo $1,000,000 C01 $1,350,000 F01 1 C02 $1,700,000 F02 1 C03 $2,050,000 F03 1 C04 $2,400,000 F04 1 C05 $2,750,000 F05 1 C06 $3,100,000 F06 1 C07 $3,450,000 F07 1 C08 $3,800,000 F08 1 C09 $4,150,000 F09 1 C010 $4,500,000 I = 8% NPV CPT $19,150,500.91 2 N 10% I/Y $1,834.16 PV PMT FV $2,219.33 CHAPTER 4 B-85 41. Enter Solve for 360 N I/Y 0.630% . 80($2,400,000) PV $13,500 PMT FV APR = 0.630% 12 = 7.56% Enter Solve for 42. Enter Solve for 3 N 7.56% NOM EFF 7.82% 13% I/Y 12 C/Y PV $97,027.02 PMT $140,000 FV Profit = $97,027.02 91,000 = $6,027.02 Enter Solve for 43. Enter Solve for Enter Solve for 44. Enter Solve for Enter Solve for 45. Enter Solve for FV = $480,979.15 = PV e.08(15); PV = $480,979.15e1.20 = $144,868.14 84 N 12% / 12 I/Y 96 N 9% / 12 I/Y 6 N 8% I/Y 24 N 3 N I/Y 15.44% 8% I/Y $91,000 PV PMT $140,000 FV PV $26,321.90 $2,500 PMT FV PV $16,587.26 PMT $26,321.90 FV PV $116,039.35 $1,700 PMT FV PV $146,607.22 $1,700 PMT $116,039.95 FV 15 12 N 8.75%/12 I/Y PV $1,300 PMT FV $480,979.15 CHAPTER 4 B-86 46. Enter Solve for 47. Enter Solve for 12 N PV@ t = 14: $2,100 / 0.082 = $25,609.76 7 N 8.2% I/Y PV $14,750.77 $15,000 PV PMT $25,609.76 FV I/Y 2.502% $1,462.50 PMT FV APR = 2.502% 12 = 30.03% Enter Solve for 48. Enter Solve for Enter Solve for Enter Solve for Enter Solve for 49. Enter Solve for 2nd BGN 2nd SET Enter Solve for 10 N 9% I/Y PV $55,961.98 $8,000 PMT FV 10 N 9% I/Y 18 N 5.11% I/Y 12 N 5.11% I/Y 8 N 5.11% I/Y 30.03% NOM EFF 34.52% 12 C/Y semiannual rate = (1.00833)6 1 = 5.11% PV $76,823.89 $10,000 PMT FV Monthly rate = .10 / 12 = .00833; 10 N 5.11% I/Y PV $51,582.02 PMT $76,823.89 FV PV $42,266.80 PMT $76,823.89 FV PV $31,350.96 PMT $76,823.89 FV PV $51,341.26 $8,000 PMT FV CHAPTER 4 B-87 Enter Solve for 10 N 9% I/Y PV $8,000 PMT FV $121,543.44 2nd BGN 2nd SET Enter Solve for 50. Enter Solve for 53. Enter Solve for 54. Enter Solve for Cost today of oldest childs expenses: Enter Solve for 14 N 7.25% I/Y PV $69,533.81 PMT $185,250.01 FV PV of college expenses: 4 N 7.25% I/Y PV $185,250.01 $55,000 PMT FV 2nd BGN 2nd SET 2 12 N 11.5% / 12 I/Y $3,500 PV PMT $162.38 FV 2nd BGN 2nd SET 60 N 6.8% / 12 I/Y $85,000 PV PMT $1,665.65 FV 10 N 9% I/Y PV $8,000 PMT FV $132,482.35 Cost today of youngest childs expenses: Enter Solve for 16 N 7.25% I/Y PV $60,450.71 PMT $185,250.01 FV Total cost today = $69,533.81 + 60,450.71 = $129,984.52 Enter Solve for 15 N 7.25% I/Y $129,984.52 PV PMT $14,497.78 FV CHAPTER 4 B-88 56. Option A: Aftertax cash flows = Pretax cash flows(1 tax rate) Aftertax cash flows = $400,000(1 .35) Aftertax cash flows = $260,000 2ND BGN 2nd SET Enter Solve for 31 N 10% I/Y PV $2,710,997.7 6 $260,000 PMT FV Option B: Aftertax cash flows = Pretax cash flows(1 tax rate) Aftertax cash flows = $290,000(1 .35) Aftertax cash flows = $188,500 Enter Solve for 30 N 10% I/Y $188,500 PMT PV $1,776,973.3 8 FV Total value = $1,776,973.38 + 900,000 = $2,676,973.38 58. Enter Solve for Enter Solve for 34 N 7.8% / 12 I/Y 5 12 N 7.8% / 12 I/Y $29,000 PV PMT $585.24 $585.24 PMT FV PV $17,801.29 FV Total payment = Amount due(1 + Prepayment penalty) + Current payment Total payment = $17,801.29(1 + .01) + 585.24 Total payment = $18,564.54 59. Enter Solve for NOM 9.57% Pre-retirement APR: 10% EFF 12 C/Y Post-retirement APR: Enter Solve for NOM 7.72% 8% EFF 12 C/Y CHAPTER 4 B-89 At retirement, he needs: Enter Solve for 240 N 7.72% / 12 I/Y PV $2,045,714.1 6 $15,000 PMT $1,000,000 FV In 10 years, his savings will be worth: Enter Solve for 120 N 9.57% / 12 I/Y PV $2,000 PMT FV $399,727.71 After purchasing the cabin, he will have: $399,727.71 300,000 = $99,727.71 Each month between years 10 and 30, he needs to save: Enter Solve for 60. Enter PV of resale: 36 8% / 12 N I/Y Solve for $30,000 14,957.84 = $15,042.16 PV of lease: 36 8% / 12 N I/Y Solve for $14,360.31 + 1,500 = $15,860.31 Buy the car. Enter 240 N 9.57% / 12 I/Y $99,727.71 PV PMT $1,914.07 $2,045,714.1 6 FV PV $14,957.84 PMT $19,000 FV PV $14,360.31 $450 PMT FV You would be indifferent when the PV of the two cash flows are equal. The present value of the purchase decision must be $15,860.31. Since the difference in the two cash flows is $30,000 15,860.31 = $14,139.69, this must be the present value of the future resale price of the car. The break-even resale price of the car is: Enter Solve for 61. Enter Solve for 5% NOM 365 C/Y 36 N 8% / 12 I/Y $14,139.69 PV PMT FV $17,960.75 EFF 5.13% CHAPTER 4 B-90 CFo $5,000,000 C01 $4,000,000 F01 1 C02 $4,800,000 F02 1 C03 $5,600,000 F03 1 C04 $6,200,000 F04 1 C05 $6,800,000 F05 1 C06 $7,300,000 F06 1 I = 5.13% NPV CPT $33,748,414.59 New contract value = $33,748,414.49 + 1,5,00,000 = $35,248,414.59 PV of payments = $35,248,414.59 8,000,000 = $27,248,414.59 Effective quarterly rate = [1 + (.05/365)]91.25 1 = 1.258% Enter Solve for 62. Enter Solve for 63. Enter Solve for Enter Solve for Enter Solve for 12 N 6.78% / 12 I/Y $39,204.27 PV $40,000 / 12 PMT NOM 6.78% 12 N 24 N 1.258% I/Y $27,248,414.59 PV PMT $1,322,389.9 1 FV 1 N I/Y 16.28% 7% EFF $17,200 PV PMT $20,000 FV 12 C/Y 6.78% / 12 I/Y PV $38,000 / 12 PMT FV $39,204.27 FV $83,216.23 CHAPTER 4 B-91 Enter Solve for 60 N 6.78% / 12 I/Y PV $190,356.23 $45,000 / 12 PMT FV Award = $83,216.23 + 190,356.23 + 200,000 + 30,000 = $503,572.47 64. Enter Solve for 65. Enter Solve for 1 N 1 N $9,800 PV $10,900 FV I/Y 11.22% PMT I/Y 16.49% $9,700 PV PMT $11,300 FV 66. Refundable fee: With the $2,500 application fee, you will need to borrow $227,500 to have $225,000 after deducting the fee. Solve for the payment under these circumstances. Enter Solve for Enter 30 12 N $225,000 PV 30 12 N 7.50% / 12 I/Y $227,500 PV PMT $1,590.71 $1,590.71 PMT FV I/Y Solve for 0.6344% APR = 0.6344% 12 = 7.61% Enter Solve for 7.61% NOM FV EFF 7.88% 12 C/Y Without refundable fee: APR = 7.50% Enter Solve for 67. Enter Solve for 36 N 7.50% NOM EFF 7.76% 12 C/Y I/Y 2.81% $2,000 PV $88.98 PMT FV APR = 2.81% 12 = 33.68% CHAPTER 4 B-92 Enter Solve for 68. Enter Solve for a. Enter Solve for b. Enter Solve for c. Enter Solve for 33.68% NOM EFF 39.39% 12 C/Y What she needs at age 65: 20 N 7% I/Y PV $1,483,161.9 9 $140,000 PMT FV 30 N 7% I/Y PV PMT $15,701.35 $1,483,161.9 9 FV 30 N 7% I/Y PV $194,838.72 $50,000 PV PMT $1,483,161.9 9 FV 10 N 7% I/Y PMT FV $98,357.57 At 65, she is short: $1,483,161.99 98,357.57 = $1,384,804.43 Enter Solve for 30 N 7% I/Y PV PMT $14,660.10 $1,384,804.4 3 FV Her employer will contribute $2,000 per year, so she must contribute: $14,660.10 2,000 = $12,660.10 per year 69. Enter Solve for Enter Solve for N 78.51 N 178.49 Without fee: 19.2% / 12 I/Y $10,000 PV $170 PMT FV 9.2% / 12 I/Y $10,000 PV $170 PMT FV CHAPTER 4 B-93 CHAPTER 4 B-94 With fee: Enter Solve for 70. Enter Solve for Enter Solve for Enter Solve for Enter Solve for Enter Solve for 1 N 10% I/Y $900 PV 2 N 10% I/Y $800 PV 3 N 10% I/Y $800 PV 4 N 10% I/Y $700 PV N 81.78 8.6% / 12 I/Y $10,300 PV $170 PMT FV Value at Year 6: 5 N 10% I/Y $700 PV PMT FV $1,127.36 PMT FV $1,024.87 PMT FV $1,064.80 PMT FV $968.00 PMT FV $990 So, at Year 5, the value is: $1,127.36 + 1,024.87 + 1,064.80 + 968.00 + 990 + 900 = $6,075.03 At Year 65, the value is: Enter Solve for 59 N 8% I/Y $6,075.03 PV PMT FV $569,573.51 The policy is not worth buying; the future value of the policy is $569,573.51but the policy contract will pay off $500,000. 71. Effective six-month rate = (1 + Daily rate)182.5 1 Effective six-month rate = (1 + .08/365)182.5 1 Effective six-month rate = .0408 or 4.08% CHAPTER 4 B-95 Enter Solve for Enter Solve for 40 N 4.08% I/Y PV $19,557,514.31 $1,000,000 PMT FV 1 N 4.08% I/Y PV $18,790,735.56 PMT $19,557,514.3 1 FV Value of winnings today = $18,790,735.56 + 4,000,000 Value of winnings today = $22,790,735.56 72. CFo C01 F01 C02 F02 IRR CPT 7.31% 76. Enter Solve for Enter Solve for 77. a. Enter Solve for b. Enter Solve for 1 N APR = 7% / 52 = 364% 365% NOM EFF 3,272.53% 52 C/Y 10 N 9% I/Y $142,553.51 PV The value at t = 6: 6 N 9% I/Y $85,000 PV PMT FV $142,553.51 $14,000 $14,000 5 $30,000 4 PMT $22,212.70 FV I/Y 7.53% $93.00 PV PMT $100.00 FV APR = 7.53% 52 = 391.40% CHAPTER 4 B-96 Enter Solve for c. Enter Solve for 391.40% NOM EFF 4,253.98% 52 C/Y 4 N I/Y 16.75% $68.92 PV $25 PMT FV APR = 16.75% 52 = 871.00% Enter Solve for 871.00% NOM EFF 314,215.72% 52 C/Y CHAPTER 5 INTEREST RATES AND BOND VALUATION Answers to Concept Questions 1. 2. 3. No. As interest rates fluctuate, the value of a Treasury security will fluctuate. Long-term Treasury securities have substantial interest rate risk. All else the same, the Treasury security will have lower coupons because of its lower default risk, so it will have greater interest rate risk. No. If the bid were higher than the ask, the implication would be that a dealer was willing to sell a bond and immediately buy it back at a higher price. How many such transactions would you like to do? Prices and yields move in opposite directions. Since the bid price must be lower, the bid yield must be higher. There are two benefits. First, the company can take advantage of interest rate declines by calling in an issue and replacing it with a lower coupon issue. Second, a company might wish to eliminate a covenant for some reason. Calling the issue does this. The cost to the company is a higher coupon. A put provision is desirable from an investors standpoint, so it helps the company by reducing the coupon rate on the bond. The cost to the company is that it may have to buy back the bond at an unattractive price. Bond issuers look at outstanding bonds of similar maturity and risk. The yields on such bonds are used to establish the coupon rate necessary for a particular issue to initially sell for par value. Bond issuers also simply ask potential purchasers what coupon rate would be necessary to attract them. The coupon rate is fixed and simply determines what the bonds coupon payments will be. The required return is what investors actually demand on the issue, and it will fluctuate through time. The coupon rate and required return are equal only if the bond sells for exactly at par. Yes. Some investors have obligations that are denominated in dollars; i.e., they are nominal. Their primary concern is that an investment provides the needed nominal dollar amounts. Pension funds, for example, often must plan for pension payments many years in the future. If those payments are fixed in dollar terms, then it is the nominal return on an investment that is important. Companies pay to have their bonds rated simply because unrated bonds can be difficult to sell; many large investors are prohibited from investing in unrated issues. Treasury bonds have no credit risk since it is backed by the U.S. government, so a rating is not necessary. Junk bonds often are not rated because there would be no point in an issuer paying a rating agency to assign its bonds a low rating (its like paying someone to kick you!). 4. 5. 6. 7. 8. 9. CHAPTER 5 B-98 10. The term structure is based on pure discount bonds. The yield curve is based on coupon-bearing issues. 11. Bond ratings have a subjective factor to them. Split ratings reflect a difference of opinion among credit agencies. 12. As a general constitutional principle, the federal government cannot tax the states without their consent if doing so would interfere with state government functions. At one time, this principle was thought to provide for the tax-exempt status of municipal interest payments. However, modern court rulings make it clear that Congress can revoke the municipal exemption, so the only basis now appears to be historical precedent. The fact that the states and the federal government do not tax each others securities is referred to as reciprocal immunity. 13. Lack of transparency means that a buyer or seller cant see recent transactions, so it is much harder to determine what the best bid and ask prices are at any point in time. 14. When the bonds are initially issued, the coupon rate is set at auction so that the bonds sell at par value. The wide range coupon of coupon rates shows the interest rate when the bond was issued. Notice that interest rates have evidently declined. Why? 15. Companies charge that bond rating agencies are pressuring them to pay for bond ratings. When a company pays for a rating, it has the opportunity to make its case for a particular rating. With an unsolicited rating, the company has no input. 16. A 100-year bond looks like a share of preferred stock. In particular, it is a loan with a life that almost certainly exceeds the life of the lender, assuming that the lender is an individual. With a junk bond, the credit risk can be so high that the borrower is almost certain to default, meaning that the creditors are very likely to end up as part owners of the business. In both cases, the equity in disguise has a significant tax advantage. 17. a. b. The bond price is the present value of the cash flows from a bond. The YTM is the interest rate used in valuing the cash flows from a bond. If the coupon rate is higher than the required return on a bond, the bond will sell at a premium, since it provides periodic income in the form of coupon payments in excess of that required by investors on other similar bonds. If the coupon rate is lower than the required return on a bond, the bond will sell at a discount since it provides insufficient coupon payments compared to that required by investors on other similar bonds. For premium bonds, the coupon rate exceeds the YTM; for discount bonds, the YTM exceeds the coupon rate, and for bonds selling at par, the YTM is equal to the coupon rate. Current yield is defined as the annual coupon payment divided by the current bond price. For premium bonds, the current yield exceeds the YTM, for discount bonds the current yield is less than the YTM, and for bonds selling at par value, the current yield is equal to the YTM. In all cases, the current yield plus the expected one-period capital gains yield of the bond must be equal to the required return. c. CHAPTER 5 B-99 18. A long-term bond has more interest rate risk compared to a short-term bond, all else the same. A low coupon bond has more interest rate risk than a high coupon bond, all else the same. When comparing a high coupon, long-term bond to a low coupon, short-term bond, we are unsure which has more interest rate risk. Generally, the maturity of a bond is a more important determinant of the interestrate risk, so the long-term, high coupon bond probably has more interest rate risk. The exception would be if the maturities are close, and the coupon rates are vastly different. Solutions to Questions and Problems NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. NOTE: Most problems do not explicitly list a par value for bonds. Even though a bond can have any par value, in general, corporate bonds in the United States will have a par value of $1,000. We will use this par value in all problems unless a different par value is explicitly stated. Basic 1. The price of a pure discount (zero coupon) bond is the present value of the par. Remember, even though there are no coupon payments, the periods are semiannual to stay consistent with coupon bond payments. So, the price of the bond for each YTM is: a. P = $476.74 b. P = $231.38 c. P = $114.22 2. The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice this problem assumes a semiannual coupon. The price of the bond at each YTM will be: a. P = $1,000.00 When the YTM and the coupon rate are equal, the bond will sell at par. b. P = $793.62 When the YTM is greater than the coupon rate, the bond will sell at a discount. c. P = $1,309.09 When the YTM is less than the coupon rate, the bond will sell at a premium. We would like to introduce shorthand notation here. Rather than write (or type, as the case may be) the entire equation for the PV of a lump sum, or the PVA equation, it is common to abbreviate the equations as: PVIFR,t = 1 / (1 + r)t which stands for Present Value Interest Factor CHAPTER 5 B-100 PVIFAR,t = ({1 [1/(1 + r)]t } / r ) which stands for Present Value Interest Factor of an Annuity These abbreviations are short hand notation for the equations in which the interest rate and the number of periods are substituted into the equation and solved. We will use this shorthand notation in the remainder of the solutions key. 3. Here we are finding the YTM of a semiannual coupon bond. The bond price equation is: YTM = 7.20% 4. Here we need to find the coupon rate of the bond. All we need to do is to set up the bond pricing equation and solve for the coupon payment as follows: Coupon rate = .0824 or 8.24% 5. The price of any bond is the PV of the interest payment, plus the PV of the par value. The fact that the bond is denominated in euros is irrelevant. Notice this problem assumes an annual coupon. The price of the bond will be: P = 867.40 6. Here we are finding the YTM of an annual coupon bond. The fact that the bond is denominated in yen is irrelevant. The bond price equation is: P = 87,000 Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial and error, we find: R = 5.45% Since the coupon payments are annual, this is the yield to maturity. 7. The approximate relationship between nominal interest rates ( R), real interest rates (r), and inflation (h) is: R=r+h Approximate r = .024 or 2.40% The Fisher equation, which shows the exact relationship between nominal interest rates, real interest rates, and inflation is: Exact r = .0235 or 2.35% 8. The Fisher equation, which shows the exact relationship between nominal interest rates, real interest rates, and inflation, is: CHAPTER 5 B-101 R = .0640 or 6.40% 9. The Fisher equation, which shows the exact relationship between nominal interest rates, real interest rates, and inflation, is: h = .0536 or 5.36% 10. The Fisher equation, which shows the exact relationship between nominal interest rates, real interest rates, and inflation, is: r = .0848 or 8.48% 11. The coupon rate, located in the first column of the quote is 5.500%. The bid price is: Bid price = 115:22 = 115 22/32 = 115.68750% $1,000 = $1,156.875 The previous days ask price is found by: Previous days asked price = Todays asked price Change = 115 22/32 17/32 = 115 5/32 The previous days price in dollars was: Previous days dollar price = $1,153.125 12. This is a discount bond because it sells for less than 100% of face value. The current yield is: Current yield = Annual coupon payment / Asked price = $35/$859.0625 = .04079 or 4.08% The YTM is located under the ASK YLD column, so the YTM is 4.3567%. The bid-ask spread is the difference between the bid price and the ask price, so: Bid-Ask spread = 85:29 85:25 = 4/32 Intermediate 13. Here we are finding the YTM of semiannual coupon bonds for various maturity lengths. The bond price equation is: Miller Corporation bond: P0 = $1,178.77 P1 = $1,169.36 P3 = $1,148.77 P8 = $1,085.30 P12 = $1,019.13 P13 = $1,000 Modigliani Company bond: P0 = $840.17 CHAPTER 5 B-102 P1 P3 P8 P12 P13 = $847.53 = $864.10 = $918.89 = $981.14 = $1,000 All else held equal, the premium over par value for a premium bond declines as maturity approaches, and the discount from par value for a discount bond declines as maturity approaches. This is called pull to par. In both cases, the largest percentage price changes occur at the shortest maturity lengths. Also, notice that the price of each bond when no time is left to maturity is the par value, even though the purchaser would receive the par value plus the coupon payment immediately. This is because we calculate the clean price of the bond. 14. Any bond that sells at par has a YTM equal to the coupon rate. Both bonds sell at par, so the initial YTM on both bonds is the coupon rate, 7 percent. If the YTM suddenly rises to 9 percent: PLaurel PHardy = $964.12 = $815.98 The percentage change in price is calculated as: Percentage change in price = (New price Original price) / Original price PLaurel% = 0.0359 or 3.59% PHardy% = 0.1840 or 18.40% If the YTM suddenly falls to 5 percent: PLaurel PHardy = $1,037.62 = $1,251.03 PLaurel% = +0.0376 or 3.76% PHardy% = +0.2510 or 25.10% All else the same, the longer the maturity of a bond, the greater is its price sensitivity to changes in interest rates. Notice also that for the same interest rate change, the gain from a decline in interest rates is larger than the loss from the same magnitude change. For a plain vanilla bond, this is always true. 15. Initially, at a YTM of 10 percent, the prices of the two bonds are: PFaulk PGonas = $702.04 = $1,297.96 CHAPTER 5 B-103 If the YTM rises from 10 percent to 12 percent: PFaulk PGonas = $597.82 = $1,134.06 The percentage change in price is calculated as: PFaulk% = 0.1485 or 14.85% PGonas% = 0.1263 or 12.63% If the YTM declines from 10 percent to 8 percent: PFaulk PGonas = $ $833.37 = $1,499.89 PFaulk% = +0.1871 or 18.71% PGonas% = +0.1556 or 15.56% All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to changes in interest rates. 16. The bond price equation for this bond is: YTM = 2 3.326% = 6.65% The current yield is: Current yield = .0705 or 7.05% Effective annual yield = .0676 or 6.76% 17. The company should set the coupon rate on its new bonds equal to the required return. The required return can be observed in the market by finding the YTM on outstanding bonds of the company. So, the YTM on the bonds currently sold in the market is: YTM = 7.44% 18. Accrued interest is the coupon payment for the period times the fraction of the period that has passed since the last coupon payment. Since we have a semiannual coupon bond, the coupon payment per six months is one-half of the annual coupon payment. There are two months until the next coupon payment, so four months have passed since the last coupon payment. The accrued interest for the bond is: Accrued interest = $25.33 CHAPTER 5 B-104 And we calculate the clean price as: Clean price = $824.67 19. Accrued interest is the coupon payment for the period times the fraction of the period that has passed since the last coupon payment. Since we have a semiannual coupon bond, the coupon payment per six months is one-half of the annual coupon payment. There are four months until the next coupon payment, so two months have passed since the last coupon payment. The accrued interest for the bond is: Accrued interest = $16.50 Dirty price = $1,076.50 20. To find the number of years to maturity for the bond, we need to find the price of the bond. Since we already have the coupon rate, we can use the bond price equation, and solve for the number of years to maturity. We are given the current yield of the bond, so we can calculate the price as: P0 = $861.01 t 25 years The bond has 25 years to maturity. 21. The bond has 10 years to maturity, so the bond price equation is: YTM = 6.73% Current yield = .0698 or 6.98% 22. We found the maturity of a bond in Problem 20. However, in this case, the maturity is indeterminate. A bond selling at par can have any length of maturity. In other words, when we solve the bond pricing equation as we did in Problem 20, the number of periods can be any positive number. Challenge 23. To find the capital gains yield and the current yield, we need to find the price of the bond. The current price of Bond P and the price of Bond P in one year is: P: P0 = $1,119.43 P1 = $1,107.79 Current yield = .0804 or 8.04% Capital gains yield = 0.0104 or 1.04% The current price of Bond D and the price of Bond D in one year is: CHAPTER 5 B-105 D: P0 = $880.57 P1 = $892.21 Current yield = 0.0568 or 5.68% Capital gains yield = 0.0132 or 1.32% All else held constant, premium bonds pay a high current income while having price depreciation as maturity nears; discount bonds pay a lower current income but have price appreciation as maturity nears. For either bond, the total return is still 7%, but this return is distributed differently between current income and capital gains. 24. a. b. R = YTM = 7.65% R = HPY = 12.95% 25. The price of any bond (or financial instrument) is the PV of the future cash flows. Even though Bond M makes different coupons payments, to find the price of the bond, we just find the PV of the cash flows. The PV of the cash flows for Bond M is: PM = $15,142.45 PN = $20,000(PVIF3.5%,40) = $5,051.45 26. To calculate this, we need to set up an equation with the callable bond equal to a weighted average of the noncallable bonds. We will invest X percent of our money in the first noncallable bond, which means our investment in Bond 3 (the other noncallable bond) will be (1 X). The equation is: X = 0.27586 CHAPTER 5 B-106 So, we invest about 28 percent of our money in Bond 1, and about 72 percent in Bond 3. This combination of bonds should have the same value as the callable bond, excluding the value of the call. So: P2 = 107.71 The call value is the difference between this implied bond value and the actual bond price. So, the call value is: Call value = = 4.210 Assuming a $1,000 par value, the call value is $42.10. 27. In general, this is not likely to happen, although it can (and did). The reason that this bond has a negative YTM is that it is a callable U.S. Treasury bond. Market participants know this. Given the high coupon rate of the bond, it is extremely likely to be called, which means the bondholder will not receive all the cash flows promised. A better measure of the return on a callable bond is the yield to call (YTC). The YTC calculation is the basically the same as the YTM calculation, but the number of periods is the number of periods until the call date. If the YTC were calculated on this bond, it would be positive. 28. To find the present value, we need to find the real weekly interest rate. To find the real return, we need to use the effective annual rates in the Fisher equation. So, we find the real EAR is: (1 + R) = (1 + r)(1 + h) 1 + .094 = (1 + r)(1 + .035) r = .0570 or 5.70% Now, to find the weekly interest rate, we need to find the APR. Using the equation for discrete compounding: EAR = [1 + (APR / m)]m 1 We can solve for the APR. Doing so, we get: APR = .0555 or 5.55% So, the weekly interest rate is: Weekly rate = .0011 or 0.11% Now we can find the present value of the cost of the roses. The real cash flows are an ordinary annuity, discounted at the real interest rate. So, the present value of the cost of the roses is: PVA = C({1 [1/(1 + r)]t } / r) PVA = $7({1 [1/(1 + .0011)]30(52)} / .0011) PVA = $5,318.46 29. To answer this question, we need to find the monthly interest rate, which is the APR divided by 12. We also must be careful to use the real interest rate. The Fisher equation uses the effective annual rate, so, the real effective annual interest rates, and the monthly interest rates for each account are: CHAPTER 5 B-107 Stock account: (1 + R) = (1 + r)(1 + h) 1 + .12 = (1 + r)(1 + .04) r = .0769 or 7.69% APR = m[(1 + EAR)1/m 1] APR = 12[(1 + .0769)1/12 1] APR = .0743 or 7.43% Monthly rate = APR / 12 Monthly rate = .0743 / 12 Monthly rate = .0062 or 0.62% Bond account: (1 + R) = (1 + r)(1 + h) 1 + .07 = (1 + r)(1 + .04) r = .0288 or 2.88% APR = m[(1 + EAR)1/m 1] APR = 12[(1 + .0288)1/12 1] APR = .0285 or 2.85% Monthly rate = APR / 12 Monthly rate = .0285 / 12 Monthly rate = .0024 or 0.24% Now we can find the future value of the retirement account in real terms. The future value of each account will be: Stock account: FVA = C {[(1 + r)t 1] / r} FVA = $950{[(1 + .0062)360 1] / .0062} FVA = $1,263,217.07 Bond account: FVA = C {[(1 + r)t 1] / r} FVA = $450{[(1 + .0024)360 1] / .0024} FVA = $255,475.17 CHAPTER 5 B-108 The total future value of the retirement account will be the sum of the two accounts, or: Account value = $1,263,217.07 + 255,475.17 Account value = $1,518,692.24 Now we need to find the monthly interest rate in retirement. We can use the same procedure that we used to find the monthly interest rates for the stock and bond accounts, so: (1 + R) = (1 + r)(1 + h) 1 + .08 = (1 + r)(1 + .04) r = .0385 or 3.85% APR = m[(1 + EAR)1/m 1] APR = 12[(1 + .0385)1/12 1] APR = .0378 or 3.78% Monthly rate = APR / 12 Monthly rate = .0378 / 12 Monthly rate = .0031 or 0.31% Now we can find the real monthly withdrawal in retirement. Using the present value of an annuity equation and solving for the payment, we find: PVA = C({1 [1/(1 + r)]t } / r ) $1,518,692.24 = C({1 [1/(1 + .0031)]300 } / .0031) C = $7,832.87 This is the real dollar amount of the monthly withdrawals. The nominal monthly withdrawals will increase by the inflation rate each month. To find the nominal dollar amount of the last withdrawal, we can increase the real dollar withdrawal by the inflation rate. We can increase the real withdrawal by the effective annual inflation rate since we are only interested in the nominal amount of the last withdrawal. So, the last withdrawal in nominal terms will be: FV = PV(1 + r)t FV = $7,832.87(1 + .04)(30 + 25) FV = $67,725.85 30. In this problem, we need to calculate the future value of the annual savings after the five years of operations. The savings are the revenues minus the costs, or: Savings = Revenue Costs Since the annual fee and the number of members are increasing, we need to calculate the effective growth rate for revenues, which is: Effective growth rate = (1 + .06)(1 + .03) 1 Effective growth rate = .0918 or 9.18% CHAPTER 5 B-109 The revenue for the current year is the number of members times the annual fee, or: Current revenue = 500($700) Current revenue = $350,000 The revenue will grow at 9.18 percent, and the costs will grow at 2 percent, so the savings each year for the next five years will be: Year 1 2 3 4 5 Revenue $382,130.00 417,209.53 455,509.37 497,325.13 542,979.58 Costs $81,600.00 83,232.00 84,896.64 86,594.57 88,326.46 Savings $300,530.00 333,977.53 370,612.73 410,730.56 454,653.11 Now we can find the value of each years savings using the future value of a lump sum equation, so: FV = PV(1 + r)t Year 1 $300,530.00(1 + .09)4 = 2 $333,977.53(1 + .09)3 = 3 $370,612.73(1 + .09)2 = 4 $410,730.56(1 + .09)1 = 5 Total future value of savings = Future Value $424,222.62 432,510.59 440,324.98 447,696.31 454,653.11 $2,199,407.62 He will spend $400,000 on a luxury boat, so the value of his account will be: Value of account = $2,199,407.62 400,000 Value of account = $1,799,407.62 Now we can use the present value of an annuity equation to find the payment. Doing so, we find: PVA = C({1 [1/(1 + r)]t } / r ) $1,799,407.62 = C({1 [1/(1 + .09)]25 } / .09) C = $183,190.94 CHAPTER 5 B-110 Calculator Solutions 1. a. Enter Solve for b. Enter Solve for c. Enter Solve for 2. a. Enter Solve for b. Enter Solve for c. Enter Solve for 3. Enter Solve for 4. Enter Solve for $82.42 / $1,000 = 8.24% $41.21 2 = $82.42 26 N 60 N 2.5% I/Y 60 N 4.5% I/Y 30 N 7.5% I/Y 30 N 5% I/Y 30 N 2.5% I/Y PV $476.74 PMT $1,000 FV PV $231.38 PMT $1,000 FV PV $114.22 PMT $1,000 FV 60 N 3.5% I/Y PV $1,000.00 $35 PMT $1,000 FV PV $793.62 $35 PMT $1,000 FV PV $1,309.09 $1,050 PV 3.601% 2 = 7.20% $35 PMT $1,000 FV I/Y 3.601% $39 PMT $1,000 FV 27 N 3.65% I/Y $1,080 PV PMT $41.21 $1,000 FV CHAPTER 5 B-111 5. Enter Solve for 6. Enter Solve for 13. P0 Enter Solve for P1 Enter Solve for P3 Enter Solve for P8 Enter Solve for P12 Enter Solve for 25 N 7.60% I/Y PV 867.40 87,000 PV 64 PMT 1,000 FV 18 N I/Y 5.45% 5,400 PMT 100,000 FV Miller Corporation 26 N 3% I/Y PV $1,178.77 $40 PMT $1,000 FV 24 N 3% I/Y PV $1,169.36 $40 PMT $1,000 FV 20 N 3% I/Y PV $1,148.77 $40 PMT $1,000 FV 10 N 3% I/Y PV $1,085.30 $40 PMT $1,000 FV 2 N Modigliani Company 3% I/Y PV $1,019.13 $40 PMT $1,000 FV P0 Enter Solve for P1 Enter Solve for 26 N 4% I/Y PV $840.17 $30 PMT $1,000 FV 24 N 4% I/Y PV $847.53 $30 PMT $1,000 FV CHAPTER 5 B-112 P3 Enter Solve for P8 Enter Solve for P12 Enter Solve for 20 N 4% I/Y PV $864.10 $30 PMT $1,000 FV 10 N 4% I/Y PV $918.89 $30 PMT $1,000 FV 2 N 4% I/Y PV $981.14 $30 PMT $1,000 FV 14. If both bonds sell at par, the initial YTM on both bonds is the coupon rate, 7 percent. If the YTM suddenly rises to 9 percent: PLaurel Enter Solve for PHardy Enter Solve for 4 N $35 PV PMT $964.12 PLaurel% = ($964.12 1,000) / $1,000 = 3.59% $35 PV PMT $815.98 PHardy% = ($815.98 1,000) / $1,000 = 18.40% 4.5% I/Y 4.5% I/Y $1,000 FV 40 N $1,000 FV If the YTM suddenly falls to 5 percent: PLaurel Enter 4 2.5% $35 N I/Y PV PMT Solve for $1,037.62 PLaurel % = ($1,037.62 1,000) / $1,000 = + 3.76% PHardy Enter Solve for 40 N 2.5% I/Y $1,000 FV $35 $1,000 PV PMT FV $1,251.03 PHardy % = ($1,251.03 1,000) / $1,000 = + 25.10% All else the same, the longer the maturity of a bond, the greater is its price sensitivity to changes in interest rates. CHAPTER 5 B-113 15. Initially, at a YTM of 10 percent, the prices of the two bonds are: PFaulk Enter Solve for PGonas Enter Solve for 28 N 5% I/Y 28 N 5% I/Y $30 PMT $1,000 FV PV $702.04 PV $1,297.96 $70 PMT $1,000 FV If the YTM rises from 10 percent to 12 percent: PFaulk Enter 28 6% N I/Y PV Solve for $597.82 PFaulk% = ($597.82 702.04) / $702.04 = 14.85% PGonas Enter 28 N 6% I/Y $30 PMT $1,000 FV PV Solve for $1,134.06 PGonas% = ($1,134.06 1,297.96) / $1,297.96 = 12.63 If the YTM declines from 10 percent to 8 percent: PFaulk Enter 28 4% N I/Y PV Solve for $833.37 PFaulk% = ($833.37 702.04) / $702.04 = +18.71% PGonas Enter 28 N 4% I/Y $70 PMT $1,000 FV $30 PMT $1,000 FV PV Solve for $1,499.89 PGonas% = ($1,499.89 1,297.96) / $1,297.96 = +15.56% $70 PMT $1,000 FV All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to changes in interest rates. 16. Enter Solve for 18 N $1,050 $37 I/Y PV PMT 3.326% YTM = 3.326% 2 = 6.65% $1,000 FV CHAPTER 5 B-114 17. The company should set the coupon rate on its new bonds equal to the required return; the required return can be observed in the market by finding the YTM on outstanding bonds of the company. Enter Solve for 50 N I/Y 3.721% $1,063 PV 3.721% 2 = 7.44% 20. Current yield = .0813 = $70/P0 ; P0 = $861.01 Enter Solve for 21. Enter Solve for 23. Bond P P0 Enter Solve for P1 Enter 7 N 7% I/Y N 25.0045 26 N 8.34% I/Y $861.02 PV $70 PMT $1,000 FV $40 PMT $1,000 FV I/Y 3.366% $1,052.13 PV 3.366% 2 = 6.73% $36.70 PMT $1,000 FV 8 N 7% I/Y PV $1,119.43 $90 PMT $1,000 FV $90 $1,000 PV PMT FV Solve for $1,107.79 Current yield = $90 / $1,119.43 = 8.04% Capital gains yield = ($1,107.79 1,119.43) / $1,119.43 = 1.04% Bond D P0 Enter Solve for P1 Enter 7 N 7% I/Y 8 N 7% I/Y PV $880.57 $50 PMT $1,000 FV Solve for Current yield = $50 / $880.57 = 5.68% Capital gains yield = ($892.21 880.57) / $88057 = 1.32% All else held constant, premium bonds pay a higher current income while having price depreciation as maturity nears; discount bonds pay a lower current income but have price appreciation as maturity nears. For either bond, the total return is still 7%, but this return is distributed differently between current income and capital gains. PV $892.21 $50 PMT $1,000 FV CHAPTER 5 B-115 24. a. Enter I/Y Solve for 7.65% This is the rate of return you expect to earn on your investment when you purchase the bond. b. Enter Solve for 19 N 6.65% I/Y $59 PMT $1,000 FV 21 N $820 PV $59 PMT $1,000 FV PV $920.53 The HPY is: Enter I/Y Solve for 12.95% The realized HPY is greater than the expected YTM when the bond was bought because interest rates dropped by 1 percent; bond prices rise when yields fall. 25. PM CFo $0 C01 $0 F01 12 C02 $800 F02 16 C03 $1,000 F03 11 C04 $21,000 F04 1 I = 3.5% NPV CPT $15,142.45 PN Enter Solve for 40 N 3.5% I/Y $20,000 FV 2 N $820 PV $59 PMT $920.53 FV PV $5,051.45 PMT 28. Real return: 1 + .094 = (1 + r)(1 + .035); r = 5.70% Enter 5.70% 12 NOM EFF C/Y Solve for 5.55% Enter Solve for 12 30 N 5.55% / 12 I/Y PV $5,318.46 $7 PMT FV CHAPTER 5 B-116 29. Real return for stock account: 1 + .12 = (1 + r)(1 + .04); r = 7.6923% Enter 7.6923% 12 NOM EFF C/Y Solve for 7.4337% Real return for bond account: 1 + .07 = (1 + r)(1 + .04); r = 2.8846% Enter 2.8846% 12 NOM EFF C/Y Solve for 2.8472% Real return post-retirement: 1 + .08 = (1 + r)(1 + .04); r = 3.8462% Enter 3.8462% 12 NOM EFF C/Y Solve for 3.7800% Stock portfolio value: Enter 12 30 N Solve for Bond portfolio value: Enter 12 30 N Solve for 7.4337% / 12 I/Y $950 PMT PV FV $1,263,217.0 7 2.8472% / 12 I/Y PV $450 PMT FV $255,475.17 Retirement value = $1,263,217.07 + 255,475.17 = $1,518,692.24 Retirement withdrawal: Enter 25 12 N Solve for The last withdrawal in real terms is: Enter 30 + 25 4% N I/Y Solve for 30. Future value of savings: Year 1: Enter 4 N Solve for Year 2: Enter 3 N $7,532.87 PV 3.7800% / 12 I/Y $1,518,692.2 4 PV PMT $7,532.87 FV PMT FV $67,725.85 9% I/Y $300,530 PV PMT FV $424,222.62 9% I/Y $333,977.53 PV PMT FV CHAPTER 5 B-117 Solve for $432,510.59 CHAPTER 5 B-118 Year 3: Enter Solve for Year 4: Enter Solve for 2 N 9% I/Y $370,612.73 PV PMT FV $440,324.98 1 N 9% I/Y $410,730.56 PV PMT FV $447,696.31 Future value = $424,222.62 + 432,510.59 + 440,324.98 + 447,696.31 + 454,653.11 Future value = $2,199,407.62 He will spend $400,000 on a luxury boat, so the value of his account will be: Value of account = $2,199,407.62 400,000 Value of account = $1,799,407.62 Enter Solve for 25 N 9% I/Y $1,799,407.6 2 PV PMT $183,190.94 FV CHAPTER 6 STOCK VALUATION Answers to Concept Questions 1. 2. 3. The value of any investment depends on the present value of its cash flows; i.e., what investors will actually receive. The cash flows from a share of stock are the dividends. Investors believe the company will eventually start paying dividends (or be sold to another company). In general, companies that need the cash will often forgo dividends since dividends are a cash expense. Young, growing companies with profitable investment opportunities are one example; another example is a company in financial distress. This question is examined in depth in a later chapter. The general method for valuing a share of stock is to find the present value of all expected future dividends. The dividend growth model presented in the text is only valid (i) if dividends are expected to occur forever; that is, the stock provides dividends in perpetuity, and (ii) if a constant growth rate of dividends occurs forever. A violation of the first assumption might be a company that is expected to cease operations and dissolve itself some finite number of years from now. The stock of such a company would be valued by applying the general method of valuation explained in this chapter. A violation of the second assumption might be a start-up firm that isnt currently paying any dividends, but is expected to eventually start making dividend payments some number of years from now. This stock would also be valued by the general dividend valuation method explained in this chapter. The common stock probably has a higher price because the dividend can grow, whereas it is fixed on the preferred. However, the preferred is less risky because of the dividend and liquidation preference, so it is possible the preferred could be worth more, depending on the circumstances. The two components are the dividend yield and the capital gains yield. For most companies, the capital gains yield is larger. This is easy to see for companies that pay no dividends. For companies that do pay dividends, the dividend yields are rarely over five percent and are often much less. Yes. If the dividend grows at a steady rate, so does the stock price. In other words, the dividend growth rate and the capital gains yield are the same. The three factors are: 1) The companys future growth opportunities. 2) The companys level of risk, which determines the interest rate used to discount cash flows. 3) The accounting method used. In a corporate election, you can buy votes (by buying shares), so money can be used to influence or even determine the outcome. Many would argue the same is true in political elections, but, in principle at least, no one has more than one vote. 4. 5. 6. 7. 8. 9. 10. It wouldnt seem to be. Investors who dont like the voting features of a particular class of stock are under no obligation to buy it. CHAPTER 6 B-120 11. Investors buy such stock because they want it, recognizing that the shares have no voting power. Presumably, investors pay a little less for such shares than they would otherwise. 12. Presumably, the current stock value reflects the risk, timing and magnitude of all future cash flows, both short-term and long-term. If this is correct, then the statement is false. Solutions to Questions and Problems NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. The constant dividend growth model is: Pt = Dt (1 + g) / (R g) So, the price of the stock today is: P0 = D0 (1 + g) / (R g) = $2.15 (1.04) / (.12 .04) = $27.95 The dividend at year 4 is the dividend today times the FVIF for the growth rate in dividends and four years, so: P3 = D3 (1 + g) / (R g) = D0 (1 + g)4 / (R g) = $2.15 (1.04)4 / (.12 .04) = $31.44 We can do the same thing to find the dividend in Year 16, which gives us the price in Year 15, so: P15 = D15 (1 + g) / (R g) = D0 (1 + g)16 / (R g) = $2.15 (1.04)16 / (.12 .04) = $50.34 There is another feature of the constant dividend growth model: The stock price grows at the dividend growth rate. So, if we know the stock price today, we can find the future value for any time in the future we want to calculate the stock price. In this problem, we want to know the stock price in three years, and we have already calculated the stock price today. The stock price in three years will be: P3 = P0(1 + g)3 = $27.95(1 + .04)3 = $31.44 And the stock price in 15 years will be: P15 = P0(1 + g)15 = $27.95(1 + .04)15 = $50.35 2. We need to find the required return of the stock. Using the constant growth model, we can solve the equation for R. Doing so, we find: R = (D1 / P0) + g = ($2.85 / $84) + .045 = .0789 or 7.89% CHAPTER 6 B-121 3. The dividend yield is the dividend next year divided by the current price, so the dividend yield is: Dividend yield = D1 / P0 = $2.85 / $84 = .0339 or 3.39% The capital gains yield, or percentage increase in the stock price, is the same as the dividend growth rate, so: Capital gains yield = 4.5% 4. Using the constant growth model, we find the price of the stock today is: P0 = D1 / (R g) = $2.90 / (.11 .0475) = $46.40 5. The required return of a stock is made up of two parts: The dividend yield and the capital gains yield. So, the required return of this stock is: R = Dividend yield + Capital gains yield = .044 + .052 = .0960 or 9.60% 6. We know the stock has a required return of 12 percent, and the dividend and capital gains yield are equal, so: Dividend yield = 1/2(.12) = .06 = Capital gains yield Now we know both the dividend yield and capital gains yield. The dividend is simply the stock price times the dividend yield, so: D1 = .06 ($73) = $4.38 This is the dividend next year. The question asks for the dividend this year. Using the relationship between the dividend this year and the dividend next year: D1 = D0(1 + g) We can solve for the dividend that was just paid: $4.38 = D0 (1 + .06) D0 = $4.38 / 1.06 = $4.13 7. The price of any financial instrument is the PV of the future cash flows. The future dividends of this stock are an annuity for 8 years, so the price of the stock is the PVA, which will be: P0 = $14(PVIFA11%,8) = $72.05 8. The price of a share of preferred stock is the dividend divided by the required return. This is the same equation as the constant growth model, with a dividend growth rate of zero percent. Remember that most preferred stock pays a fixed dividend, so the growth rate is zero. Using this equation, we find the price per share of the preferred stock is: R = D/P0 = $4.70/$103 = .0456 or 4.56% CHAPTER 6 B-122 9. The growth rate of earnings is the return on equity times the retention ratio, so: g = ROE b g = .14(.70) g = .0980 or 9.80% To find next years earnings, we simply multiply the current earnings times one plus the growth rate, so: Next years earnings = Current earnings(1 + g) Next years earnings = $30,000,000(1 + .0980) Next years earnings = $32,940,000 Intermediate 10. This stock has a constant growth rate of dividends, but the required return changes twice. To find the value of the stock today, we will begin by finding the price of the stock at Year 6, when both the dividend growth rate and the required return are stable forever. The price of the stock in Year 6 will be the dividend in Year 7, divided by the required return minus the growth rate in dividends. So: P6 = D6 (1 + g) / (R g) = D0 (1 + g)7 / (R g) = $2.40(1.05)7 / (.11 .05) = $56.28 Now we can find the price of the stock in Year 3. We need to find the price here since the required return changes at that time. The price of the stock in Year 3 is the PV of the dividends in Years 4, 5, and 6, plus the PV of the stock price in Year 6. The price of the stock in Year 3 is: P3 = $2.40(1.05)4 / 1.14 + $2.40(1.05)5 / 1.142 + $2.40(1.05)6 / 1.143 + $56.28 / 1.143 P3 = $45.08 Finally, we can find the price of the stock today. The price today will be the PV of the dividends in Years 1, 2, and 3, plus the PV of the stock in Year 3. The price of the stock today is: P0 = $2.40(1.05) / 1.16 + $2.40(1.05)2 / (1.16)2 + $2.40(1.05)3 / (1.16)3 + $45.08 / (1.16)3 P0 = $34.80 11. Here we have a stock that pays no dividends for 10 years. Once the stock begins paying dividends, it will have a constant growth rate of dividends. We can use the constant growth model at that point. It is important to remember that general form of the constant dividend growth formula is: Pt = [Dt (1 + g)] / (R g) This means that since we will use the dividend in Year 13, we will be finding the stock price in Year 12. The dividend growth model is similar to the PVA and the PV of a perpetuity: The equation gives you the PV one period before the first payment. So, the price of the stock in Year 12 will be: P12 = D13 / (R g) = $11.00 / (.13 .055) = $146.67 The price of the stock today is simply the PV of the stock price in the future. We simply discount the future stock price at the required return. The price of the stock today will be: P0 = $146.67 / 1.1312 = $33.84 CHAPTER 6 B-123 12. The price of a stock is the PV of the future dividends. This stock is paying five dividends, so the price of the stock is the PV of these dividends using the required return. The price of the stock is: P0 = $15 / 1.13 + $18 / 1.132 + $21 / 1.133 + $24 / 1.134 + $27 / 1.135 = $71.30 13. With supernormal dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the supernormal growth period. The stock begins constant growth in Year 5, so we can find the price of the stock in Year 4, one year before the constant dividend growth begins, as: P4 = D4 (1 + g) / (R g) = $3(1.05) / (.13 .05) = $39.38 The price of the stock today is the PV of the first four dividends, plus the PV of the Year 4 stock price. So, the price of the stock today will be: P0 = $10 / 1.13 + $8 / 1.132 + $5 / 1.133 + ($3.00 + 39.38) / 1.134 = $44.57 14. With supernormal dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the supernormal growth period. The stock begins constant growth in Year 4, so we can find the price of the stock in Year 3, one year before the constant dividend growth begins as: P3 = D3 (1 + g) / (R g) = D0 (1 + g1)3 (1 + g2) / (R g2) = $2.40(1.30)3(1.07) / (.10 .07) = $188.06 The price of the stock today is the PV of the first three dividends, plus the PV of the Year 3 stock price. The price of the stock today will be: P0 = $2.40(1.30) / 1.10 + $2.40(1.30)2 / 1.102 + $2.40(1.30)3 / 1.103 + $188.06 / 1.103 P0 = $151.44 15. Here we need to find the dividend next year for a stock experiencing supernormal growth. We know the stock price, the dividend growth rates, and the required return, but not the dividend. First, we need to realize that the dividend in Year 3 is the current dividend times the FVIF. The dividend in Year 3 will be: D3 = D0 (1.27)3 And the dividend in Year 4 will be the dividend in Year 3 times one plus the growth rate, or: D4 = D0 (1.27)3 (1.17) The stock begins constant growth after the 4th dividend is paid, so we can find the price of the stock in Year 4 as the dividend in Year 5, divided by the required return minus the growth rate. The equation for the price of the stock in Year 4 is: P4 = D4 (1 + g) / (R g) CHAPTER 6 B-124 Now we can substitute the previous dividend in Year 4 into this equation as follows: P4 = D0 (1 + g1)3 (1 + g2) (1 + g3) / (R g3) P4 = D0 (1.27)3 (1.17) (1.07) / (.13 .07) = 51.29D0 When we solve this equation, we find that the stock price in Year 4 is 56.00 times as large as the dividend today. Now we need to find the equation for the stock price today. The stock price today is the PV of the dividends in Years 1, 2, 3, and 4, plus the PV of the Year 4 price. So: P0 = D0(1.27)/1.13 + D0(1.27)2/1.132 + D0(1.27)3/1.133+ D0(1.27)3(1.17)/1.134 + 51.29D0/1.134 We can factor out D0 in the equation, and combine the last two terms. Doing so, we get: P0 = $65.00 = D0{1.27/1.13 + 1.272/1.132 + 1.273/1.133 + [(1.27)3(1.17) + 51.29] / 1.134} Reducing the equation even further by solving all of the terms in the braces, we get: $65 = $37.93D0 D0 = $65.00 / $37.93 = $1.71 This is the dividend today, so the projected dividend for the next year will be: D1 = $1.71(1.27) = $2.18 16. The constant growth model can be applied even if the dividends are declining by a constant percentage, just make sure to recognize the negative growth. So, the price of the stock today will be: P0 = D0 (1 + g) / (R g) = $12(1 .04) / [(.09 (.04)] = $88.62 17. We are given the stock price, the dividend growth rate, and the required return, and are asked to find the dividend. Using the constant dividend growth model, we get: P0 = $57.25 = D0 (1 + g) / (R g) Solving this equation for the dividend gives us: D0 = $57.25(.11 .05) / (1.05) = $3.27 18. The price of a share of preferred stock is the dividend payment divided by the required return. We know the dividend payment in Year 10, so we can find the price of the stock in Year 9, one year before the first dividend payment. Doing so, we get: P4 = $10 / .06 = $166.67 The price of the stock today is the PV of the stock price in the future, so the price today will be: P0 = $166.67 / (1.06)9 = $98.65 CHAPTER 6 B-125 19. The dividend yield is the dividend divided by the stock price, so: Dividend yield = Dividend / Stock price .024 = Dividend / $24.90 Dividend = $0.60 The Net Chg of the stock shows the stock increased by $0.13 on this day, so the closing stock price yesterday was: Yesterdays closing price = $24.90 0.13 = $24.77 To find the net income, we need to find the EPS. The stock quote tells us the P/E ratio for the stock is 26. Since we know the stock price as well, we can use the P/E ratio to solve for EPS as follows: P/E = 26 = Stock price / EPS = $24.90 / EPS EPS = $24.90 / 26 = $0.958 We know that EPS is just the total net income divided by the number of shares outstanding, so: EPS = NI / Shares = $0.958 = NI / 25,000,000 NI = $0.958(25,000,000) = $23,942,308 20. To find the number of shares owned, we can divide the amount invested by the stock price. The share price of any financial asset is the present value of the cash flows, so, to find the price of the stock we need to find the cash flows. The cash flows are the two dividend payments plus the sale price. We also need to find the aftertax dividends since the assumption is all dividends are taxed at the same rate for all investors. The aftertax dividends are the dividends times one minus the tax rate, so: Year 1 aftertax dividend = $1.80(1 .28) Year 1 aftertax dividend = $1.30 Year 2 aftertax dividend = $2.20(1 .28) Year 2 aftertax dividend = $1.58 We can now discount all cash flows from the stock at the required return. Doing so, we find the price of the stock is: P = $1.30/1.15 + $1.58/(1.15)2 + $75/(1+.15)3 P = $60.44 The number of shares owned is the total investment divided by the stock price, which is: Shares owned = $100,000 / $60.44 Shares owned = 1,654.64 CHAPTER 6 B-126 21. Here we have a stock paying a constant dividend for a fixed period, and an increasing dividend thereafter. We need to find the present value of the two different cash flows using the appropriate quarterly interest rate. The constant dividend is an annuity, so the present value of these dividends is: PVA = C(PVIFAR,t) PVA = $0.80(PVIFA2.5%,12) PVA = $8.21 Now we can find the present value of the dividends beyond the constant dividend phase. Using the present value of a growing annuity equation, we find: P12 = D13 / (R g) P12 = $0.80(1 + .012) / (.025 .012) P12 = $62.28 This is the price of the stock immediately after it has paid the last constant dividend. So, the present value of the future price is: PV = $62.28 / (1 + .025)12 PV = $46.31 The price today is the sum of the present value of the two cash flows, so: P0 = $8.21 + 46.31 P0 = $54.51 22. Here we need to find the dividend next year for a stock with nonconstant growth. We know the stock price, the dividend growth rates, and the required return, but not the dividend. First, we need to realize that the dividend in Year 3 is the constant dividend times the FVIF. The dividend in Year 3 will be: D3 = D(1.045) The equation for the stock price will be the present value of the constant dividends, plus the present value of the future stock price, or: P0 = D / 1.11 + D /1.112 + D(1.045)/(.11 .05)/1.112 $43 = D / 1.11 + D /1.112 + D(1.045)/(.11 .05)/1.112 We can factor out D0 in the equation. Doing so, we get: $43 = D{1/1.11 + 1/1.112 + [(1.045)/(.11 .05)] / 1.112} Reducing the equation even further by solving all of the terms in the braces, we get: $43 = D(14.7609) D = $43 / 14.7609 = $2.91 CHAPTER 6 B-127 23. The required return of a stock consists of two components, the capital gains yield and the dividend yield. In the constant dividend growth model (growing perpetuity equation), the capital gains yield is the same as the dividend growth rate, or algebraically: R = D1/P0 + g We can find the dividend growth rate by the sustainable growth rate equation, or: g = ROE b g = .14 .60 g = .0840 or 8.40% This is also the growth rate in dividends. To find the current dividend, we can use the information provided about the net income, shares outstanding, and payout ratio. The total dividends paid is the net income times the payout ratio. To find the dividend per share, we can divide the total dividends paid by the number of shares outstanding. So: Dividend per share = (Net income Payout ratio) / Shares outstanding Dividend per share = ($12,000,000 .40) / 2,000,000 Dividend per share = $2.40 Now we can use the initial equation for the required return. We must remember that the equation uses the dividend in one year, so: R = D1/P0 + g R = $2.40(1 + .0840)/$85 + .0840 R = .1146 or 11.46% 24. First, we need to find the annual dividend growth rate over the past four years. To do this, we can use the future value of a lump sum equation, and solve for the interest rate. Doing so, we find the dividend growth rate over the past four years was: FV = PV(1 + R)t $2.43 = $1.70(1 + R)4 R = ($2.43 / $1.70)1/4 1 R = .0934 or 9.34% We know the dividend will grow at this rate for five years before slowing to a constant rate indefinitely. So, the dividend amount in seven years will be: D7 = D0(1 + g1)5(1 + g2)2 D7 = $2.43(1 + .0934)5(1 + .05)2 D7 = $4.19 25. a. We can find the price of all the outstanding company stock by using the dividends the same way we would value an individual share. Since earnings are equal to dividends, and there is no growth, the value of the companys stock today is the present value of a perpetuity, so: P=D/R P = $1,100,000 / .12 P = $9,166,666.67 CHAPTER 6 B-128 The price-earnings ratio is the stock price divided by the current earnings, so the price-earnings ratio of each company with no growth is: P/E = Price / Earnings P/E = $9,166,666.67 / $1,100,000 P/E = 8.33 times b. Since the earnings have increased, the price of the stock will increase. The new price of all the outstanding company stock is: P=D/R P = ($1,100,000 + 220,000) / .12 P = $11,000,000 The price-earnings ratio is the stock price divided by the current earnings, so the price-earnings with the increased earnings is: P/E = Price / Earnings P/E = $11,000,000 / $1,100,000 P/E = 10.00 times c. Since the earnings have increased, the price of the stock will increase. The new price of the all the outstanding company stock is: P=D/R P = ($1,100,000 + 440,000) / .12 P = $12,833,333.33 The price-earnings ratio is the stock price divided by the current earnings, so the price-earnings with the increased earnings is: P/E = Price / Earnings P/E = $12,833,333.33 / $1,100,000 P/E = 11.67 times 26. a. If the company does not make any new investments, the stock price will be the present value of the constant perpetual dividends. In this case, all earnings are paid dividends, so, applying the perpetuity equation, we get: P = Dividend / R P = $7.50 / .11 P = $68.18 b. The investment is a one-time investment that creates an increase in EPS for two years. To calculate the new stock price, we need the cash cow price plus the NPVGO. In this case, the NPVGO is simply the present value of the investment plus the present value of the increases in EPS. So, the NPVGO will be: NPVGO = C1 / (1 + R) + C2 / (1 + R)2 + C3 / (1 + R)3 NPVGO = $1.10 / 1.11 + $2.30 / 1.112 + $2.60 / 1.113 NPVGO = $2.78 CHAPTER 6 B-129 So, the price of the stock if the company undertakes the investment opportunity will be: P = $68.18 + 2.78 P = $70.96 c. After the project is over, and the earnings increase no longer exists, the price of the stock will revert back to $68.18, the value of the company as a cash cow. 27. a. The price of the stock is the present value of the dividends. Since earnings are equal to dividends, we can find the present value of the earnings to calculate the stock price. Also, since we are excluding taxes, the earnings will be the revenues minus the costs. We simply need to find the present value of all future earnings to find the price of the stock. The present value of the revenues is: PVRevenue = C1 / (R g) PVRevenue = $8,000,000(1 + .05) / (.13 .05) PVRevenue = $105,000,000 And the present value of the costs will be: PVCosts = C1 / (R g) PVCosts = $3,600,000(1 + .05) / (.13 .05) PVCosts = $47,250,000 Since there are no taxes, the present value of the companys earnings and dividends will be: PVDividends = $105,000,000 47,250,000 PVDividends = $57,750,000 Note that since revenues and costs increase at the same rate, we could have found the present value of future dividends as the present value of current dividends. Doing so, we find: D0 = Revenue0 Costs0 D0 = $8,000,000 3,600,000 D0 = $4,400,000 Now, applying the growing perpetuity equation, we find: PVDividends = C1 / (R g) PVDividends = $4,400,000(1 + .05) / (.13 .05) PVDividends = $57,750,000 This is the same answer we found previously. The price per share of stock is the total value of the companys stock divided by the shares outstanding, or: P = Value of all stock / Shares outstanding P = $57,750,000 / 1,000,000 P = $57.75 CHAPTER 6 B-130 b. The value of a share of stock in a company is the present value of its current operations, plus the present value of growth opportunities. To find the present value of the growth opportunities, we need to discount the cash outlay in Year 1 back to the present, and find the value today of the increase in earnings. The increase in earnings is a perpetuity, which we must discount back to today. So, the value of the growth opportunity is: NPVGO = C0 + C1 / (1 + R) + (C2 / R) / (1 + R) NPVGO = $3,000,000 $4,000,000 / (1 + .13) + ($2,000,000 / .15) / (1 + .13) NPVGO = $7,074,880.87 To find the value of the growth opportunity on a per share basis, we must divide this amount by the number of shares outstanding, which gives us: NPVGOPer share = $7,074,880.87 / $1,000,000 NPVGOPer share = $7.07 The stock price will increase by $7.07 per share. The new stock price will be: New stock price = $57.75 + 7.07 New stock price = $64.82 28. a. If the company continues its current operations, it will not grow, so we can value the company as a cash cow. The total value of the company as a cash cow is the present value of the future earnings, which are a perpetuity, so: Cash cow value of company = C / R Cash cow value of company = $75,000,000 / .12 Cash cow value of company = $625,000,000 The value per share is the total value of the company divided by the shares outstanding, so: Share price = $625,000,000 / 14,000,000 Share price = $44.64 b. To find the value of the investment, we need to find the NPV of the growth opportunities. The initial cash flow occurs today, so it does not need to be discounted. The earnings growth is a perpetuity. Using the present value of a perpetuity equation will give us the value of the earnings growth one period from today, so we need to discount this back to today. The NPVGO of the investment opportunity is: NPVGO = C0 + C1 / (1 + R) + (C2 / R) / (1 + R) NPVGO = $9,000,000 5,000,000 / (1 + .12) + ($8,000,000 / .12) / (1 + .12) NPVGO = $46,059,523.81 c. The price of a share of stock is the cash cow value plus the NPVGO. We have already calculated the NPVGO for the entire project, so we need to find the NPVGO on a per share basis. The NPVGO on a per share basis is the NPVGO of the project divided by the shares outstanding, which is: NPVGO per share = $46,059,523.81 / 14,000,000 NPVGO per share = $3.29 CHAPTER 6 B-131 This means the per share stock price if the company undertakes the project is: Share price = Cash cow price + NPVGO per share Share price = $44.64 + 3.29 Share price = $47.93 Challenge 29. a. If the company does not make any new investments, the stock price will be the present value of the constant perpetual dividends. In this case, all earnings are paid as dividends, so, applying the perpetuity equation, we get: P = Dividend / R P = $9 / .11 P = $81.82 b. The investment occurs every year in the growth opportunity, so the opportunity is a growing perpetuity. So, we first need to find the growth rate. The growth rate is: g = Retention Ratio Return on Retained Earnings g = 0.30 0.15 g = 0.045 or 4.5% Next, we need to calculate the NPV of the investment. During year 3, 30 percent of the earnings will be reinvested. Therefore, $2.70 is invested ($9 .30). One year later, the shareholders receive a 15 percent return on the investment, or $0.41 ($2.70 .15), in perpetuity. The perpetuity formula values that stream as of year 3. Since the investment opportunity will continue indefinitely and grows at 4.5 percent, apply the growing perpetuity formula to calculate the NPV of the investment as of year 2. Discount that value back two years to today. NPVGO = [(Investment + Return / R) / (R g)] / (1 + R)2 NPVGO = [($2.70 + $0.41 / .11) / (0.11 0.045)] / (1.11)2 NPVGO = $12.26 The value of the stock is the PV of the firm without making the investment plus the NPV of the investment, or: P = PV(EPS) + NPVGO P = $81.82 + $12.26 P = $94.08 CHAPTER 6 B-132 30. We are asked to find the dividend yield and capital gains yield for each of the stocks. All of the stocks have a 17 percent required return, which is the sum of the dividend yield and the capital gains yield. To find the components of the total return, we need to find the stock price for each stock. Using this stock price and the dividend, we can calculate the dividend yield. The capital gains yield for the stock will be the total return (required return) minus the dividend yield. W: P0 = D0(1 + g) / (R g) = $4.50(1.10)/(.17 .10) = $71.70 Dividend yield = D1/P0 = 4.50(1.10)/$71.70 = .07 or 7% Capital gains yield = .17 .07 = .10 or 10% X: P0 = D0(1 + g) / (R g) = $4.50/(.17 0) = $26.47 Dividend yield = D1/P0 = $4.50/$26.47 = .17 or 17% Capital gains yield = .17 .17 = 0% Y: P0 = D0(1 + g) / (R g) = $4.50(1 .05)/(.17 + .05) = $19.43 Dividend yield = D1/P0 = $4.50(0.95)/$19.43 = .22 or 22% Capital gains yield = .17 .22 = .05 or 5% Z: P2 = D2(1 + g) / (R g) = D0(1 + g1)2(1 + g2)/(R g2) = $4.50(1.30)2(1.07)/(.17 .07) P2 = $91.26 P0 = $4.50 (1.30) / (1.17) + $4.50 (1.30)2 / (1.17)2 + $91.26 / (1.17)2 P0 = $77.22 Dividend yield = D1/P0 = $4.50(1.30)/$77.22 = .0758 or 7.58% Capital gains yield = .17 .0758 = .0942 or 9.42% In all cases, the required return is 17 percent, but the return is distributed differently between current income and capital gains. High-growth stocks have an appreciable capital gains component but a relatively small current income yield; conversely, mature, negative-growth stocks provide a high current income but also price depreciation over time. 31. a. Using the constant growth model, the price of the stock paying annual dividends will be: P0 = D0(1 + g) / (R g) = $2.80(1.05)/(.13 .05) = $36.75 CHAPTER 6 B-133 b. If the company pays quarterly dividends instead of annual dividends, the quarterly dividend will be one-fourth of annual dividend, or: Quarterly dividend: $2.80(1.05)/4 = $0.7350 To find the equivalent annual dividend, we must assume that the quarterly dividends are reinvested at the required return. We can then use this interest rate to find the equivalent annual dividend. In other words, when we receive the quarterly dividend, we reinvest it at the required return on the stock. So, the effective quarterly rate is: Effective quarterly rate: 1.13.25 1 = .0310 The effective annual dividend will be the FVA of the quarterly dividend payments at the effective quarterly required return. In this case, the effective annual dividend will be: Effective D1 = $0.7350(FVIFA3.10%,4) = $3.08 Now, we can use the constant growth model to find the current stock price as: P0 = $3.08/(.13 .05) = $38.50 Note that we cannot simply find the quarterly effective required return and growth rate to find the value of the stock. This would assume the dividends increased each quarter, not each year. 32. a. If the company does not make any new investments, the stock price will be the present value of the constant perpetual dividends. In this case, all earnings are paid as dividends, so, applying the perpetuity equation, we get: P = Dividend / R P = $7.50 / .13 P = $57.69 b. The investment occurs every year in the growth opportunity, so the opportunity is a growing perpetuity. So, we first need to find the growth rate. The growth rate is: g = Retention Ratio Return on Retained Earnings g = 0.25 0.11 g = 0.0275 or 2.75% Next, we need to calculate the NPV of the investment. During year 3, 25 percent of the earnings will be reinvested. Therefore, $1.88 is invested ($7.50 .25). One year later, the shareholders receive an 11 percent return on the investment, or $0.206 ($1.88 .11), in perpetuity. The perpetuity formula values that stream as of year 3. Since the investment opportunity will continue indefinitely and grows at 2.2 percent, apply the growing perpetuity formula to calculate the NPV of the investment as of year 2. Discount that value back two years to today. NPVGO = [(Investment + Return / R) / (R g)] / (1 + R)2 NPVGO = [($1.88 + $0.206 / .13) / (0.13 0.0275)] / (1.13)2 NPVGO = $2.20 CHAPTER 6 B-134 The value of the stock is the PV of the firm without making the investment plus the NPV of the investment, or: P = PV(EPS) + NPVGO P = $57.69 2.20 P = $55.49 c. Zero percent! There is no retention ratio which would make the project profitable for the company. If the company retains more earnings, the growth rate of the earnings on the investment will increase, but the project will still not be profitable. Since the return of the project is less than the required return on the company stock, the project is never worthwhile. In fact, the more the company retains and invests in the project, the less valuable the stock becomes. 33. Here we have a stock with supernormal growth, but the dividend growth changes every year for the first four years. We can find the price of the stock in Year 3 since the dividend growth rate is constant after the third dividend. The price of the stock in Year 3 will be the dividend in Year 4, divided by the required return minus the constant dividend growth rate. So, the price in Year 3 will be: P3 = $3.90(1.16)(1.12)(1.08)(1.04) / (.12 .04) = $71.14 The price of the stock today will be the PV of the first three dividends, plus the PV of the stock price in Year 3, so: P0 = $3.90(1.16)/(1.12) + $3.90(1.16)(1.12)/1.122 + $3.90(1.16)(1.12)(1.08)/1.123 + $71.14/1.123 P0 = $62.61 34. Here we want to find the required return that makes the PV of the dividends equal to the current stock price. The equation for the stock price is: P = $3.90(1.16)/(1 + R) + $3.90(1.16)(1.12)/(1 + R)2 + $3.90(1.16)(1.12)(1.08)/(1 + R)3 + [$3.90(1.16)(1.12)(1.08)(1.04)/(R .04)]/(1 + R)3 = $73.05 We need to find the roots of this equation. Using spreadsheet, trial and error, or a calculator with a root solving function, we find that: R = .1087 or 10.87% CHAPTER 7 NET PRESENT VALUE AND OTHER INVESTMENT CRITERIA Answers to Concept Questions 1. Assuming conventional cash flows, a payback period less than the projects life means that the NPV is positive for a zero discount rate, but nothing more definitive can be said. For discount rates greater than zero, the payback period will still be less than the projects life, but the NPV may be positive, zero, or negative, depending on whether the discount rate is less than, equal to, or greater than the IRR. The discounted payback includes the effect of the relevant discount rate. If a projects discounted payback period is less than the projects life, it must be the case that NPV is positive. Assuming conventional cash flows, if a project has a positive NPV for a certain discount rate, then it will also have a positive NPV for a zero discount rate; thus, the payback period must be less than the project life. Since discounted payback is calculated at the same discount rate as is NPV, if NPV is positive, the discounted payback period must be less than the projects life. If NPV is positive, then the present value of future cash inflows is greater than the initial investment cost; thus, PI must be greater than 1. If NPV is positive for a certain discount rate R, then it will be zero for some larger discount rate R*; thus, the IRR must be greater than the required return. a. Payback period is simply the accounting break-even point of a series of cash flows. To actually compute the payback period, it is assumed that any cash flow occurring during a given period is realized continuously throughout the period, and not at a single point in time. The payback is then the point in time for the series of cash flows when the initial cash outlays are fully recovered. Given some predetermined cutoff for the payback period, the decision rule is to accept projects that pay back before this cutoff, and reject projects that take longer to pay back. The worst problem associated with the payback period is that it ignores the time value of money. In addition, the selection of a hurdle point for the payback period is an arbitrary exercise that lacks any steadfast rule or method. The payback period is biased towards shortterm projects; it fully ignores any cash flows that occur after the cutoff point. 2. 3. CHAPTER 7 B-136 b. The average accounting return is interpreted as an average measure of the accounting performance of a project over time, computed as some average profit measure attributable to the project divided by some average balance sheet value for the project. This text computes AAR as average net income with respect to average (total) book value. Given some predetermined cutoff for AAR, the decision rule is to accept projects with an AAR in excess of the target measure, and reject all other projects. AAR is not a measure of cash flows or market value, but is rather a measure of financial statement accounts that often bear little resemblance to the relevant value of a project. In addition, the selection of a cutoff is arbitrary, and the time value of money is ignored. For a financial manager, both the reliance on accounting numbers rather than relevant market data and the exclusion of time value of money considerations are troubling. Despite these problems, AAR continues to be used in practice because (1) the accounting information is usually available, (2) analysts often use accounting ratios to analyze firm performance, and (3) managerial compensation is often tied to the attainment of target accounting ratio goals. The IRR is the discount rate that causes the NPV of a series of cash flows to be identically zero. IRR can thus be interpreted as a financial break-even rate of return; at the IRR discount rate, the net value of the project is zero. The acceptance and rejection criteria are: If C0 < 0 and all future cash flows are positive, accept the project if the internal rate of return is greater than or equal to the discount rate. If C0 < 0 and all future cash flows are positive, reject the project if the internal rate of return is less than the discount rate. If C0 > 0 and all future cash flows are negative, accept the project if the internal rate of return is less than or equal to the discount rate. If C0 > 0 and all future cash flows are negative, reject the project if the internal rate of return is greater than the discount rate. IRR is the discount rate that causes NPV for a series of cash flows to be zero. NPV is preferred in all situations to IRR; IRR can lead to ambiguous results if there are non-conventional cash flows, and it also may ambiguously rank some mutually exclusive projects. However, for standalone projects with conventional cash flows, IRR and NPV are interchangeable techniques. c. CHAPTER 7 137 d. The profitability index is the present value of cash inflows relative to the project cost. As such, it is a benefit/cost ratio, providing a measure of the relative profitability of a project. The profitability index decision rule is to accept projects with a PI greater than one, and to reject projects with a PI less than one. The profitability index can be expressed as: PI = (NPV + cost)/ cost = 1 + (NPV/cost). If a firm has a basket of positive NPV projects and is subject to capital rationing, PI may provide a good ranking measure of the projects, indicating the bang for the buck of each particular project. NPV is simply the present value of a projects cash flows, including the initial outlay. specifically NPV measures, after considering the time value of money, the net increase or decrease in firm wealth due to the project. The decision rule is to accept projects that have a positive NPV, and reject projects with a negative NPV. NPV is superior to the other methods of analysis presented in the text because it has no serious flaws. The method unambiguously ranks mutually exclusive projects, and it can differentiate between projects of different scale and time horizon. The only drawback to NPV is that it relies on cash flow and discount rate values that are often estimates and thus not certain, but this is a problem shared by the other performance criteria as well. A project with NPV = $2,500 implies that the total shareholder wealth of the firm will increase by $2,500 if the project is accepted. e. 4. For a project with future cash flows that are an annuity: Payback = I / C And the IRR is: 0 = I + C / IRR CHAPTER 7 B-138 Solving the IRR equation for IRR, we get: IRR = C / I Notice this is just the reciprocal of the payback. So: IRR = 1 / PB For long-lived projects with relatively constant cash flows, the sooner the project pays back, the greater is the IRR, and the IRR is approximately equal to the reciprocal of the payback period. 5. There are a number of reasons. Two of the most important have to do with transportation costs and exchange rates. Manufacturing in the U.S. places the finished product much closer to the point of sale, resulting in significant savings in transportation costs. It also reduces inventories because goods spend less time in transit. Higher labor costs tend to offset these savings to some degree, at least compared to other possible manufacturing locations. Of great importance is the fact that manufacturing in the U.S. means that a much higher proportion of the costs are paid in dollars. Since sales are in dollars, the net effect is to immunize profits to a large extent against fluctuations in exchange rates. This issue is discussed in greater detail in the chapter on international finance. The single biggest difficulty, by far, is coming up with reliable cash flow estimates. Determining an appropriate discount rate is also not a simple task. These issues are discussed in greater depth in the next several chapters. The payback approach is probably the simplest, followed by the AAR, but even these require revenue and cost projections. The discounted cash flow measures (discounted payback, NPV, IRR, and profitability index) are really only slightly more difficult in practice. Yes, they are. Such entities generally need to allocate available capital efficiently, just as for-profits do. However, it is frequently the case that the revenues from not-for-profit ventures are not tangible. For example, charitable giving has real opportunity costs, but the benefits are generally hard to measure. To the extent that benefits are measurable, the question of an appropriate required return remains. Payback rules are commonly used in such cases. Finally, realistic cost/benefit analysis along the lines indicated should definitely be used by the U.S. government and would go a long way toward balancing the budget! The statement is false. If the cash flows of Project B occur early and the cash flows of Project A occur late, then for a low discount rate the NPV of A can exceed the NPV of B. Observe the following example. C0 $1,000,000 $2,000,000 C1 $0 $2,400,000 C2 $1,440,000 $0 IRR 20% 20% NPV @ 0% $440,000 400,000 6. 7. 8. Project A Project B However, in one particular case, the statement is true for equally risky projects. If the lives of the two projects are equal and the cash flows of Project B are twice the cash flows of Project A in every time period, the NPV of Project B will be twice the NPV of Project A. 9. Although the profitability index (PI) is higher for Project B than for Project A, Project A should be chosen because it has the greater NPV. Confusion arises because Project B requires a smaller investment than Project A. Since the denominator of the PI ratio is lower for Project B than for Project A, B can have a higher PI yet have a lower NPV. Only in the case of capital rationing could the companys decision have been incorrect. CHAPTER 7 139 10. a. b. Project A would have a higher IRR since initial investment for Project A is less than that of Project B, if the cash flows for the two projects are identical. Yes, since both the cash flows as well as the initial investment are twice that of Project B. 11. Project Bs NPV would be more sensitive to changes in the discount rate. The reason is the time value of money. Cash flows that occur further out in the future are always more sensitive to changes in the interest rate. This sensitivity is similar to the interest rate risk of a bond. 12. The MIRR is calculated by finding the present value of all cash outflows, the future value of all cash inflows to the end of the project, and then calculating the IRR of the two cash flows. As a result, the cash flows have been discounted or compounded by one interest rate (the required return), and then the interest rate between the two remaining cash flows is calculated. As such, the MIRR is not a true interest rate. In contrast, consider the IRR. If you take the initial investment, and calculate the future value at the IRR, you can replicate the future cash flows of the project exactly. 13. The statement is incorrect. It is true that if you calculate the future value of all intermediate cash flows to the end of the project at the required return, then calculate the NPV of this future value and the initial investment, you will get the same NPV. However, NPV says nothing about reinvestment of intermediate cash flows. The NPV is the present value of the project cash flows. What is actually done with those cash flows once they are generated is not relevant. Put differently, the value of a project depends on the cash flows generated by the project, not on the future value of those cash flows. The fact that the reinvestment works only if you use the required return as the reinvestment rate is also irrelevant simply because reinvestment is not relevant in the first place to the value of the project. One caveat: Our discussion here assumes that the cash flows are truly available once they are generated, meaning that it is up to firm management to decide what to do with the cash flows. In certain cases, there may be a requirement that the cash flows be reinvested. For example, in international investing, a company may be required to reinvest the cash flows in the country in which they are generated and not repatriate the money. Such funds are said to be blocked and reinvestment becomes relevant because the cash flows are not truly available. 14. The statement is incorrect. It is true that if you calculate the future value of all intermediate cash flows to the end of the project at the IRR, then calculate the IRR of this future value and the initial investment, you will get the same IRR. However, as in the previous question, what is done with the cash flows once they are generated does not affect the IRR. Consider the following example: Project A C0 $100 C1 $10 C2 $110 IRR 10% Suppose this $100 is a deposit into a bank account. The IRR of the cash flows is 10 percent. Does the IRR change if the Year 1 cash flow is reinvested in the account, or if it is withdrawn and spent on pizza? No. Finally, consider the yield to maturity calculation on a bond. If you think about it, the YTM is the IRR on the bond, but no mention of a reinvestment assumption for the bond coupons is suggested. The reason is that reinvestment is irrelevant to the YTM calculation; in the same way, reinvestment is irrelevant in the IRR calculation. Our caveat about blocked funds applies here as well. CHAPTER 7 B-140 Solutions to Questions and Problems NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. Project A: Payback period = 1.90 years Project B: Payback period = 2.06 years Since project A has a payback period less than 2 years and project B has a payback period longer than 2 years, the company should choose project A. b. Discount each projects cash flows at 15 percent. Choose the project with the highest NPV. Project A: NPV = $515.62 Project B: NPV = $655.15 The firm should choose Project B since it has a higher NPV than Project A. 2. To calculate the payback period, we need to find the time that the project has recovered its initial investment. The cash flows in this problem are an annuity, so the calculation is simpler. If the initial cost is $3,500, the payback period is: Payback = 4.79 years There is a shortcut to calculate the payback period if the future cash flows are an annuity. Just divide the initial cost by the annual cash flow. For the $3,500 cost, the payback period is: Payback = 4.79 years For an initial cost of $5,000, the payback period is: Payback = 6.85 years The payback period for an initial cost of $6,000 is a little trickier. Notice that the total cash inflows after eight years will be: Total cash inflows = $5,840 CHAPTER 7 141 If the initial cost is $6,000, the project never pays back. Notice that if you use the shortcut for annuity cash flows, you get: Payback = = 8.22 years This answer does not make sense since the cash flows stop after eight years, so there is no payback period. 3. When we use discounted payback, we need to find the value of all cash flows today. The value today of the project cash flows for the first four years is: Value today of Year 1 cash flow = $6,400/1.12 = $5,714.29 Value today of Year 2 cash flow = $6,900/1.122 = $5,500.64 Value today of Year 3 cash flow = $7,300/1.123 = $5,196.00 Value today of Year 4 cash flow = $6,100/1.124 = $3,876.66 To find the discounted payback, we use these values to find the payback period. The discounted first year cash flow is $5,714.29, so the discounted payback for an $8,000 initial cost is: Discounted payback = 1 + ($8,000 5,714.29)/$5,500.64 = 1.42 years For an initial cost of $13,000, the discounted payback is: Discounted payback = 2 + ($13,000 5,714.29 5,500.64)/$5,196.00 = 2.34 years Notice the calculation of discounted payback. We know the payback period is between two and three years, so we subtract the discounted values of the Year 1 and Year 2 cash flows from the initial cost. This is the numerator, which is the discounted amount we still need to make to recover our initial investment. We divide this amount by the discounted amount we will earn in Year 3 to get the fractional portion of the discounted payback. If the initial cost is $18,000, the discounted payback is: Discounted payback = 3 + ($18,000 5,714.29 5,500.64 5,196.00) / $3,876.66 Discounted payback = 3.41 years 4. To calculate the discounted payback, discount all future cash flows back to the present, and use these discounted cash flows to calculate the payback period. To find the fractional year, we divide the amount we need to make in the last year to payback the project by the amount we will make. Doing so, we find: R = 0%: 3 + ($2,300 / $4,900) = 3.47 years Discounted payback = Regular payback = 3.47 years R = 7%: $4,900/1.07 + $4,900/1.072 + $4,900/1.073 + $4,900/1.074 = $16,597.34 $4,900/1.075 = $3,493.63 Discounted payback = 4 + ($17,000 16,597.34) / $3,493.63 = 4.12 years R = 21%: $4,900/1.21 + $4,900/1.212 + $4,900/1.213 + $4,900/1.214 + $4,900/1.215 + $4,900/1.216 = $15,898.61; The project never pays back. CHAPTER 7 B-142 5. a. The average accounting return is the average project earnings after taxes, divided by the average book value, or average net investment, of the machine during its life. The book value of the machine is the gross investment minus the accumulated depreciation. Average book value = (Book value0 + Book value1 + Book value2 + Book value3 + Book value4 + Book value5) / (Economic life) Average book value = ($46,000 + 34,500 + 23,000 + 11,500 + 0) / 5 Average book value = $23,000 Average project earnings = $4,700 To find the average accounting return, we divide the average project earnings by the average book value of the machine to calculate the average accounting return. Doing so, we find: Average accounting return = Average project earnings / Average book value Average accounting return = $4,700 / $23,000 Average accounting return = 0.2043 or 20.43% b. The three flaws of the AAR are: 1) The AAR does not work with the right raw materials. It uses net income and book value of the investment, both of which come from the accounting books. Accounting numbers are somewhat arbitrary. 2) AAR takes no account of timing. The AAR would be the same if the net income in the first year occurs in the last year. 3) The AAR uses an arbitrary interest rate as the cutoff rate and offers no guidance on what the right targeted rate of return should be. 6. First, we need to determine the average book value of the project. The book value is the gross investment minus accumulated depreciation. Purchase Date $18,000 0 $18,000 Year 1 $18,000 5,300 $12,700 Year 2 $18,000 13,100 $4,900 Year 3 $18,000 18,000 $0 Gross Investment Less: Accumulated depreciation Net Investment Now, we can calculate the average book value as: Average book value = ($18,000 + 12,700 + 4,900 + 0) / 4 Average book value = $8,900 To calculate the average accounting return, we must remember to use the aftertax average net income when calculating the average accounting return. So, the average aftertax net income is: Average aftertax net income = (1 tc) Annual pretax net income Average aftertax net income = (1 0.25) $2,260 Average aftertax net income = $1,695 The average accounting return is the average after-tax net income divided by the average book value, which is: Average accounting return = $1,695 / $8,900 Average accounting return = 0.1904 or 19.04% CHAPTER 7 143 7. IRR = 16.74% Since the IRR is greater than the required return we would accept the project. 8. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this Project A is: 0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3 0 = $4,900 + $1,700/(1 + IRR) + $2,900/(1 + IRR)2 + $2,100/(1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 16.79% And the IRR for Project B is: 0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3 0 = $3,200 + $1,100/(1 + IRR) + $1,400/(1 + IRR)2 + $1,700/(1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 13.82% 9. The profitability index is defined as the PV of the cash inflows divided by the PV of the cash outflows. The cash flows from this project are an annuity, so the equation for the profitability index is: PI = C(PVIFAR,t) / C0 PI = $71,000(PVIFA15%,7) / $260,000 PI = $295,389.90 / $260,000 PI = 1.1361 Therefore, the project should be accepted. 10. a. The profitability index is the present value of the future cash flows divided by the initial cost. So, for Project Alpha, the profitability index is: PIAlpha = [$500 / 1.10 + $900 / 1.102 + $800 / 1.103] / $1,200 = 1.499 And for Project Beta the profitability index is: PIBeta = [$900 / 1.10 + $2,400 / 1.102 + $1,300 / 1.103] / $2,600 = 1.453 b. According to the profitability index, you would accept Project Alpha. However, remember the profitability index rule can lead to an incorrect decision when ranking mutually exclusive projects. Intermediate 11. a. To have a payback equal to the projects life, given C is a constant cash flow for N years: CHAPTER 7 B-144 C = I/N b. c. To have a positive NPV, I < C (PVIFAR%, N). Thus, C > I / (PVIFAR%, N). Benefits = C (PVIFAR%, N) = 2 costs = 2I C = 2I / (PVIFAR%, N) 12. a. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is: 0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3 + C4 / (1 + IRR)4 0 = $10,000 $4,300 / (1 + IRR) $3,900 / (1 + IRR)2 $3,200 / (1 + IRR)3 $1,200 / (1 +IRR)4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 11.94% b. This problem differs from previous ones because the initial cash flow is positive and all future cash flows are negative. In other words, this is a financing-type project, while previous projects were investing-type projects. For financing situations, accept the project when the IRR is less than the discount rate. Reject the project when the IRR is greater than the discount rate. IRR = 11.94% Discount Rate = 10% IRR > Discount Rate Reject the offer when the discount rate is less than the IRR. c. Using the same reason as part b., we would accept the project if the discount rate is 20 percent. IRR = 11.94% Discount Rate = 20% IRR < Discount Rate Accept the offer when the discount rate is greater than the IRR. d. The NPV is the sum of the present value of all cash flows, so the NPV of the project if the discount rate is 10 percent will be: NPV = $10,000 $4,300 / 1.1 $3,900 / 1.12 $3,200 / 1.13 $1,200 / 1.14 NPV = $356.05 When the discount rate is 10 percent, the NPV of the offer is negative, so reject the offer. And the NPV of the project if the discount rate is 20 percent will be: NPV = $10,000 $4,300 / 1.2 $3,900 / 1.22 $3,200 / 1.23 $1,200 / 1.24 CHAPTER 7 145 NPV = $1,277.78 When the discount rate is 20 percent, the NPV of the offer is positive. So accept the offer. e. Yes, the decisions under the NPV rule are consistent with the choices made under the IRR rule since the signs of the cash flows change only once. 13. a. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR for each project is: Deepwater Fishing IRR: 0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3 0 = $850,000 + $480,000 / (1 + IRR) + $430,000 / (1 + IRR)2 + $320,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 22.70% Submarine Ride IRR: 0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3 0 = $2,100,000 + $1,500,000 / (1 + IRR) + $750,000 / (1 + IRR)2 + $600,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 20.66% Based on the IRR rule, the deepwater fishing project should be chosen because it has the higher IRR. b. To calculate the incremental IRR, we subtract the smaller projects cash flows from the larger projects cash flows. In this case, we subtract the deepwater fishing cash flows from the submarine ride cash flows. The incremental IRR is the IRR of these incremental cash flows. So, the incremental cash flows of the submarine ride are: Year 0 $2,100,000 850,000 $1,250,000 Year 1 $1,500,000 480,000 $1,020,000 Year 2 $750,000 430,000 $320,000 Year 3 $600,000 320,000 $280,000 Submarine Ride Deepwater Fishing Submarine Fishing Setting the present value of these incremental cash flows equal to zero, we find the incremental IRR is: 0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3 0 = $1,250,000 + $1,020,000 / (1 + IRR) + $320,000 / (1 + IRR)2 + $280,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: CHAPTER 7 B-146 Incremental IRR = 18.95% For investing-type projects, accept the larger project when the incremental IRR is greater than the discount rate. Since the incremental IRR, 18.95 percent, is greater than the required rate of return of 15 percent, choose the submarine ride project. Note that this is not the choice when evaluating only the IRR of each project. The IRR decision rule is flawed because there is a scale problem. That is, the submarine ride has a greater initial investment than does the deepwater fishing project. This problem is corrected by calculating the IRR of the incremental cash flows, or by evaluating the NPV of each project. c. The NPV is the sum of the present value of the cash flows from the project, so the NPV of each project will be: Deepwater fishing: NPV = $850,000 + $480,000 / 1.15 + $430,000 / 1.152 + $320,000 / 1.153 NPV = $102,938.28 Submarine ride: NPV = $2,100,000 + $1,500,000 / 1.15 + $750,000 / 1.152 + $600,000 / 1.153 NPV = $165,965.32 Since the NPV of the submarine ride project is greater than the NPV of the deepwater fishing project, choose the submarine ride project. The incremental IRR rule is always consistent with the NPV rule. 14. a. The profitability index is the PV of the future cash flows divided by the initial investment. The cash flows for both projects are an annuity, so: PII = $32,000(PVIFA10%,3 ) / $60,000 = 1.326 PIII = $20,000(PVIFA10%,3) / $34,000 = 1.463 The profitability index decision rule implies that we accept project II, since PIII is greater than the PII. b. The NPV of each project is: NPVI = $60,000 + $32,000(PVIFA10%,3) = $19,579.26 NPVII = $34,000 + $20,000(PVIFA10%,3) = $15,737.04 The NPV decision rule implies accepting Project I, since the NPVI is greater than the NPVII. c. Using the profitability index to compare mutually exclusive projects can be ambiguous when the magnitudes of the cash flows for the two projects are of different scale. In this problem, project I is roughly twice as large as project II and produces a larger NPV, yet the profitability index criterion implies that project II is more acceptable. CHAPTER 7 147 15. a. The equation for the NPV of the project is: NPV = $6,347,107.44 The NPV is greater than 0, so we would accept the project. b. The equation for the IRR of the project is: IRR = 31.16%, 76.40% When there are multiple IRRs, the IRR decision rule is ambiguous. Both IRRs are correct; that is, both interest rates make the NPV of the project equal to zero. If we are evaluating whether or not to accept this project, we would not want to use the IRR to make our decision. 16. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. Board game: Cumulative cash flows Year 1 = $240,000 Cumulative cash flows Year 2 = $240,000 + 130,000 = $240,000 = $370,000 Payback period = 1 + ($320,000 240,000) / $130,000 = 1.62 years CD-ROM: Cumulative cash flows Year 1 = $310,000 Cumulative cash flows Year 2 = $310,000 + 280,000 Payback period = 1 + ($550,000 310,000) / $280,000 Payback period = 1.86 years Since the board game has a shorter payback period than the CD-ROM project, the company should choose the board game. b. The NPV is the sum of the present value of the cash flows from the project, so the NPV of each project will be: Board game: NPV = $320,000 + $240,000 / 1.10 + $130,000 / 1.102 + $75,000 / 1.103 NPV = $61,968.44 = $310,000 = $590,000 CHAPTER 7 B-148 CD-ROM: NPV = $550,000 + $310,000 / 1.10 + $280,000 / 1.102 + $195,000 / 1.103 NPV = $109,729.53 Since the NPV of the CD-ROM is greater than the NPV of the board game, choose the CDROM. c. The IRR is the interest rate that makes the NPV of a project equal to zero. So, the IRR of each project is: Board game: 0 = $320,000 + $240,000 / (1 + IRR) + $130,000 / (1 + IRR)2 + $75,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 23.34% CD-ROM: 0 = $550,000 + $310,000 / (1 + IRR) + $280,000 / (1 + IRR)2 + $195,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 21.95% Since the IRR of the board game is greater than the IRR of the CD-ROM, IRR implies we choose the board game. d. To calculate the incremental IRR, we subtract the smaller projects cash flows from the larger projects cash flows. In this case, we subtract the board game cash flows from the CD-ROM cash flows. The incremental IRR is the IRR of these incremental cash flows. So, the incremental cash flows of the submarine ride are: Year 0 $550,000 320,000 $230,000 Year 1 $310,000 240,000 $70,000 Year 2 $280,000 130,000 $150,000 Year 3 $195,000 75,000 $120,000 CD-ROM Board game CD-ROM Board game Setting the present value of these incremental cash flows equal to zero, we find the incremental IRR is: 0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3 0 = $230,000 + $70,000 / (1 + IRR) + $150,000 / (1 + IRR)2 + $120,000 / (1 + IRR)3 CHAPTER 7 149 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: Incremental IRR = 20.49% For investing-type projects, accept the larger project when the incremental IRR is greater than the discount rate. Since the incremental IRR, 20.49%, is greater than the required rate of return of 10 percent, choose the CD-ROM project. Note that this is not the choice when evaluating only the IRR of each project. The IRR decision rule is flawed because there is a scale problem. That is, the CD-ROM has a greater initial investment than does the board game. This problem is corrected by calculating the IRR of the incremental cash flows, or by evaluating the NPV of each project. 17. a. The profitability index is the PV of the future cash flows divided by the initial investment. The profitability index for each project is: PICDMA = [$36,000,000 / 1.10 + $23,000,000 / 1.102 + $10,000,000 / 1.103]/$20,000,000 = 2.96 PIG4 = [$29,000,000 / 1.10 + $73,000,000 / 1.102 + $61,000,000 / 1.103]/$35,000,000 = 3.79 PIWi-Fi = [$43,000,000 / 1.10 + $105,000,000 / 1.102 + $85,000,000/1.103]/$55,000,000 = 3.45 The profitability index implies we accept the G4 project. Remember this is not necessarily correct because the profitability index does not necessarily rank projects with different initial investments correctly. b. The NPV of each project is: NPVCDMA = $20,000,000 + $36,000,000 / 1.10 + $23,000,000 / 1.102 + $10,000,000 / 1.103 NPVCDMA = $39,248,685.20 NPVG4 = $35,000,000 + $29,000,000 / 1.10 + $73,000,000 / 1.102 + $61,000,000 / 1.103 NPVG4 = $97,524,417.73 NPVWi-Fi = $55,000,000 + $43,000,000 / 1.10 + $105,000,000 / 1.102 + $85,000,000 / 1.103 NPVWi-Fi = $134,729,526.67 NPV implies we accept the Wi-Fi project since it has the highest NPV. This is the correct decision if the projects are mutually exclusive. c. We would like to invest in all three projects since each has a positive NPV. If the budget is limited to $55 million, we can only accept the CDMA project and the G4 project, or the Wi-Fi project. NPV is additive across projects and the company. The total NPV of the CDMA project and the G4 project is: NPVCDMA and G4 = $39,248,685.20 + 97,524,417.73 NPVCDMA and G4 = $136,773,102.93 This is greater than the Wi-Fi project, so we should accept the CDMA project and the G4 project. CHAPTER 7 B-150 18. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. AZM Mini-SUV: Cumulative cash flows Year 1 = $310,000 Cumulative cash flows Year 2 = $310,000 + 240,000 = $310,000 = $550,000 Payback period = 1 + ($450,000 310,000) / $240,000 = 1.58 years AZF Full-SUV: Cumulative cash flows Year 1 = $420,000 Cumulative cash flows Year 2 = $420,000 + 390,000 = $420,000 = $810,000 Payback period = 1 + ($800,000 420,000) / $390,000 = 1.97 years Since the AZM has a shorter payback period than the AZF, the company should choose the AZM. Remember the payback period does not necessarily rank projects correctly. b. The NPV of each project is: NPVAZM = $450,000 + $310,000 / 1.10 + $240,000 / 1.102 + $210,000 / 1.103 NPVAZM = $187,941.40 NPVAZF = $800,000 + $420,000 / 1.10 + $390,000 / 1.102 + $340,000 / 1.103 NPVAZF = $159,579.26 The NPV criteria implies we accept the AZM because it has the highest NPV. c. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of each AZM is: 0 = $450,000 + $310,000 / (1 + IRR) + $240,000 / (1 + IRR)2 + $210,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRAZM = 34.40% And the IRR of the AZF is: 0 = $800,000 + $420,000 / (1 + IRR) + $390,000 / (1 + IRR)2 + $340,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRAZF = 21.45% The IRR criteria implies we accept the AZM because it has the highest IRR. Remember the IRR does not necessarily rank projects correctly. CHAPTER 7 151 d. 19. a. Incremental IRR analysis is not necessary. The AZM has the smallest initial investment and the largest NPV, so it should be accepted. The profitability index is the PV of the future cash flows divided by the initial investment. The profitability index for each project is: PIA = [$125,000 / 1.12 + $125,000 / 1.122] / $200,000 = 1.06 PIB = [$215,000 / 1.12 + $215,000 / 1.122] / $350,000 = 1.04 PIC = [$130,000 / 1.12 + $115,000 / 1.122] / $200,000 = 1.04 b. The NPV of each project is: NPVA = $200,000 + $125,000 / 1.12 + $125,000 / 1.122 NPVA = $11,256.38 NPVB = $350,000 + $215,000 / 1.12 + $215,000 / 1.122 NPVB = $13,360.97 NPVC = $200,000 + $130,000 / 1.12 + $115,000 / 1.122 NPVC = $7,748.72 c. d. Accept projects A, B, and C. Since the projects are independent, accept all three projects because the respective profitability index of each is greater than one. Accept Project B. Since the Projects are mutually exclusive, choose the Project with the highest PI, while taking into account the scale of the Project. Because Projects A and C have the same initial investment, the problem of scale does not arise when comparing the profitability indices. Based on the profitability index rule, Project C can be eliminated because its PI is less than the PI of Project A. Because of the problem of scale, we cannot compare the PIs of Projects A and B. However, we can calculate the PI of the incremental cash flows of the two projects, which are: Project BA C0 $150,000 C1 $90,000 C2 $90,000 When calculating incremental cash flows, remember to subtract the cash flows of the project with the smaller initial cash outflow from those of the project with the larger initial cash outflow. This procedure insures that the incremental initial cash outflow will be negative. The incremental PI calculation is: PI(B A) = [$90,000 / 1.12 + $90,000 / 1.122] / $150,000 PI(B A) = 1.014 The company should accept Project B since the PI of the incremental cash flows is greater than one. e. Remember that the NPV is additive across projects. Since we can spend $550,000, we could take two of the projects. In this case, we should take the two projects with the highest NPVs, which are Project B and Project A. CHAPTER 7 B-152 20. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. Dry Prepeg: Cumulative cash flows Year 1 = $700,000 Cumulative cash flows Year 2 = $700,000 + 500,000 Cumulative cash flows Year 3 = $700,000 + 500,000 + 1,300,000 = $700,000 = $1,200,000 = $2,500,000 Payback period = 2 + ($1,800,000 700,000 500,000) / $1,300,000 = 2.46 years Solvent Prepeg: Cumulative cash flows Year 1 = $600,000 = $600,000 Cumulative cash flows Year 2 = $600,000 + 400,000 = $1,000,000 Payback period = 1 + ($950,000 600,000) / $400,000 = 1.88 years Since the solvent prepeg has a shorter payback period than the dry prepeg, the company should choose the solvent prepeg. Remember the payback period does not necessarily rank projects correctly. b. The NPV of each project is: NPVDry prepeg = $1,800,000 + $700,000 / 1.10 + $500,000 / 1.102 + $1,300,000 / 1.103 NPVDry prepeg = $226,296.02 NPVSolvent prepeg = $950,000 + $600,000 / 1.10 + $400,000 / 1.102 + $350,000 / 1.103 NPVSolvent prepeg = $188,993.24 The NPV criteria implies accepting the dry prepeg because it has the highest NPV. c. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of each dry prepeg is: 0 = $1,800,000 + $700,000 / (1 + IRR) + $500,000 / (1 + IRR)2 + $1,300,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRDry prepeg = 16.24% And the IRR of the solvent prepeg is: 0 = $950,000 + $600,000 / (1 + IRR) + $400,000 / (1 + IRR)2 + $350,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRSolvent prepeg = 22.25% CHAPTER 7 153 The IRR criteria implies accepting the solvent prepeg because it has the highest IRR. Remember the IRR does not necessarily rank projects correctly. d. Incremental IRR analysis is necessary. The solvent prepeg has a higher IRR, but is relatively smaller in terms of investment and NPV. In calculating the incremental cash flows, we subtract the cash flows from the project with the smaller initial investment from the cash flows of the project with the large initial investment, so the incremental cash flows are: Year 0 $1,800,000 950,000 $850,000 Year 1 $700,000 600,000 $100,000 Year 2 $500,000 400,000 $100,000 Year 3 $1,300,000 350,000 $950,000 Dry prepeg Solvent prepeg Dry prepeg Solvent prepeg Setting the present value of these incremental cash flows equal to zero, we find the incremental IRR is: 0 = $850,000 + $100,000 / (1 + IRR) + $100,000 / (1 + IRR)2 + $950,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: Incremental IRR = 11.76% For investing-type projects, we accept the larger project when the incremental IRR is greater than the discount rate. Since the incremental IRR, 11.76%, is greater than the required rate of return of 10 percent, we choose the dry prepeg. Note that this is not the choice when evaluating only the IRR of each project. The IRR decision rule is flawed because there is a scale problem. That is, the dry prepeg has a greater initial investment than does the solvent prepeg. This problem is corrected by calculating the IRR of the incremental cash flows, or by evaluating the NPV of each project. 21. a. The NPV of each project is: NPVNP-30 = $900,000 + $375,000 / 1.12 + $350,000 / 1.122 + $325,000 / 1.123 + $275,000 / 1.124 + $185,000 / 1.125 NPVNP-30 = $224,909.31 NPVNX-20 = $650,000 + $250,000 / 1.12 + $250,000 / 1.122 + $300,000 / 1.123 + $250,000 / 1.124 + $165,000 / 1.125 NPVNX-20 = $238,551.78 The NPV criteria implies accepting the NX-20. b. The IRR is the interest rate that makes the NPV of the project equal to zero, so the IRR of each project is: NP-30: 0 = $900,000 + $375,000 / (1 + IRR) + $350,000 / (1 + IRR)2 + $325,000 / (1 + IRR)3 + $275,000 / (1 + IRR)4 + $185,000 / (1 + IRR)5 CHAPTER 7 B-154 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRNP-30 = 22.81% And the IRR of the NX-20 is: 0 = $650,000 + $250,000 / (1 + IRR) + $250,000 / (1 + IRR)2 + $300,000 / (1 + IRR)3 + $250,000 / (1 + IRR)4 + $165,000 / (1 + IRR)5 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRNX-20 = 26.55% The IRR criteria implies accepting the NX-20. c. Incremental IRR analysis is not necessary. The NX-20 has a higher IRR, and is relatively smaller in terms of investment, with a larger NPV. Nonetheless, we will calculate the incremental IRR. In calculating the incremental cash flows, we subtract the cash flows from the project with the smaller initial investment from the cash flows of the project with the large initial investment, so the incremental cash flows are: Incremental cash flow $250,000 125,000 100,000 25,000 25,000 20,000 Year 0 1 2 3 4 5 Setting the present value of these incremental cash flows equal to zero, we find the incremental IRR is: 0 = $250,000 + $125,000 / (1 + IRR) + $100,000 / (1 + IRR)2 + $25,000 / (1 + IRR)3 + $25,000 / (1 + IRR)4 + $20,000 / (1 + IRR)5 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: Incremental IRR = 8.74% For investing-type projects, accept the larger project when the incremental IRR is greater than the discount rate. Since the incremental IRR, 8.74 percent, is less than the required rate of return of 12 percent, we reject the larger project and choose the NX-20. CHAPTER 7 155 d. The profitability index is the present value of all subsequent cash flows, divided by the initial investment, so the profitability index of each project is: PINP-30 = [$375,000 / 1.12 + $350,000 / 1.122 + $325,000 / 1.123 + $275,000 / 1.124 + $185,000 / 1.125] / $900,000 PINP-30 = 1.250 PINX-20 = [$250,000 / 1.12 + $250,000 / 1.122 + $300,000 / 1.123 + $250,000 / 1.124 + $165,000 / 1.125] / $650,000 PINX-20 = 1.367 The PI criteria implies accepting the NX-20. 22. a. The NPV of each project is: NPVA = $650,000 + $320,000 / 1.15 + $320,000 / 1.152 + $230,000 / 1.153 + $175,000 / 1.154 + $120,000 / 1.155 NPVA = $181,173.60 NPVB = $975,000 + $260,000 / 1.15 + $350,000 / 1.152 + $360,000 / 1.153 + $400,000 / 1.154 + $500,000 / 1.155 NPVB = $229,732.75 The NPV criteria implies accepting Project B. b. The IRR is the interest rate that makes the NPV of the project equal to zero, so the IRR of each project is: Project A: 0 = $650,000 + $320,000 / (1 + IRR) + $320,000 / (1 + IRR)2 + $230,000 / (1 + IRR)3 + $175,000 / (1 + IRR)4 + $120,000 / (1 + IRR)5 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRA = 28.47% And the IRR of the Project B is: 0 = $975,000 + $260,000 / (1 + IRR) + $350,000 / (1 + IRR)2 + $360,000 / (1 + IRR)3 + $400,000 / (1 + IRR)4 + $500,000 / (1 + IRR)5 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRB = 23.59% The IRR criteria implies accepting Project A. CHAPTER 7 B-156 c. In calculating the incremental cash flows, we subtract the cash flows from the project with the smaller initial investment from the cash flows of the project with the large initial investment, so the incremental cash flows are: Incremental cash flow $325,000 60,000 30,000 130,000 225,000 380,000 Year 0 1 2 3 4 5 Setting the present value of these incremental cash flows equal to zero, we find the incremental IRR is: 0 = $325,000 $60,000 / (1 + IRR) + $30,000 / (1 + IRR)2 + $130,000 / (1 + IRR)3 + $225,000 / (1 + IRR)4 + $380,000 / (1 + IRR)5 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: Incremental IRR = 18.55% For investing-type projects, accept the larger project when the incremental IRR is greater than the discount rate. Since the incremental IRR, 18.55 percent, is greater than the required rate of return of 15 percent, choose the Project B. d. The profitability index is the present value of all subsequent cash flows, divided by the initial investment, so the profitability index of each project is: PIA = [320,000 / 1.15 + $320,000 / 1.152 + $230,000 / 1.153 + $175,000 / 1.154 + $120,000 / 1.155] / $650,000 PIA = 1.279 PIB = [$260,000 / 1.15 + $350,000 / 1.152 + $360,000 / 1.153 + $400,000 / 1.154 + $500,000 / 1.155] / $975,000 PIB = 1.236 The PI criteria implies accepting Project A. CHAPTER 7 157 23. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. Project A: Cumulative cash flows Year 1 = $175,000 Cumulative cash flows Year 2 = $175,000 + 280,000 = $175,000 = $455,000 Payback period = 1 + ($400,000 175,000)/$280,000 = 1.80 years Project B: Cumulative cash flows Year 1 = $195,000 Cumulative cash flows Year 2 = $195,000 + 105,000 = $195,000 = $300,000 Payback period = 1 + ($200,000 195,000)/$105,000 = 1.05 years Project C: Cumulative cash flows Year 1 = $160,000 Payback period = $150,000 / 160,000 = 0.94 years Payback implies that we accept Project C since it has the shortest payback period. Regardless, the payback period does not necessarily rank projects correctly. b. The IRR is the interest rate that makes the NPV of the project equal to zero, so the IRR of each project is: Project A: 0 = $400,000 + $175,000 / (1 + IRR) + $280,000 / (1 + IRR)2 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRA = 8.35% And the IRR of the Project B is: 0 = $200,000 + $195,000 / (1 + IRR) + $105,000 / (1 + IRR)2 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRB = 36.08% And the IRR of the Project C is: 0 = $150,000 + $160,000 / (1 + IRR) + $50,000 / (1 + IRR)2 = $160,000 CHAPTER 7 B-158 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRC = 31.93% The IRR criteria implies accepting Project B. c. Project A can be excluded from the incremental IRR analysis. Since the project has a negative NPV, and an IRR less than its required return, the project is rejected. We need to calculate the incremental IRR between Project B and Project C. In calculating the incremental cash flows, we subtract the cash flows from the project with the smaller initial investment from the cash flows of the project with the large initial investment, so the incremental cash flows are: Incremental cash flow $50,000 35,000 55,000 Year 0 1 2 Setting the present value of these incremental cash flows equal to zero, we find the incremental IRR is: 0 = $50,000 + $35,000 / (1 + IRR) + $55,000 / (1 + IRR)2 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: Incremental IRR = 45.57% For investing-type projects, accept the larger project when the incremental IRR is greater than the discount rate. Since the incremental IRR, 45.57 percent, is greater than the required rate of return of 20 percent, choose the Project B. d. The profitability index is the present value of all subsequent cash flows, divided by the initial investment. We need to discount the cash flows of each project by the required return of each project. The profitability index of each project is: PIA = [$175,000 / 1.10 + $280,000 / 1.102] / $400,000 PIA = 0.98 PIB = [$195,000 / 1.20 + $105,000 / 1.202] / $200,000 PIB = 1.18 PIC = [$160,000 / 1.20 + $50,000 / 1.202] / $150,000 PIC = 1.12 The PI criteria implies accepting Project B. CHAPTER 7 159 e. We need to discount the cash flows of each project by the required return of each project. The NPV of each project is: NPVA = $400,000 + $175,000 / 1.10 + $280,000 / 1.102 NPVA = $9,504.13 NPVB = $200,000 + $195,000 / 1.20 + $105,000 / 1.202 NPVB = $35,416.67 NPVC = $150,000 + $160,000 / 1.20 + $50,000 / 1.202 NPVC = $18,055.56 The NPV criteria implies accepting Project B. Challenge 24. Given the six-year payback, the worst case is that the payback occurs at the end of the sixth year. Thus, the worst case: NPV = $647,000 + $647,000/1.126 = $319,209.66 The best case has infinite cash flows beyond the payback point. Thus, the best-case NPV is infinite. 25. The equation for the IRR of the project is: 0 = $252 + $1,431/(1 + IRR) $3,035/(1 + IRR)2 + $2,850/(1 + IRR)3 $1,000/(1 + IRR)4 Using Descartes rule of signs, from looking at the cash flows we know there are four IRRs for this project. Even with most computer spreadsheets, we have to do some trial and error. From trial and error, IRRs of 25%, 33.33%, 42.86%, and 66.67% are found. We would accept the project when the NPV is greater than zero. See for yourself if that the NPV is greater than zero for required returns between 25% and 33.33% or between 42.86% and 66.67%. 26. a. Here the cash inflows of the project go on forever, which is a perpetuity. Unlike ordinary perpetuity cash flows, the cash flows here grow at a constant rate forever, which is a growing perpetuity. If you remember back to the chapter on time value of money, we presented a formula for growing perpetuity, so we can use it here. The PV of the future cash flows from the project is: PV of cash inflows = C1/(R g) PV of cash inflows = $52,000/(.11 .05) = $866,666.67 NPV is the PV of the outflows minus by the PV of the inflows, so the NPV is: NPV of the project = $900,000 + 866,666.67 = $33,333.33 The NPV is negative, so we would reject the project. CHAPTER 7 B-160 b. Here we want to know the minimum growth rate in cash flows necessary to accept the project. The minimum growth rate is the growth rate at which we would have a zero NPV. The equation for a zero NPV, using the equation for the PV of a growing perpetuity is: 0 = $900,000 + $52,000/(.11 g) Solving for g, we get: g = .0522 or 5.22% 27. a. The project involves three cash flows: the initial investment, the annual cash inflows, and the abandonment costs. The mine will generate cash inflows over its 11-year economic life. To express the PV of the annual cash inflows, apply the growing annuity formula, discounted at the IRR and growing at eight percent. PV(Cash Inflows) = C {[1/(r g)] [1/(r g)] [(1 + g)/(1 + r)]t} PV(Cash Inflows) = $250,000{[1/(IRR .08)] [1/(IRR .08)] [(1 + .08)/(1 + IRR)]11} At the end of 11 years, the Utah Mining Corporate will abandon the mine, incurring a $300,000 charge. Discounting the abandonment costs back 11 years at the IRR to express its present value, we get: PV(Abandonment) = C11 / (1 + IRR)11 PV(Abandonment) = $300,000 / (1+IRR)11 So, the IRR equation for this project is: 0 = $1,400,000 + $225,000{[1/(IRR .08)] [1/(IRR .08)] [(1 + .08)/(1 + IRR)]11} $300,000 / (1 + IRR)11 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 17.17% b. Yes. Since the mines IRR exceeds the required return of 13 percent, the mine should be opened. The correct decision rule for an investment-type project is to accept the project if the discount rate is above the IRR. Although it appears there is a sign change at the end of the project because of the abandonment costs, the last cash flow is actually positive because of the operating cash in the last year. 28. a. We can apply the growing perpetuity formula to find the PV of stream A. The perpetuity formula values the stream as of one year before the first payment. Therefore, the growing perpetuity formula values the stream of cash flows as of year 2. Next, discount the PV as of the end of year 2 back two years to find the PV as of today, year 0. Doing so, we find: PV(A) = [C3 / (R g)] / (1 + R)2 PV(A) = [$7,000 / (0.12 0.03)] / (1.12)2 PV(A) = $62,003.97 CHAPTER 7 161 We can apply the perpetuity formula to find the PV of stream B. The perpetuity formula discounts the stream back to year 1, one period prior to the first cash flow. Discount the PV as of the end of year 1 back one year to find the PV as of today, year 0. Doing so, we find: PV(B) = [C2 / R] / (1 + R) PV(B) = [$8,000 / 0.12] / (1.12) PV(B) = $59,523.81 b. If we combine the cash flow streams to form Project C, we get: Project A = [C3 / (R g)] / (1 + R)2 Project B = [C2 / R] / (1 + R) Project C = Project A + Project B Project C = [C3 / (R g)] / (1 + R)2 + [C2 / R] / (1 + R) 0 = [$7,000 / (IRR .03)] / (1 + IRR)2 + [$8,000 / IRR] / (1 + IRR) Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 13.21% c. The correct decision rule for an investing-type project is to accept the project if the discount rate is below the IRR. Since there is one IRR, a decision can be made. At a point in the future, the cash flows from stream A will be greater than those from stream B. Therefore, although there are many cash flows, there will be only one change in sign. When the sign of the cash flows change more than once over the life of the project, there may be multiple internal rates of return. In such cases, there is no correct decision rule for accepting and rejecting projects using the internal rate of return. 29. To answer this question, we need to examine the incremental cash flows. To make the projects equally attractive, Project Billion must have a larger initial investment. We know this because the subsequent cash flows from Project Billion are larger than the subsequent cash flows from Project Million. So, subtracting the Project Million cash flows from the Project Billion cash flows, we find the incremental cash flows are: Year 0 1 2 3 Incremental cash flows I + $1,500 300 300 500 Now we can find the present value of the subsequent incremental cash flows at the discount rate, 12 percent. The present value of the incremental cash flows is: PV = $1,500 + $300 / 1.12 + $300 / 1.122 + $500 / 1.123 PV = $2,362.91 CHAPTER 7 B-162 So, if I0 is greater than $2,362.91, the incremental cash flows will be negative. Since we are subtracting Project Million from Project Billion, this implies that for any value over $2,362.91 the NPV of Project Billion will be less than that of Project Million, so I0 must be less than $2,362.91. 30. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of the project is: 0 = $50,000 $61,000 / (1 + IRR) + $41,000 / (1 + IRR)2 Even though it appears there are two IRRs, a spreadsheet, financial calculator, or trial and error will not give an answer. The reason is that there is no real IRR for this set of cash flows. If you examine the IRR equation, what we are really doing is solving for the roots of the equation. Going back to high school algebra, in this problem we are solving a quadratic equation. In case you dont remember, the quadratic equation is: 2 x = b b 4ac 2a In this case, the equation is: x= ( 61,000) (61,000) 2 4(50,000)(41,000) 2(50,000) The square root term works out to be: 3,721,000,000 8,200,000,000 = 4,479,000,000 The square root of a negative number is a complex number, so there is no real number solution, meaning the project has no real IRR. CHAPTER 7 163 Calculator Solutions 1. b. Project A CFo C01 F01 C02 F02 C03 F03 I = 15% NPV CPT $515.62 CFo C01 F01 C02 F02 C03 F03 IRR CPT 16.74% 8. Project A CFo C01 F01 C02 F02 C03 F03 IRR CPT 16.79% $10,500 $6,000 1 $5,000 1 $1,500 1 CFo C01 F01 C02 F02 C03 F03 I = 15% NPV CPT $655.15 $8,400 $4,300 1 $3,900 1 $3,600 1 7. $10,500 $6,300 1 $4,900 1 $2,400 1 $4,900 $1,700 1 $2,900 1 $2,100 1 Project B CFo C01 F01 C02 F02 C03 F03 IRR CPT 13.82% $3,200 $1,100 1 $1,400 1 $1,700 1 CHAPTER 7 B-164 9. CFo 0 C01 $71,000 F01 7 I = 15% NPV CPT $295,389.90 PI = $295,389.90 / $260,000 = 1.1361 12. CFo C01 F01 C02 F02 C03 F03 C04 F04 IRR CPT 11.94% $10,000 $4,300 1 $3,900 1 $3,200 1 $1,200 1 CFo C01 F01 C02 F02 C03 F03 C04 F04 I = 10% NPV CPT $356.05 13. a. $10,000 $4,300 1 $3,900 1 $3,200 1 $1,200 1 CFo C01 F01 C02 F02 C03 F03 C04 F04 I = 20% NPV CPT $1,277.78 $10,000 $4,300 1 $3,900 1 $3,200 1 $1,200 1 Deepwater fishing CFo $850,000 C01 $480,000 F01 1 C02 $430,000 F02 1 C03 $320,000 F03 1 IRR CPT 22.70% Submarine ride CFo $2,100,000 C01 $1,500,000 F01 1 C02 $750,000 F02 1 C03 $600,000 F03 1 IRR CPT 20.66% CHAPTER 7 165 b. CFo C01 F01 C02 F02 C03 F03 IRR CPT 18.95% c. $1,250,000 $1,020,000 1 $320,000 1 $280,000 1 Deepwater fishing CFo $850,000 C01 $480,000 F01 1 C02 $430,000 F02 1 C03 $320,000 F03 1 I = 15% NPV CPT $102,938.28 Project I CFo C01 F01 I = 10% NPV CPT $79,579.26 $0 $32,000 3 Submarine ride CFo $2,100,000 C01 $1,500,000 F01 1 C02 $750,000 F02 1 C03 $600,000 F03 1 I = 15% NPV CPT $165,965.32 CFo C01 F01 I = 10% NPV CPT $19,579.26 $60,000 $32,000 3 14. PI = $79,579.26 / $60,000 = 1.326 Project II CFo C01 F01 I = 10% NPV CPT $49,737.04 $0 $20,000 3 CFo C01 F01 I = 10% NPV CPT $15,737.04 $34,000 $20,000 3 PI = $49,737.04 / $34,000 = 1.463 CHAPTER 7 B-166 15. CFo $42,000,000 C01 $65,000,000 F01 1 C02 $13,000,000 F02 1 I = 10% NPV CPT $6,347,107.44 CFo C01 F01 C02 F02 IRR CPT 31.16% $42,000,000 $65,000,000 1 $13,000,000 1 Financial calculators will only give you one IRR, even if there are multiple IRRs. Using trial and error, or a root solving calculator, the other IRR is 76.40%. 16. b. Board game CFo C01 F01 C02 F02 C03 F03 I = 10% NPV CPT $61,968.44 Board game CFo C01 F01 C02 F02 C03 F03 IRR CPT 23.34% CFo C01 F01 C02 F02 C03 F03 IRR CPT 20.49% $320,000 $240,000 1 $130,000 1 $75,000 1 CD-ROM CFo C01 F01 C02 F02 C03 F03 I = 10% NPV CPT $109,729.53 CD-ROM CFo C01 F01 C02 F02 C03 F03 IRR CPT 21.95% $550,000 $310,000 1 $280,000 1 $195,000 1 c. $320,000 $240,000 1 $130,000 1 $75,000 1 $550,000 $310,000 1 $280,000 1 $195,000 1 d. $230,000 $70,000 1 $150,000 1 $120,000 1 CHAPTER 7 167 17. a. CDMA CFo C01 F01 C02 0 $36,000,000 1 $23,000,000 G4 CFo C01 F01 C02 0 $29,000,000 1 $73,000,000 Wi-Fi CFo C01 F01 C02 F02 1 C03 $10,000,000 F03 1 I = 10% NPV CPT $59,248,685.20 F02 1 C03 $61,000,000 F03 1 I = 10% NPV CPT $132,524,417.73 F02 C03 F03 I = 10% NPV CPT $189,729,526.67 0 $43,000,000 1 $100,000,00 0 1 $85,000,000 1 PICDMA = $59,248,685.20 / $20,000,000 = 2.96 PIG4 = $132,524,417.73 / $35,000,000 = 3.79 PIWi-Fi = $189,729,526.67 / $45,000,000 = 3.45 b. CDMA CFo $20,000,000 C01 $36,000,000 F01 1 C02 $23,000,000 F02 1 C03 $10,000,000 F03 1 I = 10% NPV CPT $39,248,685.20 AZM CFo C01 F01 C02 F02 C03 F03 I = 10% NPV CPT $187,941.40 AZM CFo C01 F01 C02 F02 C03 F03 IRR CPT $450,000 $310,000 1 $240,000 1 $210,000 1 G4 Wi-Fi CFo $35,000,000 CFo $55,000,000 C01 $29,000,000 C01 $43,000,000 F01 1 F01 1 C02 $73,000,000 C02 $100,000,000 F02 1 F02 1 C03 $61,000,000 C03 $85,000,000 F03 1 F03 1 I = 10% I = 10% NPV CPT NPV CPT $97,524,417.73 $134,729,526.67 AZF CFo C01 F01 C02 F02 C03 F03 I = 10% NPV CPT $159,579.26 AZF CFo C01 F01 C02 F02 C03 F03 IRR CPT $800,000 $420,000 1 $390,000 1 $340,0000 1 18. b. c. $450,000 $310,000 1 $240,000 1 $210,000 1 $800,000 $420,000 1 $390,000 1 $340,0000 1 CHAPTER 7 B-168 34.40% 21.45% CHAPTER 7 169 19. a. Project A CFo 0 C01 $125,000 F01 2 C02 F02 I = 12% NPV CPT $211,256.38 Project B CFo 0 C01 $215,000 F01 2 C02 F02 I = 12% NPV CPT $363,360.97 Project C CFo C01 F01 C02 F02 I = 12% NPV CPT $207,748.72 0 $130,000 1 $115,000 1 PIA = $211,256.38 / $200,000 = 1.06 PIB = $363,360.97 / $350,000 = 1.04 PIC = $207,748.78 / $200,000 = 1.04 b. Project A CFo C01 F01 C02 F02 I = 12% NPV CPT $11,256.38 $200,000 $125,000 2 Project B CFo C01 F01 C02 F02 I = 12% NPV CPT $13,360.97 $350,000 $215,000 2 Project C CFo C01 F01 C02 F02 I = 12% NPV CPT $7,748.72 $200,000 $130,000 1 $115,000 1 d. Project B A CFo $150,000 C01 $90,000 F01 2 C02 F02 I = 12% NPV CPT $152,104.59 PI(B A) = $152,104.59 / $150,000 = 1.014 20. b. Dry prepeg CFo C01 F01 C02 F02 C03 F03 I = 10% NPV CPT $226,296.02 $1,800,000 $700,000 1 $500,000 1 $1,300,000 1 Solvent prepeg CFo $950,000 C01 $600,000 F01 1 C02 $400,000 F02 1 C03 $350,0000 F03 1 I = 10% NPV CPT $188,993.24 CHAPTER 7 B-170 c. Dry prepeg CFo C01 F01 C02 F02 C03 F03 IRR CPT 16.24% CFo C01 F01 C03 F03 IRR CPT 11.76% $1,800,000 $700,000 1 $500,000 1 $1,300,000 1 Solvent prepeg CFo $950,000 C01 $600,000 F01 1 C02 $400,000 F02 1 C03 $350,0000 F03 1 IRR CPT 22.25% d. $850,000 $100,000 2 $950,000 1 21. a. NP-30 CFo C01 F01 C02 F02 C03 F03 C04 F04 C05 F05 I = 12% NPV CPT $224,909.31 $900,000 $375,000 1 $350,000 1 $325,000 1 $275,000 1 $185,000 1 NX-20 CFo C01 F01 C02 F02 C03 F03 C04 F04 C05 F05 I = 12% NPV CPT $238,551.78 $650,000 $250,000 2 $300,000 1 $250,000 1 $165,000 1 CHAPTER 7 171 b. NP-30 CFo C01 F01 C02 F02 C03 F03 C04 F04 C05 F05 IRR CPT 22.81% CFo C01 F01 C02 F02 C03 F03 C04 F04 IRR CPT 8.74% $900,000 $375,000 1 $350,000 1 $325,000 1 $275,000 1 $185,000 1 NX-20 CFo C01 F01 C02 F02 C03 F03 C04 F04 C05 F05 IRR CPT 26.55% $650,000 $250,000 2 $300,000 1 $250,000 1 $165,000 1 c. $250,000 $125,000 1 $100,000 1 $25,000 2 $20,000 1 d. NP-30 CFo 0 C01 $375,000 F01 1 C02 $350,000 F02 1 C03 $325,000 F03 1 C04 $275,000 F04 1 C05 $185,000 F05 1 I = 12% NPV CPT $1,124,909.31 NX-20 CFo C01 F01 C02 F02 C03 F03 C04 F04 C05 F05 I = 12% NPV CPT $888,551.78 0 $250,000 2 $300,000 1 $250,000 1 $165,000 1 PINP-30 = $1,124,909.31 / $900,000 = 1.250 PINX-20 = $888,551.78 / $650,000 = 1.367 CHAPTER 7 B-172 22. a. Project A CFo C01 F01 C02 F02 C03 F03 C04 F04 C05 F05 I = 15% NPV CPT $181,173.60 Project A CFo C01 F01 C02 F02 C03 F03 C04 F04 C05 F05 IRR CPT 28.47% CFo C01 F01 C02 F02 C03 F03 C04 F04 C05 F05 IRR CPT 18.55% $650,000 $320,000 2 $230,000 1 $175,000 1 $120,000 1 Project B CFo C01 F01 C02 F02 C03 F03 C04 F04 C05 F05 I = 15% NPV CPT $229,732.75 Project B CFo C01 F01 C02 F02 C03 F03 C04 F04 C05 F05 I = 15% 23.59% $975,000 $260,000 1 $350,000 1 $360,000 1 $400,000 1 $500,000 1 b. $650,000 $320,000 2 $230,000 1 $175,000 1 $120,000 1 $975,000 $260,000 1 $350,000 1 $360,000 1 $400,000 1 $500,000 1 c. $325,000 $60,000 1 $30,000 1 $130,000 1 $225,000 1 $380,000 1 CHAPTER 7 173 d. Project A CFo C01 F01 C02 F02 C03 F03 C04 F04 C05 F05 I = 15% NPV CPT $831,173.60 0 $320,000 2 $230,000 1 $175,000 1 $120,000 1 Project B CFo 0 C01 $260,000 F01 1 C02 $350,000 F02 1 C03 $360,000 F03 1 C04 $400,000 F04 1 C05 $500,000 F05 1 I = 15% NPV CPT $1,204,732.75 PIA = $831,173.60 / $650,000 = 1.279 PIB = $1,204,732.75 / $975,000 = 1.236 23. b. Project A CFo C01 F01 C02 F02 IRR CPT 8.35% Project B C CFo C01 F01 C02 F02 IRR CPT 45.57% Project A CFo C01 F01 C02 F02 I = 10% NPV CPT $390,495.87 $400,000 $175,000 1 $280,000 1 Project B CFo C01 F01 C02 F02 IRR CPT 36.08% $200,000 $195,000 1 $105,000 1 Project C CFo C01 F01 C02 F02 IRR CPT 31.93% $150,000 $160,000 1 $50,000 1 c. $50,000 $35,000 1 $55,000 1 d. 0 $175,000 1 $280,000 1 Project B CFo C01 F01 C02 F02 I = 00% NPV CPT $235,416.67 0 $195,000 1 $105,000 1 Project C CFo C01 F01 C02 F02 I = 00% NPV CPT $168,055.56 0 $160,000 1 $50,000 1 PIA = $390,495.87 / $400,000 = 0.98 PIB = $235,416.67 / $200,000 = 1.18 PIC = $168,055.56 / $150,000 = 1.12 CHAPTER 7 B-174 e. Project A CFo C01 F01 C02 F02 I = 10% NPV CPT $9,504.13 CFo C01 F01 C02 F02 IRR CPT ERROR 7 $400,000 $175,000 1 $280,000 1 Project B CFo C01 F01 C02 F02 I = 20% NPV CPT $35,416.67 $200,000 $195,000 1 $105,000 1 Project C CFo C01 F01 C02 F02 I = 20% NPV CPT $18,055.56 $150,000 $160,000 1 $50,000 1 30. $50,000 $61,000 1 $41,000 1 CHAPTER 8 MAKING CAPITAL INVESTMENT DECISIONS Answers to Concept Questions 1. In this context, an opportunity cost refers to the value of an asset or other input that will be used in a project. The relevant cost is what the asset or input is actually worth today, not, for example, what it cost to acquire. a. Yes, the reduction in the sales of the companys other products, referred to as erosion, should be treated as an incremental cash flow. These lost sales are included because they are a cost (a revenue reduction) that the firm must bear if it chooses to produce the new product. Yes, expenditures on plant and equipment should be treated as incremental cash flows. These are costs of the new product line. However, if these expenditures have already occurred (and cannot be recaptured through a sale of the plant and equipment), they are sunk costs and are not included as incremental cash flows. No, the research and development costs should not be treated as incremental cash flows. The costs of research and development undertaken on the product during the past three years are sunk costs and should not be included in the evaluation of the project. Decisions made and costs incurred in the past cannot be changed. They should not affect the decision to accept or reject the project. Yes, the annual depreciation expense must be taken into account when calculating the cash flows related to a given project. While depreciation is not a cash expense that directly affects cash flow, it decreases a firms net income and hence, lowers its tax bill for the year. Because of this depreciation tax shield, the firm has more cash on hand at the end of the year than it would have had without expensing depreciation. No, dividend payments should not be treated as incremental cash flows. A firms decision to pay or not pay dividends is independent of the decision to accept or reject any given investment project. For this reason, dividends are not an incremental cash flow to a given project. Dividend policy is discussed in more detail in later chapters. Yes, the resale value of plant and equipment at the end of a projects life should be treated as an incremental cash flow. The price at which the firm sells the equipment is a cash inflow, and any difference between the book value of the equipment and its sale price will create accounting gains or losses that result in either a tax credit or liability. Yes, salary and medical costs for production employees hired for a project should be treated as incremental cash flows. The salaries of all personnel connected to the project must be included as costs of that project. 2. b. c. d. e. f. g. CHAPTER 10 B-176 3. Item I is a relevant cost because the opportunity to sell the land is lost if the new golf club is produced. Item II is also relevant because the firm must take into account the erosion of sales of existing products when a new product is introduced. If the firm produces the new club, the earnings from the existing clubs will decrease, effectively creating a cost that must be included in the decision. Item III is not relevant because the costs of research and development are sunk costs. Decisions made in the past cannot be changed. They are not relevant to the production of the new clubs. For tax purposes, a firm would choose MACRS because it provides for larger depreciation deductions earlier. These larger deductions reduce taxes, but have no other cash consequences. Notice that the choice between MACRS and straight-line is purely a time value issue; the total depreciation is the same; only the timing differs. Its probably only a mild over-simplification. Current liabilities will all be paid, presumably. The cash portion of current assets will be retrieved. Some receivables wont be collected, and some inventory will not be sold, of course. Counterbalancing these losses is the fact that inventory sold above cost (and not replaced at the end of the projects life) acts to increase working capital. These effects tend to offset one another. Managements discretion to set the firms capital structure is applicable at the firm level. Since any one particular project could be financed entirely with equity, another project could be financed with debt, and the firms overall capital structure would remain unchanged, financing costs are not relevant in the analysis of a projects incremental cash flows according to the stand-alone principle. The EAC approach is appropriate when comparing mutually exclusive projects with different lives that will be replaced when they wear out. This type of analysis is necessary so that the projects have a common life span over which they can be compared. For example, if one project has a three-year life and the other has a five-year life, then a 15-year horizon is the minimum necessary to place the two projects on an equal footing, implying that one project will be repeated five times and the other will be repeated three times. Note the shortest common life may be quite long when there are more than two alternatives and/or the individual project lives are relatively long. Assuming this type of analysis is valid implies that the project cash flows remain the same over the common life, thus ignoring the possible effects of, among other things: (1) inflation, (2) changing economic conditions, (3) the increasing unreliability of cash flow estimates that occur far into the future, and (4) the possible effects of future technology improvement that could alter the project cash flows. Depreciation is a non-cash expense, but it is tax-deductible on the income statement. Thus depreciation causes taxes paid, an actual cash outflow, to be reduced by an amount equal to the depreciation tax shield, tcD. A reduction in taxes that would otherwise be paid is the same thing as a cash inflow, so the effects of the depreciation tax shield must be added in to get the total incremental aftertax cash flows. There are two particularly important considerations. The first is erosion. Will the essentialized book simply displace copies of the existing book that would have otherwise been sold? This is of special concern given the lower price. The second consideration is competition. Will other publishers step in and produce such a product? If so, then any erosion is much less relevant. A particular concern to book publishers (and producers of a variety of other product types) is that the publisher only makes money from the sale of new books. Thus, it is important to examine whether the new book would displace sales of used books (good from the publishers perspective) or new books (not good). The concern arises any time there is an active market for used product. 4. 5. 6. 7. 8. 9. CHAPTER 10 177 10. Definitely. The damage to Porsches reputation is a factor the company needed to consider. If the reputation was damaged, the company would have lost sales of its existing car lines. 11. One company may be able to produce at lower incremental cost or market better. Also, of course, one of the two may have made a mistake! 12. Porsche would recognize that the outsized profits would dwindle as more products come to market and competition becomes more intense. Solutions to Questions and Problems NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. NPV = $1,675.46 Since the NPV is positive, the company should accept the project. 2. We will use the bottom-up approach to calculate the operating cash flow for each year. We also must be sure to include the net working capital cash flows each year. So, the total cash flow each year will be: Year 1 $11,500 2,000 6,000 $3,500 1,190 $2,310 $8,310 0 350 $7,960 Year 2 $12,800 2,000 6,000 $4,800 1,632 $3,168 $9,168 0 300 $8,868 Year 3 $13,700 2,000 6,000 $5,700 1,938 $3,762 $9,762 0 250 $9,512 Year 4 $10,300 2,000 6,000 $2,300 782 $1,518 $7,518 0 1,200 $8,718 Sales Costs Depreciation EBT Tax Net income OCF Capital spending NWC Incremental cash flow $24,000 300 $24,300 CHAPTER 10 B-178 The NPV for the project is: NPV = $2,187.56 3. Using the tax shield approach to calculating OCF, we get: NPV = $7,472.45 4. The cash outflow at the beginning of the project will increase because of the spending on NWC. At the end of the project, the company will recover the NWC, so it will be a cash inflow. The sale of the equipment will result in a cash inflow, but we also must account for the taxes which will be paid on this sale. So, the cash flows for each year of the project will be: Year 0 1 2 3 Cash Flow $3,100,000 1,122,917 1,122,917 1,715,417 = $2,800,000 300,000 = $1,122,917 + 300,000 + 450,000 + (0 450,000)(.35) And the NPV of the project is: NPV = $137,681.57 5. First we will calculate the annual depreciation for the equipment necessary for the project. The depreciation amount each year will be: Year 1 depreciation = $2,800,000(0.3333) = $933,240 Year 2 depreciation = $2,800,000(0.4445) = $1,244,600 Year 3 depreciation = $2,800,000(0.1481) = $414,680 So, the book value of the equipment at the end of three years, which will be the initial investment minus the accumulated depreciation, is: Book value in 3 years = $2,800,000 ($933,240 + 1,244,600 + 414,680) Book value in 3 years = $207,480 The asset is sold at a gain to book value, so this gain is taxable. Aftertax salvage value = $450,000 + ($207,480 450,000)(0.35) Aftertax salvage value = $365,118 To calculate the OCF, we will use the tax shield approach, so the cash flow each year is: OCF = (Sales Costs)(1 tC) + tCDepreciation Year 0 1 2 3 Cash Flow $3,100,000 1,122,884 1,231,860 1,606,506 = $2,800,000 300,000 = $1,475,000(.65) + 0.35($933,240) = $1,475,000(.65) + 0.35($1,244,600) = $1,475,000(.65) + 0.35($414,680) + $300,000 + 365,118 CHAPTER 10 179 Remember to include the NWC cost in Year 0, and the recovery of the NWC at the end of the project. The NPV of the project with these assumptions is: NPV = $3,100,000 + ($1,112,884/1.10) + ($1,231,860/1.102) + ($1,606,506/1.103) NPV = $145,861.49 6. First, we will calculate the annual depreciation of the new equipment. It will be: IRR = 23.16% 7. 8. NPV = $54,822.06 To find the BV at the end of four years, we need to find the accumulated depreciation for the first four years. We could calculate a table with the depreciation each year, but an easier way is to add the MACRS depreciation amounts for each of the first four years and multiply this percentage times the cost of the asset. We can then subtract this from the asset cost. Doing so, we get: BV4 = $7,400,000 7,400,000(0.2000 + 0.3200 + 0.1920 + 0.1152) BV4 = $1,278,720 The asset is sold at a gain to book value, so this gain is taxable. Aftertax salvage value = $1,750,000 + ($1,278,720 1,750,000)(.35) Aftertax salvage value = $1,585,052 9. NPV = $116,246.23 10. We will need the aftertax salvage value of the equipment to compute the EAC. Even though the equipment for each product has a different initial cost, both have the same salvage value. The aftertax salvage value for both is: Aftertax salvage value = $76,000(1 0.35) = $49,400 To calculate the EAC, we first need the OCF and NPV of each option. The OCF and NPV for Techron I are: OCF = $85,000(1 0.35) + 0.35($450,000/3) = $2,750 NPV = $450,000 $2,750(PVIFA14%,3) + ($49,400/1.143) = $423,040.90 EAC = $423,040.90 / (PVIFA14%,3) = $182,217.03 And the OCF and NPV for Techron II are: OCF = $91,000(1 0.35) + 0.35($580,000/5) = $18,550 NPV = $580,000 $18,550(PVIFA14%,5) + ($49,400/1.145) = $618,026.84 EAC = $618,026.84 / (PVIFA14%,5) = $180,021.05 CHAPTER 10 B-180 The two milling machines have unequal lives, so they can only be compared by expressing both on an equivalent annual basis, which is what the EAC method does. Thus, you prefer the Techron II because it has the lower (less negative) annual cost. Intermediate 11. First, we will calculate the depreciation each year, which will be: D1 = $475,000(0.2000) = $95,000 D2 = $475,000(0.3200) = $152,000 D3 = $475,000(0.1920) = $91,200 D4 = $475,000(0.1152) = $54,720 The book value of the equipment at the end of the project is: BV4 = $475,000 ($95,000 + 152,000 + 91,200 + 54,720) = $82,080 The asset is sold at a loss to book value, so this creates a tax refund. Aftertax salvage value = $45,000 + ($82,080 45,000)(0.35) = $57,978 So, the OCF for each year will be: OCF1 = $183,000(1 0.35) + 0.35($95,000) = $152,200 OCF2 = $183,000(1 0.35) + 0.35($152,000) = $172,150 OCF3 = $183,000(1 0.35) + 0.35($91,200) = $150,870 OCF4 = $183,000(1 0.35) + 0.35($54,720) = $138,102 Now we have all the necessary information to calculate the project NPV. We need to be careful with the NWC in this project. Notice the project requires $20,000 of NWC at the beginning, and $4,000 more in NWC each successive year. We will subtract the $20,000 from the initial cash flow, and subtract $4,000 each year from the OCF to account for this spending. In Year 4, we will add back the total spent on NWC, which is $32,000. The $4,000 spent on NWC capital during Year 4 is irrelevant. Why? Well, during this year the project required an additional $4,000, but we would get the money back immediately. So, the net cash flow for additional NWC would be zero. With all this, the equation for the NPV of the project is: NPV = $475,000 20,000 + ($152,200 4,000)/1.09 + ($172,150 4,000)/1.092 + ($150,870 4,000)/1.093 + ($138,102 + 32,000 + 57,978)1.094 NPV = $57,480.00 12. If we are trying to decide between two projects that will not be replaced when they wear out, the proper capital budgeting method to use is NPV. Both projects only have costs associated with them, not sales, so we will use these to calculate the NPV of each project. Using the tax shield approach to calculate the OCF, the NPV of System A is: OCFA = $141,000(1 0.34) + 0.34($530,000/4) OCFA = $48,010 NPVA = $530,000 $48,010(PVIFA11%,4) NPVA = $678,948.42 CHAPTER 10 181 And the NPV of System B is: OCFB = $120,000(1 0.34) + 0.34($720,000/6) OCFB = $38,400 NPVB = $720,000 $38,400(PVIFA11%,6) NPVB = $882,452.65 If the system will not be replaced when it wears out, then System A should be chosen, because it has the less negative NPV. 13. If the equipment will be replaced at the end of its useful life, the correct capital budgeting technique is EAC. Using the NPVs we calculated in the previous problem, the EAC for each system is: EACA = $678,948.42 / (PVIFA11%,4) EACA = $218,842.97 EACB = $882,452.65 / (PVIFA11%,6) EACB = $208,591.13 If the conveyor belt system will be continually replaced, we should choose System B since it has the more positive EAC. 14. Since we need to calculate the EAC for each machine, revenue is irrelevant. The sales figure is only used to calculate the variable costs since EAC only uses the costs of operating the equipment, not the sales. Using the bottom up approach, or net income plus depreciation, method to calculate OCF, we get: Machine A Machine B Variable costs $4,200,000 $3,600,000 Fixed costs 2,100,000 2,400,000 Depreciation 650,000 600,000 EBT $6,950,000 $6,600,000 Tax 2,432,500 2,310,000 Net income $4,517,500 $4,290,000 + Depreciation 650,000 600,000 OCF $3,867,500 $3,690,000 The NPV and EAC for Machine A is: NPVA = $3,900,000 $3,867,500(PVIFA10%,6) NPVA = $20,743,970.76 EACA = $20,743,970.76 / (PVIFA10%,6) EACA = $4,762,968.78 And the NPV and EAC for Machine B is: CHAPTER 10 B-182 NPVB = $5,400,000 3,690,000(PVIFA10%,9) NPVB = $26,650,797.88 EACB = $26,650,797.88 / (PVIFA10%,9) EACB = $4,627,658.91 You should choose Machine B since it has a more positive EAC. 15. When we are dealing with nominal cash flows, we must be careful to discount cash flows at the nominal interest rate, and we must discount real cash flows using the real interest rate. Project As cash flows are in real terms, so we need to find the real interest rate. Using the Fisher equation, the real interest rate is: 1 + R = (1 + r)(1 + h) 1.11 = (1 + r)(1 + .04) r = .0673 or 6.73% So, the NPV of Project As real cash flows, discounting at the real interest rate, is: NPV = $46,000 + $22,000 / 1.0673 + $30,000 / 1.06732 + $16,000 / 1.06733 NPV = $14,107.99 Project Bs cash flow are in nominal terms, so the NPV discount at the nominal interest rate is: NPV = $56,000 + $29,000 / 1.11 + $36,000 / 1.112 + $18,000 / 1.113 NPV = $12,505.98 We should accept Project A if the projects are mutually exclusive since it has the highest NPV. 16. To determine the value of a firm, we can simply find the present value of the firms future cash flows. No depreciation is given, so we can assume depreciation is zero. Using the tax shield approach, we can find the present value of the aftertax revenues, and the present value of the aftertax costs. The required return, growth rates, price, and costs are all given in real terms. Subtracting the costs from the revenues will give us the value of the firms cash flows. We must calculate the present value of each separately since each is growing at a different rate. First, we will find the present value of the revenues. The revenues in year 1 will be the number of bottles sold, times the price per bottle, or: Aftertax revenue in year 1 in real terms = (5,000,000 $1.10)(1 0.34) Aftertax revenue in year 1 in real terms = $3,630,000 Revenues will grow at 2 percent per year in real terms forever. Apply the growing perpetuity formula, we find the present value of the revenues is: PV of revenues = C1 / (R g) PV of revenues = $3,630,000/ (0.06 0.02) PV of revenues = $90,750,000 The real aftertax costs in year 1 will be: Aftertax costs in year 1 in real terms = (5,000,000 $0.89)(1 0.34) CHAPTER 10 183 Aftertax costs in year 1 in real terms = $2,937,000 Costs will grow at 1.5 percent per year in real terms forever. Applying the growing perpetuity formula, we find the present value of the costs is: PV of costs = C1 / (R g) PV of costs = $2,937,000 / (0.06 0.015) PV of costs = $65,266,667 Now we can find the value of the firm, which is: Value of the firm = PV of revenues PV of costs Value of the firm = $90,750,000 65,266,667 Value of the firm = $25,483,333 17. To calculate the nominal cash flows, we simple increase each item in the income statement by the inflation rate, except for depreciation. Depreciation is a nominal cash flow, so it does not need to be adjusted for inflation in nominal cash flow analysis. Since the resale value is given in nominal terms as of the end of year 5, it does not need to be adjusted for inflation. Also, no inflation adjustment is needed for either the depreciation charge or the recovery of net working capital since these items are already expressed in nominal terms. Note that an increase in required net working capital is a negative cash flow whereas a decrease in required net working capital is a positive cash flow. The nominal aftertax salvage value is: Market price Tax on sale Aftertax salvage value $60,000 20,400 $39,600 Remember, to calculate the taxes paid (or tax credit) on the salvage value, we take the book value minus the market value, times the tax rate, which, in this case, would be: Taxes on salvage value = (BV MV)tC Taxes on salvage value = ($0 60,000)(.34) Taxes on salvage value = $20,400 Now we can find the nominal cash flows each year using the income statement. Doing so, we find: Year 0 Sales Expenses Depreciation EBT Tax Net income OCF Capital spending $450,000 Year 1 $230,000 72,000 90,000 $68,000 23,120 $44,880 $134,880 Year 2 $239,200 74,880 90,000 $74,320 25,269 $49,051 $139,051 Year 3 $248,768 77,875 90,000 $80,893 27,504 $53,389 $143,389 Year 4 $258,719 80,990 90,000 $87,729 29,828 $57,901 $147,901 Year 5 $269,067 84,230 90,000 $94,838 32,245 $62,593 $152,593 $39,600 CHAPTER 10 B-184 NWC Total cash flow 25,000 $475,000 25,000 $217,193 $134,880 $139,051 $143,389 $147,901 18. The present value of the company is the present value of the future cash flows generated by the company. Here we have real cash flows, a real interest rate, and a real growth rate. The cash flows are a growing perpetuity, with a negative growth rate. Using the growing perpetuity equation, the present value of the cash flows are: PV = C1 / (R g) PV = $210,000 / [.065 (.03)] PV = $2,210,526.32 19. To find the EAC, we first need to calculate the NPV of the incremental cash flows. We will begin with the aftertax salvage value, which is: Taxes on salvage value = (BV MV)tC Taxes on salvage value = ($0 9,000)(.34) Taxes on salvage value = $3,060 Market price Tax on sale Aftertax salvage value $9,000 3,060 $5,940 Now we can find the operating cash flows. Using the tax shield approach, the operating cash flow each year will be: OCF = $6,500(1 0.34) + 0.34($71,000/7) OCF = $841.43 So, the NPV of the cost of the decision to buy is: NPV = $71,000 + $841.43(PVIFA12%,7) + ($5,940/1.127) NPV = $72,153.12 In order to calculate the equivalent annual cost, set the NPV of the equipment equal to an annuity with the same economic life. Since the project has an economic life of three years and is discounted at 12 percent, set the NPV equal to a three-year annuity, discounted at 12 percent. EAC = $72,153.12 / (PVIFA12%,7) EAC = $15,810.03 20. We will find the EAC of the EVF first. There are no taxes since the university is tax-exempt, so the maintenance costs are the operating cash flows. The NPV of the decision to buy one EVF is: NPV = $8,500 $1,800(PVIFA9%,4) NPV = $14,331.50 In order to calculate the equivalent annual cost, set the NPV of the equipment equal to an annuity with the same economic life. Since the project has an economic life of four years and is discounted at 9 percent, set the NPV equal to a three-year annuity, discounted at 9 percent. So, the EAC per unit is: CHAPTER 10 185 EAC = $14,331.50 / (PVIFA9%,4) EAC = $4,423.68 Since the university must buy 10 of the mowers, the total EAC of the decision to buy the EVF mower is: Total EAC = 10($4,423.68) Total EAC = $44,236.84 Note, we could have found the total EAC for this decision by multiplying the initial cost by the number of mowers needed, and multiplying the annual maintenance cost of each by the same number. We would have arrived at the same EAC. We can find the EAC of the AEH mowers using the same method, but we need to include the salvage value as well. There are no taxes on the salvage value since the university is tax-exempt, so the NPV of buying one AEH will be: NPV = $5,300 $2,300(PVIFA9%,3) + ($800/1.093) NPV = $10,504.23 So, the EAC per mower is: EAC = $10,504.23 / (PVIFA9%,3) EAC = $4,149.75 Since the university must buy 11 of the mowers, the total EAC of the decision to buy the AEH mowers is: Total EAC = 11($4,149.75) Total EAC = $45,647.21 The university should buy the EVF mowers since the EAC is lower. Notice that the EAC of the AEH is lower on a per mower basis, but because the university needs more of these mowers, the total EAC is higher. 21. NPV = $48,925.36 22. a. NPV = $208,956.36 IRR = 11.29% So, this analysis still tells us the company should purchase the new machine. This is really the same type of analysis we originally did. Consider this: Subtract the NPV of the decision to keep the old machine from the NPV of the decision to purchase the new machine. You will get: b. Even though the saved expenses are less than the cost of the machine, the cash flows are also increased because of the higher depreciation of the new machine. The depreciation tax shield increases the cash flows enough to make the NPV positive. CHAPTER 10 B-186 23. We can find the NPV of a project using nominal cash flows or real cash flows. Either method will result in the same NPV. For this problem, we will calculate the NPV using both nominal and real cash flows. The initial investment in either case is $1,140,000 since it will be spent today. We will begin with the nominal cash flows. The revenues and production costs increase at different rates, so we must be careful to increase each at the appropriate growth rate. The nominal cash flows for each year will be: Year 0 Revenues Costs Depreciation EBT Taxes Net income OCF Capital spending Total cash flow $1,140,000 $1,140,000 Year 4 Revenues Costs Depreciation EBT Taxes Net income OCF $868,218.75 461,194.24 162,857.14 $244,167.37 83,016.90 $161,150.46 $324,007.61 Year 1 $750,000.00 $410,000.00 162,857.14 $177,142.86 60,228.57 $116,914.29 $279,771.43 Year 2 $787,500.00 426,400.00 162,857.14 $198,242.86 67,402.57 $130,840.29 $293,697.43 Year 3 $826,875.00 443,456.00 162,857.14 $220,561.86 74,991.03 $145,570.83 $308,427.97 $279,771.43 Year 5 $911,629.69 479,642.01 162,857.14 $269,130.54 91,504.38 $177,626.15 $340,483.30 $293,697.43 Year 6 $957,211.17 498,827.69 162,857.14 $295,526.34 100,478.96 $195,047.38 $357,904.53 $308,427.97 Year 7 $1,005,071.7 3 518,780.80 162,857.14 $323,433.79 109,967.49 $213,466.30 $376,323.44 Now that we have the nominal cash flows, we can find the NPV. We must use the nominal required return with nominal cash flows. Using the Fisher equation to find the nominal required return, we get: (1 + R) = (1 + r)(1 + h) (1 + R) = (1 + .11)(1 + .05) R = .1655 or 16.55% So, the NPV of the project using nominal cash flows is: NPV = $1,140,000 + $279,771.43 / 1.1655 + $293,697.43 / 1.16552 + $308,427.97 / 1.16553 + $324,007.61 / 1.16554 + $340,483.30 / 1.16555 + $357,904.53 / 1.16556 + $376,323.44 / 1.16557 NPV = $116,548.87 CHAPTER 10 187 We can also find the NPV using real cash flows and the real required return. This will allow us to find the operating cash flow using the tax shield approach. Both the revenues and expenses are growing annuities, but growing at different rates. This means we must find the present value of each separately. We also need to account for the effect of taxes, so we will multiply by one minus the tax rate. So, the present value of the aftertax revenues using the growing annuity equation is: PV of aftertax revenues = C {[1/(r g)] [1/(r g)] [(1 + g)/(1 + r)]t}(1 tC) PV of aftertax revenues = $750,000{[1/(.11 .05)] [1/(.11 .05)] [(1 + .05)/(1+.11)]7}(1.34) PV of aftertax revenues = $2,221,463.95 And the present value of the aftertax costs will be: PV of aftertax costs = C {[1/(r g)] [1/(r g)] [(1 + g)/(1 + r)]t}(1 tC) PV of aftertax costs = $410,000{[1/(.11 .04)] [1/(.11 .04)] [(1 + .04)/(1 + .11)]7}(1 .34) PV of aftertax costs = $1,184,924.26 Now we need to find the present value of the depreciation tax shield. The depreciation amount in the first year is a real value, so we can find the present value of the depreciation tax shield as an ordinary annuity using the real required return. So, the present value of the depreciation tax shield will be: PV of depreciation tax shield = ($1,140,000/7)(.34)(PVIFA11%,7) PV of depreciation tax shield = $220,045.17 Using the present value of the real cash flows to find the NPV, we get: NPV = Initial cost + PV of aftertax revenues PV of aftertax costs + PV of depreciation tax shield NPV = $1,140,000 + 2,221,463.95 1,184,924.26 + 220,045.17 NPV = $116,584.87 Notice, the NPV using nominal cash flows or real cash flows is identical, which is what we would expect. 24. Here we have a project in which the quantity sold each year increases. First, we need to calculate the quantity sold each year by increasing the current years quantity by the growth rate. So, the quantity sold each year will be: Year 1 quantity = 8,000 Year 2 quantity = 8,000(1 + .08) Year 3 quantity = 8,640(1 + .08) Year 4 quantity = 9,331(1 + .08) Year 5 quantity = 10,078(1 + .08) = 8,640 = 9,331 = 10,078 = 10,884 Now we can calculate the sales revenue and variable costs each year. The pro forma income statements and operating cash flow each year will be: CHAPTER 10 B-188 Year 0 Revenues Fixed costs Variable costs Depreciation EBT Taxes Net income OCF Equipment NWC Total CF $320,000 50,000 $370,000 Year 1 $440,000.0 0 190,000.00 184,000.00 64,000.00 $2,000.00 680.00 $1,320.00 $65,320.00 Year 2 $475,200.00 190,000.00 198,720.00 64,000.00 $22,480.00 7,643.20 $14,836.80 $78,836.80 Year 3 $513,216.00 190,000.00 214,617.60 64,000.00 $44,598.40 15,163.46 $29,434.94 $93,434.94 Year 4 $554,273.28 190,000.00 231,787.01 64,000.00 $68,486.27 23,285.33 $45,200.94 $109,200.94 Year 5 $598,615.14 190,000.00 250,329.97 64,000.00 $94,285.17 32,056.96 $62,228.21 $126,228.21 50,000 $65,320.00 $78,836.80 $93,434.94 $109,200.94 $176,228.21 So, the NPV of the project is: NPV = $370,000 + $65,230 / 1.17 + $78,836.80 / 1.172 + $93,434.94 / 1.173 + $109,200.94 / 1.174 + $176,228.21 / 1.175 NPV = $59,586.82 We could also have calculated the cash flows using the tax shield approach, with growing annuities and ordinary annuities. The sales and variable costs increase at the same rate as sales, so both are growing annuities. The fixed costs and depreciation are both ordinary annuities. Using the growing annuity equation, the present value of the revenues is: PV of revenues = C {[1/(r g)] [1/(r g)] [(1 + g)/(1 + r)]t}(1 tC) PV of revenues = $440,000{[1/(.17 .08)] [1/(.17 .08)] [(1 + .08)/(1 + .17)]5} PV of revenues = $1,612,468.38 And the present value of the variable costs will be: PV of variable costs = C{[1/(r g)] [1/(r g)] [(1 + g)/(1 + r)]t}(1 tC) PV of variable costs = $184,000{[1/(.17 .08)] [1/(.17 .08)] [(1 + .08)/(1 + .17)]5} PV of variable costs = $674,304.96 The fixed costs and depreciation are both ordinary annuities. The present value of each is: PV of fixed costs = C({1 [1/(1 + r)]t } / r ) PV of fixed costs = $190,000({1 [1/(1 + .17)]5 } / .17) PV of fixed costs = $607,875.77 PV of depreciation = C({1 [1/(1 + r)]t } / r ) PV of depreciation = $64,000({1 [1/(1 + .17)]5 } / .17) PV of depreciation = $204,758.15 CHAPTER 10 189 Now, we can use the depreciation tax shield approach to find the NPV of the project, which is: NPV = $370,000 + ($1,612,468.38 674,304.96 607,875.77)(1 .34) + ($204,758.15)(.34) + $50,000 / 1.175 NPV = $59,586.82 25. We will begin by calculating the aftertax salvage value of the equipment at the end of the projects life. The aftertax salvage value is the market value of the equipment minus any taxes paid (or refunded), so the aftertax salvage value in four years will be: Taxes on salvage value = (BV MV)tC Taxes on salvage value = ($0 400,000)(.38) Taxes on salvage value = $152,000 Market price Tax on sale Aftertax salvage value $400,000 152,000 $248,000 Now we need to calculate the operating cash flow each year. Using the bottom up approach to calculating operating cash flow, we find: Year 0 Revenues Fixed costs Variable costs Depreciation EBT Taxes Net income OCF Capital spending Land NWC Total cash flow $4,300,000 $1,150,000 $150,000 $5,600,000 $1,255,752 $2,006,613 $1,743,635 Year 1 $2,220,000 740,000 333,000 1,433,190 $286,190 108,752 $177,438 $1,255,752 Year 2 $3,300,000 740,000 495,000 1,911,350 $153,650 58,387 $95,263 $2,006,613 Year 3 $3,720,000 740,000 558,000 636,830 $1,785,170 678,365 $1,106,805 $1,743,635 Year 4 $2,460,000 740,000 369,000 318,630 $1,032,370 392,301 $640,069 $958,699 248,000 1,250,000 150,000 $2,606,699 Notice the calculation of the cash flow at time 0. The capital spending on equipment and investment in net working capital are cash outflows are both cash outflows. The aftertax selling price of the land is also a cash outflow. Even though no cash is actually spent on the land because the company already owns it, the aftertax cash flow from selling the land is an opportunity cost, so we need to include it in the analysis. Additionally, at the end of the project, the land can be sold. With all the project cash flows, we can calculate the NPV, which is: CHAPTER 10 B-190 NPV = $5,600,000 + $1,255,752 / 1.13 + $2,006,613 / 1.132 + $1,743,635 / 1.133 + $2,606,699 / 1.134 NPV = $110,078.20 The company should not proceed with the manufacture of the zithers. 26. Replacement decision analysis is the same as the analysis of two competing projects, in this case, keep the current equipment, or purchase the new equipment. We will consider the purchase of the new machine first. Purchase new machine: The initial cash outlay for the new machine is the cost of the new machine. We can calculate the operating cash flow created if the company purchases the new machine. The maintenance cost is an incremental cash flow, so using the pro forma income statement, and adding depreciation to net income, the operating cash flow created by purchasing the new machine each year will be: Maintenance cost Depreciation EBT Taxes Net income OCF $265,000 840,000 $1,105,000 375,700 $729,300 $110,700 Notice the taxes are negative, implying a tax credit. The new machine also has a salvage value at the end of five years, so we need to include this in the cash flows analysis. The aftertax salvage value will be: Sell machine Taxes Total The NPV of purchasing the new machine is: NPV = $4,200,000 $110,700(PVIFA12%,5) + $396,000 / 1.125 NPV = $3,576,250.24 Notice the NPV is negative. This does not necessarily mean we should not purchase the new machine. In this analysis, we are only dealing with costs, so we would expect a negative NPV. The revenue is not included in the analysis since it is not incremental to the machine. Similar to an EAC analysis, we will use the machine with the least negative NPV. Now we can calculate the decision to keep the old machine: $600,000 204,000 $396,000 CHAPTER 10 191 Keep old machine: The initial cash outlay for keeping the old machine is the market value of the old machine, including any potential tax. The decision to keep the old machine has an opportunity cost, namely, the company could sell the old machine. Also, if the company sells the old machine at its current value, it will incur taxes. Both of these cash flows need to be included in the analysis. So, the initial cash flow of keeping the old machine will be: Keep machine Taxes Total $1,800,000 238,000 $1,562,000 Next, we can calculate the operating cash flow created if the company keeps the old machine. We need to account for the cost of maintenance, as well as the cash flow effects of depreciation. The incomes statement, adding depreciation to net income to calculate the operating cash flow will be: Maintenance cost Depreciation EBT Taxes Net income OCF $750,000 220,000 $970,000 329,800 $640,200 $420,200 The old machine also has a salvage value at the end of five years, so we need to include this in the cash flows analysis. The aftertax salvage value will be: Sell machine Taxes Total $175,000 59,500 $115,500 So, the NPV of the decision to keep the old machine will be: NPV = $1,562,000 $420,200(PVIFA12%,5) + $115,500 / 1.125 NPV = $3,011,189.16 The company should keep the old machine since it has a greater (less negative) NPV. There is another way to analyze a replacement decision that is often used. It is an incremental cash flow analysis of the change in cash flows from the existing machine to the new machine, assuming the new machine is purchased. In this type of analysis, the initial cash outlay would be the cost of the new machine, and the cash inflow (including any applicable taxes) of selling the old machine. In this case, the initial cash flow under this method would be: CHAPTER 10 B-192 Purchase new machine Sell old machine Taxes on old machine Total $4,200,000 1,800,000 238,000 $2,638,000 The cash flows from purchasing the new machine would be the difference in the operating expenses. We would also need to include only the change in depreciation. The old machine has a depreciation of $220,000 per year, and the new machine has a depreciation of $840,000 per year, so the increased depreciation will be $620,000 per year. The pro forma income statement and operating cash flow under this approach will be: Maintenance cost Depreciation EBT Taxes Net income OCF $485,000 620,000 $135,000 45,900 $89,100 $530,900 The salvage value of the differential cash flow approach is more complicated. The company will sell the new machine, and incur taxes on the sale in five years. However, we must also include the lost sale of the old machine. Since we assumed we sold the old machine in the initial cash outlay, we lose the ability to sell the machine in five years. This is an opportunity loss that must be accounted for. So, the salvage value is: Sell machine Taxes Lost sale of old Taxes on lost sale of old Total The NPV under this method is: NPV = $2,638,000 + $530,900(PVIFA12%,5) + $280,500 / 1.125 NPV = $565,061.08 So, this analysis still tells us the company should not purchase the new machine. This is really the same type of analysis as we did considering the replacement decision as mutually exclusive projects. Consider this: Subtract the NPV of the decision to keep the old machine from the NPV of the decision to purchase the new machine. You will get: Differential NPV = $3,576,250.24 (3,011,189.16) = $565,061.08 This is the exact same NPV we calculated when using the second analysis method. 27. A kilowatt hour is 1,000 watts for 1 hour. A 60-watt bulb burning for 500 hours per year uses 30,000 watts, or 30 kilowatts. Since the cost of a kilowatt hour is $0.101, the cost per year is: $600,000 204,000 175,000 59,500 $280,500 CHAPTER 10 193 Cost per year = 30($0.101) Cost per year = $3.03 The 60-watt bulb will last for 1,000 hours, which is 2 years of use at 500 hours per year. So, the NPV of the 60-watt bulb is: NPV = $0.50 $3.03(PVIFA10%,2) NPV = $5.76 And the EAC is: EAC = $5.76 / (PVIFA10%,2) EAC = $3.32 Now we can find the EAC for the 15-watt CFL. A 15-watt bulb burning for 500 hours per year uses 7,500 watts, or 7.5 kilowatts. And, since the cost of a kilowatt hour is $0.101, the cost per year is: Cost per year = 7.5($0.101) Cost per year = $0.7575 The 15-watt CFL will last for 12,000 hours, which is 24 years of use at 500 hours per year. So, the NPV of the CFL is: NPV = $3.50 $0.7575(PVIFA10%,24) NPV = $10.31 And the EAC is: EAC = $10.31 / (PVIFA10%,24) EAC = $1.15 Thus, the CFL is much cheaper. But: see our next two questions. 28. To solve the EAC algebraically for each bulb, we can set up the variables as follows: W = light bulb wattage C = cost per kilowatt hour H = hours burned per year P = price the light bulb The number of watts use by the bulb per hour is: WPH = W / 1,000 And the kilowatt hours used per year is: KPY = WPH H The electricity cost per year is therefore: ECY = KPY C CHAPTER 10 B-194 The NPV of the decision to but the light bulb is: NPV = P ECY(PVIFAR%,t) And the EAC is: EAC = NPV / (PVIFAR%,t) Substituting, we get: EAC = [P (W / 1,000 H C)PVIFAR%,t] / PFIVAR%,t We need to set the EAC of the two light bulbs equal to each other and solve for C, the cost per kilowatt hour. Doing so, we find: [$0.50 (60 / 1,000 500 C)PVIFA10%,2] / PVIFA10%,2 = [$3.50 (15 / 1,000 500 C)PVIFA10%,24] / PVIFA10%,24 C = $0.004509 So, unless the cost per kilowatt hour is extremely low, it makes sense to use the CFL. But when should you replace the incandescent bulb? See the next question. 29. We are again solving for the breakeven kilowatt hour cost, but now the incandescent bulb has only 500 hours of useful life. In this case, the incandescent bulb has only one year of life left. The breakeven electricity cost under these circumstances is: [$0.50 (60 / 1,000 500 C)PVIFA10%,1] / PVIFA10%,1 = [$3.50 (15 / 1,000 500 C)PVIFA10%,24] / PVIFA10%,24 C = $0.007131 Unless the electricity cost is negative (Not very likely!), it does not make financial sense to replace the incandescent bulb until it burns out. 30. The debate between incandescent bulbs and CFLs is not just a financial debate, but an environmental one as well. The numbers below correspond to the numbered items in the question: 1. The extra heat generated by an incandescent bulb is waste, but not necessarily in a heated structure, especially in northern climates. 2. Since CFLs last so long, from a financial viewpoint, it might make sense to wait if prices are declining. 3. Because of the nontrivial health and disposal issues, CFLs are not as attractive as our previous analysis suggests. 4. From a companys perspective, the cost of replacing working incandescent bulbs may outweigh the financial benefit. However, since CFLs last longer, the cost of replacing the bulbs will be lower in the long run. CHAPTER 10 195 5. Because incandescent bulbs use more power, more coal has to be burned, which generates more mercury in the environment, potentially offsetting the mercury concern with CFLs. 6. As in the previous question, if CO2 production is an environmental concern, the lower power consumption from CFLs is a benefit. 7. CFLs require more energy to make, potentially offsetting (at least partially) the energy savings from their use. Worker safety and site contamination are also negatives for CFLs. 8. This fact favors the incandescent bulb because the purchasers will only receive part of the benefit from the CFL. 9. This fact favors waiting for new technology. 10. This fact also favors waiting for new technology. While there is always a best answer, this question shows that the analysis of the best answer is not always easy and may not be completely possible because of incomplete data. As for how to better legislate the use of incandescent bulbs, our analysis suggests that requiring them in new construction might make sense. Rental properties in general should probably be required to use CFLs (why rentals?). Another piece of legislation that makes sense is requiring the producers of CFLs to supply a disposal kit and proper disposal instructions with each one sold. Finally, we need much better research on the hazards associated with broken bulbs in the home and workplace and proper procedures for dealing with broken bulbs. 31. Here we have a situation where a company is going to buy one of two assets, so we need to calculate the EAC of each asset. To calculate the EAC, we can calculate the EAC of the combined costs of each computer, or calculate the EAC of an individual computer, then multiply by the number of computers the company is purchasing. In this instance, we will calculate the EAC of each individual computer. For the SAL 5000, we will begin by calculating the aftertax salvage value, then the operating cash flows. So: SAL 5000: Taxes on salvage value = (BV MV)tC Taxes on salvage value = ($0 200)(.34) Taxes on salvage value = $68 Market price Tax on sale Aftertax salvage value $200 68 $132 The incremental costs will include the maintenance costs, depreciation, and taxes. Notice the taxes are negative, signifying a lower tax bill. So, the incremental cash flows will be: Maintenance cost $500.00 CHAPTER 10 B-196 Depreciation EBT Tax Net income OCF 462.50 $962.50 327.25 $635.25 $172.75 So, the NPV of the decision to buy one unit is: NPV = $3,700 $172.75(PVIFA11%,8) + $132 / 1.118 NPV = $4,531.71 And the EAC on a per unit basis is: $4,531.71 = EAC(PVIFA11%,8) EAC = $880.61 Since the company must buy 9 units, the total EAC of the decision is: Total EAC = 9($880.61) Total EAC = $7,925.47 And the EAC for the HAL 1000: Taxes on salvage value = (BV MV)tC Taxes on salvage value = ($0 220)(.34) Taxes on salvage value = $74.80 Market price Tax on sale Aftertax salvage value $220.00 74.80 $145.20 The incremental costs will include the maintenance costs, depreciation, and taxes. Notice the taxes are negative, signifying a lower tax bill. So, the incremental cash flows will be: Maintenance cost Depreciation EBT Tax Net income OCF $550.00 733.33 $1,283.33 436.33 $847.00 $113.67 So, the NPV of the decision to buy one unit is: NPV = $4,400 $113.67(PVIFA11%,6) + $145.20 / 1.116 NPV = $4,803.24 CHAPTER 10 197 And the EAC on a per unit basis is: $4,803.24 = EAC(PVIFA11%,6) EAC = $1,135.37 Since the company must buy 7 units, the total EAC of the decision is: Total EAC = 7($1,135.37) Total EAC = $7,947.62 The company should choose the SAL 5000 since the total EAC is lower. 32. Here we are comparing two mutually exclusive projects with inflation. Since each will be replaced when it wears out, we need to calculate the EAC for each. We have real cash flows. Similar to other capital budgeting projects, when calculating the EAC, we can use real cash flows with the real interest rate, or nominal cash flows and the nominal interest rate. Using the Fisher equation to find the real required return, we get: (1 + R) = (1 + r)(1 + h) (1 + .16) = (1 + r)(1 + .05) r = .0667 or 6.67% This is the interest rate we need to use with real cash flows. We are given the real aftertax cash flows for each asset, so the NPV for the XX40 is: NPV = $1,800 $135(PVIFA6.67%,3) NPV = $2,156.45 So, the EAC for the XX40 is: $2,156.45 = EAC(PVIFA6.67%,3) EAC = $816.72 And the EAC for the RH45 is: NPV = $2,200 $195(PVIFA6.67%,5) NPV = $3,006.73 $3,006.73 = EAC(PVIFA6.67%,5) EAC = $726.78 The company should choose the RH45 because it has the lower (less negative) EAC. 33. The project has a sales price that increases at five percent per year, and a variable cost per unit that increases at 7 percent per year. First, we need to find the sales price and variable cost for each year. The table below shows the price per unit and the variable cost per unit each year. Year 1 $75.00 Year 2 $78.75 Year 3 $82.69 Year 4 $86.82 Year 5 $91.16 Sales price CHAPTER 10 B-198 Cost per unit $20.00 $21.40 $22.90 $24.50 $26.22 Using the sales price and variable cost, we can now construct the pro forma income statement for each year. We can use this income statement to calculate the cash flow each year. We must also make sure to include the net working capital outlay at the beginning of the project, and the recovery of the net working capital at the end of the project. The pro forma income statement and cash flows for each year will be: Year 0 Revenues Fixed costs VC Dep. EBT Taxes Net income OCF Equipment NWC Total CF $820,000 130,000 $950,000 $250,460.00 $267,521.00 $285,231.77 $303,610.57 Year 1 $825,000.00 310,000.00 220,000.00 164,000.00 $131,000.00 44,540.00 $86,460.00 $250,460.00 Year 2 $866,250.00 310,000.00 235,400.00 164,000.00 $156,850.00 53,329.00 $103,521.00 $267,521.00 Year 3 $909,562.50 310,000.00 251,878.00 164,000.00 $183,684.50 62,452.73 $121,231.77 $285,231.77 Year 4 $955,040.63 310,000.00 269,509.46 164,000.00 $211,531.17 71,920.60 $139,610.57 $303,610.57 Year 5 $1,002,792.6 6 310,000.00 288,375.12 164,000.00 $240,417.53 81,741.96 $158,675.57 $322,675.57 130,000 $452,675.57 With these cash flows, the NPV of the project is: NPV = $950,000 + $250,460 / 1.11 + $267,521 / 1.112 + $285,231.77 / 1.113 + $303,610.57 / 1.114 + $452,675.57 / 1.115 NPV = $169,963.30 We could also answer this problem using the depreciation tax shield approach. The revenues and variable costs are growing annuities, growing at different rates. The fixed costs and depreciation are ordinary annuities. Using the growing annuity equation, the present value of the revenues is: PV of revenues = C {[1/(r g)] [1/(r g)] [(1 + g)/(1 + r)]t} PV of revenues = $825,000{[1/(.11 .05)] [1/(.11 .05)] [(1 + .05)/(1 + .11)]5} PV of revenues = $3,335,598.91 And the present value of the variable costs will be: PV of variable costs = C {[1/(r g)] [1/(r g)] [(1 + g)/(1 + r)]t} PV of variable costs = $220,000{[1/(.11 .07)] [1/(.11 .07)] [(1 + .10)/(1 + .07)]5} PV of variable costs = $922,095.97 The fixed costs and depreciation are both ordinary annuities. The present value of each is: PV of fixed costs = C({1 [1/(1 + r)]t } / r ) CHAPTER 10 199 PV of fixed costs = $310,000({1 [1/(1 + .11)]5 } / .11) PV of fixed costs = $1,145,728.08 PV of depreciation = C({1 [1/(1 + r)]t } / r ) PV of depreciation = $164,000({1 [1/(1 + .11)]5 } / .11) PV of depreciation = $606,127.11 Now, we can use the depreciation tax shield approach to find the NPV of the project, which is: NPV = $950,000 + ($3,335,598.91 922,095.97 1,145,728.08)(1 .34) + ($606,127.11)(.34) + $130,000 / 1.115 NPV = $169,963.30 Challenge 34. Probably the easiest OCF calculation for this problem is the bottom up approach, so we will construct an income statement for each year. Beginning with the initial cash flow at time zero, the project will require an investment in equipment. The project will also require an investment in NWC. So, the cash flow required for the project today will be: Capital spending Change in NWC Total cash flow $25,000,000 1,500,000 $26,500,000 Now we can begin the remaining calculations. Sales figures are given for each year, along with the price per unit. The variable costs per unit are used to calculate total variable costs, and fixed costs are given at $2,400,000 per year. To calculate depreciation each year, we use the initial equipment cost of $25 million, times the appropriate MACRS depreciation each year. The remainder of each income statement is calculated below. Notice at the bottom of the income statement we added back depreciation to get the OCF for each year. The section labeled Net cash flows will be discussed below: Year Ending book value 0 Year 1 $21,427,50 0 $28,475,00 0 16,575,000 2,400,000 3,572,500 5,927,500 2,074,625 3,852,875 3,572,500 $7,425,375 Year 2 $15,305,00 0 $30,820,00 0 17,940,000 2,400,000 6,122,500 4,357,500 1,525,125 2,832,375 6,122,500 $8,954,875 Year 3 $10,932,50 0 $41,875,00 0 24,375,000 2,400,000 4,372,500 10,727,500 3,754,625 6,972,875 4,372,500 $11,345,37 5 Year 4 $7,810,000 $36,850,00 0 21,450,000 2,400,000 3,122,500 9,877,500 3,457,125 6,420,375 3,122,500 $9,542,875 Year 5 $5,577,500 $29,145,00 0 16,965,000 2,400,000 2,232,500 7,547,500 2,641,625 4,905,875 2,232,500 $7,138,375 Sales Variable costs Fixed costs Depreciation EBIT Taxes Net income Depreciation OCF CHAPTER 10 B-200 Net cash flows OCF Change in NWC Capital spending Total cash flow $0 1,500,000 25,000,000 $26,500,000 $7,425,375 351,750 0 $7,073,625 $8,954,875 1,658,250 0 $7,296,625 $11,345,37 5 753,750 0 $12,099,12 5 $9,542,875 1,155,750 0 $10,698,62 5 $7,138,375 1,600,500 5,202,125 $13,941,00 0 After we calculate the OCF for each year, we need to account for any other cash flows. The other cash flows in this case are NWC cash flows and capital spending, which is the aftertax salvage of the equipment. The required NWC capital is 15 percent of the sales change. We will work through the NWC cash flow for Year 1. The total NWC in Year 1 will be 15 percent of sales increase from Year 1 to Year 2, or: Increase in NWC for Year 1 = .15($28,475,000 30,820,000) Increase in NWC for Year 1 = $351,750 Notice that the NWC cash flow is negative. Since the sales are increasing, we will have to spend more money to increase NWC. In Year 3 and Year 4, the NWC cash flow is positive since sales are declining. And, in Year 5, the NWC cash flow is the recovery of all NWC the company still has in the project. To calculate the aftertax salvage value, we first need the book value of the equipment. The book value at the end of the five years will be the purchase price, minus the total depreciation. So, the ending book value is: Ending book value = $25,000,000 ($3,572,500 + 6,122,500 + 4,372,500 + 3,122,500 + 2,232,500) Ending book value = $5,577,500 The market value of the used equipment is 20 percent of the purchase price, or $5 million, so the aftertax salvage value will be: Aftertax salvage value = $5,000,000 + ($5,577,500 5,000,000)(.35) Aftertax salvage value = $5,202,125 The aftertax salvage value is included in the total cash flows are capital spending. Now we have all of the cash flows for the project. The NPV of the project is: NPV = $26,500,000 + $7,073,625/1.17 + $7,296,625/1.172 + $12,099,125/1.173 + $10,698,625/1.174 + $13,941,000/1.175 NPV = $4,498,421.50 And the IRR is: NPV = 0 = $26,500,000 + $7,073,625/(1 + IRR) + $7,296,625/(1 + IRR)2 + $12,099,125/(1 + IRR)3 + $10,698,625/(1 + IRR)4 + $13,941,000/(1 + IRR)5 IRR = 23.32% We should accept the project. CHAPTER 10 201 35. To find the initial pretax cost savings necessary to buy the new machine, we should use the tax shield approach to find the OCF. We begin by calculating the depreciation each year using the MACRS depreciation schedule. The depreciation each year is: D1 = $740,000(0.3333) = $246,642 D2 = $740,000(0.4445) = $328,930 D3 = $740,000(0.1481) = $109,594 D4 = $740,000(0.0741) = $54,834 Using the tax shield approach, the OCF each year is: OCF1 = (S C)(1 0.35) + 0.35($246,642) OCF2 = (S C)(1 0.35) + 0.35($328,930) OCF3 = (S C)(1 0.35) + 0.35($109,594) OCF4 = (S C)(1 0.35) + 0.35($54,834) OCF5 = (S C)(1 0.35) Now we need the aftertax salvage value of the equipment. The aftertax salvage value is: Aftertax salvage value = $70,000(1 0.35) = $45,500 To find the necessary cost reduction, we must realize that we can split the cash flows each year. The OCF in any given year net of the depreciation tax shield is the cost reduction (S C) times one minus the tax rate, which is an annuity for the project life, and the depreciation tax shield. To calculate the necessary cost reduction, we would require a zero NPV. The equation for the NPV of the project is: NPV = 0 = $740,000 40,000 + (S C)(0.65)(PVIFA12%,5) + 0.35($246,642/1.12) + $328,930/1.122 + $109,594/1.123 + $54,834/1.124) + ($40,000 + 45,500)/1.125 Solving this equation for the sales minus costs, we get: (S C)(0.65)(PVIFA12%,5) = $523,132.84 S C = $223,264.83 36. To find the bid price, we need to calculate all other cash flows for the project, and then solve for the bid price. The aftertax salvage value of the equipment is: Aftertax salvage value = $150,000(1 0.35) = $97,500 Now we can solve for the necessary OCF that will give the project a zero NPV. The equation for the NPV of the project is: NPV = 0 = $1,300,000 175,000 + OCF(PVIFA13%,5) + [($175,000 + 97,500) / 1.135] Solving for the OCF, we find the OCF that makes the project NPV equal to zero is: OCF = $1,327,097.92 / PVIFA13%,5 = $377,313.24 The easiest way to calculate the bid price is the tax shield approach, so: CHAPTER 10 B-202 OCF = $377,313.24 = [(P v)Q FC ](1 tc) + tcD $377,313.24 = [(P $10.40)(200,000) $330,000 ](1 0.35) + 0.35($1,300,000/5) P = $14.25 37. a. This problem is basically the same as the previous problem, except that we are given a sales price. The cash flow at Time 0 for all three parts of this question will be: Capital spending Change in NWC Total cash flow $1,300,000 175,000 $1,475,000 We will use the initial cash flow and the salvage value we already found in that problem. Using the bottom up approach to calculating the OCF, we get: Assume price per unit = $15 and units/year = 200,000 Year 1 2 3 Sales $3,000,000 $3,000,000 $3,000,000 Variable costs 2,080,000 2,080,000 2,080,000 Fixed costs 330,000 330,000 330,000 Depreciation 260,000 260,000 260,000 EBIT $330,000 $330,000 $330,000 Taxes (35%) 115,500 115,500 115,500 Net Income $214,500 $214,500 $214,500 Depreciation 260,000 260,000 260,000 Operating CF $474,500 $474,500 $474,500 Year Operating CF Change in NWC Capital spending Total CF 1 $474,500 2 $474,500 3 $474,500 4 $3,000,000 2,080,000 330,000 260,000 $330,000 115,500 $214,500 260,000 $474,500 4 $474,500 5 $3,000,000 2,080,000 330,000 260,000 $330,000 115,500 $214,500 260,000 $474,500 5 $474,500 175,000 97,500 $747,000 $474,500 $474,500 $474,500 $474,500 With these cash flows, the NPV of the project is: NPV = $1,475,000 + $474,500(PVIFA13%,5) + [($175,000 + 97,500) / 1.135] NPV = $341,828.32 If the actual price is above the bid price that results in a zero NPV, the project will have a positive NPV. As for the cartons sold, if the number of cartons sold increases, the NPV will increase, and if the costs increase, the NPV will decrease. b. To find the minimum number of cartons sold to still breakeven, we need to use the tax shield approach to calculating OCF, and solve the problem similar to finding a bid price. Using the initial cash flow and salvage value we already calculated, the equation for a zero NPV of the project is: NPV = 0 = $1,475,000 + OCF(PVIFA13%,5) + [($175,000 + 97,500) / 1.135] CHAPTER 10 203 So, the necessary OCF for a zero NPV is: OCF = $1,327,097.92 / PVIFA13%,5 = $377,313.24 Now we can use the tax shield approach to solve for the minimum quantity as follows: OCF = $377,313.24 = [(P v)Q FC ](1 tc) + tcD $377,313.24 = [($15.00 10.40)Q 330,000 ](1 0.35) + 0.35($1,300,000/5) Q = 167,496 As a check, we can calculate the NPV of the project with this quantity. The calculations are: Year Sales Variable costs Fixed costs Depreciation EBIT Taxes (35%) Net Income Depreciation Operating CF Year Operating CF Change in NWC Capital spending Total CF 1 $2,512,441 1,741,959 330,000 260,000 $180,482 63,169 $117,313 260,000 $377,313 1 $377,313 2 $2,512,441 1,741,959 330,000 260,000 $180,482 63,169 $117,313 260,000 $377,313 2 $377,313 3 $2,512,441 1,741,959 330,000 260,000 $180,482 63,169 $117,313 260,000 $377,313 3 $377,313 4 $2,512,441 1,741,959 330,000 260,000 $180,482 63,169 $117,313 260,000 $377,313 4 $377,313 5 $2,512,441 1,741,959 330,000 260,000 $180,482 63,169 $117,313 260,000 $377,313 5 $377,313 175,000 97,500 $649,813 $377,313 $377,313 $377,313 $377,313 NPV = $1,475,000 + $377,313(PVIFA13%,5) + [($175,000 + 97,500) / 1.135] $0 Note that the NPV is not exactly equal to zero because we had to round the number of cartons sold; you cannot sell one-half of a carton. c. To find the highest level of fixed costs and still breakeven, we need to use the tax shield approach to calculating OCF, and solve the problem similar to finding a bid price. Using the initial cash flow and salvage value we already calculated, the equation for a zero NPV of the project is: NPV = 0 = 1,475,000 + OCF(PVIFA13%,5) + [($175,000 + 97,500) / 1.135] OCF = $1,327,097.92 / PVIFA13%,5 = $377,313.24 Notice this is the same OCF we calculated in part b. Now we can use the tax shield approach to solve for the maximum level of fixed costs as follows: OCF = $377,313.24 = [(P v)Q FC ](1 tC) + tCD $377,313.24 = [($15.00 $10.40)(200,000) FC](1 0.35) + 0.35($1,300,000/5) FC = $479,518.09 CHAPTER 10 B-204 As a check, we can calculate the NPV of the project with this quantity. The calculations are: Year Sales Variable costs Fixed costs Depreciation EBIT Taxes (35%) Net Income Depreciation Operating CF Year Operating CF Change in NWC Capital spending Total CF 1 $3,000,000 2,080,000 479,518 260,000 $180,482 63,169 $117,313 260,000 $377,313 1 $377,313 2 $3,000,000 2,080,000 479,518 260,000 $180,482 63,169 $117,313 260,000 $377,313 2 $377,313 3 $3,000,000 2,080,000 479,518 260,000 $180,482 63,169 $117,313 260,000 $377,313 3 $377,313 4 $3,000,000 2,080,000 479,518 260,000 $180,482 63,169 $117,313 260,000 $377,313 4 $377,313 5 $3,000,000 2,080,000 479,518 260,000 $180,482 63,169 $117,313 260,000 $377,313 5 $377,313 175,000 97,500 $649,813 $377,313 $377,313 $377,313 $377,313 NPV = $1,010,000 + $377,313(PVIFA13%,5) + [($175,000 + 97,500) / 1.135] $0 38. We need to find the bid price for a project, but the project has extra cash flows. Since we dont already produce the keyboard, the sales of the keyboard outside the contract are relevant cash flows. Since we know the extra sales number and price, we can calculate the cash flows generated by these sales. The cash flow generated from the sale of the keyboard outside the contract is: Year 1 $1,056,00 0 704,000 $352,000 140,800 $211,200 Year 2 $1,896,00 0 1,264,000 $632,000 252,800 $379,200 Year 3 $2,328,00 0 1,552,000 $776,000 310,400 $465,600 Year 4 $1,416,00 0 944,000 $472,000 188,800 $283,200 Sales Variable costs EBT Tax Net income (and OCF) So, the addition to NPV of these market sales is: NPV of market sales = $211,200/1.13 + $379,200/1.132 + $465,600/1.133 + $283,200/1.134 NPV of market sales = $980,247.90 You may have noticed that we did not include the initial cash outlay, depreciation, or fixed costs in the calculation of cash flows from the market sales. The reason is that it is irrelevant whether or not we include these here. Remember that we are not only trying to determine the bid price, but we are also determining whether or not the project is feasible. In other words, we are trying to calculate the NPV of the project, not just the NPV of the bid price. We will include these cash flows in the bid price calculation. The reason we stated earlier that whether we included these costs in this initial calculation was irrelevant is that you will come up with the same bid price if you include these costs in this calculation, or if you include them in the bid price calculation. CHAPTER 10 205 Next, we need to calculate the aftertax salvage value, which is: Aftertax salvage value = $350,000(1 .40) = $210,000 Instead of solving for a zero NPV as is usual in setting a bid price, the company president requires an NPV of $100,000, so we will solve for a NPV of that amount. The NPV equation for this project is (remember to include the NWC cash flow at the beginning of the project, and the NWC recovery at the end): NPV = $100,000 = $3,400,000 150,000 + 980,247.90 + OCF (PVIFA13%,4) + [($150,000 + 210,000) / 1.134] Solving for the OCF, we get: OCF = $2,448,957.36 / PVIFA13%,4 = $823,325.25 Now we can solve for the bid price as follows: OCF = $823,325.25 = [(P v)Q FC ](1 tC) + tCD $823,325.25 = [(P $160)(24,000) $625,000](1 0.40) + 0.40($3,400,000/4) P = $219.61 39. Since the two computers have unequal lives, the correct method to analyze the decision is the EAC. We will begin with the EAC of the new computer. Using the depreciation tax shield approach, the OCF for the new computer system is: OCF = ($60,000)(1 .38) + ($465,000 / 5)(.38) = $72,540 Notice that the costs are positive, which represents a cash inflow. The costs are positive in this case since the new computer will generate a cost savings. The only initial cash flow for the new computer is cost of $465,000. We next need to calculate the aftertax salvage value, which is: Aftertax salvage value = $45,000(1 .38) = $27,900 Now we can calculate the NPV of the new computer as: NPV = $465,000 + $72,540(PVIFA11%,5) + $27,900 / 1.115 NPV = $180,342.34 And the EAC of the new computer is: EAC = $180,342.34 / (PVIFA11%,5) = $48,795.28 Analyzing the old computer, the only OCF is the depreciation tax shield, so: OCF = $40,000(.38) = $15,200 The initial cost of the old computer is a little trickier. You might assume that since we already own the old computer there is no initial cost, but we can sell the old computer, so there is an opportunity cost. We need to account for this opportunity cost. To do so, we will calculate the aftertax salvage value of the old computer today. We need the book value of the old computer to do so. The book CHAPTER 10 B-206 value is not given directly, but we are told that the old computer has depreciation of $40,000 per year for the next three years, so we can assume the book value is the total amount of depreciation over the remaining life of the system, or $120,000. So, the aftertax salvage value of the old computer is: Aftertax salvage value = $165,000 + ($120,000 165,000)(.38) = $147,900 This is the initial cost of the old computer system today because we are forgoing the opportunity to sell it today. We next need to calculate the aftertax salvage value of the computer system in two years since we are buying it today. The aftertax salvage value in two years is: Aftertax salvage value = $25,000 + ($25,000 40,000)(.38) = $30,700 Now we can calculate the NPV of the old computer as: NPV = $147,900 + $15,200(PVIFA11%,2) + 30,700 / 1.112 NPV = $96,952.84 And the EAC of the old computer is: EAC = $96,952.84 / (PVIFA11%,2) = $56,614.02 If we are going to replace the system in two years no matter what our decision today, we should instead replace it today since the EAC is lower. b. If we are only concerned with whether or not to replace the machine now, and are not worrying about what will happen in two years, the correct analysis is NPV. To calculate the NPV of the decision on the computer system now, we need the difference in the total cash flows of the old computer system and the new computer system. From our previous calculations, we can say the cash flows for each computer system are: t 0 1 2 3 4 5 New computer $465,000 72,540 72,540 72,540 72,540 100,440 Old computer $147,900 15,200 45,900 0 0 0 Difference $317,100 57,340 26,640 72,540 72,540 100,440 Since we are only concerned with marginal cash flows, the cash flows of the decision to replace the old computer system with the new computer system are the differential cash flows. The NPV of the decision to replace, ignoring what will happen in two years is: NPV = $317,100 + $57,340/1.11 + $26,640/1.112 + $72,540/1.113 + $72,540/1.114 + $100,440/1.115 NPV = $83,389.50 If we are not concerned with what will happen in two years, we should not replace the old computer system. CHAPTER 10 207 40. To answer this question, we need to compute the NPV of all three alternatives, specifically, continue to rent the building, Project A, or Project B. If all three of the projects have a positive NPV, the project that is more favorable is the one with the highest NPV There are several important cash flows we should not consider in the incremental cash flow analysis. The remaining fraction of the value of the building and depreciation are not incremental and should not be included in the analysis of the two alternatives. The $1,500,000 purchase price of the building is a sunk cost and should be ignored. In effect, what we are doing is finding the NPV of the future cash flows of each option, so the only cash flow today would be the building modifications needed for Project A and Project B. If we did include these costs, the effect would be to lower the NPV of all three options by the same amount, thereby leading to the same conclusion. The cash flows from renting the building after year 15 are also irrelevant. No matter what the company chooses today, it will rent the building after year 15, so these cash flows are not incremental to any project. We will begin by calculating the NPV of the decision of continuing to rent the building first. Continue to rent: Rent Taxes Net income $35,000 11,900 $23,100 Since there is no incremental depreciation, the operating cash flow is simply the net income. So, the NPV of the decision to continue to rent is: NPV = $23,100(PVIFA12%,15) NPV = $157,330.97 Product A: Next, we will calculate the NPV of the decision to modify the building to produce Product A. The income statement for this modification is the same for the first 14 years, and in year 15, the company will have an additional expense to convert the building back to its original form. This will be an expense in year 15, so the income statement for that year will be slightly different. The cash flow at time zero will be the cost of the equipment, and the cost of the initial building modifications, both of which are depreciable on a straight-line basis. So, the pro forma cash flows for Product A are: Initial cash outlay: Building modifications Equipment Total cash flow $75,000 190,000 $265,000 Years 1-14 $225,000 135,000 17,667 0 $72,333 Year 15 $225,000 135,000 17,667 45,000 $27,333 Revenue Expenditures Depreciation Restoration cost EBT CHAPTER 10 B-208 Tax NI OCF 24,593 $47,740 $65,407 9,293 $18,040 $35,707 The OCF each year is net income plus depreciation. So, the NPV for modifying the building to manufacture Product A is: NPV = $265,000 + $65,407(PVIFA12%,14) + $35,707 / 1.1215 NPV = $175,049.86 Product B: Now we will calculate the NPV of the decision to modify the building to produce Product B. The income statement for this modification is the same for the first 14 years, and in year 15, the company will have an additional expense to convert the building back to its original form. This will be an expense in year 15, so the income statement for that year will be slightly different. The cash flow at time zero will be the cost of the equipment, and the cost of the initial building modifications, both of which are depreciable on a straight-line basis. So, the pro forma cash flows for Product A are: Initial cash outlay: Building modifications Equipment Total cash flow $100,000 245,000 $345,000 Years 1-14 $275,000 173,000 23,000 0 $79,000 26,860 $52,140 $75,140 Year 15 $275,000 173,000 23,000 55,000 $24,000 8,160 $15,840 $38,840 Revenue Expenditures Depreciation Restoration cost EBT Tax NI OCF The OCF each year is net income plus depreciation. So, the NPV for modifying the building to manufacture Product B is: NPV = $345,000 + $75,140(PVIFA12%,14) + $38,840 / 1.1215 NPV = $160,136.48 We could have also done the analysis as the incremental cash flows between Product A and continuing to rent the building, and the incremental cash flows between Product B and continuing to rent the building. The results of this type of analysis would be: NPV of differential cash flows between Product A and continuing to rent: NPV = NPVProduct A NPVRent CHAPTER 10 209 NPV = $175,049.86 157,330.97 NPV = $17,718.89 NPV of differential cash flows between Product B and continuing to rent: NPV = NPVProduct B NPVRent NPV = $160,136.48 157,330.97 NPV = $2,805.51 Both of these incremental analyses have a positive NPV, so the company should choose Product A since it has the highest marginal NPV, which is the same as our original result. 41. The discount rate is expressed in real terms, and the cash flows are expressed in nominal terms. We can answer this question by converting all of the cash flows to real dollars. We can then use the real interest rate. The real value of each cash flow is the present value of the year 1 nominal cash flows, discounted back to the present at the inflation rate. So, the real value of the revenue and costs will be: Revenue in real terms = $235,000 / 1.04 = $225,961.54 Labor costs in real terms = $165,000 / 1.04 = $158,653.85 Other costs in real terms = $55,000 / 1.04 = $52,884.62 Lease payment in real terms = $60,000 / 1.04 = $57,692.31 Revenues, labor costs, and other costs are all growing perpetuities. Each has a different growth rate, so we must calculate the present value of each separately. Other costs are a growing perpetuity with a negative growth rate. Using the real required return, the present value of each of these is: PVRevenue = $225,961.54 / (0.07 0.02) = $4,519,230.77 PVLabor costs = $158,653.85 / (0.07 0.01) = $2,644,230.77 PVOther costs = $52,884.62 / [0.07 (0.01)] = $661,057.69 The lease payments are constant in nominal terms, so they are declining in real terms by the inflation rate. Therefore, the lease payments form a growing perpetuity with a negative growth rate. The real present value of the lease payments is: PVLease payments = $60,000 / [0.07 (0.04)] = $524,475.52 Now we can use the tax shield approach to calculate the net present value. Since there is no investment in equipment, there is no depreciation; therefore, no depreciation tax shield, so we will ignore this in our calculation. This means the cash flows each year are equal to net income. There is also no initial cash outlay, so the NPV is the present value of the future aftertax cash flows. The NPV of the project is: NPV = PVRevenue PVLabor costs PVOther costs PVLease payments NPV = ($4,519,230.77 2,644,230.77 661,057.69 524,475.52)(1 .34) NPV = $455,048.08 Alternatively, we could have solved this problem by expressing everything in nominal terms. This approach yields the same answer as given above. However, in this case, the computation would have been much more difficult. The reason is that we are dealing with growing perpetuities. In other problems, when calculating the NPV of nominal cash flows, we could simply calculate the nominal CHAPTER 10 B-210 cash flow each year since the cash flows were finite. Because of the perpetual nature of the cash flows in this problem, we cannot calculate the nominal cash flows each year until the end of the project. In this case, using real cash flow is the only practical method. 42. We are given the real revenue and costs, and the real growth rates, so the simplest way to solve this problem is to calculate the NPV with real values. While we could calculate the NPV using nominal values, we would need to find the nominal growth rates, and convert all values to nominal terms. The real labor costs will increase at a real rate of two percent per year, and the real energy costs will increase at a real rate of three percent per year, so the real costs each year will be: Year 1 $18.50 $6.40 Year 2 $18.87 $6.59 Year 3 $19.25 $6.79 Year 4 $19.63 $6.99 Real labor cost each year Real energy cost each year Remember that the depreciation tax shield also affects a firms aftertax cash flows. The present value of the depreciation tax shield must be added to the present value of a firms revenues and expenses to find the present value of the cash flows related to the project. The depreciation the firm will recognize each year is: Annual depreciation = Investment / Economic Life Annual depreciation = $75,000,000 / 4 Annual depreciation = $18,750,000 Depreciation is a nominal cash flow, so to find the real value of depreciation each year, we discount the real depreciation amount by the inflation rate. Doing so, we find the real depreciation each year is: Year 1 real depreciation = $18,750,000 / 1.04 = $18,028,846.15 Year 2 real depreciation = $18,750,000 / 1.042 = $17,335,428.99 Year 3 real depreciation = $18,750,000 / 1.043 = $16,668,681.73 Year 4 real depreciation = $18,750,000 / 1.044 = $16,027,578.58 Now we can calculate the pro forma income statement each year in real terms. We can then add back depreciation to net income to find the operating cash flow each year. Doing so, we find the cash flow of the project each year is: Year 0 Revenues Labor cost Energy cost Depreciation EBT Taxes Net income OCF Year 1 $81,000,000. 00 43,475,000.0 0 1,219,200.00 18,028,846.1 5 $18,276,953. 85 6,214,164.31 $12,062,789. 54 $30,091,635. Year 2 $94,500,000. 00 54,156,900.0 0 1,516,160.00 17,335,428.9 9 $21,491,511. 01 7,307,113.74 $14,184,397. 26 $31,519,826. Year 3 $121,500,000. 00 59,666,940.00 1,833,235.20 16,668,681.73 $43,331,143.0 7 14,732,588.65 $28,598,554.4 3 $45,267,236.1 Year 4 $67,500,000. 00 38,283,078.6 0 1,192,383.70 16,027,578.5 8 $11,996,959. 12 4,078,966.10 $7,917,993.0 2 $23,945,571. CHAPTER 10 211 69 Capital sp. $75,000,000 $30,091,635. 69 $31,519,826. 26 $45,267,236.1 5 $23,945,571. 60 26 5 60 Total cash flow $75,000,000 We can use the total cash flows each year to calculate the NPV, which is: NPV = $75,000,000 + $30,091,635.69 / 1.08 + $31,519,826.26 / 1.082 + $45,267,236.15 / 1.083 + $23,945,571.60 / 1.084 NPV = $33,421,097.79 43. Here we have the sales price and production costs in real terms. The simplest method to calculate the project cash flows is to use the real cash flows. In doing so, we must be sure to adjust the depreciation, which is in nominal terms. We could analyze the cash flows using nominal values, which would require calculating the nominal discount rate, nominal price, and nominal production costs. This method would be more complicated, so we will use the real numbers. We will calculate the NPV of the headache only pill first. Headache only: We can find the real revenue and production costs by multiplying each by the units sold. We must be sure to discount the depreciation, which is in nominal terms. We can then find the pro forma net income, and add back depreciation to find the operating cash flow. Discounting the depreciation each year by the inflation rate, we find the following cash flows each year: Year 1 $34,200,000 18,600,000 8,974,359 $6,625,641 2,252,718 $4,372,923 $13,347,282 Year 2 $34,200,000 18,600,000 8,629,191 $6,970,809 2,370,075 $4,600,734 $13,229,925 Year 3 $34,200,000 18,600,000 8,297,299 $7,302,701 2,482,918 $4,819,782 $13,117,082 Sales Production costs Depreciation EBT Tax Net income OCF And the NPV of the headache only pill is: NPV = $28,000,000 + $13,347,282 / 1.06 + $13,299,925 / 1.062 + $13,117,082 / 1.063 NPV = $7,379,716.51 Headache and arthritis: For the headache and arthritis pill project, the equipment has a salvage value. We will find the aftertax salvage value of the equipment first, which will be: Market value $1,000,000 CHAPTER 10 B-212 Taxes Total 340,000 $660,000 Remember, to calculate the taxes on the equipment salvage value, we take the book value minus the market value, times the tax rate. Using the same method as the headache only pill, the cash flows each year for the headache and arthritis pill will be: Year 1 $62,700,000 42,900,000 11,858,974 $7,941,026 2,699,949 $5,241,077 $17,100,051 Year 2 $62,700,000 42,900,000 11,402,860 $8,397,140 2,855,028 $5,542,112 $16,944,972 Year 3 $62,700,000 42,900,000 10,964,288 $8,835,712 3,004,142 $5,831,570 $16,795,858 Sales Production costs Depreciation EBT Tax Net income OCF So, the NPV of the headache and arthritis pill is: NPV = $37,000,000 + $17,100,051 / 1.06 + $16,944,972 / 1.062 + ($16,795,858 + 660,000) / 1.063 NPV = $8,869,363.98 The company should manufacture the headache and arthritis remedy since the project has a higher NPV. 44. Since the project requires an initial investment in inventory as a percentage of sales, we will calculate the sales figures for each year first. The incremental sales will include the sales of the new table, but we also need to include the lost sales of the existing model. This is an erosion cost of the new table. The lost sales of the existing table are constant for every year, but the sales of the new table change every year. So, the total incremental sales figure for the five years of the project will be: Year 1 $10,400,000 1,200,000 $9,200,000 Year 2 $11,245,000 1,200,000 $10,045,000 Year 3 $12,025,000 1,200,000 $10,825,000 Year 4 $10,530,000 1,200,000 $9,330,000 Year 5 $8,970,000 1,200,000 $7,770,000 New Lost sales Total Now we will calculate the initial cash outlay that will occur today. The company has the necessary production capacity to manufacture the new table without adding equipment today. So, the equipment will not be purchased today, but rather in two years. The reason is that the existing capacity is not being used. If the existing capacity were being used, the new equipment would be required, so it would be a cash flow today. The old equipment would have an opportunity cost of it could be sold. As there is no discussion that the existing equipment could be sold, we must assume it cannot be sold. The only initial cash flow is the cost of the inventory. The company will have to spend money for inventory in the new table, but will be able to reduce inventory of the existing table. So, the initial cash flow today is: New table $1,040,000 CHAPTER 10 213 Old table Total 120,000 $920,000 In year 2, the company will have a cash outflow to pay for the cost of the new equipment. Since the equipment will be purchased in two years rather than now, the equipment will have the higher salvage value. The book value of the equipment in five years will be the initial cost, minus the accumulated depreciation, or: Book value = $9,500,000 1,357,550 2,326,550 1,661,550 Book value = $4,154,350 The taxes on the salvage value will be: Taxes on salvage = ($4,154,350 6,100,000)(.38) Taxes on salvage = $737,347 So, the aftertax salvage value of the equipment in five years will be: Sell equipment Taxes Total $6,100,000 737,347 $5,360,653 Next, we need to calculate the variable costs each year. The variable costs of the lost sales are included as a variable cost savings, so the variable costs will be: Year 1 $4,680,000 480,000 $4,200,000 Year 2 $5,060,250 480,000 $4,580,250 Year 3 $5,411,250 480,000 $4,931,250 Year 4 $4,738,500 480,000 $4,258,500 Year 5 $4,036,500 480,000 $3,556,500 New Lost sales Total Now we can prepare the rest of the pro forma income statements for each year. The project will have no incremental depreciation for the first two years as the equipment is not purchased for two years. Adding back depreciation to net income to calculate the operating cash flow, we get: Year 1 $9,200,000 4,200,000 2,200,000 0 $2,800,000 1,064,000 $1,736,000 0 $1,736,000 Year 2 $10,045,000 4,580,250 2,200,000 0 $3,264,750 1,240,605 $2,024,145 0 $2,024,145 Year 3 $10,825,000 4,931,250 2,200,000 1,357,550 $2,336,200 887,756 $1,448,444 1,357,550 $2,805,994 Year 4 $9,330,000 4,258,500 2,200,000 2,326,550 $544,950 207,081 $337,869 2,326,550 $2,664,419 Year 5 $7,770,000 3,556,500 2,200,000 1,661,550 $351,950 133,741 $218,209 1,661,550 $1,879,759 Sales VC Fixed costs Dep. EBT Tax NI Dep. OCF CHAPTER 10 B-214 Next, we need to account for the changes in inventory each year. The inventory is a percentage of sales. The way we will calculate the change in inventory is the beginning of period inventory minus the end of period inventory. The sign of this calculation will tell us whether the inventory change is a cash inflow, or a cash outflow. The inventory each year, and the inventory change, will be: Beginning Ending Change $1,040,000 1,124,500 $84,500 $1,124,500 1,202,500 $78,000 $1,202,500 1,053,000 $149,500 $1,053,000 897,000 $156,000 $897,000 0 $897,000 Notice that we recover the remaining inventory at the end of the project. The total cash flows for the project will be the operating cash flow, the capital spending, and the inventory cash flows, so: Year 1 $1,736,000 0 84,500 $1,651,500 Year 2 $2,024,145 9,500,000 78,000 $7,553,855 Year 3 $2,805,994 0 149,500 $2,955,494 Year 4 $2,664,419 0 156,000 $2,820,419 Year 5 $1,879,759 5,360,653 897,000 $8,137,412 OCF Equipment Inventory Total The NPV of the project, including the inventory cash flow at the beginning of the project, will be: NPV = $920,000 + $1,651,500 / 1.14 $7,553,855 / 1.132 + $2,955,494 / 1.133 + $2,820,419 / 1.144 + $8,137,412 / 1.135 NPV = $2,607,343.61 The company should go ahead with the new table. b. c. You can perform an IRR analysis, and would expect to find up to three IRRs since the cash flows change signs three times. The profitability index is intended as a bang for the buck measure; that is, it shows how much shareholder wealth is created for every dollar of initial investment. In this case, the largest investment is not at the beginning of the project, but later in its life. However, since the future negative cash flow is discounted, the profitability index will still measure the amount of shareholder wealth created for every dollar spent today. CHAPTER 10 RISK AND RETURN LESSONS FROM MARKET HISTORY Answers to Concept Questions CHAPTER 10 215 1. 2. 3. 4. 5. 6. They all wish they had! Since they didnt, it must have been the case that the stellar performance was not foreseeable, at least not by most. As in the previous question, its easy to see after the fact that the investment was terrible, but it probably wasnt so easy ahead of time. No, stocks are riskier. Some investors are highly risk averse, and the extra possible return doesnt attract them relative to the extra risk. Unlike gambling, the stock market is a positive sum game; everybody can win. Also, speculators provide liquidity to markets and thus help to promote efficiency. T-bill rates were highest in the early eighties. This was during a period of high inflation and is consistent with the Fisher effect. Before the fact, for most assets the risk premium will be positive; investors demand compensation over and above the risk-free return to invest their money in the risky asset. After the fact, the observed risk premium can be negative if the assets nominal return is unexpectedly low, the riskfree return is unexpectedly high, or if some combination of these two events occurs. Yes, the stock prices are currently the same. Below is a table that depicts the stocks price movements. Two years ago, each stock had the same price, P0. Over the first year, General Materials stock price increased by 10 percent, or (1.1) P0. Standard Fixtures stock price declined by 10 percent, or (0.9) P0. Over the second year, General Materials stock price decreased by 10 percent, or (0.9)(1.1) P0, while Standard Fixtures stock price increased by 10 percent, or (1.1) (0.9) P0. Today, each of the stocks is worth 99 percent of its original value. 2 years ago P0 P0 1 year ago (1.1)P0 (0.9)P0 Today (1.1)(0.9)P0 = (0.99)P0 (0.9)(1.1)P0 = (0.99)P0 7. General Materials Standard Fixtures 8. The stock prices are not the same. The return quoted for each stock is the arithmetic return, not the geometric return. The geometric return tells you the wealth increase from the beginning of the period to the end of the period, assuming the asset had the same return each year. As such, it is a better measure of ending wealth. To see this, assuming each stock had a beginning price of $100 per share, the ending price for each stock would be: Lake Minerals ending price = $100(1.10)(1.10) = $121.20 Small Town Furniture ending price = $100(1.25)(.95) = $118.75 Whenever there is any variance in returns, the asset with the larger variance will always have the greater difference between the arithmetic and geometric return. 9. To calculate an arithmetic return, you simply sum the returns and divide by the number of returns. As such, arithmetic returns do not account for the effects of compounding. Geometric returns do account for the effects of compounding. As an investor, the more important return of an asset is the geometric return. CHAPTER 10 B-216 10. Risk premiums are about the same whether or not we account for inflation. The reason is that risk premiums are the difference between two returns, so inflation essentially nets out. Returns, risk premiums, and volatility would all be lower than we estimated because aftertax returns are smaller than pretax returns. Solutions to Questions and Problems NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. The return of any asset is the increase in price, plus any dividends or cash flows, all divided by the initial price. The return of this stock is: R = [($96 84) + 1.40] / $84 R = .1595 or 15.95% 2. The dividend yield is the dividend divided by price at the beginning of the period price, so: Dividend yield = $1.40 / $84 Dividend yield = .0167 or 1.67% And the capital gains yield is the increase in price divided by the initial price, so: Capital gains yield = ($96 84) / $84 Capital gains yield = .1429 or 14.29% 3. Using the equation for total return, we find: R = [($71 84) + 1.40] / $84 R = .1381 or 13.81% And the dividend yield and capital gains yield are: Dividend yield = $1.40 / $84 Dividend yield = .0167 or 1.67% Capital gains yield = ($71 84) / $84 Capital gains yield = .1548 or 15.48% Heres a question for you: Can the dividend yield ever be negative? No, that would mean you were paying the company for the privilege of owning the stock. It has happened on bonds issued by Berkshire Hathaway, the company operated by the noted investor Warren Buffett. 4. a. The total dollar return is the change in price plus the coupon payment, so: Total dollar return = $970 1,030 + 75 CHAPTER 10 217 Total dollar return = $15 b. The total percentage return of the bond is: R = [($970 1,030) + 75] / $1,030 R = .0146 or 1.46% Notice here that we could have simply used the total dollar return of $15 in the numerator of this equation. c. Using the Fisher equation, the real return was: (1 + R) = (1 + r)(1 + h) r = (1.046 / 1.030) 1 r = 0.0150 or 1.50% 5. The nominal return is the stated return, which is 11.80 percent. Using the Fisher equation, the real return was: (1 + R) = (1 + r)(1 + h) r = (1.1180)/(1.031) 1 r = .0844 or 8.44% 6. Using the Fisher equation, the real returns for government and corporate bonds were: (1 + R) = (1 + r)(1 + h) rG = 1.058/1.031 1 rG = .0262 or 2.62% rC = 1.062/1.031 1 rC = .0301 or 3.01% 7. The average return is the sum of the returns, divided by the number of returns. The average return for each stock was: N [.21 + .02 + .13 .19 + .08] = .0500 or 5.00% X = xi N = 5 i =1 N [.31 + .11 + .08 .35 + .16] = .0620 or 6.20% Y = yi N = 5 i =1 We calculate the variance of each stock as: CHAPTER 10 B-218 N X 2 = ( xi x ) 2 ( N 1) i =1 1 ( .21 .05) 2 + ( .02 .05) 2 + ( .13 .05) 2 + ( .19 .05) 2 + ( .08 .05) 2 = .022850 X2 = 5 1 1 ( .31 .062) 2 + ( .11 .062) 2 + ( .08 .062) 2 + ( .35 .062) 2 + ( .16 .062) 2 = .060870 Y2 = 5 1 { { } } The standard deviation is the square root of the variance, so the standard deviation of each stock is: X = (.022850)1/2 X = .1512 or 15.12% Y = (.060870)1/2 Y = .2467 or 24.67% 8. We will calculate the sum of the returns for each asset and the observed risk premium first. Doing so, we get: Year 1973 1974 1975 1976 1977 1978 a. Large co. stock return 14.69% 26.47 37.23 23.93 7.16 6.57 19.41 T-bill return 7.29% 7.99 5.87 5.07 5.45 7.64 39.31 Risk premium 21.98% 34.46 31.36 18.86 12.61 1.07 19.90 The average return for large company stocks over this period was: Large company stock average return = 19.41% /6 Large company stock average return = 3.24% And the average return for T-bills over this period was: T V b. T-bills average return = 39.31% / 6 U T-bills average return = 6.55% Using the equation for variance, we find the variance for large company stocks over this period was: Variance = 1/5[(.1469 .0324)2 + (.2647 .0324)2 + (.3723 .0324)2 + (.2393 .0324)2 + (.0716 .0324)2 + (.0657 .0324)2] Variance = 0.058136 And the standard deviation for large company stocks over this period was: Standard deviation = (0.058136)1/2 Standard deviation = 0.2411 or 24.11% CHAPTER 10 219 Using the equation for variance, we find the variance for T-bills over this period was: Variance = 1/5[(.0729 .0655)2 + (.0799 .0655)2 + (.0587 .0655)2 + (.0507 .0655)2 + (.0545 .0655)2 + (.0764 .0655)2] Variance = 0.000153 And the standard deviation for T-bills over this period was: Standard deviation = (0.000153)1/2 Standard deviation = 0.0124 or 1.24% CHAPTER 10 B-220 c. The average observed risk premium over this period was: Average observed risk premium = 19.90% / 6 Average observed risk premium = 3.32% The variance of the observed risk premium was: Variance = 1/5[(.2198 .0332)2 + (.3446 .0332)2 + (.3136 .0332)2 + (.1886 .0332)2 + (.1261 .0332)2 + (.0107 .0332)2] Variance = 0.062078 And the standard deviation of the observed risk premium was: Standard deviation = (0.06278)1/2 Standard deviation = 0.2492 or 24.92% 9. a. b. 10. a. Arithmetic average return = .1380 or 13.80% Standard deviation = 0.1717 or 17.17% To calculate the average real return, we can use the average return of the asset, and the average inflation rate in the Fisher equation. Doing so, we find: (1 + R) = (1 + r)(1 + h) r = (1.1380/1.028) 1 r = .1070 or 10.70% b. The average risk premium is simply the average return of the asset, minus the average risk-free rate, so, the average risk premium for this asset would be: RP = R R f RP = .1380 .036 RP = .1020 or 10.20% 11. We can find the average real risk-free rate using the Fisher equation. The average real risk-free rate was: (1 + R) = (1 + r)(1 + h) r f = (1.036/1.028) 1 r f = .0078 or 0.78% And to calculate the average real risk premium, we can subtract the average real risk-free rate from the average real return. So, the average real risk premium was: rp = r r f = 10.70% 0.78% rp = .0992 or 9.92% CHAPTER 10 221 12. Apply the five-year holding-period return formula to calculate the total return of the stock over the five-year period, we find: 5-year holding-period return = [(1 + R1)(1 + R2)(1 +R3)(1 +R4)(1 +R5)] 1 5-year holding-period return = [(1 .1762)(1 + .1538)(1 + .1095)(1 + .2683)(1 + .0531)] 1 5-year holding-period return = 0.4085 or 40.85% 13. To find the return on the zero coupon bond, we first need to find the price of the bond today. Remember that a zero coupon bond price is calculated with semiannual periods. Since one year has elapsed, the bond now has 19 years to maturity, so the price today is: P1 = $1,000/1.0438 P1 = $225.29 There are no intermediate cash flows on a zero coupon bond, so the return is the capital gains, or: R = ($225.29 215.81) / $215.81 R = .0439 or 4.39% 14. The return of any asset is the increase in price, plus any dividends or cash flows, all divided by the initial price. This preferred stock paid a dividend of $5.50, so the return for the year was: R = ($95.89 92.73 + 5.50) / $92.73 R = .0934 or 9.34% 15. The return of any asset is the increase in price, plus any dividends or cash flows, all divided by the initial price. This stock paid no dividend, so the return was: R = ($37.53 32.81) / $32.81 R = .1439 or 14.39% This is the return for three months, so the APR is: APR = 4(14.39%) APR = 57.54% And the EAR is: EAR = (1 + .1439)4 1 EAR = .7119 or 71.19% 16. To find the real return each year, we will use the Fisher equation, which is: 1 + R = (1 + r)(1 + h) Using this relationship for each year, we find: T-bills 0.0330 0.0315 Inflation (0.0112) (0.0226) Real Return 0.0447 0.0554 1926 1927 CHAPTER 10 B-222 1928 1929 1930 1931 1932 0.0405 0.0447 0.0227 0.0115 0.0088 (0.0116) 0.0058 (0.0640) (0.0932) (0.1027) 0.0527 0.0387 0.0926 0.1155 0.1243 So, the average real return was: Average = (.0447 + .0554 + .0527 + .0387 + .0926 + .1155 + .1243) / 7 Average = .0748 or 7.48% Notice the real return was higher than the nominal return during this period because of deflation, or negative inflation. 17. Looking at the long-term corporate bond return history in Table 10.2, we see that the mean return was 6.2 percent, with a standard deviation of 8.3 percent. The range of returns you would expect to see 68 percent of the time is the mean plus or minus 1 standard deviation, or: R r 1 = 6.2% 8.3% = 2.10% to 14.50% The range of returns you would expect to see 95 percent of the time is the mean plus or minus 2 standard deviations, or: R r 2 = 6.2% 2(8.3%) = 10.40% to 22.80% 18. Looking at the large-company stock return history in Figure 10.2, we see that the mean return was 11.8 percent, with a standard deviation of 20.5 percent. The range of returns you would expect to see 68 percent of the time is the mean plus or minus 1 standard deviation, or: R r 1 = 11.8% 20.5% = 8.70% to 32.30% The range of returns you would expect to see 95 percent of the time is the mean plus or minus 2 standard deviations, or: R r 2 = 11.8% 2(20.5%) = 29.20% to 52.80% 19. To find the best forecast, we apply Blumes formula as follows: 5 -1 30 - 5 11.7% + 13.4% = 13.17% 29 29 10 - 1 30 - 10 R(10) = 11.7% + 13.4% = 12.87% 29 29 20 - 1 30 - 20 R(20) = 11.7% + 13.4% = 12.29% 29 29 R(5) = 20. The best forecast for a one year return is the arithmetic average, which is 11.8 percent. The geometric average, found in Table 10.3 is 9.8 percent. To find the best forecast for other periods, we apply Blumes formula as follows: CHAPTER 10 223 5 -1 84 - 5 9.8% + 11.8% = 11.70% 83 83 20 - 1 84 - 20 R(20) = 9.8% + 11.8% = 11.34% 83 83 30 - 1 84 - 30 R(30) = 9.8% + 11.8% = 11.10% 83 83 R(5) = Intermediate 21. Standard deviation = 0.1691 or 16.91% 22. The arithmetic average return is the sum of the known returns divided by the number of returns, so: Arithmetic average return = (.17 + .42 + .31 .24 + .17 +.21) / 6 Arithmetic average return = .1167 or 11.67% Using the equation for the geometric return, we find: Geometric average return = [(1 + R1) (1 + R2) (1 + RT)]1/T 1 Geometric average return = [(1 .17)(1 + .42)(1 + .31)(1 .24)(1 + .17)(1 + .21)](1/6) 1 Geometric average return = .0883 or 8.83% Remember, the geometric average return will always be less than the arithmetic average return if the returns have any variation. 23. To calculate the arithmetic and geometric average returns, we must first calculate the return for each year. The return for each year is: R1 = ($29.63 30.06 + 0.88) / $30.06 = .0150 or 1.50% R2 = ($32.40 29.63 + 1.00) / $29.63 = .1272 or 12.72% R3 = ($33.27 32.40 + 1.12) / $32.40 = .0614 or 6.14% R4 = ($15.32 33.27 + 1.24)/ $33.27 = .5023 or 50.23% R5 = ($15.04 15.32 + 0.40) / $15.32 = .0078 or 0.78% The arithmetic average return was: RA = (0.0150 + 0.1272 + 0.0614 0.5023 + 0.0078)/5 RA = 0.0582 or 5.82% And the geometric average return was: RG = [(1 + .0150)(1 + .1272)(1 + .0614)(1 .5023)(1 + .0078)]1/5 1 RG = 0.0944 or 9.44% 24. To find the real return we need to use the Fisher equation. Re-writing the Fisher equation to solve for the real return, we get: r = [(1 + R)/(1 + h)] 1 So, the real return each year was: CHAPTER 10 B-224 Year 1973 1974 1975 1976 1977 1978 1979 1980 T-bill return 0.0729 0.0799 0.0587 0.0507 0.0545 0.0764 0.1056 0.1210 0.6197 Inflation 0.0871 0.1234 0.0694 0.0486 0.0670 0.0902 0.1329 0.1252 0.7438 Real return 0.0131 0.0387 0.0100 0.0020 0.0117 0.0127 0.0241 0.0037 0.1120 a. The average return for T-bills over this period was: Average return = 0.6197 / 8 Average return = .0775 or 7.75% And the average inflation rate was: W Y b. Average inflation = 0.7438 / 8 X Average inflation = .0930 or 9.30% Using the equation for variance, we find the variance for T-bills over this period was: Variance = 1/7[(.0729 .0775)2 + (.0799 .0775)2 + (.0587 .0775)2 + (.0507 .0775)2 + (.0545 .0775)2 + (.0764 .0775)2 + (.1056 .0775)2 + (.1210 .0775)2] Variance = 0.000616 And the standard deviation for T-bills was: Standard deviation = (0.000616)1/2 Standard deviation = 0.0248 or 2.48% The variance of inflation over this period was: Variance = 1/7[(.0871 .0930)2 + (.1234 .0930)2 + (.0694 .0930)2 + (.0486 .0930)2 + (.0670 .0930)2 + (.0902 .0930)2 + (.1329 .0930)2 + (.1252 .0930)2] Variance = 0.000971 And the standard deviation of inflation was: Standard deviation = (0.000971)1/2 Standard deviation = 0.0312 or 3.12% c. The average observed real return over this period was: Average observed real return = .1120 / 8 Average observed real return = .0140 or 1.40% CHAPTER 10 225 d. The statement that T-bills have no risk refers to the fact that there is only an extremely small chance of the government defaulting, so there is little default risk. Since T-bills are short term, there is also very limited interest rate risk. However, as this example shows, there is inflation risk, i.e. the purchasing power of the investment can actually decline over time even if the investor is earning a positive return. 25. To find the return on the coupon bond, we first need to find the price of the bond today. Since one year has elapsed, the bond now has 8 years to maturity, so the price today is: P1 = $70(PVIFA6.8%,8) + $1,000/1.0688 P1 = $1,012.04 You received the coupon payments on the bond, so the nominal return was: R = ($1,012.04 979 + 70) / $979 R = .1052 or 10.52% And using the Fisher equation to find the real return, we get: r = (1.1052 / 1.034) 1 r = .0689 or 6.89% 26. Looking at the long-term government bond return history in Table 10.2, we see that the mean return was 5.8 percent, with a standard deviation of 9.6 percent. In the normal probability distribution, approximately 2/3 of the observations are within one standard deviation of the mean. This means that 1/3 of the observations are outside one standard deviation away from the mean. Or: Pr(R< 3.8 or R>15.4) 1/3 But we are only interested in one tail here, that is, returns less than 3.8 percent, so: Pr(R< 3.8) 1/6 You can use the z-statistic and the cumulative normal distribution table to find the answer as well. Doing so, we find: z = (X )/ z = (3.8% 5.8)/9.6% = 1.00 Looking at the z-table, this gives a probability of 15.87%, or: Pr(R< 3.4) .1587 or 15.87% The range of returns you would expect to see 95 percent of the time is the mean plus or minus 2 standard deviations, or: 95% level: R r 2 = 5.8% 2(9.6%) = 13.40% to 25.00% CHAPTER 10 B-226 The range of returns you would expect to see 99 percent of the time is the mean plus or minus 3 standard deviations, or: 99% level: R 3 = 5.8% 3(9.6%) = 23.00% to 34.60% Challenge 27. The mean return for small company stocks was 16.6 percent, with a standard deviation of 32.8 percent. Doubling your money is a 100% return, so if the return distribution is normal, we can use the z-statistic. So: z = (X )/ z = (100% 16.6)/32.8% = 2.543 standard deviations above the mean This corresponds to a probability of 0.550%, or about once every 200 years. Tripling your money would be: z = (200% 16.6)/32.8% = 5.5915 standard deviations above the mean. This corresponds to a probability of (much) less than 0.5%, or once every 200 years. The actual answer is .00000113%, or about once every 1 million years. 28. It is impossible to lose more than 100 percent of your investment. Therefore, return distributions are truncated on the lower tail at 100 percent. 29. Using the z-statistic, we find: z = (X )/ z = (0% 11.8)/20.5% = 0.5756 Pr(R=0) 28.24% 30. For each of the questions asked here, we need to use the z-statistic, which is: z = (X )/ a. z1 = (10% 6.2)/8.3% = 0.4578 This z-statistic gives us the probability that the return is less than 10 percent, but we are looking for the probability the return is greater than 10 percent. Given that the total probability is 100 percent (or 1), the probability of a return greater than 10 percent is 1 minus the probability of a return less than 10 percent. Using the cumulative normal distribution table, we get: Pr(R=10%) = 1 Pr(R=10%) = 1 .6765 32.35% For a return less than 0 percent: Pr(R<0%) = 1 Pr(R>0%) = 1 .7725 22.75% CHAPTER 10 227 b. The probability that T-bill returns will be greater than 10 percent is: z3 = (10% 3.7)/3.1% = 2.0323 Pr(R=10%) = 1 Pr(R=10%) = 1 .9789 2.11% And the probability that T-bill returns will be less than 0 percent is: z4 = (0% 3.7)/3.1% = 1.1935 Pr(R=0) 11.63% c. The probability that the return on long-term corporate bonds will be less than 4.18 percent is: z5 = (4.18% 6.2)/8.3% = 1.2506 Pr(R=4.18%) 10.55% And the probability that T-bill returns will be greater than 10.32 percent is: z6 = (10.32% 3.7)/3.1% = 2.1355 Pr(R=10.38%) = 1 Pr(R=10.38%) = 1 .9836 1.64% CHAPTER 11 RISK AND RETURN: THE CAPITAL ASSET PRICING MODEL (CAPM) Answers to Concept Questions 1. Some of the risk in holding any asset is unique to the asset in question. By investing in a variety of assets, this unique portion of the total risk can be eliminated at little cost. On the other hand, there are some risks that affect all investments. This portion of the total risk of an asset cannot be costlessly eliminated. In other words, systematic risk can be controlled, but only by a costly reduction in expected returns. If the market expected the growth rate in the coming year to be 2 percent, then there would be no change in security prices if this expectation had been fully anticipated and priced. However, if the market had been expecting a growth rate other than 2 percent and the expectation was incorporated into security prices, then the governments announcement would most likely cause security prices in general to change; prices would drop if the anticipated growth rate had been more than 2 percent, and prices would rise if the anticipated growth rate had been less than 2 percent. a. b. c. d. e. f. a. b. c. d. e. systematic unsystematic both; probably mostly systematic unsystematic unsystematic systematic a change in systematic risk has occurred; market prices in general will most likely decline. no change in unsystematic risk; company price will most likely stay constant. no change in systematic risk; market prices in general will most likely stay constant. a change in unsystematic risk has occurred; company price will most likely decline. no change in systematic risk; market prices in general will most likely stay constant. 2. 3. 4. 5. 6. 7. 8. No to both questions. The portfolio expected return is a weighted average of the asset returns, so it must be less than the largest asset return and greater than the smallest asset return. False. The variance of the individual assets is a measure of the total risk. The variance on a welldiversified portfolio is a function of systematic risk only. Yes, the standard deviation can be less than that of every asset in the portfolio. However, p cannot be less than the smallest beta because p is a weighted average of the individual asset betas. Yes. It is possible, in theory, to construct a zero beta portfolio of risky assets whose return would be equal to the risk-free rate. It is also possible to have a negative beta; the return would be less than the risk-free rate. A negative beta asset would carry a negative risk premium because of its value as a diversification instrument. CHAPTER 20 B-229 9. Such layoffs generally occur in the context of corporate restructurings. To the extent that the market views a restructuring as value-creating, stock prices will rise. So, its not layoffs per se that are being cheered on. Nonetheless, Wall Street does encourage corporations to takes actions to create value, even if such actions involve layoffs. 10. Earnings contain information about recent sales and costs. This information is useful for projecting future growth rates and cash flows. Thus, unexpectedly low earnings often lead market participants to reduce estimates of future growth rates and cash flows; price drops are the result. The reverse is often true for unexpectedly high earnings. 11. The covariance is a more appropriate measure of a securitys risk in a well-diversified portfolio because the covariance reflects the effect of the security on the variance of the portfolio. Investors are concerned with the variance of their portfolios and not the variance of the individual securities. Since covariance measures the impact of an individual security on the variance of the portfolio, covariance is the appropriate measure of risk. 12. If we assume that the market has not stayed constant during the past three years, then the lack in movement of Southern Co.s stock price only indicates that the stock either has a standard deviation or a beta that is very near to zero. The large amount of movement in Texas Instrument stock price does not imply that the firms beta is high. Total volatility (the price fluctuation) is a function of both systematic and unsystematic risk. The beta only reflects the systematic risk. Observing the standard deviation of price movements does not indicate whether the price changes were due to systematic factors or firm specific factors. Thus, if you observe large stock price movements like that of TI, you cannot claim that the beta of the stock is high. All you know is that the total risk of TI is high. 13. The wide fluctuations in the price of oil stocks do not indicate that these stocks are a poor investment. If an oil stock is purchased as part of a well-diversified portfolio, only its contribution to the risk of the entire portfolio matters. This contribution is measured by systematic risk or beta. Since price fluctuations in oil stocks reflect diversifiable plus non-diversifiable risk, observing the standard deviation of price movements is not an adequate measure of the appropriateness of adding oil stocks to a portfolio. 14. The statement is false. If a security has a negative beta, investors would want to hold the asset to reduce the variability of their portfolios. Those assets will have expected returns that are lower than the risk-free rate. To see this, examine the Capital Asset Pricing Model: E(RS) = Rf + S[E(RM) Rf] If S < 0, then the E(RS) < Rf. CHAPTER 20 B-230 Solutions to Questions and Problems NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. The portfolio weight for each stock is: WeightA = 85($38)/$7,550 = .4278 WeightB = 160($27)/$7,550 = .5722 2. 3. 4. E(Rp) = .1262 or 12.62% E(Rp) = .1265 or 12.65% XX = 0.3889 Investment in X = 0.3889($10,000) = $3,888.89 Investment in Y = (1 0.3889)($10,000) = $6,111.11 5. The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of the asset is: E(R) = .1165 or 11.65% 6. The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of each stock asset is: E(RA) = .15(.03) + .50(.07) + .35(.11) = .0780 or 7.80% E(RB) = .15(.20) + .50(.13) + .35(.33) = .1505 or 15.05% To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance and standard deviation of each stock are: A2 =.15(.03 .0780)2 + .50(.07 .0780)2 + .35(.11 .0780)2 = .00074 A = (.00074)1/2 = .0271 or 2.71% B2 =.15(.20 .1505)2 + .50(.13 .1505)2 + .35(.33 .1505)2 = .02991 B = (.02991)1/2 = .1730 or 17.30% 7. The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of the stock is: CHAPTER 20 B-231 E(RA) = .05(.245) + .15(.085) + .50(.140) + .30(.321) = .1413 or 14.13% To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance and standard deviation of the stock are: 2 =.05(.245 .1413)2 + .15(.085 .1413)2 + .50(.14 .1413)2 + .30(.321 .1413)2 = .02483 = (.02483)1/2 = .1576 or 15.76% 8. E(Rp) = .1255 or 12.55% If we own this portfolio, we would expect to get a return of 12.55 percent. 9. a. This portfolio does not have an equal weight in each asset. We first need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: Boom: Good: Poor: Bust: E(Rp) = .40(.30) + .20(.45) + .40(.33) = .3420 or 34.20% E(Rp) = .40(.12) + .20(.10) + .40(.15) = .1280 or 12.80% E(Rp) = .40(.01) + .20(.15) + .40(.05) = .0460 or 4.60% E(Rp) = .40(.06) + .20(.30) + .40(.09) = .1200 or 12.00% And the expected return of the portfolio is: E(Rp) = .15(.3420) + .45(.1280) + .35(.0460) + .05(.1200) = .0868 or 8.68% b. To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance and standard deviation of each stock are: p2 = .15(.3420 .0868)2 + .45(.1280 .0868)2 + .35(.0460 .0868)2 + .05(.1200 .0868)2 2 p = .01884 p = (.01884)1/2 = .1373 or 13.73% 10. 11. p = 1.23 X = 1.40 12. E(Ri) = .1353 or 13.53% 13. i = 1.19 CHAPTER 20 B-232 14. E(Ri) = .098 E(RM) = .1089 or 10.89% 15. Rf = .0372 or 3.72% 16. a. b. E(Rp) = .0910 or 9.10% XS = .6522 And, the weight of the risk-free asset is: XRf = 1 .6522 = .3478 c. d. p = 0.4459 XS = 2.3/1.15 = 2 XRf = 1 2 = 1 The portfolio is invested 200% in the stock and 100% in the risk-free asset. This represents borrowing at the risk-free rate to buy more of the stock. 17. First, we need to find the of the portfolio. The of the risk-free asset is zero, and the weight of the risk-free asset is one minus the weight of the stock, the of the portfolio is: p = XW(1.30) + (1 XW)(0) = 1.30XW So, to find the of the portfolio for any weight of the stock, we simply multiply the weight of the stock times its . Even though we are solving for the and expected return of a portfolio of one stock and the riskfree asset for different portfolio weights, we are really solving for the SML. Any combination of this stock, and the risk-free asset will fall on the SML. For that matter, a portfolio of any stock and the risk-free asset, or any portfolio of stocks, will fall on the SML. We know the slope of the SML line is the market risk premium, so using the CAPM and the information concerning this stock, the market risk premium is: E(RW) = .1290 = .041 + MRP(1.30) MRP = .088/1.30 = .0677 or 6.77% So, now we know the CAPM equation for any stock is: E(Rp) = .041 + .0677 p The slope of the SML is equal to the market risk premium, which is 0.0677. Using these equations to fill in the table, we get the following results: XW 0% E(Rp) .0410 p 0 CHAPTER 20 B-233 25 50 75 100 125 150 .0630 .0850 .1070 .1290 .1510 .1730 0.325 0.650 0.975 1.300 1.625 1.950 18. E(RY) = .1375 or 13.75% E(RZ) = .0955 or 9.55% Reward-to-risk ratio Y = (.142 .043) / 1.35 = .0733 The reward-to-risk ratio for Stock Y is too high, which means the stock plots above the SML, and the stock is undervalued. Its price must increase until its reward-to-risk ratio is equal to the market reward-to-risk ratio. For Stock Z, we find: Reward-to-risk ratio Z = (.091 .043) / .75 = .0640 The reward-to-risk ratio for Stock Z is too low, which means the stock plots below the SML, and the stock is overvalued. Its price must decrease until its reward-to-risk ratio is equal to the market reward-to-risk ratio. 19. We need to set the reward-to-risk ratios of the two assets equal to each other (see the previous problem), which is: (.142 Rf)/1.35 = (.091 Rf)/0.75 We can cross multiply to get: 0.75(.142 Rf) = 1.35(.091 Rf) Solving for the risk-free rate, we find: 0.1065 0.75Rf = 0.12285 1.35Rf Rf = .0273 or 2.73% Intermediate 20. For a portfolio that is equally invested in large-company stocks and long-term bonds: Return = (11.8% + 5.8%)/2 = 8.80% For a portfolio that is equally invested in small stocks and Treasury bills: Return = (16.6% + 3.7%)/2 = 10.15% 21. We know that the reward-to-risk ratios for all assets must be equal (See Question 19). This can be expressed as: CHAPTER 20 B-234 [E(RA) Rf]/ A = [E(RB) Rf]/B The numerator of each equation is the risk premium of the asset, so: RPA/ A = RPB/ B We can rearrange this equation to get: B/ A = RPB/RPA If the reward-to-risk ratios are the same, the ratio of the betas of the assets is equal to the ratio of the risk premiums of the assets. 22. a. We need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: Boom: E(Rp) = .30(.20) + .30(.35) + .40(.60) = .4050 or 40.50% Normal: E(Rp) = .30(.15) + .30(.12) + .40(.05) = .1010 or 10.10% Bust: E(Rp) = .30(.01) + .30(.25) + .40(.50) = .2720 or 27.20% And the expected return of the portfolio is: E(Rp) = .25(.4050) + .60(.1010) + .15(.2720) = .1211 or 12.11% To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, than add all of these up. The result is the variance. So, the variance and standard deviation of the portfolio is: 2p = .30(.4050 .1211)2 + .30(.1010 .1211)2 + .40(.2720 .1211)2 2p = .04357 p = (.04357)1/2 = .2087 or 20.87% b. The risk premium is the return of a risky asset, minus the risk-free rate. T-bills are often used as the risk-free rate, so: RPi = E(Rp) Rf = .1211 .038 = .0831 or 8.31% c. The approximate expected real return is the expected nominal return minus the inflation rate, so: Approximate expected real return = .1211 .031 = .0901 or 9.01% To find the exact real return, we will use the Fisher equation. Doing so, we get: 1 + E(Ri) = (1 + h)[1 + E(ri)] 1.1211 = (1.0310)[1 + E(ri)] E(ri) = (1.1211/1.031) 1 = .0873 or 8.73% CHAPTER 20 B-235 The approximate real risk premium is the expected nominal risk premium minus the inflation rate, so: Approximate expected real risk premium = .0831 .031 = .0521 or 5.21% To find the exact expected real risk premium we use the Fisher effect. Doing do, we find: Exact expected real risk premium = (1.0831/1.031) 1 = .0505 or 5.05% 23. XA = .28 XB = .34 XC = .285692 Invest in Stock C = .285692($1,000,000) = $285,692.31 We also know the total portfolio weight must be one, so the weight of the risk-free asset must be one minus the asset weight we know, or: 1 = XA + XB + XC + XRf = 1 .28 .34 .285692 XRf XRf = .094308 So, the dollar investment in the risk-free asset must be: Invest in risk-free asset = .094308($1,000,000) = $94,307.69 24. We are given the expected return of the assets in the portfolio. We also know the sum of the weights of each asset must be equal to one. Using this relationship, we can express the expected return of the portfolio as: E(Rp) = .185 = XX(.172) + XY(.136) .185 = XX(.172) + (1 XX)(.136) .185 = .172XX + .136 .136XX .049 = .036XX XX = 1.36111 And the weight of Stock Y is: XY = 1 1.36111 XY = .36111 The amount to invest in Stock Y is: Investment in Stock Y = .36111($100,000) Investment in Stock Y = $36,111.11 A negative portfolio weight means that you short sell the stock. If you are not familiar with short selling, it means you borrow a stock today and sell it. You must then purchase the stock at a later CHAPTER 20 B-236 date to repay the borrowed stock. If you short sell a stock, you make a profit if the stock decreases in value. To find the beta of the portfolio, we can multiply the portfolio weight of each asset times its beta and sum. So, the beta of the portfolio is: P = 1.36111(1.40) + (.36111)(0.95) P = 1.56 25. The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of each stock is: E(RA) = .33(.041) + .33(.113) + .33(.153) = .1023 or 10.23% E(RB) = .33(.089) + .33(.025) + .33(.416) = .1007 or 10.07% To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance and standard deviation of Stock A are: A =.33(.041 .1023)2 + .33(.113 .1023)2 + .33(.153 .1023)2 = .00215 = (.00215)1/2 = .0463 or 4.63% And the standard deviation of Stock B is: B =.33(.089 .1007)2 + .33(.025 .1007)2 + .33(.416 .1007)2 = .05040 = (.05040)1/2 = .2245 or 22.45% To find the covariance, we multiply probability of each state times the product of each assets deviation from the mean in that state. The sum of these products is the covariance. So, the covariance is: Cov(A,B) = .33(.041 .1023)(.089 .1007) + .33(.113 .1023)(.025 .1007) + .33(.153 .1023)(.416 .1007) Cov(A,B) = .008756 And the correlation is: A,B = Cov(A,B) / A B A,B = .008756 / (.0463)(.2245) A,B = .8417 26. The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of each stock is: E(RJ) = .15(.080) + .60(.130) + .25(.347) = .1528 or 15.28% E(RK) = .15(.080) + .60(.091) + .25(.062) = .0821 or 8.21% 2 2 CHAPTER 20 B-237 To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance and standard deviation of Stock A is: 2 =.15(.080 .1528)2 + .60(.130 .1528)2 + .25(.347 .1528)2 = .01787 J J = (.01787)1/2 = .1337 or 13.37% And the standard deviation of Stock B is: 2 =.15(.080 .0821)2 + .60(.091 .0821)2 + .25(.062 .0821)2 = .00015 K K = (.00015)1/2 = .0122 or 1.22% To find the covariance, we multiply the probability of each state times the product of each assets deviation from the mean in that state. The sum of these products is the covariance. So, the covariance is: Cov(J,K) = .15(.080 .1528)(.080 .0821) + .60(.130 .1528)(.091 .0821) + .25(.347 .1528)(.062 .0821) Cov(J,K) = .001024 And the correlation is: J,K = Cov(J,K) / J K J,K = .001024 / (.1337)(.0122) J.K = .6273 27. a. E(RP) = .1210 or 12.10% b. = .3297 or 32.97% 28. a. E(RP) = .1280 or 12.80% = .5258 or 52.58% b. E(RP) = .1280 or 12.80% = .3380 or 33.80% c. 29. a. As Stock A and Stock B become less correlated, or more negatively correlated, the standard deviation of the portfolio decreases. (i) Using the equation to calculate beta, we find: CHAPTER 20 B-238 I = ( I,M)( I) / M 0.7 = ( I,M)(0.33) / 0.22 I,M = 0.47 (ii) Using the equation to calculate beta, we find: I = ( I,M)( I) / M 1.35 = (.36)( I) / 0.22 I = 0.83 (iii) Using the equation to calculate beta, we find: I = ( I,M)( I) / M I = (.37)(.40) / 0.22 I = 0.67 (iv) The market has a correlation of 1 with itself. (v) The beta of the market is 1. (vi) The risk-free asset has zero standard deviation. (vii) The risk-free asset has zero correlation with the market portfolio. (viii) The beta of the risk-free asset is 0. b. Using the CAPM to find the expected return of the stock, we find: Firm A: E(RA) = Rf + A[E(RM) Rf] E(RA) = 0.05 + 0.7(0.12 0.05) E(RA) = .0990 or 9.90% According to the CAPM, the expected return on Firm As stock should be 9.9 percent. However, the expected return on Firm As stock given in the table is 11 percent. Therefore, Firm As stock is underpriced, and you should buy it. Firm B: E(RB) = Rf + B[E(RM) Rf] E(RB) = 0.05 + 1.35(0.12 0.05) E(RB) = .1445 or 14.45% According to the CAPM, the expected return on Firm Bs stock should be 14.45 percent. However, the expected return on Firm Bs stock given in the table is 14 percent. Therefore, Firm Bs stock is overpriced, and you should sell it. Firm C: E(RC) = Rf + C[E(RM) Rf] E(RC) = 0.05 + 0.67(0.12 0.05) E(RC) = .0971 or 9.71% CHAPTER 20 B-239 According to the CAPM, the expected return on Firm Cs stock should be 9.71 percent. However, the expected return on Firm Cs stock given in the table is 10 percent. Therefore, Firm Cs stock is underpriced, and you should buy it. 30. Because a well-diversified portfolio has no unsystematic risk, this portfolio should lie on the Capital Market Line (CML). The slope of the CML equals: SlopeCML = [E(RM) Rf] / M SlopeCML = (0.11 0.045) / 0.21 SlopeCML = 0.30952 a. The expected return on the portfolio equals: E(RP) = Rf + SlopeCML( P) E(RP) = .045 + .30952(.24) E(RP) = .1193 or 11.93% The expected return on the portfolio equals: E(RP) = Rf + SlopeCML( P) .17 = .045 + .30952( P) P = .4038 or 40.38% 31. First, we can calculate the standard deviation of the market portfolio using the Capital Market Line (CML). We know that the risk-free rate asset has a return of 4.2 percent and a standard deviation of zero and the portfolio has an expected return of 9 percent and a standard deviation of 11 percent. These two points must lie on the Capital Market Line. The slope of the Capital Market Line equals: SlopeCML = Rise / Run SlopeCML = Increase in expected return / Increase in standard deviation SlopeCML = (.09 .042) / (.11 0) SlopeCML = .436 According to the Capital Market Line: E(RI) = Rf + SlopeCML( I) Since we know the expected return on the market portfolio the risk-free rate, and the slope of the Capital Market Line, we can solve for the standard deviation of the market portfolio which is: E(RM) = Rf + SlopeCML( M) .13 = .042 + (.436)( M) M = (.13 .042) / .436 M = .2017 or 20.17% Next, we can use the standard deviation of the market portfolio to solve for the beta of a security using the beta equation. Doing so, we find the beta of the security is: I = ( I,M)( I) / M I = (.55)(.50) / .2017 b. CHAPTER 20 B-240 I = 1.36 Now we can use the beta of the security in the CAPM to find its expected return, which is: E(RI) = Rf + I[E(RM) Rf] E(RI) = 0.042 + 1.36(.13 0.042) E(RI) = .1620 or 16.20% 32. First, we need to find the standard deviation of the market and the portfolio, which are: M = (.0460)1/2 M = .2145 or 21.45% Z = (.3017)1/2 Z = .5493 or 54.93% Now we can use the equation for beta to find the beta of the portfolio, which is: Z = ( Z,M)( Z) / M Z = (.55)(.5493) / .2145 Z = 1.41 Now, we can use the CAPM to find the expected return of the portfolio, which is: E(RZ) = Rf + Z[E(RM) Rf] E(RZ) = 0.039 + 1.41(.112 .039) E(RZ) = .1418 or 14.18% Challenge 33. The amount of systematic risk is measured by the of an asset. Since we know the market risk premium and the risk-free rate, if we know the expected return of the asset we can use the CAPM to solve for the of the asset. The expected return of Stock I is: E(RI) = .20(.04) + .50(.21) + .30(.12) = .1490 or 14.90% Using the CAPM to find the of Stock I, we find: .1490 = .04 + .075 I I = 1.45 The total risk of the asset is measured by its standard deviation, so we need to calculate the standard deviation of Stock I. Beginning with the calculation of the stocks variance, we find: I2 = .20(.04 .1490)2 + .50(.21 .1490)2 + .30(.12 .1490)2 I2 = .00449 I = (.00449)1/2 = .0670 or 6.70% CHAPTER 20 B-241 CHAPTER 20 B-242 Using the same procedure for Stock II, we find the expected return to be: E(RII) = .20(.30) + .50(.12) + .30(.44) = .1320 Using the CAPM to find the of Stock II, we find: .1320 = .04 + .075 II II = 1.23 And the standard deviation of Stock II is: II2 = .20(.30 .1320)2 + .50(.12 .1320)2 + .30(.44 .1320)2 II2 = .06586 II = (.06586)1/2 = .2566 or 25.66% Although Stock II has more total risk than I, it has less systematic risk, since its beta is smaller than Is. Thus, I has more systematic risk, and II has more unsystematic and more total risk. Since unsystematic risk can be diversified away, I is actually the riskier stock despite the lack of volatility in its returns. Stock I will have a higher risk premium and a greater expected return. 34. RM = .1138 or 11.38% 35. a. RM = .1138 or 11.38% The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the expected return and standard deviation of each stock are: Asset 1: E(R1) = .10(.20) + .40(.15) + .40(.10) + .10(.05) = .1250 or 12.50% 2 1 =.10(.20 .1250)2 + .40(.15 .1250)2 + .40(.10 .1250)2 + .10(.05 .1250)2 = .00163 1 = (.00163)1/2 = .0403 or 4.03% CHAPTER 20 B-243 Asset 2: E(R2) = .10(.20) + .40(.10) + .40(.15) + .10(.05) = .1250 or 12.50% 2 =.10(.20 .1250)2 + .40(.10 .1250)2 + .40(.15 .1250)2 + .10(.05 .1250)2 = .00163 2 2 = (.00163)1/2 = .0403 or 4.03% Asset 3: E(R3) = .10(.05) + .40(.10) + .40(.15) + .10(.20) = .1250 or 12.50% 2 3 =.10(.05 .1250)2 + .40(.10 .1250)2 + .40(.15 .1250)2 + .10(.20 .1250)2 = .00163 3 = (.00163)1/2 = .0403 or 4.03% b. To find the covariance, we multiply each possible state times the product of each assets deviation from the mean in that state. The sum of these products is the covariance. The correlation is the covariance divided by the product of the two standard deviations. So, the covariance and correlation between each possible set of assets are: Asset 1 and Asset 2: Cov(1,2) = .10(.20 .1250)(.20 .1250) + .40(.15 .1250)(.10 .1250) + .40(.10 .1250)(.15 .1250) + .10(.05 .1250)(.05 .1250) Cov(1,2) = .000625 1,2 = Cov(1,2) / 1 2 1,2 = .000625 / (.0403)(.0403) 1,2 = .3846 Asset 1 and Asset 3: Cov(1,3) = .10(.20 .1250)(.05 .1250) + .40(.15 .1250)(.10 .1250) + .40(.10 .1250)(.15 .1250) + .10(.05 .1250)(.20 .1250) Cov(1,3) = .001625 1,3 = Cov(1,3) / 1 3 1,3 = .001625 / (.0403)(.0403) 1,3 = 1 Asset 2 and Asset 3: Cov(2,3) = .10(.20 .1250)(.05 .1250) + .40(.10 .1250)(.10 .1250) + .40(.15 .1250)(.15 .1250) + .20(.05 .1250)(.20 .1250) Cov(2,3) = .000625 2,3 = Cov(2,3) / 2 3 2,3 = .000625 / (.0403)(.0403) 2,3 = .3846 c. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so, for a portfolio of Asset 1 and Asset 2: E(RP) = w1E(R1) + w2E(R2) E(RP) = .50(.1250) + .50(.1250) E(RP) = .1250 or 12.50% CHAPTER 20 B-244 The variance of a portfolio of two assets can be expressed as: 2 2 2 = X 1 1 + X 2 2 + 2X1X2 1 2 1,2 P 2 2 2 = .502(.04032) + .502(.04032) + 2(.50)(.50)(.0403)(.0403)(.3846) P 2 = .001125 P And the standard deviation of the portfolio is: P = (.001125)1/2 P = .0335 or 3.35% d. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so, for a portfolio of Asset 1 and Asset 3: E(RP) = X1E(R1) + X3E(R3) E(RP) = .50(.1250) + .50(.1250) E(RP) = .1250 or 12.50% The variance of a portfolio of two assets can be expressed as: 2 2 2 2 2 = X 1 1 + X 3 3 + 2X1X3 1 3 1,3 P 2 = .502(.04032) + .502(.04032) + 2(.50)(.50)(.0403)(.0403)(1) P 2 = .000000 P Since the variance is zero, the standard deviation is also zero. e. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so, for a portfolio of Asset 2 and Asset 3: E(RP) = X2E(R2) + X3E(R3) E(RP) = .50(.1250) + .50(.1250) E(RP) = .1250 or 12.50% The variance of a portfolio of two assets can be expressed as: 2 2 2 = X 2 2 + X 3 3 + 2X2X3 2 3 2,3 P 2 2 2 = .502(.04032) + .502(.04032) + 2(.50)(.50)(.3846) P 2 = .000500 P And the standard deviation of the portfolio is: P = (.000500)1/2 P = .0224 or 2.24% CHAPTER 20 B-245 f. As long as the correlation between the returns on two securities is below 1, there is a benefit to diversification. A portfolio with negatively correlated stocks can achieve greater risk reduction than a portfolio with positively correlated stocks, holding the expected return on each stock constant. Applying proper weights on perfectly negatively correlated stocks can reduce portfolio variance to 0. 36. a. The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of each stock is: E(RA) = .20(.08) + .60(.13) + .20(.48) = .1580 or 15.80% E(RB) = .20(.05) + .60(.14) + .20(.29) = .1320 or 13.20% b. We can use the expected returns we calculated to find the slope of the Security Market Line. We know that the beta of Stock A is .40 greater than the beta of Stock B. Therefore, as beta increases by .40, the expected return on a security increases by .026 (= .1580 .1320). The slope of the security market line (SML) equals: SlopeSML = Rise / Run SlopeSML = Increase in expected return / Increase in beta SlopeSML = (.1580 .1320) / .40 SlopeSML = .0650 or 6.50% Since the markets beta is 1 and the risk-free rate has a beta of zero, the slope of the Security Market Line equals the expected market risk premium. So, the expected market risk premium must be 6.5 percent. We could also solve this problem using CAPM. The equations for the expected returns of the two stocks are: .158 = Rf + ( B + .40)(MRP) .132 = Rf + B(MRP) We can rewrite the CAPM equation for Stock A as: .132 = Rf + B(MRP) + .40(MRP) Subtracting the CAPM equation for Stock B from this equation yields: .026 = .40MRP MRP = .065 or 6.5% which is the same answer as our previous result. CHAPTER 20 B-246 37. a. A typical, risk-averse investor seeks high returns and low risks. For a risk-averse investor holding a well-diversified portfolio, beta is the appropriate measure of the risk of an individual security. To assess the two stocks, we need to find the expected return and beta of each of the two securities. Stock A: Since Stock A pays no dividends, the return on Stock A is simply: (P 1 P0) / P0. So, the return for each state of the economy is: RRecession = ($40 52) / $52 = .2308 or 23.08% RNormal = ($59 52) / $52 = .1346 or 13.46% RExpanding = ($68 52) / $52 = .3077 or 30.77% The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return the stock is: E(RA) = .10(.2308) + .65(.1346) + .25(.3077) = .1413 or 14.13% And the variance of the stock is: 2 = .10(0.2308 0.1413)2 + .65(.1346 .1413)2 + .25(.3077 .1413)2 A 2 = 0.0208 A Which means the standard deviation is: A = (0.0208)1/2 A = .1442 or 14.42% Now we can calculate the stocks beta, which is: A = ( A,M)( A) / M A = (.45)(.1442) / .20 A = .324 For Stock B, we can directly calculate the beta from the information provided. So, the beta for Stock B is: Stock B: B = ( B,M)( B) / M B = (.40)(.508) / .20 B = 1.02 Using the CAPM to find the expected return for Stock B, we get: E(RB) = .04 + 1.02(.075) E(RB) = .1165 or 11.65% CHAPTER 20 B-247 The expected return on Stock B is lower than the expected return on Stock A. The risk of Stock B, as measured by its beta, is higher than the risk of Stock A. Thus, a typical risk-averse investor holding a well-diversified portfolio will prefer Stock A. b. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so: E(RP) = wAE(RA) + wBE(RB) E(RP) = .70(.1413) + .30(.1165) E(RP) = .1339 or 13.39% To find the standard deviation of the portfolio, we first need to calculate the variance. The variance of the portfolio is: 2 = X 2 2 + X 2 2 + 2XAXB A B A,B P A A B B 2 = (.70)2(.1442)2 + (.30)2(.51)2 + 2(.70)(.30)(.1442)(.51)(.50) P 2 = .04904 P And the standard deviation of the portfolio is: P = (0.04904)1/2 P = .2215 or 22.15% c. The beta of a portfolio is the weighted average of the betas of its individual securities. So the beta of the portfolio is: P = .70(.324) + .30(1.02) P = .533 38. a. The variance of a portfolio of two assets equals: 2 = X 2 2 + X 2 2 + 2XAXB A BCov(A,B) P A A B B Since the weights of the assets must sum to one, we can write the variance of the portfolio as: 2 = X 2 2 + (1 XA) P A A 2 B + 2XA(1 XA) A BCov(A,B) To find the minimum for any function, we find the derivative and set the derivative equal to zero. Finding the derivative of the variance function, setting the derivative equal to zero, and solving for the weight of Asset A, we find: XA = [ 2 Cov(A,B)] / [ 2 + 2 2Cov(A,B)] B A B Using this expression, we find the weight of Asset A must be: XA = (.592 .01) / [.482 + .592 2(.01)] XA = .6054 CHAPTER 20 B-248 This implies the weight of Stock B is: XB = 1 XA XB = 1 .6054 XB = .3946 b. Using the weights calculated in part a, determine the expected return of the portfolio, we find: E(RP) = XAE(RA) + XBE(RB) E(RP) = .6054(.12) + .3946(0.10) E(RP) = .1121 or 11.21% c. Using the derivative from part a, with the new covariance, the weight of each stock in the minimum variance portfolio is: XA = [ 2 + Cov(A,B)] / [ 2 + 2 2Cov(A,B)] B A B XA = (.592 .15) / [.482 + .592 2(.15)] XA = .5670 This implies the weight of Stock B is: XB = 1 XA XB = 1 .5670 XB = .4330 d. The variance of the portfolio with the weights on part c is: 2 = X 2 2 + X 2 2 + 2XAXB A BCov(A,B) P A A B B 2 = (.5670)2(.48)2 + (.4330)2(.59)2 + 2(.5670)(.4330)(.48)(.59)(.15) P 2 = .0657 P And the standard deviation of the portfolio is: P = (0.0657)1/2 P = .2563 or 25.63% ... View Full Document