Learning Module 1 - Free Cash Flow Valuation (EOC)

Practice Problems PRACTICE PROBLEMS The following information relates to questions 1-2 Shimotsuke Co. LTD. has FCFF of 1.7 billion Japanese yen (JPY) and FCFE of JPY1.3 billion. Shimotsuke Co.’s WACC is 11%, and its required rate of return for equity is 13%. FCFF is expected to grow forever at 7%, and FCFE is expected to grow forever at 7.5%. Shimotsuke Co. has debt outstanding of JPY15 billion. 1. What is the total value of Shimotsuke Co.’s equity using the FCFF valuation approach? 2. What is the total value of Shimotsuke Co.’s equity using the FCFE valuation approach? The following information relates to questions 3-6 Elina Kuznetsova is planning to value BCC Corporation, a provider of a variety of industrial metals and minerals. Kuznetsova uses a single-stage FCFF approach. The financial information Kuznetsova has assembled for her valuation is as follows: ■ The company has 1,852 million shares outstanding. ■ The market value of its debt is $3.192 billion. ■ The FCFF is currently $1.1559 billion. ■ The equity beta is 0.90; the equity risk premium is 5.5%; the risk-free rate is 5.5%. ■ The before-tax cost of debt is 7.0%. ■ The tax rate is 40%. ■ To calculate WACC, he will assume the company is financed 25% with debt. ■ The FCFF growth rate is 4%. Using Kuznetsova’s information, calculate the following: 61 62 Learning Module 1 Free Cash Flow Valuation 3. WACC. 4. Value of the firm. 5. Total market value of equity. 6. Value per share. The following information relates to questions 7-13 Yandie Izzo manages a dividend growth strategy for a large asset management firm. Izzo meets with her investment team to discuss potential investments in three companies: Company A, Company B, and Company C. Statements of cash flow for the three companies are presented in Exhibit 1. Exhibit 1: Statements of Cash Flow, Most Recent Fiscal Year End (Amounts in Millions of Dollars) Company A Company B Company C 4,844 1,212 15,409 500 288 3,746 1,000 — — Cash Flow from Operating Activities Net Income Adjustments Depreciation Other noncash expenses Changes in working capital (Increase) Decrease accounts receivable (Increase) Decrease inventories Increase (Decrease) accounts payable Increase (Decrease) other current liabilities Net cash from operating activities (452) (150) (536) — (200) (803) (210) 100 (3) 540 14 350 6,222 1,264 18,163 Cash Flow from Investing Activities (Purchase) Sale of fixed assets Net cash from investing activities 2,379 (1,000) (3,463) 2,379 (1,000) (3,463) Cash Flow from Financing Activities 25 3000 1,238 Increase (Decrease) long-term debt Increase (Decrease) notes payable (1,500) (1,000) (1,379) Payment of common stock dividends (1,000) (237) (15,000) Net cash from financing activities (2,475) 1,763 (15,141) Net change in cash and cash equivalents 6,126 2,027 (441) Cash and equivalents at beginning of year 50 100 3,000 6,176 2,127 2,559 (353) (50) (552) Cash and equivalents at end of year Supplemental Cash Flow Disclosures Interest Practice Problems Income taxes 63 Company A Company B Company C (1,605) (648) (3,787) Izzo’s team first discusses key characteristics of Company A. The company has a history of paying modest dividends relative to FCFE, has a stable capital structure, and is owned by a controlling investor. The team also considers the impact of Company A’s three noncash transactions in the most recent year on its FCFE, including the following: Transaction 1: A $900 million loss on a sale of equipment Transaction 2: An impairment of intangibles of $400 million Transaction 3: A $300 million reversal of a previously recorded restructuring charge In addition, Company A’s annual report indicates that the firm expects to incur additional noncash charges related to restructuring over the next few years. To value the three companies’ shares, one team member suggests valuing the companies’ shares using net income as a proxy for FCFE. Another team member proposes forecasting FCFE using a sales-based methodology based on the following equation: FCFE = NI – (1 – DR)(FCInv – Dep) – (1 – DR)(WCInv). Izzo’s team ultimately decides to use actual free cash flow to value the three companies’ shares. Selected data and assumptions are provided in Exhibit 2. Exhibit 2: Supplemental Data and Valuation Assumptions Company A Company B Company C Tax rate 35% 35% 30% Beta 1.00 0.90 1.10 Before-tax cost of debt 6% 7% 6% Target debt ratio 50% 30% 40% Market data: Risk-free rate: 3% Market risk premium: 7% The team calculates the intrinsic value of Company B using a two-stage FCFE model. FCFE growth rates for the first four years are estimated at 10%, 9%, 8%, and 7%, respectively, before declining to a constant 6% starting in the fifth year. To calculate the intrinsic value of Company C’s equity, the team uses the FCFF approach assuming a single-stage model where FCFF is expected to grow at 5% indefinitely. 7. Based on Company A’s key characteristics, which discounted cash flow model would most likely be used by the investment team to value Company A’s shares? A. DDM B. FCFE C. FCFF 64 Learning Module 1 Free Cash Flow Valuation 8. Which noncash transaction should be subtracted from net income in arriving at Company A’s FCFE? A. Transaction 1 B. Transaction 2 C. Transaction 3 9. Based on Exhibit 1, Company A’s FCFE for the most recent year is closest to: A. $5,318 million. B. $6,126 million. C. $7,126 million. 10. Based on Exhibit 1, using net income as a proxy for Company B’s FCFE would result in an intrinsic value that is: A. lower than the intrinsic value if actual FCFE were used. B. equal to the intrinsic value if actual FCFE were used. C. higher than the intrinsic value if actual FCFE were used. 11. Based on Exhibit 1, using the proposed sales-based methodology to forecast FCFE would produce an inaccurate FCFE projection for which company? A. Company A B. Company B C. Company C 12. Based on Exhibits 1 and 2 and the proposed two-stage FCFE model, the intrinsic value of Company B’s equity is closest to: A. $70,602 million. B. $73,588 million. C. $79,596 million. 13. Based on Exhibits 1 and 2 and the proposed single-stage FCFF model, the intrinsic value of Company C’s equity is closest to: A. $277,907 million. B. $295,876 million. C. $306,595 million. Practice Problems The following information relates to questions 14-15 The term “free cash flow” is frequently applied to cash flows that differ from the definition for FCFF that should be used to value a firm. Two such definitions of free cash flow are given below. Compare these two definitions for free cash flow with the technically correct definition of FCFF used in our coverage of the topic. 14. FCF = Net income + Depreciation and amortization – Cash dividends – Capital expenditures. 15. FCF = Cash flow from operations (from the statement of cash flows) – Capital expenditures. The following information relates to questions 16-18 LaForge Systems, Inc., has net income of $285 million for the year 2020. Using information from the company’s financial statements given here, show the adjustments to net income that would be required to find: 16. FCFF. 17. FCFE. 18. In addition, show the adjustments to FCFF that would result in FCFE. The following information relates to questions 19-20 Do Pham is evaluating Phaneuf Accelerateur by using the FCFF and FCFE valuation approaches. Pham has collected the following information (currency in euros): ■ Phaneuf has net income of €250 million, depreciation of €90 million, capital expenditures of €170 million, and an increase in working capital of €40 million. ■ Phaneuf will finance 40% of the increase in net fixed assets (capital expenditures less depreciation) and 40% of the increase in working capital with debt financing. ■ Interest expenses are €150 million. The current market value of Phaneuf ’s outstanding debt is €1,800 million. ■ FCFF is expected to grow at 6.0% indefinitely, and FCFE is expected to grow at 7.0%. ■ The tax rate is 30%. ■ Phaneuf is financed with 40% debt and 60% equity. The before-tax cost of debt is 9%, and the before-tax cost of equity is 13%. ■ Phaneuf has 10 million outstanding shares. 65 66 Learning Module 1 Free Cash Flow Valuation 19. Using the FCFF valuation approach, estimate the total value of the firm, the total market value of equity, and the per-share value of equity. 20. Using the FCFE valuation approach, estimate the total market value of equity and the per-share value of equity. The following information relates to questions 21-22 LaForge Systems, Inc., Balance Sheet (in Millions) Years Ended 31 December 2019 2020 Cash and equivalents $210 $248 Accounts receivable 474 513 Inventory 520 564 Assets Current assets 1,204 1,325 Gross fixed assets Total current assets 2,501 2,850 Accumulated depreciation (604) (784) Net fixed assets 1,897 2,066 $3,101 $3,391 Accounts payable $295 $317 Notes payable 300 310 Accrued taxes and expenses 76 99 Total assets Liabilities and shareholders’ equity Current liabilities 671 726 Long-term debt Total current liabilities 1,010 1,050 Common stock 50 50 Additional paid-in capital Retained earnings Total shareholders’ equity Total liabilities and shareholders’ equity Statement of Income In Millions, except Per-Share Data 300 300 1,070 1,265 1,420 1,615 $3,101 $3,391 31 December 2020 Total revenues $2,215 Operating costs and expenses 1,430 EBITDA 785 Depreciation 180 EBIT 605 Practice Problems Statement of Income In Millions, except Per-Share Data 67 31 December 2020 Interest expense 130 Income before tax 475 Taxes (at 40%) 190 Net income 285 Dividends 90 Addition to retained earnings 195 Statement of Cash Flows In Millions 31 December 2020 Operating activities Net income $285 Adjustments Depreciation 180 Changes in working capital Accounts receivable (39) Inventories (44) Accounts payable Accrued taxes and expenses Cash provided by operating activities 22 23 $427 Investing activities Purchases of fixed assets Cash used for investing activities 349 $349 Financing activities Notes payable $(10) Long-term financing issuances (40) Common stock dividends Cash used for financing activities 90 $40 Cash and equivalents increase (decrease) 38 Cash and equivalents at beginning of year 210 Cash and equivalents at end of year $248 Supplemental cash flow disclosures Interest paid $130 Income taxes paid $190 Note: The statement of cash flows shows the use of a convention by which the positive numbers of $349 and $40 for cash used for investing activities and cash used for financing activities, respectively, are understood to be subtractions, because “cash used” is an outflow. For LaForge Systems, whose financial statements are given in Problem 2, show the adjustments from the current levels of CFO (which is $427 million), EBIT ($605 million), and EBITDA ($785 million) to find: 68 Learning Module 1 Free Cash Flow Valuation 21. FCFF. 22. FCFE. The following information relates to questions 23-28 Ryan Leigh is preparing a presentation that analyzes the valuation of the common stock of two companies under consideration as additions to his firm’s recommended list, Emerald Corporation and Holt Corporation. Leigh has prepared preliminary valuations of both companies using an FCFE model and is also preparing a value estimate for Emerald using a dividend discount model. Holt’s 2019 and 2020 financial statements, contained in Exhibits 1 and 2, are prepared in accordance with US GAAP. Exhibit 1: Holt Corporation Consolidated Balance Sheets (US$ Millions) As of 31 December Assets 2020 2019 $ 372 $ 315 770 711 Current assets Cash and cash equivalents Accounts receivable Inventories Total current assets Gross fixed assets 4,275 Less: Accumulated depreciation 1,176 Total assets 846 780 1,988 1,806 3,752 3,099 906 2,846 $5,087 $4,652 $ 476 $ 443 149 114 Liabilities and shareholders’ equity Current liabilities Accounts payable Accrued taxes and expenses Notes payable Total current liabilities Long-term debt 465 450 1,090 1,007 1,575 1,515 525 525 Retained earnings 1,897 1,605 Total liabilities and shareholders’ equity $5,087 $4,652 Common stock Exhibit 2: Holt Corporation Consolidated Income Statement for the Year Ended 31 December 2020 (US$ Millions) Total revenues $3,323 Cost of goods sold 1,287 Selling, general, and administrative expenses 858 Practice Problems Earnings before interest, taxes, depreciation, and amortization (EBITDA) 69 1,178 Depreciation expense 270 Operating income 908 Interest expense 195 Pretax income 713 Income tax (at 32%) Net income 228 $ 485 Leigh presents his valuations of the common stock of Emerald and Holt to his supervisor, Alice Smith. Smith has the following questions and comments: 1. “I estimate that Emerald’s long-term expected dividend payout rate is 20% and its return on equity is 10% over the long term.” 2. “Why did you use an FCFE model to value Holt’s common stock? Can you use a DDM instead?” 3. “How did Holt’s FCFE for 2008 compare with its FCFF for the same year? I recommend you use an FCFF model to value Holt’s common stock instead of using an FCFE model because Holt has had a history of leverage changes in the past.” 4. “In the last three years, about 5% of Holt’s growth in FCFE has come from decreases in inventory.” Leigh responds to each of Smith’s points as follows: 1. “I will use your estimates and calculate Emerald’s long-term, sustainable dividend growth rate.” 2. “There are two reasons why I used the FCFE model to value Holt’s common stock instead of using a DDM. The first reason is that Holt’s dividends have differed significantly from its capacity to pay dividends. The second reason is that Holt is a takeover target and once the company is taken over, the new owners will have discretion over the uses of free cash flow.” 3. “I will calculate Holt’s FCFF for 2020 and estimate the value of Holt’s common stock using an FCFF model.” 4. “Holt is a growing company. In forecasting either Holt’s FCFE or FCFF growth rates, I will not consider decreases in inventory to be a long-term source of growth.” 23. Which of the following long-term FCFE growth rates is most consistent with the facts and stated policies of Emerald? A. 5% or lower B. 2% or higher C. 8% or higher 24. Do the reasons provided by Leigh support his use of the FCFE model to value Holt’s common stock instead of using a DDM? A. Yes B. No, because Holt’s dividend situation argues in favor of using the DDM 70 Learning Module 1 Free Cash Flow Valuation C. No, because FCFE is not appropriate for investors taking a control perspective 25. Holt’s FCFF (in millions) for 2020 is closest to: A. $308. B. $370. C. $422. 26. Holt’s FCFE (in millions) for 2020 is closest to: A. $175. B. $250. C. $364. 27. Leigh’s comment about not considering decreases in inventory to be a source of long-term growth in free cash flow for Holt is: A. inconsistent with a forecasting perspective. B. mistaken because decreases in inventory are a use rather than a source of cash. C. consistent with a forecasting perspective because inventory reduction has a limit, particularly for a growing firm. 28. Smith’s recommendation to use an FCFF model to value Holt is: A. logical, given the prospect of Holt changing capital structure. B. not logical because an FCFF model is used only to value the total firm. C. not logical because FCFE represents a more direct approach to free cash flow valuation. 29. Indicate the effect on this period’s FCFF and FCFE of a change in each of the items listed here. Assume a $100 increase in each case and a 40% tax rate. A. Net income. B. Cash operating expenses. C. Depreciation. D. Interest expense. E. EBIT. F. Accounts receivable. G. Accounts payable. H. Property, plant, and equipment. I. Notes payable. J. Cash dividends paid. Practice Problems 71 K. Proceeds from issuing new common shares. L. Common shares repurchased. The following information relates to questions 30-32 The management of Telluride, an international diversified conglomerate, believes that the recent strong performance of its wholly owned medical supply subsidiary, Sundanci, has gone unnoticed. To realize Sundanci’s full value, Telluride has announced that it will divest Sundanci in a tax-free spin-off. Sue Carroll is director of research at Kesson and Associates. In developing an investment recommendation for Sundanci, Carroll has gathered the information shown in Exhibit 1 and Exhibit 2. Exhibit 1: Sundanci Actual 2019 and 2020 Financial Statements for Fiscal Years Ending 31 May (Dollars in Millions except Per-Share Data) Income Statement 2019 2020 Revenue $474 $598 Depreciation 20 23 Other operating costs 368 460 Income before taxes 86 115 Taxes 26 35 Net income 60 80 Dividends 18 24 EPS $0.714 $0.952 Dividends per share $0.214 $0.286 Common shares outstanding 84.0 84.0 Balance Sheet 2019 2020 Current assets (includes $5 cash in 2007 and 2008) $201 $326 Net property, plant, and equipment 474 489 675 815 Current liabilities (all non-interest-bearing) 57 141 Long-term debt 0 0 57 141 618 674 675 815 Total assets Total liabilities Shareholders’ equity Total liabilities and equity 72 Learning Module 1 Free Cash Flow Valuation Balance Sheet Capital expenditures 2019 2020 34 38 Exhibit 2: Selected Financial Information Required rate of return on equity 14% Industry growth rate 13% Industry P/E 26 Abbey Naylor has been directed by Carroll to determine the value of Sundanci’s stock by using the FCFE model. Naylor believes that Sundanci’s FCFE will grow at 27% for two years and at 13% thereafter. Capital expenditures, depreciation, and working capital are all expected to increase proportionately with FCFE. 30. Calculate the amount of FCFE per share for 2020 by using the data from Exhibit 22. 31. Calculate the current value of a share of Sundanci stock based on the two-stage FCFE model. 32. Describe limitations that the two-stage DDM and FCFE models have in common. The following information relates to questions 33-38 Gurmeet Singh, an equity portfolio manager at a wealth management company, meets with junior research analyst Cindy Ho to discuss potential investments in three companies: Sienna Limited, Colanari Manufacturing, and Bern Pharmaceutical. Singh and Ho review key financial data from Sienna’s most recent annual report, which are presented in Exhibits 1 and 2, to assess the company’s ability to generate free cash flow. Exhibit 1: Selected Data from Sienna Limited’s Statement of Income for the Year Ended 31 December 2019 (Amounts in Millions of Euros) EBITDA Depreciation expense Operating income (EBIT) 4,000 800 3,200 Interest expense 440 Tax rate 35% Practice Problems 73 Exhibit 2: Sienna Limited’s Statement of Cash Flows for the Year Ended 31 December 2019 (Amounts in Millions of Euros) Cash flow from operations Net income 1,794 Plus: Depreciation 800 Increase in accounts receivable (2,000) Increase in inventory (200) Increase in accounts payable 1,000 Cash flow from operations 1,394 Cash flow from investing activities Purchases of PP&E (1,000) Cash flow from financing activities Borrowing (repayment) 500 Total cash flow 894 Singh and Ho also discuss the impact of dividends, share repurchases, and leverage on Sienna’s free cash flow. Ho tells Singh the following: Statement 1 Changes in leverage do not impact free cash flow to equity. Statement 2 Transactions between the company and its shareholders, such as the payment of dividends or share repurchases, do affect free cash flow. Singh and Ho next analyze Colanari. Last year, Colanari had FCFF of €140 million. Singh instructs Ho to perform an FCFF sensitivity analysis of Colanari’s firm value using the three sets of estimates presented in Exhibit 3. In her analysis, Ho assumes a tax rate of 35% and a stable capital structure of 30% debt and 70% equity. Exhibit 3: Sensitivity Analysis for Colanari Valuation Variable Base-Case Estimate Low Estimate High Estimate FCFF growth rate 4.6% 4.2% 5.0% Before-tax cost of debt 4.9% 3.9% 5.9% Cost of equity 11.0% 10.0% 12.0% Finally, Singh and Ho analyze Bern. Selected financial information on Bern is presented in Exhibit 4. Exhibit 4: Selected Financial Data on Bern Pharmaceutical Debt Market Value Required Return €15,400 million 6.0% Preferred stock €4,000 million 5.5% Common stock €18,100 million 11.0% 74 Learning Module 1 Free Cash Flow Valuation Market Value FCFF, most recent year Required Return €3,226 million Corporate tax rate 26.9% Singh notes that Bern has two new drugs that are currently in clinical trials awaiting regulatory approval. In addition to its operating assets, Bern owns a parcel of land from a decommissioned manufacturing facility with a current market value of €50 million that is being held for investment. Singh and Ho elect to value Bern under two scenarios: Scenario 1 Value Bern assuming the two new drugs receive regulatory approval. In this scenario, FCFF is forecast to grow at 4.5% into perpetuity. Scenario 2 Value Bern assuming the two new drugs do not receive regulatory approval. In this scenario, FCFF is forecast using a stable growth in FCFF of 1.5% for the next three years and then 0.75% thereafter into perpetuity. 33. Based on Exhibits 1 and 2, Sienna’s FCFF in 2019 is: A. €680 million. B. €1,200 million. C. €3,080 million. 34. Based on Exhibits 1 and 2, Sienna’s FCFE in 2019 is: A. €894 million. B. €1,466 million. C. €2,894 million. 35. Which of Ho’s statements regarding free cash flow is (are) correct? A. Statement 1 only B. Statement 2 only C. Neither Statement 1 nor Statement 2 36. Based on Exhibit 3, Ho’s FCFF sensitivity analysis should conclude that Colanari’s value is most sensitive to the: A. FCFF growth rate. B. before-tax cost of debt. C. required rate of return for equity. 37. Based on Exhibit 4, Bern’s firm value under Scenario 1 is closest to: A. €100,951.3 million. Practice Problems B. €105,349.1 million. C. €105,399.1 million. 38. Based on Exhibit 4, Singh and Ho should conclude that under Scenario 2, shares of Bern are: A. undervalued. B. fairly valued. C. overvalued. 39. PHB Company currently sells for £32.50 per share. In an attempt to determine whether PHB is fairly priced, an analyst has assembled the following information: ■ The before-tax required rates of return on PHB debt, preferred stock, and common stock are, respectively, 7.0%, 6.8%, and 11.0%. ■ The company’s target capital structure is 30% debt, 15% preferred stock, and 55% common stock. ■ The market value of the company’s debt is £145 million, and its preferred stock is valued at £65 million. ■ PHB’s FCFF for the year just ended is £28 million. FCFF is expected to grow at a constant rate of 4% for the foreseeable future. ■ The tax rate is 35%. ■ PHB has 8 million outstanding common shares. What is PHB’s estimated value per share? Is PHB’s stock underpriced? The following information relates to questions 40-41 An aggressive financial planner who claims to have a superior method for picking undervalued stocks is trying to steal one of your clients. The planner claims that the best way to find the value of a stock is to divide EBITDA by the risk-free bond rate. The planner is urging your client to invest in NewMarket, Inc. The planner says that NewMarket’s EBITDA of $1,580 million divided by the long-term government bond rate of 7% gives a total value of $22,571.4 million. With 318 million outstanding shares, NewMarket’s value per share found by using this method is $70.98. Shares of NewMarket currently trade for $36.50. 40. Provide your client with an alternative estimate of NewMarket’s value per share based on a two-stage FCFE valuation approach. Use the following assumptions: ■ Net income is currently $600 million. Net income will grow by 20% annually for the next three years. ■ The net investment in operating assets (capital expenditures less depreciation plus investment in working capital) will be $1,150 million next year and grow at 15% for the following two years. ■ 40% of the net investment in operating assets will be financed with net new debt financing. ■ NewMarket’s beta is 1.3; the risk-free bond rate is 7%; the equity risk premium is 4%. 75 76 Learning Module 1 Free Cash Flow Valuation ■ After three years, the growth rate of net income will be 8% and the net investment in operating assets (capital expenditures minus depreciation plus increase in working capital) each year will drop to 30% of net income. ■ Debt is, and will continue to be, 40% of total assets. ■ NewMarket has 318 million shares outstanding. 41. Criticize the valuation approach that the aggressive financial planner used. The following information relates to questions 42-44 John Jones is head of the research department of Peninsular Research and is estimating the value of Mackinac Inc. The company has released its June 2019 financial statements, shown in Exhibit 1, Exhibit 2, and Exhibit 3. Exhibit 1: Mackinac Inc. Annual Income Statement 30 June 2019 (in Thousands, except Per-Share Data) Sales $250,000 Cost of goods sold 125,000 Gross operating profit 125,000 Selling, general, and administrative expenses 50,000 EBITDA 75,000 Depreciation and amortization 10,500 EBIT 64,500 Interest expense 11,000 Pretax income 53,500 Income taxes 16,050 Net income $37,450 Shares outstanding 13,000 EPS $2.88 Exhibit 2: Mackinac Inc. Balance Sheet 30 June 2019 (in Thousands) Current Assets Cash and equivalents $20,000 Receivables 40,000 Inventories 29,000 Other current assets 23,000 Total current assets $112,000 Noncurrent Assets Property, plant, and equipment $145,000 Less: Accumulated depreciation 43,000 Net property, plant, and equipment 102,000 Practice Problems 77 Investments 70,000 Other noncurrent assets 36,000 Total noncurrent assets 208,000 Total assets $320,000 Current Liabilities Accounts payable $41,000 Short-term debt 12,000 Other current liabilities 17,000 Total current liabilities $ 70,000 Noncurrent Liabilities Long-term debt 100,000 Total noncurrent liabilities 100,000 Total liabilities 170,000 Shareholders’ Equity Common equity 40,000 Retained earnings 110,000 Total equity 150,000 Total liabilities and equity $320,000 Exhibit 3: Mackinac Inc. Statement of Cash Flows 30 June 2019 (in Thousands) Cash Flow from Operating Activities Net income $37,450 Depreciation and amortization 10,500 Change in Working Capital (Increase) decrease in receivables ($5,000) (Increase) decrease in inventories (8,000) Increase (decrease) in payables 6,000 Increase (decrease) in other current liabilities 1,500 Net change in working capital (5,500) Net cash from operating activities $42,450 Cash Flow from Investing Activities Purchase of property, plant, and equipment ($15,000) Net cash from investing activities ($15,000) Cash Flow from Financing Activities Change in debt outstanding $4,000 Payment of cash dividends (22,470) Net cash from financing activities (18,470) Net change in cash and cash equivalents $8,980 Cash at beginning of period 11,020 Cash at end of period $20,000 Mackinac has announced that it has finalized an agreement to handle North American production of a successful product currently marketed by a company headquartered outside North America. Jones decides to value Mackinac by using 78 Learning Module 1 Free Cash Flow Valuation the DDM and FCFE models. After reviewing Mackinac’s financial statements and forecasts related to the new production agreement, Jones concludes the following: ■ Mackinac’s earnings and FCFE are expected to grow 17% a year over the next three years before stabilizing at an annual growth rate of 9%. ■ Mackinac will maintain the current payout ratio. ■ Mackinac’s beta is 1.25. ■ The government bond yield is 6%, and the market equity risk premium is 5%. 42. Calculate the value of a share of Mackinac’s common stock by using the two-stage DDM. 43. Calculate the value of a share of Mackinac’s common stock by using the two-stage FCFE model. 44. Jones is discussing with a corporate client the possibility of that client acquiring a 70% interest in Mackinac. Discuss whether the DDM or FCFE model is more appropriate for this client’s valuation purposes. The following information relates to questions 45-46 James Smith is valuing McInish Corporation and performing a sensitivity analysis on his valuation. He uses a single-stage FCFE growth model. The base-case values for each of the parameters in the model are given, together with possible low and high estimates for each variable, in the following table. Variable Normalized FCFE0 Base-Case Value Low Estimate High Estimate £0.88 £0.70 £1.14 Risk-free rate 5.08% 5.00% 5.20% Equity risk premium 5.50% 4.50% 6.50% Beta FCFE growth rate 0.70 0.60 0.80 6.40% 4.00% 7.00% 45. Use the base-case values to estimate the current value of McInish Corporation. 46. Calculate the range of stock prices that would occur if the base-case value for FCFE0 were replaced by the low estimate and the high estimate for FCFE0. Similarly, using the base-case values for all other variables, calculate the range of stock prices caused by using the low and high values for beta, the risk-free rate, the equity risk premium, and the growth rate. Based on these ranges, rank the sensitivity of the stock price to each of the five variables. Practice Problems 79 The following information relates to questions 47-48 KMobile Telecom is an Asian mobile network operator headquartered in Seoul, South Korea. Sol Kim has estimated the normalized FCFE per share for KMobile to be 1,300 Korean won (KRW) for the year just ended. The real country return for South Korea is 6.50%. To estimate the required return for KMobile, Kim makes the following adjustments to the real country return: an industry adjustment of +0.60%, a size adjustment of –0.10%, and a leverage adjustment of +0.25%. The long-term real growth rate for South Korea is estimated to be 3.5%, and Kim expects the real growth rate of KMobile to track the country rate. 47. What is the real required rate of return for KMobile Telecom? 48. Using the single-stage FCFE valuation model and real values for the discount rate and FCFE growth rate, estimate the value of one share of KMobile. 49. Hugo Dubois is evaluating NYL Manufacturing Company, Ltd. In 2020, when Dubois is performing his analysis, the company is unprofitable. Furthermore, NYL pays no dividends on its common shares. Dubois decides to value NYL Manufacturing by using his forecasts of FCFE. Dubois gathers the following facts and assumptions: ■ The company has 17.0 billion shares outstanding. ■ Sales will be €5.5 billion in 2021, increasing at 28% annually for the next four years (through 2025). ■ Net income will be 32% of sales. ■ Investment in fixed assets will be 35% of sales; investment in working capital will be 6% of sales; depreciation will be 9% of sales. ■ 20% of the net investment in assets will be financed with debt. ■ Interest expenses will be only 2% of sales. ■ The tax rate will be 10%. NYL Manufacturing’s beta is 2.1; the risk-free government bond rate is 6.4%; the equity risk premium is 5.0%. ■ At the end of 2025, Dubois projects NYL terminal stock value at 18 times earnings. What is the value of one ordinary share of NYL Manufacturing Company? 50. Bron has EPS of $3.00 in 2019 and expects EPS to increase by 21% in 2020. EPS are expected to grow at a decreasing rate for the following five years, as shown in the following table. 2020 2021 2022 2023 2024 2025 Growth rate for EPS 21% 18% 15% 12% 9% 6% Net capital expenditures per share $5.00 $5.00 $4.50 $4.00 $3.50 $1.50 In 2025, the growth rate will be 6%, and it is expected to stay at that rate thereafter. Net capital expenditures (capital expenditures minus depreciation) will be $5.00 per share in 2019 and then follow the pattern predicted in the table. In 2025, net capital expenditures are expected to be $1.50, and they will then grow at 6% annually. The investment in working capital parallels the increase in net capital expenditures and is predicted to equal 25% of net capital expenditures each year. In 2025, investment in working capital will be $0.375, and it is pre- 80 Learning Module 1 Free Cash Flow Valuation dicted to grow at 6% thereafter. Bron will use debt financing to fund 40% of net capital expenditures and 40% of the investment in working capital. The required rate of return for Bron is 12%. Estimate the value of a Bron share using a two-stage FCFE valuation approach. 51. Minsuh Park is preparing a valuation of QuickChange Auto Centers, Inc. Park has decided to use a three-stage FCFE valuation model and the following estimates. The FCFE per share for the current year is $0.75. The FCFE is expected to grow at 10% for next year, then at 26% annually for the following three years, and then at 6% in Year 5 and thereafter. QuickChange’s estimated beta is 2.00, and Park believes that current market conditions dictate a 4.5% risk-free rate of return and a 5.0% equity risk premium. Given Park’s assumptions and approach, estimate the value of a share of QuickChange. 52. Astrid Nilsson has valued the operating assets of Gothenburg Extrusion AB at 720 million Swedish kronor (SEK). The company also has short-term cash and securities with a market value of SEK60 million that are not needed for Gothenburg’s operations. The noncurrent investments have a book value of SEK30 million and a market value of SEK45 million. The company also has an overfunded pension plan, with plan assets of SEK210 million and plan liabilities of SEK170 million. Gothenburg Extrusion has SEK215 million of notes and bonds outstanding and 100 million outstanding shares. What is the value per share of Gothenburg Extrusion stock? Solutions 81 SOLUTIONS 1. The firm value is the present value of FCFF discounted at the WACC, or ​FCFF​0​(​ ​1 + g​)​​ ​FCFF​ ​ 1.7​(​ ​1.07​)​​ Firm value = _ ​WACC −1 g ​ = ___________    ​ WACC − g ​ = _ ​0.11 − 0.07       ​​ 1.819 =_ ​0.04 ​ = JPY45.475 billion. The market value of equity is the value of the firm minus the value of debt: Equity = 45.475 – 15 = JPY30.475 billion. 2. Using the FCFE valuation approach, we find the present value of FCFE discounted at the required rate of return on equity to be ​FCFE​0​(​ ​1 + g​)​​ ​FCFE​ ​ 1.3​(​ ​1.07​)​​ 1.3975 PV = _ ​ r − g 1 ​ = ___________    ​ r−g ​= _ ​0.13 − 0.075 ​ = _ ​0.055 ​​       = JPY25.409 billion. The value of equity using this approach is JPY25.409 billion. 3. The required return on equity is ​r = E​(​ ​Ri​​)​​ = ​RF​ ​+ β​ i​​​[​E​(​ ​RM ​ ​)​​− R ​ F​ ​]​​ = 5 . 5 %   + 0.90​(​ ​5.5%​)​​ = 10.45 % . The weighted-average cost of capital is WACC = 0.25(7.0%)(1 – 0.40) + 0.75(10.45%) = 8.89%. 4. ​FCFF​0​​(​1 + g​)​​ ___________ Firm value =    ​ WACC − g ​.     1.1559​(​ ​1.04​)​​ _ Firm value = ​0.0889 − 0.04 ​ = $24.583. 5. Equity value = Firm value – Market value of debt. Equity value = 24.583 − 3.192 = $21.391 billion. 6. Value per share = Equity value/Number of shares. Value per share = $21.391 billion/1.852 billion = $11.55. 7. B is correct. Company A has a history of paying modest dividends relative to FCFE. An FCFF or FCFE model provides a better estimate of value over a DDM model when dividends paid differ significantly from the company’s capacity to pay dividends. Also, Company A has a controlling investor; with control comes discretion over the uses of free cash flow. Therefore, there is the possibility that the controlling shareholder could change the dividend policy. Finally, Company A has a stable capital structure; using FCFE is a more direct and simpler method to value a company’s equity than using FCFF when a company’s capital structure is stable. 8. C is correct. The applicable noncash adjustments to net income in arriving at FCFE are as follows: Noncash Item Transaction 1: Loss on sale of equipment Adjustment to Net Income Amount (millions) Added back +900 82 Learning Module 1 Free Cash Flow Valuation Adjustment to Net Income Amount (millions) Transaction 2: Impairment of intangibles Added back +400 Transaction 3: Reversal of restructuring charge Subtracted –300 Noncash Item In the case of Transaction 1, a loss reduces net income and thus must be added back in arriving at FCFE. Similarly, an impairment of intangibles (Transaction 2) reduces net income and thus must be added back in arriving at FCFE. Transaction 3 (reversal of a restructuring charge) would increase net income and thus must be subtracted in arriving at FCFE. 9. C is correct. FCFE for Company A for the most recent year is calculated as follows: Net income $4,844 Plus: Net noncash charges 1,500 Less: Investment in working capital Plus: Proceeds from sale of fixed capital 122 2,379 Less: Net borrowing repayment 1,475 FCFE (millions) $7,126 Net noncash charges are found by adding depreciation to other noncash expenses: $500 million + $1,000 million = $1,500 million. Investment in working capital is calculated by netting the increase in accounts receivable, the decrease in accounts payable, and the increase in other current liabilities: –$452 million – $210 million + $540 million = –$122 million (outflow). Net borrowing repayment is calculated by netting the increase in notes payable and the decrease in long-term debt: $25 million – $1,500 million = –$1,475 million (outflow). 10. A is correct. FCFE is significantly higher than net income for Company B: Net income = $1,212 million. FCFE for Company B is calculated as follows: Net income Plus: Net noncash charges Less: Investment in WC Less: Investment in fixed assets $1,212 288 236 1,000 Plus: Net borrowing 2,000 FCFE (millions) $2,264 Investment in working capital is calculated by adding the increase in accounts receivable, the increase in inventories, the increase in accounts payable, and the increase in other current liabilities: –$150 million – $200 million + $100 million + $14 million = –$236 million. Net borrowing is calculated by adding the increase in notes payable to the decrease in long-term debt: $3,000 million – $1,000 million = $2,000 million. Solutions 83 Therefore, using net income of $1,212 million as a proxy for FCFE ($2,264 million) for Company B would result in a much lower valuation estimate than if actual FCFE were used. 11. A is correct. In addition to significant noncash charges other than depreciation in the most recent year, the annual report indicates that Company A expects to recognize additional noncash charges related to restructuring over the next few years. The given equation for forecasting assumes that the only noncash charge is depreciation. When the company being analyzed has significant noncash charges other than depreciation expense, this sales-based methodology will result in a less accurate estimate of FCFE than one obtained by forecasting all the individual components of FCFE. 12. C is correct. FCFE for the most recent year for Company B is calculated as follows: Net income $1,212 Plus: Net noncash charges 288 Less: Investment in WC 236 Less: Investment in fixed assets 1,000 Plus: Net borrowing 2,000 FCFE (millions) $2,264 The required rate of return on equity for Company B is r = E(Ri) = RF + βi[E(RM) – RF] = 3% + 0.90(7%) = 9.3%. The most recent FCFE grows for the next four years at annual growth rates of 10%, 9%, 8%, and 7%, respectively, and then 6% thereafter: t g Calculation FCFE (millions) 1 10% $2,264.00 × 1.10 $2,490.40 2 9% $2,490.40 × 1.09 $2,714.54 3 8% $2,714.54 × 1.08 $2,931.70 4 7% $2,931.70 × 1.07 $3,136.92 5 6% $3,136.92 × 1.06 $3,325.13 The present value of FCFE for the first four years is calculated as follows: 2, 490.40 ​1.093​​ ​ 2, 714.54 ​1.093​​ ​ 2, 931.70 ​1.093​​ ​ 3, 136.92 ​1.093​​ ​ _ _ _ PV = _ ​ 1 ​+ ​ 2 ​+ ​ 3 ​+ ​ 4 ​. PV = 2,278.50 + 2,272.25 + 2,245.22 + 2.197.97 = 8,993.94. The present value of the terminal value is calculated as follows: 3, 325.13 ​​ ​0.093 − 0.06​ ​​ ​1.093​ ​ ​ PV of TV4 = _________________    ​   ​ = 70, 601.58.​ ( )( )4 So, the estimated total market value of the equity is 8,993.94 + 70,601.58 = 79,595.52 ≈ $79,596 million. 13. C is correct. Company C’s firm value is calculated as follows: The required rate of return on equity for Company C is r = E(Ri) = RF + βi[E(RM) – RF] = 3% + 1.1(7%) = 10.7%. MV​​(​Equity​)​​ MV​​(​Debt​)​​ ___________________    ​r​ ​d​​(​1 − Tax rate​)​​+ ___________________       ​ ( ​​r​ ​. ​WACC =    ​ ( ) ( ) MV​​ ​Debt​ ​​+ MV​​ ​Equity​ ​​ MV​​ ​Debt​)​​+ MV​​(​Equity​)​​ e 84 Learning Module 1 Free Cash Flow Valuation WACC = 0.40(6%)(1 – 0.30) + 0.60(10.7%) = 1.68% + 6.42% = 8.10%. FCFF for the most recent year for Company C is calculated as follows: Net income Plus: Net noncash charges Less: Investment in working capital Less: Investment in fixed capital Plus: Interest expense × (1 – Tax rate) FCFF (in millions) $15,409.00 3,746.00 992.00 3,463.00 386.40 $15,086.40 Investment in working capital is found by adding the increase in accounts receivable, the increase in inventories, the decrease in accounts payable, and the increase in other current liabilities: –$536 million – $803 million – $3 million + $350 million = –$992 million. FCFF is expected to grow at 5.0% indefinitely. Thus, ​FCFF​0​(​ ​1 + g​)​​ ​FCFF​ ​ 15, 086.4​(​ ​1.05​)​​ ___________ ​Firm value = _ ​WACC −1 g ​ =    ​ WACC − g ​ = ____________    ​ 0.081 − 0.05 ​ = $510, 990.97 million. The value of equity is the value of the firm minus the value of debt. The value of debt is found by multiplying the target debt ratio by the total firm value: Debt value = 0.40($510,990.97) = $204,396.39. Therefore, equity value = $510,990.97 – $204,396.39 = $306,594.58 million. 14. FCF = Net income + Depreciation and amortization – Cash dividends − Capital expenditures. This definition of free cash flow is sometimes used to determine how much “discretionary” cash flow the management has at its disposal. Management discretion concerning dividends is limited by investor expectations that dividends will be maintained. Comparing this definition with Equation 7, FCFF = NI + NCC + Int(1 – Tax rate) – FCInv – WCInv, we find that FCFF includes a reduction for investments in working capital and the addition of after-tax interest expense. Common stock dividends are not subtracted from FCFF because dividends represent a distribution of the cash available to investors. (If a company pays preferred dividends and they were previously taken out when net income available to common shareholders was calculated, they are added back in Equation 7 to include them in FCFF.) 15. FCF = Cash flow from operations (from the statement of cash flows) – Capital expenditures. Comparing this definition of free cash flow with Equation 8, FCFF = CFO + Int(1 – Tax rate) – FCInv, highlights the relationship of CFO to FCFF: The primary point is that when Equation 8 is used, after-tax interest is added back to CFO to arrive at the cash flow to all investors. Then FCInv is subtracted to arrive at the amount of that cash flow that is “free” in the sense of available for distribution to those investors after taking care of capital investment needs. If preferred dividends were subtracted to obtain net income (in CFO), they would also have to be added back in. This definition is commonly used to approximate FCFF, but it generally understates the actual FCFF by the amount of after-tax interest expense. 16. Free cash flow to the firm, found with Equation 7, is FCFF = NI + NCC + Int​(​ ​1 − Tax rate​)​​− FCInv − WCInv. FCFF = 285 + 180 + 130​​(​1 − 0.40​)​​− 349 − ​(​ ​39 + 44 − 22​ − 23​)​​.​              ​​ FCFF = 285 + 180 + 78 − 349 − 38 = $156 million. Solutions 17. Free cash flow to equity, found with Equation 10, is FCFE = NI + NCC − FCInv − WFCInv + Net borrowing.              ​​ ​ )​​.​ FCFE = 285 + 180 − 349 − ​(​ ​39 + 44 − 22 − 23​)​​+ ​(​ ​10 + 40​ FCFE = 285 + 180 − 349 − 38 + 50 = $128 million. 18. To find FCFE from FCFF, one uses the relationship in Equation 9: FCFE = FCFF − Int​​(​1 − Tax rate​)​​+ Net borrowing.            ​​ FCFE = 156 − 130​​(​1 − 0.40​)​​+ ​(​ ​10 + 40​)​​. FCFE = 156 − 78 + 50 = $128 million. 19. The FCFF is (in euros) FCFF = NI + NCC + Int​​(​1 − Tax rate​)​​− FCInv − WCInv.            ​​ FCFF = 250 + 90 + 150​​(​1 − 0.30​)​​− 170 − 40. FCFF = 250 + 90 + 105 − 170 − 40 = 235 million. The weighted-average cost of capital is WACC = 9%(1 – 0.30)(0.40) + 13%(0.60) = 10.32%. The value of the firm (in euros) is ​FCFF​0​​(​1 + g​)​​ ​FCFF​ ​ 235​(​ ​1.06​)​​ ___________ Firm value = _ ​WACC −1 g ​ =    ​ WACC − g ​ = _ ​0.1032 − 0.06       ​​ 249.1 =_ ​0.0432 ​ = 5, 766.20 million. The total value of equity is the total firm value minus the value of debt: Equity = €5,766.20 million – €1,800 million = €3,966.20 million. Dividing by the number of shares gives the per-share estimate of V0 = €3,966.20 million/10 million = €396.62 per share. 20. The free cash flow to equity is FCFE = NI + NCC − FCInv − WCInv + Net borrowing. ​​             FCFE = 250 + 90 − 170 − 40 + 0.40​(​ ​170 − 90 + 40​)​​. FCFE = 250 + 90 − 170 − 40 + 48 = €178 million. Because the company is borrowing 40% of the increase in net capital expenditures (170 – 90) and working capital (40), net borrowing is €48 million. The total value of equity is the FCFE discounted at the required rate of return of equity: ​FCFE​0​​(​1 + g​)​​ ​FCFE​ ​ 178​(​ ​1.07​)​​ ___________ ​ r−g ​= _ ​0.13 − 0.07 Equity value = _ ​ r − g 1 ​ =          ​ 190.46 =_ ​ 0.06 ​ = €3, 174.33 million. The value per share is V0 = €3,174.33 million/10 million = €317.43 per share. 21. To find FCFF from CFO, EBIT, or EBITDA, the analyst can use Equations 8, 12, and 13. To find FCFF from CFO: FCFF = CFO + Int​​(​1 − Tax rate​)​​− FCInv. ​​ ​      FCFF = 427 + 130​​(​1 − 0.40​)​​− 349 = 427 + 78 − 349 = $156 million. To find FCFF from EBIT: 85 86 Learning Module 1 Free Cash Flow Valuation FCFF = EBIT​​(​1 − Tax rate​)​​+ Dep − FCInv − WCInv.            ​​ FCFF = 605​​(​1 − 0.40​)​​+ 180 − 349 − 38.​ FCFF = 363 + 180 − 349 − 38 = $156 million. Finally, to obtain FCFF from EBITDA: FCFF = EBITDA​​(​1 − Tax rate​)​​+ Dep​(​ ​Tax rate​)​​− FCInv − WCInv. ​​              FCFF = 785​​(​1 − 0.40​)​​+ 180​(​ ​0.40​)​​− 349 − 38.​ FCFF = 471 + 72 − 349 − 38 = $156 million. 22. The simplest approach is to calculate FCFF from CFO, EBIT, or EBITDA as was done in Part A and then to find FCFE by making the appropriate adjustments to FCFF: FCFE = FCFF − Int​​(​1 − Tax rate​)​​+ Net borrowing. ​       ​ FCFE = 156 − 130​​(​1 − 0.40​)​​+ 50 = 156 − 78 + 50 = $128 million. The analyst can also find FCFE by using CFO, EBIT, or EBITDA directly. Starting with CFO and using Equation 11, FCFE is found to be FCFE = CFO − FCInv + Net borrowing. FCFE = 427 − 349 + 50 = $128 million. Starting with EBIT, on the basis of Equations 9 and 12, FCFE is FCFE = EBIT​(​ ​1 − Tax rate​)​​+ Dep − Int​(​ ​1 − Tax rate​)​​− FCInv − WCInv + Net borrowing.                    ( FCFE = 605​​ ​1 − 0.40​)​​+ 180 − 130​(​ ​1 − 0.40​)​​− 349 − 38 + 50. FCFE = 363 + 180 − 78 − 349 − 38 + 50 = $128 million. Finally, starting with EBITDA, on the basis of Equations 9 and 13, FCFE is FCFE = EBITDA​​(​1 − Tax rate​)​​+ Dep​(​ ​Tax rate​) − Int​(​ ​1 − Tax rate​)​​− FCInv − WCInv + Net borrowing. ​ ​ ​                     FCFE = 785​​(​1 − 0.40​)​​+ 180​(​ ​0.40​)​​− 130​(​ ​1 − 0.40​)​​− 349 − 38 + 50. FCFE = 471 + 72 − 78 − 349 − 38 + 50 = $128 million. 23. C is correct. The sustainable growth rate is return on equity (ROE) multiplied by the retention ratio. ROE is 10%, and the retention ratio is 1 – Payout ratio, or 1.0 – 0.2 = 0.8. The sustainable growth rate is 0.8 × 10% = 8%. FCFE growth should be at least 8% per year in the long term. 24. A is correct. Justifications for choosing the FCFE model over the DDM include the following: ■ The company pays dividends, but its dividends differ significantly from the company’s capacity to pay dividends (the first reason given by Leigh). ■ The investor takes a control perspective (the second reason given by Leigh). 25. A is correct. FCFF = NI + NCC + Interest expense(1 – Tax rate) – FCInv – WCInv. In this case: NI = $485 million NCC = Depreciation expense = $270 million Interest expense(1 – Tax rate) = 195(1 – 0.32) = $132.6 million FCInv = Net purchase of fixed assets = Increase in gross fixed assets = 4,275 – 3,752 = $523 million WCInv = Increase in accounts receivable + Increase in inventory – Increase in accounts payable – Increase in accrued liabilities Solutions 87 = (770 – 711) + (846 – 780) – (476 – 443) – (149 – 114) = $57 million FCFF = 485 + 270 + 132.6 – 523 – 57 = 307.6, or $308 million 26. B is correct. FCFE = NI + NCC – FCInv – WCInv + Net borrowing. In this case: NI = $485 million. NCC = Depreciation expense = $270 million. FCInv = Net purchase of fixed assets = Increase in gross fixed assets = 4,275 − 3,752 = $523 million. WCInv = Increase in accounts receivable + Increase in inventory ​ ​ in accounts ​                                                         ​ −​ Increase ​ payable − Increase ​ in accrued liabilities = ​(​ ​770 − 711​)​​+ ​(​ ​846 − 780​)​​− ​(​ ​476 − 443​)​​− ​(​ ​149 − 114​)​​ = $57 million. Net borrowing = Increase in notes payable + Increase in long-term debt = ​(​ ​465 − 450​)​​+ ​(​ ​1,575 − 1,515​)​​ = $75 million. FCFE = 485 + 270 − 523 − 57 + 75 = $250 million. An alternative calculation is FCFE = FCFF – Int(1 – Tax rate) + Net borrowing. FCFE = 307.6 – 195(1 – 0.32) + (15 +60) = $250 million. 27. C is correct. Inventory cannot be reduced below zero. Furthermore, sales growth tends to increase inventory. 28. A is correct. The FCFF model is often selected when the capital structure is expected to change because FCFF estimation may be easier than FCFE estimation in the presence of changing financial leverage. 29. Change in FCFF (in US Dollars) Change in FCFE (in US Dollars) A. Net income +100 +100 B. Cash operating expenses –60 –60 C. Depreciation +40 +40 0 –60 E. EBIT +60 +60 F. Accounts receivable –100 –100 G. Accounts payable +100 +100 H. Property, plant, and equipment –100 –100 For a $100 increase in: D. Interest expense I. Notes payable 0 +100 J. Cash dividends paid 0 0 K. Proceeds from issuing new common shares 0 0 L. Common shares repurchased 0 0 88 Learning Module 1 Free Cash Flow Valuation 30. FCFE is defined as the cash flow remaining after the company meets all financial obligations, including debt payment, and covers all capital expenditure and working capital needs. Sundanci’s FCFE for the year 2020 is calculated as follows: Net income = $80 million Plus: Depreciation expense = 23 Less: Capital expenditures = 38 Less: Investment in WC = 41 Equals: FCFE = $24 million Thus, FCFE per share equals ($24 million)/(84 million shares) = $0.286. 31. The FCFE model requires forecasts of FCFE for the high-growth years (2021 and 2022) plus a forecast for the first year of stable growth (2023) to allow for an estimate of the terminal value in 2022 based on constant perpetual growth. Because all of the components of FCFE are expected to grow at the same rate, the values can be obtained by projecting the FCFE at the common rate. (Alternatively, the components of FCFE can be projected and aggregated for each year.) The following table provides the process for estimating Sundanci’s current value on a per-share basis. Free Cash Flow to Equity Base assumptions: Shares outstanding (millions) 84 Required return on equity, r 14% Actual 2020 Total Projected 2021 Projected 2022 Projected 2023 g = 27% g = 27% g = 13% Per share Earnings after tax $80 $0.952 $1.2090 $1.5355 $1.7351 Plus: Depreciation expense $23 $0.274 $0.3480 $0.4419 $0.4994 Less: Capital expenditures $38 $0.452 $0.5740 $0.7290 $0.8238 Less: Increase in net working capital $41 $0.488 $0.6198 $0.7871 $0.8894 Equals: FCFE $24 $0.286 $0.3632 $0.4613 $0.5213 Terminal valuea $52.1300 Total cash flows to equityb $0.3632 $52.5913 Discounted $0.3186 $40.4673 valuec Current value per shared $40.7859 aProjected 2022 terminal value = Projected 2023 FCFE/(r – g). 2022 total cash flows to equity = Projected 2022 FCFE + Projected 2022 terminal value. cDiscounted values obtained by using r = 14%. dCurrent value per share = Discounted value 2021 + Discounted value 2022. bProjected 32. The following limitations of the DDM are addressed by the FCFE model: The DDM uses a strict definition of cash flow to equity; that is, cash flows to equity are the dividends on the common stock. The FCFE model expands the definition of cash flow to include the balance of residual cash flows after all financial obligations and investment needs have been met. Thus, the FCFE model explicitly recognizes the company’s investment and financing policies as well as its dividend policy. In instances of a change of corporate control, and thus the possibility Solutions of changing dividend policy, the FCFE model provides a better estimate of value. Both two-stage valuation models allow for two distinct phases of growth—an initial finite period when the growth is abnormal followed by a stable growth period that is expected to last forever. These two-stage models share the same limitations with respect to the growth assumptions: First, the analyst must confront the difficulty of defining the duration of the extraordinary growth period. A long period of high growth will produce a higher valuation, and the analyst may be tempted to assume an unrealistically long period of extraordinary growth. Second, the analyst must realize that assuming a sudden shift from high growth to lower, stable growth is unrealistic. The transformation is more likely to occur gradually over time. Third, because value is quite sensitive to the steady-state growth assumption, overestimating or underestimating this rate can lead to large errors in value. The two models also share other limitations—notably, difficulties in accurately estimating required rates of return. 33. A is correct. Sienna’s FCFF in 2019 is calculated as FCFF = EBIT(1 – Tax rate) + Dep – FCInv – WCInv. FCInv = Purchases of PP&E = 1,000 (outflow). WCInv = Increase in accounts receivable (outflow) + Increase in inventory (outflow) + Increase in accounts payable (inflow). WCInv = –2,000 (outflow) + –200 (outflow) + 1,000 (inflow) = –1,200 (outflow). FCFF = 3,200(1 – 0.35) + 800 – 1,000 –1,200. FCFF = €680 million. FCFF can also be computed from CFO: FCFF = CFO + Int(1 – Tax rate) – FCInv. FCFF = 1,394 + 440(1 – 0.35) – 1,000. FCFF = €680 million. 34. A is correct. Sienna’s FCFE in 2019 is calculated as FCFE = CFO – FCInv + Net borrowing = 1,394 – 1,000 + 500 = €894 million. Alternatively, FCFE may be calculated as FCFE = FCFF – Int(1 – Tax rate) + Net borrowing. = 680 – 440(1 – 0.35) + 500 = €894 million. 35. C is correct. Transactions between the company and its shareholders (through cash dividends, share repurchases, and share issuances) do not affect free cash flow. However, leverage changes, such as the use of more debt financing, have some impact on free cash flow because they increase the interest tax shield (re- 89 90 Learning Module 1 Free Cash Flow Valuation duce corporate taxes because of the tax deductibility of interest) and reduce the cash flow available to equity. 36. C is correct. Colanari’s valuation is most sensitive to the cost of equity (re) because the range of estimated values is larger than the valuation ranges estimated from the sensitivity analysis of both the FCFF growth rate (GFCFF) and the before-tax cost of debt (rd). Variable Base Case Low Estimate High Estimate Valuation with Low Estimate (€ millions) GFCFF 4.6% 4.2% 5.0% 3,274.16 4,021.34 747.18 rd 4.9% 3.9% 5.9% 3,793.29 3,445.24 348.05 11.0% 10.0% 12.0% 4,364.18 3,079.38 1,284.80 re Valuation with High Estimate (€ millions) Range (€ millions) WACC = [wd × rd(1 – Tax rate)] + (we × re). Firm value = FCFF0(1 + g)/(WACC – g). Cost of equity sensitivity Using the base case estimates for the FCFF growth rate and the before-tax cost of debt and using the low estimate for the cost of equity (re) of 10.0%, the valuation estimate is WACC = [(0.30)(0.049)(1 – 0.35)] + (0.70)(0.10) = 7.96%. Firm value = 140 million(1 + 0.046)/(0.0796 – 0.046) = €4,364.18 million. Using the base case estimates for the FCFF growth rate and the before-tax cost of debt and using the high estimate for the cost of equity (re) of 12.0%, the valuation estimate is WACC = [(0.30)(0.049)(1 – 0.35)] + (0.70)(0.120) = 9.36%. Firm value = 140 million(1 + 0.046)/(0.0936 – 0.046) = €3,079.38 million. Therefore, the range in valuation estimates from using the highest and lowest estimates of the cost of equity is €1,284.80 million. FCFF growth rate sensitivity Using the base case estimates for the cost of equity and the before-tax cost of debt and using the low estimate for the FCFF growth rate (GFCFF) of 4.2%, the valuation estimate is WACC = [(0.30)(0.049)(1 – 0.35)] + (0.70)(0.11) = 8.66%. Firm value = 140 million(1 + 0.042)/(0.0866 – 0.042) = €3,274.16 million. Using the base case estimates for the cost of equity and the before-tax cost of debt and using the high estimate for the FCFF growth rate (GFCFF) of 5.0%, the valuation estimate is WACC = [(0.30)(0.049)(1 – 0.35)] + (0.70)(0.11) = 8.66%. Firm value = 140 million(1 + 0.05)/(0.0866 – 0.05) = €4,021.34 million. Therefore, the range in valuation estimates from using the highest and lowest estimates of the FCFF growth rate is €747.18 million. Before-tax cost of debt sensitivity Solutions 91 Using the base case estimates for the FCFF growth rate and the cost of equity and using the low estimate for the before-tax cost of debt (rd) of 3.9%, the valuation estimate is WACC = [(0.30)(0.039)(1 – 0.35)] + (0.70)(0.11) = 8.46%. Firm value = 140 million(1 + 0.046)/(0.0846 – 0.046) = €3,793.29 million. Using the base case estimates for the FCFF growth rate and the cost of equity and using the high estimate for the before-tax cost of debt (rd) of 5.9%, the valuation estimate is WACC = [(0.30)(0.059)(1 – 0.35)] + (0.70)(0.11) = 8.85%. Firm value = 140 million(1 + 0.046)/(0.0885 – 0.046) = €3,445.24 million. Therefore, the range in valuation estimates from using the highest and lowest estimates of the before-tax cost of debt is €348.05 million. 37. C is correct. Based on Scenario 1, where Bern receives regulatory approval for its new drugs, the growth rate in FCFF for Bern will be constant at 4.5%. Therefore, a constant-growth valuation model can be used to calculate firm value. Bern’s weighted average cost of capital is calculated as WACC = [wd × rd(1 – Tax rate)] + (wp × rp) + (we × re). The total market value of the firm is the sum of the debt, preferred stock, and common stock market values: 15,400 + 4,000 + 18,100 = 37,500. WACC = [(15,400/37,500)(0.060)(1 – 0.269] + (4,000/37,500)(0.055) + (18,100/37,500)(0.11) = 7.70%. Value of operating assets = FCFF0(1 + g)/(WACC – g). Value of operating assets = 3,226 million(1 + 0.045)/(0.0770 – 0.045) = €105,349.06 million. Total value of the company = Value of operating assets + Value of non-operating assets. Total value of the company = 105,349.06 million + 50 million = €105,399.06 million. 38. A is correct. The total market value of the firm is the sum of the debt, preferred stock, and common stock market values: 15,400 + 4,000 + 18,100 = 37,500 million. WACC = [wd × rd(1 – Tax rate)] + (wp × rp) + (we × re) = [(15,400/37,500)(0.060)(1 – 0.269] + (4,000/37,500)(0.055) + (18,100/37,500) (0.11) = 7.70%. Under the assumption that Bern has a low growth rate because it did not receive regulatory approval for its new drugs, the value of Bern can be analyzed using a two-stage valuation model. n ​FCFF​​ ​FCFF​ ​ t n+1 _ 1 ​( Company value = ​∑ _ ​+ _ ​ ​ ​. )​t​ ​(​WACC − g​)​(​ ​1 + WACC​)​n​ ​ 1 ​ + WACC​ t=1 92 Learning Module 1 Free Cash Flow Valuation Year 0 1 1.50% 1.50% 1.50% 0.75% 3,226 3,274.39 3,323.51 3,373.36 3,398.66 Present Value Factor 0.928529 0.862167 0.800547 Present Value (€ millions) 3,040.37 2,865.42 2,700.53 g FCFFn (€ millions) 2 3 4 The terminal value at the end of Year 3 is TV3 = FCFF4/(WACC – g4). TV3 = 3,398.66/(0.0770 – 0.0075) = €48,901.58 million. The total value of operating assets = (3,040.37 + 2,865.42 + 2,700.53) + 48,901.58/(1 + 0.0770)3 = 8,606.32 + 39,144.95 = €47,751.27 million. Value of Bern’s common stock = Value of operating assets + Value of non-operating assets – Market value of debt – Preferred stock = 47,751.27 + 50.00 – 15,400 – 4,000 = €28,401.27 million. Since the current market value of Bern’s common stock (€18,100 million) is less than the estimated value (€28,401.27 million), the shares are undervalued. 39. The WACC for PHB Company is ​WACC = 0.30​​(​7.0%​)​​(​ ​1 − 0.35​)​​+ 0.15​(​ ​6.8%​)​​+ 0.55​(​ ​11.0%​)​​ = 8.435 % . The firm value is ​FCFF​0​​(​1 + g​)​​    Firm value = ___________ ​ WACC − g ​. 28​(​ ​1.04​)​​ ___________ Firm value ​ = ​           ​​ 0.08435 − 0.04 29.12 =_ ​0.04435 = £656.60 million. The value of equity is the firm value minus the value of debt minus the value of preferred stock: Equity = 656.60 − 145 − 65 = £446.60 million. Dividing this amount by the number of shares gives the estimated value per share of £446.60 million/8 million shares = £55.82. The estimated value for the stock is greater than the market price of £32.50, so the stock appears to be undervalued. 40. Using the CAPM, the required rate of return for NewMarket is ​r = E​​(​Ri​​)​​ = ​RF​ ​+ ​βi​​​[​E​(​ ​RM ​ ​)​​− ​RF​ ​]​​ = 7 %   + 1 . 3​(​ ​4%​)​​ = 12 . 2 % . To estimate FCFE, we use Equation 15: Solutions 93 FCFE = Net income − ​​(​1 − DR​)​​(​ ​FCInv − Depreciation​)​​       ​​ ​​ − ​(​ ​1 − DR​)​​(​ ​WCInv​)​​, which can be written FCFE = Net income − ​​(​1 − DR​)​​(​ ​FCInv − Depreciation + WCInv​)​​        ​​ ​​ = Net income − ​​(​1 − DR​)​​(​ ​Net investment in operating assets​)​​. The following table shows that net income grows at 20% annually for Years 1, 2, and 3 and then grows at 8% for Year 4. The net investment in operating assets is $1,150 million in Year 1 and grows at 15% annually for Years 2 and 3. Debt financing is 40% of this investment. FCFE is NI – Net investment in operating assets + New debt financing. Finally, the present value of FCFE for Years 1, 2, and 3 is found by discounting at 12.2%. Year (in $ Millions) Net income Net investment in operating assets 1 2 3 4 720.00 864.00 1,036.80 1,119.74 1,150.00 1,322.50 1,520.88 335.92 New debt financing 460.00 529.00 608.35 134.37 FCFE 30.00 70.50 124.27 918.19 PV of FCFE discounted at 12.2% 26.74 56.00 87.98 In Year 4, net income is 8% larger than in Year 3. In Year 4, the investment in operating assets is 30% of net income and debt financing is 40% of this investment. The FCFE in Year 4 is $918.19 million. The value of FCFE after Year 3 is found by using the constant-growth model: ​FCFE​ ​ 918.19 ​ r − g 4​ = _ ​0.122 − 0.08 ​ = $21, 861.67 million. ​​V3​ ​ = _ The present value of V3 discounted at 12.2% is $15,477.64 million. The total value of equity, the present value of the first three years’ FCFE plus the present value of V3, is $15,648.36 million. Dividing this amount by the number of outstanding shares (318 million) gives a value per share of $49.21. For the first three years, NewMarket has a small FCFE because of the large investments it is making during the high-growth phase. In the normal-growth phase, FCFE is much larger because the investments required are much smaller. 41. The planner’s estimate of the share value of $70.98 is much higher than the FCFE model estimate of $49.21 for several reasons. First, taxes and interest expenses have a prior claim to the company’s cash flow and should be taken out of the cash flows used in estimating the value of equity because these amounts are not available to equity holders. The planner did not do this. Second, EBITDA does not account for the company’s reinvestments in operating assets. So, EBITDA overstates the funds available to stockholders if reinvestment needs exceed depreciation charges, which is the case for growing companies such as NewMarket. Third, EBITDA does not account for the company’s capital structure. Using EBITDA to represent a benefit to stockholders (as opposed to stockholders and bondholders combined) is a mistake. Finally, dividing EBITDA by the bond rate is a major error. The risk-free bond rate is an inappropriate discount rate for risky equity cash flows; the proper measure is the required rate of return on the company’s equity. Dividing by a fixed rate also assumes, erroneously, that the cash flow stream is a fixed perpetuity. EBITDA cannot be a perpetual stream because if it were distributed, the stream 94 Learning Module 1 Free Cash Flow Valuation would eventually decline to zero (lacking capital investments). NewMarket is actually a growing company, so assuming it to be a nongrowing perpetuity is a mistake. 42. When a two-stage DDM is used, the value of a share of Mackinac, dividends per share (DPS), is calculated as follows: ​DPS​0​ = Cash dividends / Shares outstanding = $22,470 / 13,000 = $1.7285. ​DPS​1​ = ​DPS​0​× 1.17 = $2.0223. ​​ PS​ ​ = ​DPS​ ​× 1​​.17​​2​ = $2.3661.​                       ​D 2 0 ​DPS​3​ = ​DPS​0​× 1 ​.17​​3​ = $2.7683. ​DPS​4​ = ​DPS​0​× 1 ​.17​​3​× 1.09 = $3.0175. When the CAPM is used, the required return on equity, r, is    r = Government bond rate + (Beta × Equity risk premium)      =    0.06 + (1.25 ​ × 0.05) = 0.1225, or 12.25%.​ ​ ​                      Value per share = ​DPS​1​/ ​​(​1 + r​)​​​+ ​DPS​2​/ ​​(​1 + r​)​​2​+ D ​ PS​3​/ ​​(​1 + r​)​​3​ + ​[​ ​DPS​4​/ ​​(​r − ​g​stable​)​​]​​/ ​(​1 + r​)​3​. Value per share = $2.0223 / 1.1225 + $2.3661 / ​1.1225​​2​ + $2.7683 / 1 .​1225​​3                    + ​[​ ​$3.0175 / ​(​ ​0.1225​ − 0.09​)​]​ ​​/ 1 .​1225​​3 = $1.8016 + $1.8778 + $1.9573 + $65.6450 = $71.28. 43. When the two-stage FCFE model is used, the value of a share of Mackinac is calculated as follows (in $ thousands except per-share data): Net income = $37, 450. Depreciation = $10, 500. Capital expenditures = $15, 000. Change in working capital = $5, 500. New debt issuance − Principal repayments = Change in debt outstanding = $4, 000 ​FCFE​0​ = Net income + Depreciation − Capital expenditures− Change in working capital − Principal repayments + ​ ​​​​ ​debt                                                          ​       ​ ​ issues. New ​FCFE​0​ = $37, 450 + $10, 500 − $15, 000 − $5, 500 + $4, 000 = $31, 450. FCF ​E0​ ​pershare = $31, 450 / 13, 000 = $2.4192. FCF E ​ 1​ ​ = FCF ​E0​ ​× 1.17 = $2.8305. FCF ​E2​ ​ = FCF ​E0​ ​× ​1.17​​2​ = $3.3117. FCF E ​ 3​ ​ = FCF ​E0​ ​× ​1.17​​3​ = $3.8747. FCF ​E4​ ​ = FCF E ​ 0​ ​× 1​ .17​​3​× 1.09 = $4.2234. From the answer to A, r = 12.25%. Value per share = ​FCFE​1​/ ​​(​1+r​)​​+ ​FCFE​2​/ ​(​1+r​)​2​+ ​FCFE​3​/ ​(​1+r​)​3 ​​        ​ + ​[​ ​FCFE​4​/ ​​(​r − ​g​stable​)​​]​​/ ​(​1+r​)​3​. Solutions 95 Value per share = $2.8305 / 1.1225 + $3.3117 / 1 ​.1225​​2​ + $3.8747 / ​1.1225​​3                    + ​[​ ​$4.2234 / ​(​ ​0.1225​ − 0.09​)​​]​​/ ​1.1225​​3 = $2.5216 + $2.6283 + $2.7395 + $91.8798 = $99.77. 44. The FCFE model is best for valuing companies for takeovers or in situations that have a reasonable chance of a change in corporate control. Because controlling stockholders can change the dividend policy, they are interested in estimating the maximum residual cash flow after meeting all financial obligations and investment needs. The DDM is based on the premise that the only cash flows received by stockholders are dividends. FCFE uses a more expansive definition to measure what a company can afford to pay out as dividends. 45. The required rate of return for McInish found with the CAPM is ​r = E​(​ ​Ri​​)​​ = ​RF​ ​+ β​ i​​​[​E​(​ ​RM ​ ​)​​− R ​ F​ ​]​​ = 5.08 %   + 0.70​(​ ​5 . 50%​)​​ = 8 . 93 % . The value per share is ​FCFE​0​​(​1 + g​)​​ 0.88​(​ ​1.064​)​​ ___________ ​​V0​ ​ =    ​ r−g ​ = ___________ ​0.0893    − 0.064 ​ = $37.01. 46. The following table shows the calculated price for McInish based on the base-case values for all values except the variable being changed from the base-case value. Estimated Price with Low Value (£) Estimated Price with High Value (£) Range (Rank) (£) Normalized FCFE0 29.44 47.94 18.50 (3) Risk-free rate 38.22 35.33 2.89 (5) Equity risk premium 51.17 28.99 22.18 (2) Beta 47.29 30.40 16.89 (4) FCFE growth rate 18.56 48.79 30.23 (1) Variable As the table shows, the value of McInish is most sensitive to the changes in the FCFE growth rate, with the price moving over a wide range. McInish’s stock price is least sensitive to alternative values of the risk-free rate. Alternative values of beta, the equity risk premium, or the initial FCFE value also have a large impact on the value of the stock, although the effects of these variables are smaller than the effect of the growth rate. 47. The real required rate of return for KMobile is Country return (real) 6.50% Industry adjustment +0.60% Size adjustment –0.10% Leverage adjustment +0.25% Required rate of return 7.25% 48. The real growth rate of FCFE is expected to be the same as the country rate of 3.5%. The value of one share is 96 Learning Module 1 Free Cash Flow Valuation ​FCFE​0​​(​1 + g​ ​real​)​​ ___________ 1, 300​(​ ​1.035​)​​ ______________ ​​V0​ ​ =    ​       ​ ​r​real​− ​g​real​ ​ =    0.0725 − 0.035 ​ = KRW35, 880. 49. The required rate of return found with the CAPM is ​r = E​(​ ​Ri​​)​​ = ​RF​ ​+ ​βi​​​[​E​(​ ​RM ​ ​)​​− ​RF​ ​]​​ = 6.4 %   + 2.1​(​ ​5.0%​)​​ = 16.9 % . The following table shows the values of sales, net income, capital expenditures less depreciation, and investments in working capital. FCFE equals net income less the investments financed with equity: FCFE = Net income − ​​(​1 − DR​)​​(​ ​Capital expenditures − Depreciation​)        − ​(​ ​1 − DR​)​​(​ ​Investment in working capital​)​​, where DR is the debt ratio (debt financing as a percentage of debt and equity). Because 20% of net new investments are financed with debt, 80% of the investments are financed with equity, which reduces FCFE by 80% of (Capital expenditures – Depreciation) and 80% of the investment in working capital. (All Data in Billions of Euros) 2021 2022 2023 2024 2025 Sales (growing at 28%) 5.500 7.040 9.011 11.534 14.764 Net income = 32% of sales 1.760 2.253 2.884 3.691 4.724 FCInv – Dep = (35% – 9%) × Sales 1.430 1.830 2.343 2.999 3.839 WCInv = (6% of Sales) 0.330 0.422 0.541 0.692 0.886 0.80 × (FCInv – Dep + WCInv) 1.408 1.802 2.307 2.953 3.780 FCFE = NI – 0.80 × (FCInv – Dep + WCInv) 0.352 0.451 0.577 0.738 0.945 PV of FCFE discounted at 16.9% 0.301 0.330 0.361 0.395 0.433 Terminal stock value 85.032 PV of terminal value discounted at 16.9% 38.950 PV of FCFE (first five years) 1.820 Total value of equity 40.770 The terminal stock value is 18.0 times the earnings in 2025, or 18 × 4.724 = €85.03 billion. The present value of the terminal value (€38.95 billion) plus the present value of the first five years’ FCFE (€1.82 billion) is €40.77 billion. Because NYL Manufacturing has 17 billion outstanding shares, the value per ordinary share is €2.398. 50. The following table develops the information to calculate FCFE per share (amounts are in US dollars). 2020 2021 2022 2023 2024 2025 Growth rate for EPS 21% 18% 15% 12% 9% 6% EPS 3.630 4.283 4.926 5.517 6.014 6.374 Net capital expenditure per share 5.000 5.000 4.500 4.000 3.500 1.500 Investment in WC per share 1.250 1.250 1.125 1.000 0.875 0.375 New debt financing = 40% of (Capital expenditure + WCInv) 2.500 2.500 2.250 2.000 1.750 0.750 FCFE = NI – Net capital expenditure – WCInv + New debt financing –0.120 0.533 1.551 2.517 3.389 5.249 PV of FCFE discounted at 12% –0.107 0.425 1.104 1.600 1.923 Solutions 97 Earnings per share for 2019 are $3.00, and the EPS estimates for 2020 through 2025 in the table are found by increasing the previous year’s EPS by that year’s growth rate. The net capital expenditures each year were specified by the analyst. The increase in working capital per share is equal to 25% of net capital expenditures. Finally, debt financing is 40% of that year’s total net capital expenditures and investment in working capital. For example, in 2020, the per-share amount for net capital expenditures plus investment in working capital is $5.00 + $1.25 = $6.25. Debt financing is 40% of $6.25, or $2.50. Debt financing for 2021 through 2025 is found in the same way. FCFE equals net income minus net capital expenditures minus investment in working capital plus new debt financing. Notice that FCFE is negative in 2020 because of large capital investments and investments in working capital. As these investments decline relative to net income, FCFE becomes positive and substantial. The present values of FCFE from 2020 through 2024 are given in the bottom row of the table. These five present values sum to $4.944 per share. Because FCFE from 2025 onward will grow at a constant 6%, the constant-growth model can be used to value these cash flows. ​FCFE​2025​ 5.249 ​​V2024 ​ ​= _ ​ r−g ​= _ ​0.12 − 0.06 ​ = $87.483. The present value of this stream is $87.483/(1.12)5 = $49.640. The value per share is the present value of the first five FCFEs (2020–2024) plus the present value of the FCFE after 2024, or $4.944 + $ 49.640 = $54.58. 51. The required return for QuickChange, found by using the CAPM, is r = E(Ri) = RF + βi[E(RM) – RF] = 4.5% + 2.0(5.0%) = 14.5%. The estimated future values of FCFE per share are given in the following exhibit (amounts in US dollars): Year t 1 2 3 4 4 0 Variable Calculation FCFE1 0.75(1.10) FCFE3 0.75(1.10)(1.26)2 FCFE2 0.75(1.10)(1.26) FCFE4 0.75(1.10)(1.26)3 TV4 FCFE5/(r – g) = 0.75(1.10)(1.26)3(1.06)/(0.145 – 0.06) = 1.749/0.085. Total value = PV of FCFE for Years 1–4 + PV of terminal value The FCFE grows at 10% for Year 1 and then at 26% for Years 2–4. These calculated values for FCFE are shown in the exhibit. The present values of the FCFE for the first four years discounted at the required rate of return are given in the last column of the table. After Year 4, FCFE will grow at 6% forever, so the constant-growth FCFE model is used to find the terminal value at Time 4, which is TV4 = FCFE5/(r – g). TV4 is discounted at the required return for four periods to find its present value, as shown in the table. Finally, the total value of the stock, $15.32, is the sum of the present values of the first four years’ FCFE per share plus the present value of the terminal value per share. 52. The total value of non-operating assets is SEK60 million short-term securities SEK45 million market value of noncurrent assets Value in Year t Present Value at 14.5% 0.825 0.721 1.040 0.793 1.310 0.873 1.650 0.960 20.580 11.974 15.32 98 Learning Module 1 Free Cash Flow Valuation SEK40 million pension fund surplus SEK145 million non-operating assets The total value of the firm is the value of the operating assets plus the value of the non-operating assets, or SEK720 million plus SEK145 million = SEK865 million. The equity value is the value of the firm minus the value of debt, or SEK865 million − SEK215 million = SEK650 million. The value per share is SEK650 million/100 million shares = SEK6.50 per share. Practice Problems 189 PRACTICE PROBLEMS The following information relates to questions 1-3 As of February 2020, you are researching Jonash International, a hypothetical company subject to cyclical demand for its services. Jonash shares closed at $57.98 on 2 February 2019. You believe the 2015–18 period reasonably captures average profitability: Measure 2019 2018 2017 2016 2015 EPS E$3.03 $1.45 $0.23 $2.13 $2.55 BV per share E$19.20 $16.21 $14.52 $13.17 $11.84 ROE E16.0% 8.9% 1.6% 16.3% 21.8% 1. Define normalized EPS. 2. Calculate a normalized EPS for Jonash based on the method of historical average EPS, and then calculate the P/E based on normalized EPS. 3. Calculate a normalized EPS for Jonash based on the method of average ROE and the P/E based on normalized EPS. The following information relates to questions 4-5 An analyst plans to use P/E and the method of comparables as a basis for recommending purchasing shares of one of two peer-group companies in the business of manufacturing personal digital assistants. Neither company has been profitable to date, and neither is expected to have positive EPS over the next year. Data on the companies’ prices, trailing EPS, and expected growth rates in sales (five-year compounded rates) are given in the following table: Company Price Trailing EPS P/E Expected Growth (Sales) Hand $22 –$2.20 NMF 45% Somersault $10 –$1.25 NMF 40% Unfortunately, because the earnings for both companies have been negative, their P/Es are not meaningful. On the basis of this information, address the following: 4. Discuss how the analyst might make a relative valuation in this case. 5. State which stock the analyst should recommend. 190 Learning Module 2 Market-Based Valuation: Price and Enterprise Value Multiples The following information relates to questions 6-7 May Stewart, CFA, a retail analyst, is performing a P/E-based comparison of two hypothetical jewelry stores as of early 2020. She has the following data for Hallwhite Stores (HS) and Ruffany (RUF). ■ HS is priced at $44. RUF is priced at $22.50. ■ HS has a simple capital structure, earned $2.00 per share (basic and diluted) in 2019, and is expected to earn $2.20 (basic and diluted) in 2020. ■ RUF has a complex capital structure as a result of its outstanding stock options. Moreover, it had several unusual items that reduced its basic EPS in 2019 to $0.50 (versus the $0.75 that it earned in 2018). ■ For 2020, Stewart expects RUF to achieve net income of $30 million. RUF has 30 million shares outstanding and options outstanding for an additional 33,333,333 shares. 6. Which P/E (trailing or forward) should Stewart use to compare the two companies’ valuation? 7. Which of the two stocks is relatively more attractive when valued on the basis of P/Es (assuming that all other factors are approximately the same for both stocks)? The following information relates to questions 8-9 You are researching the valuation of the stock of a company in the food-processing industry. Suppose you intend to use the mean value of the forward P/Es for the food-processing industry stocks as the benchmark value of the multiple. This mean P/E is 18.0. The forward or expected EPS for the next year for the stock you are studying is $2.00. You calculate 18.0 × $2.00 = $36, which you take to be the intrinsic value of the stock based only on the information given here. Comparing $36 with the stock’s current market price of $30, you conclude the stock is undervalued. 8. Give two reasons why your conclusion that the stock is undervalued may be in error. 9. What additional information about the stock and the peer group would support your original conclusion? The following information relates to questions 10-16 Mark Cannan is updating research reports on two well-established consumer companies before first quarter 2021 earnings reports are released. His supervisor, Sharolyn Ritter, has asked Cannan to use market-based valuations when updating the reports. Delite Beverage is a manufacturer and distributor of soft drinks and recently Practice Problems 191 acquired a major water bottling company in order to offer a broader product line. The acquisition will have a significant impact on Delite’s future results. You Fix It is a US retail distributor of products for home improvement, primarily for those consumers who choose to do the work themselves. The home improvement industry is cyclical; the industry was adversely affected by the recent downturn in the economy, the level of foreclosures, and slow home sales. Although sales and earnings at You Fix It weakened, same store sales are beginning to improve as consumers undertake more home improvement projects. Poor performing stores were closed, resulting in significant restructuring charges in 2020. Before approving Cannan’s work, Ritter wants to discuss the calculations and choices of ratios used in the valuation of Delite and You Fix It. The data used by Cannan in his analysis are summarized in Exhibit 1. Exhibit 1: Select Financial Data for Delite Beverage and You Fix It 2020 earnings per share (EPS) Delite Beverage You Fix It $3.44 $1.77 2021 estimated EPS $3.50 $1.99 Book value per share end of year $62.05 $11.64 Current share price $65.50 $37.23 Sales (billions) $32.13 $67.44 Free cash flow per share $2.68 $0.21 2,322,034,000 1,638,821,000 Shares outstanding end of year Cannan advises Ritter that he is considering three different approaches to value the shares of You Fix It: Approach 1 Price-to-book ratio (P/B) Approach 2 Price-to-earnings ratio (P/E) using trailing earnings Approach 3 Price-to-earnings ratio using normalized earnings Cannan tells Ritter that he calculated the price-to-sales ratio (P/S) for You Fix It but chose not to use it in the valuation of the shares. Cannan states to Ritter that it is more appropriate to use the P/E than the P/S because Reason 1 Earnings are more stable than sales. Reason 2 Earnings are less easily manipulated than sales. Reason 3 The P/E reflects financial leverage, whereas the P/S does not. Cannan also informs Ritter that he did not use a price-to-cash-flow multiple in valuing the shares of Delite or You Fix It. The reason is that he could not identify a cash flow measure that would both account for working capital and noncash revenues and be after interest expense and thus not be mismatched with share price. Ritter advises Cannan that such a cash flow measure does exist. Ritter provides Cannan with financial data on three close competitors as well as the overall beverage sector, which includes other competitors, in Exhibit 2. She asks Cannan to determine, based on the P/E-to-growth (PEG) ratio, whether Delite shares are overvalued, fairly valued, or undervalued. 192 Learning Module 2 Market-Based Valuation: Price and Enterprise Value Multiples Exhibit 2: Beverage Sector Data Delite Forward P/E Earnings Growth — 12.41% Fresh Iced Tea Company 16.59 9.52% Nonutter Soda 15.64 11.94% Tasty Root Beer 44.10 20% Beverage sector average 16.40 10.80% After providing Ritter his answer, Cannan is concerned about the inclusion of Tasty Root Beer in the comparables analysis. Specifically, Cannan says to Ritter: “I feel we should mitigate the effect of large outliers but not the impact of small outliers (i.e., those close to zero) when calculating the beverage sector P/E. What measure of central tendency would you suggest we use to address this concern?” Ritter requests that Cannan incorporate their discussion points before submitting the reports for final approval. 10. Based on the information in Exhibit 1, the mostappropriate price-to-earnings ratio to use in the valuation of Delite is closest to: A. 18.71. B. 19.04. C. 24.44. 11. Based on the information in Exhibit 1, the price-to-sales ratio for You Fix It is closest to: A. 0.28. B. 0.55. C. 0.90. 12. Which valuation approach would be most appropriate in valuing shares of You Fix It? A. Approach 1 B. Approach 2 C. Approach 3 13. Cannan’s preference to use the P/E over the P/S is best supported by: A. Reason 1. B. Reason 2. C. Reason 3. 14. The cash flow measure that Ritter would most likely recommend to address Cannan’s concern is: A. free cash flow to equity. B. earnings plus noncash charges. Practice Problems 193 C. earnings before interest, tax, depreciation, and amortization. 15. Based on the information in Exhibits 1 and 2, Cannan would most likely conclude that Delite’s shares are: A. overvalued. B. undervalued. C. fairly valued. 16. The measure of central tendency that Ritter will most likely recommend is the: A. median. B. harmonic mean. C. arithmetic mean. The following information relates to questions 17-22 Andrea Risso is a junior analyst with AquistareFianco, an independent equity research firm. Risso’s supervisor asks her to update, as of 1 January 2020, a quarterly research report for Centralino S.p.A., a telecommunications company headquartered in Italy. On that date, Centralino’s common share price is €50 and its preferred shares trade for €5.25 per share. Risso gathers information on Centralino. Exhibit 1 presents earnings and dividend data, and Exhibit 2 presents balance sheet data. Net sales were €3.182 billion in 2019. Risso estimates a required return of 15% for Centralino and forecasts growth in dividends of 6% into perpetuity. Exhibit 1: Earnings and Dividends for Centralino, 2016–2020 Earnings per share (EPS, €) Dividends per share (DPS, €) Return on equity (ROE) 2016 2017 2018 2019 2020(E) 4.93 5.25 4.46 5.64 6.00 2.45 2.60 2.60 2.75 2.91 13.01% 13.71% 11.58% 14.21% 14.96% Note: The data for 2016–2019 are actual and for 2020 are estimated. Exhibit 2: Summary Balance Sheet for Centralino, Year Ended 31 December 2019 Assets (€ millions) Liabilities and Shareholders’ Equity (€ millions) Cash and cash equivalents 102 Current liabilities 259 Accounts receivable 305 Long-term debt 367 Inventory 333 Total liabilities 626 Total current assets 740 Preferred shares 80 194 Learning Module 2 Market-Based Valuation: Price and Enterprise Value Multiples Assets (€ millions) Liabilities and Shareholders’ Equity (€ millions) Property and equipment, net 913 Total assets 1,653 Common shares 826 Retained earnings 121 Total shareholders’ equity 1,027 Total liabilities and shareholders’ equity 1,653 Notes: The market value of long-term debt is equal to its book value. Shares outstanding are 41.94 million common shares and 16.00 million preferred shares. Exhibit 3 presents forward price-to-earnings ratios (P/Es) for Centralino’s peer group. Risso assumes no differences in fundamentals among the peer-group companies. Exhibit 3: Peer Group Forward P/Es Company Forward P/E Brinaregalo 5.9 Camporio 8.3 Esperto 3.0 Fornodissione 15.0 Radoresto 4.6 Risso also wants to calculate normalized EPS using the average return on equity method. She determines that the 2016–19 time period in Exhibit 1 represents a full business cycle for Centralino. 17. Based on Exhibit 1, the trailing P/E for Centralino as of 1 January 2020, ignoring any business-cycle influence, is closest to: A. 8.3. B. 8.9. C. 9.9. 18. Based on Exhibit 1 and Risso’s estimates of return and dividend growth, Centralino’s justified forward P/E based on the Gordon growth dividend discount model is closest to: A. 5.4. B. 5.7. C. 8.3. 19. Based on Exhibit 2, the price-to-book multiple for Centralino is closest to: A. 2.0. B. 2.2. C. 2.5. 20. Based on Exhibit 2, the multiple of enterprise value to sales for Centralino as of Practice Problems 195 31 December 2019 is closest to: A. 0.67. B. 0.74. C. 0.77. 21. Based on Exhibit 1 and using the harmonic mean of the peer group forward P/Es shown in Exhibit 3 as a valuation indicator, the common shares of Centralino are: A. undervalued. B. fairly valued. C. overvalued. 22. Based on Exhibits 1 and 2, the normalized earnings per share for Centralino as calculated by Risso should be closest to: A. €2.94. B. €3.21. C. €5.07. The following information relates to questions 23-29 Cátia Pinho is a supervisor in the equity research division of Suite Securities. Pinho asks Flávia Silveira, a junior analyst, to complete an analysis of Adesivo S.A., Enviado S.A., and Gesticular S.A. Pinho directs Silveira to use a valuation metric that would allow for a meaningful ranking of relative value of the three companies’ shares. Exhibit 1 provides selected financial information for the three companies. Exhibit 1: Selected Financial Information for Adesivo, Enviado, and Gesticular (Brazilian Real, BRL) Adesivo Enviado Gesticular Stock’s current price 14.72 72.20 132.16 Diluted EPS (last four quarters) 0.81 2.92 –0.05 Diluted EPS (next four quarters) 0.91 3.10 2.85 Dividend rate (annualized most recent dividend) 0.44 1.24 0.00 Silveira reviews underlying trailing EPS for Adesivo. Adesivo has basic trailing EPS of BRL0.84. Silveira finds the following note in Adesivo’s financial statements: “On a per share basis, Adesivo incurred in the last four quarters i. from a lawsuit, a nonrecurring gain of BRL0.04; and ii. from factory integration, a nonrecurring cost of BRL0.03 and a recurring cost of BRL0.01 in increased depreciation.” 196 Learning Module 2 Market-Based Valuation: Price and Enterprise Value Multiples Silveira notes that Adesivo is forecasted to pay semiannual dividends of BRL0.24 next year. Silveira estimates five-year earnings growth rates for the three companies, which are presented in Exhibit 2. Exhibit 2: Earnings Growth Rate Estimates over Five Years Company Earnings Growth Rate Estimate (%) Adesivo 16.67 Enviado 21.91 Gesticular 32.33 Pinho asks Silveira about the possible use of the price-to-sales ratio (P/S) in assessing the relative value of the three companies. Silveira tells Pinho: Statement 1 The P/S is not affected by revenue recognition practices. Statement 2 The P/S is less subject to distortion from expense accounting than is the P/E. Pinho asks Silveira about using the Fed and Yardeni models to assess the value of the equity market. Silveira states: Statement 1 The Fed model concludes that the market is undervalued when the market’s current earnings yield is greater than the 10-year Treasury bond yield. Statement 2 The Yardeni model includes the consensus five-year earnings growth rate forecast for the market index. Silveira also analyzes the three companies using the enterprising value (EV)-to-EBITDA multiple. Silveira notes that the EBITDA for Gesticular for the most recent year is BRL560 million and gathers other selected information on Gesticular, which is presented in Exhibit 4. Exhibit 3: Selected Information on Gesticular at Year End (BRL Millions) Market Value of Debt 1,733 Market Value of Common Equity Market Value of Preferred Equity Cash Short-Term Investments 6,766 275 581 495 Pinho asks Silveira about the use of momentum indicators in assessing the shares of the three companies. Silveira states: Statement 1 Relative-strength indicators compare an equity’s performance during a period with the performance of some group of equities or its own past performance. Statement 2 In the calculation of standardized unexpected earnings (SUE), the magnitude of unexpected earnings is typically scaled by the standard deviation of analysts’ earnings forecasts. Practice Problems 23. Based on Pinho’s directive and the data from the last four quarters presented in Exhibit 1, the valuation metric that Silveira should use is the: A. price-to-earnings ratio (P/E). B. production-to-demand ratio (P/D). C. earnings-to-price ratio (E/P). 24. Based on Exhibit 1 and the note to Adesivo’s financial statements, the trailing P/E for Adesivo using underlying EPS is closest to: A. 17.7. B. 18.2. C. 18.4. 25. Based on Exhibits 1 and 2, which company’s shares are the most attractively priced based on the five-year forward P/E-to-growth (PEG) ratio? A. Adesivo B. Enviado C. Gesticular 26. Which of Silveira’s statements concerning the use of the P/S is correct? A. Statement 1 only B. Statement 2 only C. Both Statement 1 and Statement 2 27. Which of Silveira’s statements concerning the Fed and Yardeni models is correct? A. Statement 3 only B. Statement 4 only C. Both Statement 3 and Statement 4 28. Based on Exhibit 4, Gesticular’s EV/EBITDA multiple is closest to: A. 11.4. B. 13.7. C. 14.6. 29. Which of Silveira’s statements concerning momentum indicators is correct? A. Statement 5 only B. Statement 6 only C. Both Statement 5 and Statement 6 197 198 Learning Module 2 Market-Based Valuation: Price and Enterprise Value Multiples The following information relates to questions 30-31 Christie Johnson, CFA, has been assigned to analyze Sundanci. Johnson assumes that Sundanci’s earnings and dividends will grow at a constant rate of 13%. Exhibits 1 and 2 provide financial statements for the most recent two years (2020 and 2021) and other information for Sundanci. Exhibit 1: Sundanci Actual 2020 and 2021 Financial Statements for Fiscal Years Ending 31 May (in Millions except Per-Share Data) Income Statement 2020 2021 Revenue $474 $598 Depreciation 20 23 Other operating costs 368 460 Income before taxes 86 115 Taxes 26 35 Net income 60 80 Dividends 18 24 Earnings per share $0.714 $0.952 Dividends per share $0.214 $0.286 Common shares outstanding 84.0 84.0 Balance Sheet 2020 2021 Current assets $201 $326 Net property, plant, and equipment 474 489 Total assets 675 815 Current liabilities 57 141 Long-term debt 0 0 Total liabilities 57 141 Shareholders’ equity 618 674 Total liabilities and equity 675 815 34 38 Other Information Capital expenditures Practice Problems 199 Exhibit 2: Selected Financial Information Required rate of ROE 14% Growth rate of industry 13% Industry P/E 26 30. Based on information in Exhibits 1 and 2 and on Johnson’s assumptions for Sundanci, calculate justified trailing and forward P/Es for this company. 31. Identify, within the context of the constant dividend growth model, how each of the following fundamental factors would affect the P/E: i. The risk (beta) of Sundanci increases substantially. ii. The estimated growth rate of Sundanci’s earnings and dividends increases. iii. The equity risk premium increases. Note: A change in a fundamental factor is assumed to happen in isolation; interactive effects between factors are ignored. That is, every other item of the company is unchanged. 32. Suppose an analyst uses an equity index as a comparison asset in valuing a stock. In making a decision to recommend purchase of an individual stock, which price multiple(s) would cause concern about the impact of potential overvaluation of the equity index? The following information relates to questions 33-34 Tom Smithfield is valuing the stock of a food-processing business. He feels confident explicitly projecting earnings and dividends to three years (to t = 3). Other information and estimates are as follows: ■ Required rate of return = 0.09. ■ Average dividend payout rate for mature companies in the market = 0.45. ■ Industry average ROE = 0.10. ■ E3 = $3.00. ■ Industry average P/E = 12. On the basis of this information, answer the following questions: 33. Compute terminal value (V3) based on comparables. 34. Contrast your answer in Part A to an estimate of terminal value based on the Gordon growth model. 35. Discuss three types of stocks or investment situations for which an analyst could appropriately use P/B in valuation. 200 Learning Module 2 Market-Based Valuation: Price and Enterprise Value Multiples The following information relates to questions 36-37 Aratatech is a multinational distributor of semiconductor chips and related products to businesses. Its leading competitor around the world is Trymye Electronics. Aratatech has a current market price of $10.00, 20 million shares outstanding, annual sales of $1 billion, and a 5% profit margin. Trymye has a market price of $20.00, 30 million shares outstanding, annual sales of $1.6 billion, and a profit margin of 4.9%. Based on the information given, answer the following questions: 36. Which of the two companies has a more attractive valuation based on P/S? 37. Identify and explain one advantage of P/S over P/E as a valuation tool. The following information relates to questions 38-41 GN Growing AG (GG) is currently selling for €240, with TTM EPS and dividends per share of €1.5 and €0.9, respectively. The company’s trailing P/E is 16.0, P/B is 3.2. P/Sales based on forecast sales, is 1.5. ROE is 20%, and for the profit margin on sales is 10.0%. The Treasury bond rate is 4.9%, the equity risk premium is 5.5%, and GG’s beta is 1.2. 38. What is GG’s required rate of return, based on the capital asset pricing model (CAPM)? 39. Assume that the dividend and earnings growth rates are 8%. What trailing P/E and P/B multiples would be justified in light of the required rate of return in Part A and current values of the dividend payout ratio and ROE ? 40. Calculate the justified P/Sales ratio based on the forward-looking margin of 10% and current values of dividend payout. 41. Given that the assumptions and constant growth model are appropriate, state and justify whether GG, based on fundamentals, appears to be fairly valued, overvalued, or undervalued. 42. Define the major alternative cash flow concepts, and state one limitation of each. 43. Data for two hypothetical companies in the pharmaceutical industry, DriveMed and MAT Technology, are given in the following table. For both companies, expenditures on fixed capital and working capital during the previous year reflect anticipated average expenditures over the foreseeable horizon. Measure DriveMed MAT Technology Current price $46.00 $78.00 Trailing CF per share $3.60 $6.00 P/CF 12.8 13.0 Trailing FCFE per share $1.00 $5.00 P/FCFE 46.0 15.6 Consensus five-year growth forecast 15% 20% Practice Problems 201 Measure Beta DriveMed MAT Technology 1.25 1.25 On the basis of the information supplied, discuss the valuation of MAT Technology relative to DriveMed. Justify your conclusion. The following information relates to questions 44-46 Jorge Zaldys, CFA, is researching the relative valuation of two companies in the aerospace/defense industry, NCI Heavy Industries (NCI) and Relay Group International (RGI). He has gathered relevant information on the companies in the following table. EBITDA Comparisons (in € Millions except Per-Share and Share-Count Data) Company RGI Price per share NCI 150 100 5 million 2 million Market value of debt 50 100 Book value of debt 52 112 Cash and investments 5 2 Net income 49.5 12 Net income from continuing operations 49.5 8 Interest expense 3 5 Depreciation and amortization 8 4 Taxes 2 3 Shares outstanding Using the information in the table, answer the following questions: 44. Calculate P/EBITDA for NCI and RGI. 45. Calculate EV/EBITDA for NCI and RGI. 46. Which company should Zaldys recommend as relatively undervalued? Justify the selection. The following information relates to questions 47-48 Wilhelm Müller, CFA, has organized the selected data on four food companies that appear below (TTM stands for trailing 12-month): Measure Stock price Hoppelli Foods Telli Foods Drisket Co. Whiteline Foods €25.70 €11.77 €23.65 €24.61 202 Learning Module 2 Market-Based Valuation: Price and Enterprise Value Multiples Measure Hoppelli Foods Telli Foods Drisket Co. Whiteline Foods Shares outstanding (thousands) 138,923 220,662 108,170 103,803 Market cap (€ millions) 3,570 2,597 2,558 2,555 Enterprise value (€ millions) 3,779 4,056 3,846 4,258 Sales (€ millions) 4,124 10,751 17,388 6,354 Operating income (€ millions) 285 135 186 396 Operating profit margin 6.91% 1.26% 1.07% 6.23% Net income (€ millions) 182 88 122 252 TTM EPS €1.30 €0.40 €1.14 €2.43 Return on equity 19.20% 4.10% 6.40% 23.00% Net profit margin 4.41% 0.82% 0.70% 3.97% On the basis of the data given, answer the following questions: 47. Calculate the trailing P/E and EV/sales for each company. 48. Explain, on the basis of fundamentals, why these stocks have different EV/S multiples. 49. John Jones, CFA, is head of the research department at Peninsular Research. Peninsular has a client who has inquired about the valuation method best suited for comparing companies in an industry with the following characteristics: ■ Principal competitors within the industry are located in the United States, France, Japan, and Brazil. ■ The industry is currently operating at a cyclical low, with many companies reporting losses. Jones recommends that the client consider the following valuation ratios: 1. P/E 2. P/B 3. EV/S Determine which one of the three valuation ratios is most appropriate for comparing companies in this industry. Support your answer with one reason that makes that ratio superior to either of the other two ratios in this case. The following information relates to questions 50-51 Your value-oriented investment management firm recently hired a new analyst, Bob Westard, because of his expertise in the life sciences and biotechnology areas. At the firm’s weekly meeting, during which each analyst proposes a stock idea for inclusion in the firm’s approved list, Westard recommends Hitech Clothing International (HCI). He bases his recommendation on two considerations. First, HCI has pending patent applications but a P/E that he judges to be low in light of the potential earnings from the patented products. Second, HCI has had high relative strength versus the S&P 500 over the past month. Practice Problems 203 50. Explain the difference between Westard’s two approaches—that is, the use of price multiples and the relative-strength approach. 51. State which, if any, of the bases for Westard’s recommendation is consistent with the investment orientation of your firm. The following information relates to questions 52-53 Kirstin Kruse, a portfolio manager, has an important client who wants to alter the composition of her equity portfolio, which is currently a diversified portfolio of 60 global common stocks. Because of concerns about the economy and based on the thesis that the consumer staples sector will be less hurt than others in a recession, the client wants to add stocks trading in the United States (including ADRs) from the consumer staples sector. In addition, the client wants the stocks to meet the following criteria: ■ Stocks must be considered large cap (i.e., have a large market capitalization). ■ Stocks must have a dividend yield of at least 4.0%. ■ Stocks must have a forward P/E no greater than 15. The following table shows how many stocks satisfied each screen, which was run in June 2019. Screen Number Satisfying Consumer staples sector 424 Large cap 361 Dividend yield of at least 4.0% 887 P/E less than 15 5,409 All four screens 3 The stocks meeting all four screens were Altria Group, Inc.; British American Tobacco PLC (the company’s ADR); and Kraft Heinz Co. 52. Critique the construction of the screen. 53. Do these criteria identify appropriate additions to this client’s portfolio? 204 Learning Module 2 Market-Based Valuation: Price and Enterprise Value Multiples SOLUTIONS 1. Normalized EPS is the level of earnings per share that the company could currently achieve under mid-cyclical conditions. 2. Averaging EPS over the 2015–18 period, we find that ($2.55 + $2.13 + $0.23 + $1.45)/4 = $1.59. According to the method of historical average EPS, Jonash’s normalized EPS is $1.59. The P/E based on this estimate is $57.98/$1.59 = 36.5. 3. Averaging ROE over the 2015–18 period, we find that (0.218 + 0.163 + 0.016 + 0.089)/4 = 0.1215. For current BV per share, you would use the estimated value of $19.20 for year end 2019. According to the method of average ROE, 0.1215 × $19.20 = $2.33 is the normalized EPS. The P/E based on this estimate is $57.98/$2.33 = 24.9. 4. The analyst can rank the two stocks by earnings yield (E/P). Whether EPS is positive or negative, a lower E/P reflects a richer (higher) valuation and a ranking from high to low E/P has a meaningful interpretation. In some cases, an analyst might handle negative EPS by using normalized EPS in its place. Neither business, however, has a history of profitability. When year-ahead EPS is expected to be positive, forward P/E is positive. Thus, the use of forward P/Es sometimes addresses the problem of trailing negative EPS. Forward P/E is not meaningful in this case, however, because next year’s earnings are expected to be negative. 5. Hand has an E/P of –0.100, and Somersault has an E/P of –0.125. A higher earnings yield has an interpretation that is similar to that of a lower P/E, so Hand appears to be relatively undervalued. The difference in earnings yield cannot be explained by differences in sales growth forecasts. In fact, Hand has a higher expected sales growth rate than Somersault. Therefore, the analyst should recommend Hand. 6. Because investing looks to the future, analysts often favor forward P/E when earnings forecasts are available, as they are here. A specific reason to use forward P/Es is the fact given that RUF had some unusual items affecting EPS for 2008. The data to make appropriate adjustments to RUF’s 2008 EPS are not given. In summary, Stewart should use forward P/Es. 7. Because RUF has a complex capital structure, the P/Es of the two companies must be compared on the basis of diluted EPS. For HS, forward P/E = $44/2.20 = 20. For RUF, forward P/E per diluted share = $22.50/($30,000,000/33,333,333) = $22.50/$0.90 = 25. Therefore, HS has the more attractive valuation at present. The problem illustrates some of the considerations that should be taken into account in using P/Es and the method of comparables. 8. Your conclusion may be in error because of the following: ■ The peer-group stocks themselves may be overvalued; that is, the mean P/E of 18.0 may be too high in terms of intrinsic value. If so, using 18.0 as a multiplier of the stock’s expected EPS will lead to an estimate of stock value in excess of intrinsic value. Solutions ■ The stock’s fundamentals may differ from those of the mean food-processing industry stock. For example, if the stock’s expected growth rate is lower than the mean industry growth rate and its risk is higher than the mean, the stock may deserve a lower P/E than the industry mean. In addition, mean P/E may be influenced by outliers. 9. The following additional evidence would support the original conclusion: ■ Evidence that stocks in the industry are, at least on average, fairly valued (that stock prices reflect fundamentals) ■ Evidence that no significant differences exist in the fundamental drivers of P/E for the stock being compared and the average industry stock 10. A is correct. The forward P/E should be used given the recent significant acquisition of the water bottling company. Since a major change such as an acquisition or divestiture can affect results, the forward P/E, also known as the leading P/E or prospective P/E, is the most appropriate P/E to use for Delite. Earnings estimates for 2021 should incorporate the performance of the water bottling company. The forward P/E is calculated as the current price divided by the projected earnings per share, or $65.50/$3.50 = 18.71. 11. C is correct. The price-to-sales ratio is calculated as price per share divided by annual net sales per share. Price per share = $37.23. Annual net sales per share = $67.44 billion/1.638821 billion shares = $41.15. Price-to-sales ratio (P/S) = $37.23/$41.15 = 0.90. 12. C is correct. You Fix It is in the cyclical home improvement industry. The use of normalized earnings should address the problem of cyclicality in You Fix It earnings by estimating the level of earnings per share that the company could achieve currently under mid-cyclical conditions. 13. C is correct. The price to sales ratio (P/S) fails to consider differences in cost structures. Also, while share price reflects the effect of debt financing on profitability and risk, sales is a pre-financing income measure and does not incorporate the impact of debt in the firm’s capital structure. Earnings reflect operating and financial leverage, and thus the price-to-earnings ratio (P/E) incorporates the impact of debt in the firm’s capital structure. 14. A is correct. Free cash flow to equity (FCFE) is defined as cash flow available to shareholders after deducting all operating expenses, interest and debt payments, and investments in working and fixed capital. Cannan’s requirement that the cash flows include interest expense, working capital, and noncash revenue is satisfied by FCFE. 15. C is correct. The P/E-to-growth (PEG) ratio is calculated by dividing a stock’s P/E by the expected earnings growth rate, expressed as a percentage. To calculate Delite’s PEG ratio, first calculate the P/E: $65.50/$3.50 = 18.71. In this case, the forward earnings should be used given the recent acquisition of the water bottling company. Next, calculate Delite’s PEG ratio: 18.71/12.41 = 1.51. Comparing Delite’s PEG ratio of 1.51 with the PEG ratios of 1.74 (16.59/9.52) for Fresh Iced Tea and 1.31 (15.64/11.94) for Nonutter Soda and with the beverage sector average of 1.52 (16.40/10.80), it appears that Delite’s shares are fairly valued. This is determined by the fact that Delite’s PEG ratio is in the middle of 205 206 Learning Module 2 Market-Based Valuation: Price and Enterprise Value Multiples the range of PEG ratios and very close to the sector average. Therefore, the shares appear to be fairly valued. 16. B is correct. The harmonic mean is sometimes used to reduce the impact of large outliers—which are typically the major concern in using the arithmetic mean multiple—but not the impact of small outliers (i.e., those close to zero). The harmonic mean may aggravate the impact of small outliers, but such outliers are bounded by zero on the downside. 17. B is correct. The trailing P/E is calculated as follows: Stock’s current price/Most recent four quarters’ EPS = €50/€5.64 = 8.9. 18. A is correct. The justified forward P/E is calculated as follows: ​P0​ ​ ​D​ ​/ ​E​ ​ _ ​​E​ ​​ = _ ​r 1− g1 ​1 (​ ​2.91 / 6.00​)​ =_ ​(​ ​0.15 − 0.06​)​​ = 5.4. 19. B is correct. Price to book is calculated as the current market price per share divided by book value per share. Book value per share is common shareholders’ equity divided by the number of common shares outstanding. Common shareholders’ equity is calculated as total shareholders’ equity minus the value of preferred stock. Thus, Common shareholders’ equity = €1,027 – €80 = €947 million. Book value per share = €947 million/41.94 million = €22.58. Price-to-book ratio (P/B) for Centralino = €50/€22.58 = 2.2. 20. C is correct. Enterprise value (EV) is calculated as follows: EV = Market value of common equity + Market value of preferred stock + Market value of debt – Cash, cash equivalents, and short-term investments = (€50 × 41.94 million) + (€5.25 × 16.00 million) + €367 million – €102 million = €2,446 million (or €2.446 billion). So, EV/sales = €2.446 billion/€3.182 billion = 0.77. 21. C is correct. The harmonic mean is calculated as follows: n 5 ​​x​H​ = _ ​ n _1 ​ = ________________________________    ​     1 1 1 1 1 ​ = 5.5. _ _ _ _ _ ​∑ ( ​ ​​x​​​)​ i=1 i ​​(​5.9 ​)​​+ ​( ​ ​8.3 ​)​​+ ​( ​ ​3.0 ​)​​+ ​( ​ ​15.0 ​)​​+ ​( ​ ​4.6 ) ​ ​​ The forward P/E for Centralino is €50/€6.00 = 8.3. Because Centralino’s forward P/E is higher than the harmonic mean of the peer group, the shares of Centralino appear relatively overvalued. 22. A is correct. Based on the method of average ROE, normalized EPS are calculated as the average ROE from the most recent full business cycle multiplied by current book value per share. The most recent business cycle was 2011–2014, and the average ROE over that period was Solutions 0.1301 + 0.1371 + 0.1158 + 0.1421 _________________________     ​ 4  ​ = 0.131. The book value of (common) equity, or simply book value, is the value of shareholders’ equity less any value attributable to the preferred stock: €1,027 million – €84 million = €943 million. Current book value per share (BVPS) is calculated as €943 million/41.94 million = €22.48. So, normalized EPS is calculated as Average ROE × BVPS = 0.131 × €22.48 = €2.94. 23. C is correct. The E/P based on trailing earnings would offer the most meaningful ranking of the shares. Using E/P places Gesticular’s negative EPS in the numerator rather than the denominator, leading to a more meaningful ranking. 24. C is correct. The EPS figure that Silveira should use is diluted trailing EPS of BRL0.81, adjusted as follows: 1. Subtract the BRL0.04 nonrecurring legal gain. 2. Add BRL0.03 for the nonrecurring factory integration charge. No adjustment needs to be made for the BRL0.01 charge related to depreciation because it is a recurring charge. Therefore, underlying trailing EPS = BRL0.81 – BRL0.04 + BRL0.03 = BRL0.80 and trailing P/E using underlying trailing EPS = BRL14.72/BRL0.80 = 18.4. 25. A is correct. The forward PEG ratios for the three companies are calculated as follows: Forward P/E = Stock’s current price/Forecasted EPS. Forward PEG ratio = Forward P/E ÷ Expected earnings growth rate (in percentage terms). Adesivo forward P/E = BRL14.72/BRL0.91 = 16.18. Adesivo forward PEG ratio = 16.18/16.67 = 0.97. Enviado forward P/E = BRL72.20/BRL3.10 = 23.29. Enviado forward PEG ratio = 23.29/21.91 = 1.06. Gesticular forward P/E = BRL132.16/BRL2.85 = 46.37. Gesticular forward PEG ratio = 46.37/32.33 = 1.43. Adesivo has the lowest forward PEG ratio, 0.97, indicating that it is the most undervalued of the three equities based on the forward PEG ratio. 26. B is correct. Statement 2 is correct because sales, as the top line of the income statement, are less subject to accounting distortion or manipulation than are other fundamentals, such as earnings. Statement 1 is incorrect because sales figures can be distorted by revenue recognition practices, in particular those tending to speed up the recognition of revenues. 27. C is correct. The Fed model considers the equity market to be undervalued when the market’s current earnings yield is greater than the 10-year Treasury bond yield. The Yardeni model incorporates the consensus five-year earnings growth rate forecast for the market index, a variable missing in the Fed model. 207 208 Learning Module 2 Market-Based Valuation: Price and Enterprise Value Multiples 28. B is correct. The EV for Gesticular is calculated as follows: EV = Market value of debt + Market value of common equity + Market value of preferred equity – Cash and short-term investments. EV = BRL1,733 million + BRL6,766 million + BRL275 million – BRL581 million – BRL495 million = BRL7,698 million. EV/EBITDA = BRL7,698 million/BRL560 million = 13.7. 29. A is correct. Relative-strength indicators compare an equity’s performance with the performance of a group of equities or with its own past performance. SUE is unexpected earnings scaled by the standard deviation in past unexpected earnings (not the standard deviation of analysts’ earnings forecasts, which is used in the calculation of the scaled earnings surprise). 30. The formula for calculating the justified forward P/E for a stable-growth company is the payout ratio divided by the difference between the required rate of return and the growth rate of dividends. If the P/E is being calculated on trailing earnings (Year 0), the payout ratio is increased by 1 plus the growth rate. According to the 2020 income statement, the payout ratio is 18/60 = 0.30; the 2021 income statement gives the same number (24/80 = 0.30). Thus, we can find the following: P/E based on trailing earnings: P/E = ​[​ ​Payout ratio × ​​(​1 + g​)​]​ ​​/ ​​(​r − g​) = ​(​ ​0.30 × 1.13​)​​/ ​​(​0.14 − 0.13​)​​ = 33.9. P/E based on next year’s earnings: P/E = Payout ratio / ​​(​r − g​) = 0.30 / ​​(​0.14 − 0.13​)​​ = 30. 31. Fundamental Factor Effect on P/E Explanation (Not Required in Question) The risk (beta) of Sundanci increases substantially. Decrease P/E is a decreasing function of risk; that is, as risk increases, P/E decreases. Increases in the risk of Sundanci stock would be expected to lower its P/E. The estimated growth rate of Sundanci’s earnings and dividends increases. Increase P/E is an increasing function of the growth rate of the company; that is, the higher the expected growth, the higher the P/E. Sundanci would command a higher P/E if the market price were to incorporate expectations of a higher growth rate. The equity risk premium increases. Decrease P/E is a decreasing function of the equity risk premium. An increased equity risk premium increases the required rate of return, which lowers the price of a stock relative to its earnings. A higher equity risk premium would be expected to lower Sundanci’s P/E. 32. In principle, the use of any price multiple for valuation is subject to the concern stated. If the stock market is overvalued, an asset that appears to be fairly or even Solutions 209 undervalued in relation to an equity index may also be overvalued. 33. Vn = Benchmark value of P/E × En = 12 × $3.00 = $36.0. 34. In the expression for the sustainable growth rate, g = b × ROE, you can use (1 − 0.45) = 0.55 = b and ROE = 0.10 (the industry average), obtaining 0.55 × 0.10 = 0.055. Given the required rate of return of 0.09, you obtain the estimate $3.00(0.45)(1.055)/(0.09 − 0.055) = $40.69. In this case, the estimate of terminal value obtained from the Gordon growth model is higher than the estimate based on multiples. The two estimates may differ for a number of reasons, including the sensitivity of the Gordon growth model to the values of the inputs. 35. Although the measurement of book value has a number of widely recognized shortcomings, P/B may still be applied fruitfully in several circumstances: ■ The company is not expected to continue as a going concern. When a company is likely to be liquidated (so ongoing earnings and cash flow are not relevant), the value of its assets less its liabilities is of utmost importance. Naturally, the analyst must establish the fair value of these assets. ■ The company is composed mainly of liquid assets, which is the case for finance, investment, insurance, and banking institutions. ■ The company’s EPS is highly variable or negative. 36. Aratatech: P/S = ($10 price per share)/[($1 billion sales)/(20 million shares)] = $10/($1,000,000,000/20,000,000) = 0.2. Trymye: P/S = ($20 price per share)/[($1.6 billion sales)/(30 million shares)] = $20/($1,600,000,000/30,000,000) = 0.375. Aratatech has a more attractive valuation than Trymye based on its lower P/S but a comparable profit margin. 37. One advantage of P/S over P/E is that companies’ accounting decisions typically have a much greater impact on reported earnings than they are likely to have on reported sales. Although companies are able to make a number of legitimate business and accounting decisions that affect earnings, their discretion over reported sales (revenue recognition) is limited. Another advantage is that sales are almost always positive, so using P/S eliminates issues that arise when EPS is zero or negative. 38. Based on the CAPM, the required rate of return is 4.9% + 1.2 × 5.5% = 11.5%. 39. The dividend payout ratio is €0.9/€1.50 = 0.6. The justified values for the trailing P/E and P/BV ratios should be ​P0​ ​ ​​(​1 − b​)​​× ​(​ ​1 + g​)​​ 0.6 × ​(​ ​1 + 0.08​)​​ _ ____________    ​ ​ =    ​ 0.115    ​​​E​ ​​ = _____________ r−g − 0.08 ​ = 18.5 0 ​P0​ ​ ROE − g 0.20 − 0.08 _ ​ r−g ​= _ ​0.115 − 0.08 ​ = 3.4 ​​​E​ ​​ = _ 0 40. The justified P/S ratio based on assumed profit margin of 10% should be ​​(_ ​S​ ​1​) ​ ​​​(​1 − b​)​​ ​P0​ ​ _ 0.10 × 0.6 1 _ ​​​S​ ​​ = ​ r − g ​= _ ​0.115 − 0.08 ​ = 1.7 1 ​E​ ​ 41. The justified trailing P/E is higher than the trailing P/E (18.5 versus 16), the justified trailing P/B is higher than the actual trailing P/B (3.4 versus 3.2). The justified P/S based on forward looking margin assumptions is higher than the actual P/S 210 Learning Module 2 Market-Based Valuation: Price and Enterprise Value Multiples based of forecast sales (1.7 versus 1.5). Therefore, based on these three measures, GG appears to be slightly undervalued. 42. The major concepts are as follows: ■ EPS plus per-share depreciation, amortization, and depletion (CF) Limitation: Ignores changes in working capital and noncash revenue; not a free cash flow concept. ■ Cash flow from operations (CFO) Limitation: Not a free cash flow concept, so not directly linked to theory. ■ Free cash flow to equity (FCFE) Limitation: Often more variable and more frequently negative than other cash flow concepts. ■ Earnings before interest, taxes, depreciation, and amortization (EBITDA) Limitation: Ignores changes in working capital and noncash revenue; not a free cash flow concept. Relative to its use in P/EBITDA, EBITDA is mismatched with the numerator because it is a pre-interest concept. 43. MAT Technology is relatively undervalued compared with DriveMed on the basis of P/FCFE. MAT Technology’s P/FCFE multiple is 34% the size of DriveMed’s FCFE multiple (15.6/46 = 0.34, or 34%). The only comparison slightly in DriveMed’s favor, or approximately equal for both companies, is the comparison based on P/CF (i.e., 12.8 for DriveMed versus 13.0 for MAT Technology). However, FCFE is more strongly grounded in valuation theory than P/CF. Because DriveMed’s and MAT Technology’s expenditures for fixed capital and working capital during the previous year reflected anticipated average expenditures over the foreseeable horizon, you would have additional confidence in the P/FCFE comparison. 44. EBITDA = Net income (from continuing operations) + Interest expense + Taxes + Depreciation + Amortization. EBITDA for RGI = €49.5 million + €3 million + €2 million + €8 million = €62.5 million. Per-share EBITDA = (€62.5 million)/(5 million shares) = €12.5. P/EBITDA for RGI = €150/€12.5 = 12. EBITDA for NCI = €8 million + €5 million + €3 million + €4 million = €20 million. Per-share EBITDA = (€20 million)/(2 million shares) = €10. P/EBITDA for NCI = €100/€10 = 10. 45. For RGI: Market value of equity = €150 × 5 million = €750 million. Market value of debt = €50 million. Total market value = €750 million + €50 million = €800 million. EV = €800 million − €5 million (cash and investments) = €795 million. Now, Zaldys would divide EV by total (as opposed to per-share) EBITDA: EV/EBITDA for RGI = (€795 million)/(€62.5 million) = 12.72. For NCI: Solutions 211 Market value of equity = €100 × 2 million = €200 million. Market value of debt = €100 million. Total market value = €200 million + €100 million = €300 million. EV = €300 million − €2 million (cash and investments) = €298 million. Now, Zaldys would divide EV by total (as opposed to per-share) EBITDA: EV/EBITDA for NCI = (€298 million)/(€20 million) = 14.9. 46. Zaldys should select RGI as relatively undervalued. First, it is correct that NCI appears to be relatively undervalued based on P/ EBITDA, because NCI has a lower P/EBITDA multiple: ■ P/EBITDA = €150/€12.5 = 12 for RGI. ■ P/EBITDA = €100/€10 = 10 for NCI. RGI is relatively undervalued on the basis of EV/EBITDA; however, because RGI has the lower EV/EBITDA multiple, ■ EV/EBITDA = (€795 million)/(€62.5 million) = 12.72 for RGI. ■ EV/EBITDA = (€298 million)/(€20 million) = 14.9 for NCI. EBITDA is a pre-interest flow; therefore, it is a flow to both debt and equity and the EV/EBITDA multiple is more appropriate than the P/EBITDA multiple. Zaldys would rely on EV/EBITDA to reach his decision if the two ratios conflicted. Note that P/EBITDA does not take into account differences in the use of financial leverage. Substantial differences in leverage exist in this case (NCI uses much more debt), so the preference for using EV/EBITDA rather than P/EBITDA is supported. 47. The P/Es are as follows: Hoppelli 25.70/1.30 = 19.8. Telli 11.77/0.40 = 29.4. Drisket 23.65/1.14 = 20.7. Whiteline 24.61/2.43 = 10.1. The EV/S multiples for each company are as follows: Hoppelli 3,779/4,124 = 0.916. Telli 4,056/10,751 = 0.377. Drisket 3,846/17,388 = 0.221. Whiteline 4,258/6,354 = 0.670. 48. The data for the problem include measures of profitability, such as operating profit margin, ROE, and net profit margin. Because EV includes the market values of both debt and equity, logically the ranking based on EV/S should be compared with a pre-interest measure of profitability—namely, operating profit margin. The ranking of the stocks by EV/S from highest to lowest and the companies’ operating margins are shown below: Company EV/S Operating Profit Margin (%) Hoppelli 0.916 6.91 212 Learning Module 2 Market-Based Valuation: Price and Enterprise Value Multiples Company EV/S Operating Profit Margin (%) Whiteline 0.670 6.23 Telli 0.377 1.26 Drisket 0.221 1.07 The differences in EV/S appear to be explained, at least in part, by differences in cost structure as measured by operating profit margin. 49. For companies in the industry described, EV/S would be superior to either of the other two ratios. Among other considerations, EV/S is: ■ more useful than P/E in valuing companies with negative earnings; ■ better than either P/E or P/B for comparing companies in different countries that are likely to use different accounting standards (a consequence of the multinational nature of the industry); ■ less subject to manipulation than earnings (i.e., through aggressive accounting decisions by management, who may be more motivated to manage earnings when a company is in a cyclical low, rather than in a high, and thus likely to report losses). 50. Relative strength is based strictly on price movement (a technical indicator). As used by Westard, the comparison is between the returns on HCI and the returns on the S&P 500. In contrast, the price multiple approaches are based on the relationship of current price not to past prices but to some measure of value, such as EPS, book value, sales, or cash flow. 51. Only the reference to the P/E in relationship to the pending patent applications in Westard’s recommendation is consistent with the company’s value orientation. High relative strength would be relevant for a portfolio managed with a growth/ momentum investment style. 52. As a rule, a screen that includes a maximum P/E should include criteria requiring positive earnings; otherwise, the screen could select companies with negative P/ Es. The screen may be too narrowly focused on value measures. It did not include criteria related to expected growth, required rate of return, risk, or financial strength. 53. The screen results in a very concentrated portfolio. The screen selected only three companies, including two tobacco companies, which typically pay high dividends. Owning these three stocks would provide little diversification. Practice Problems PRACTICE PROBLEMS 1. Based on the following information, determine whether Vertically Integrated Manufacturing (VIM) earned any residual income for its shareholders: ■ VIM had total assets of $3,000,000, financed with twice as much debt capital as equity capital. ■ VIM’s pretax cost of debt is 6% and cost of equity capital is 10%. ■ VIM had EBIT of $300,000 and was taxed at a rate of 40%. Calculate residual income by using the method based on deducting an equity charge. 2. Because New Market Products (NMP) markets consumer staples, it is able to make use of considerable debt in its capital structure; specifically, 90% of the company’s total assets of $450,000,000 are financed with debt capital. Its cost of debt is 8% before taxes, and its cost of equity capital is 12%. NMP achieved a pretax income of $5.1 million in 2006 and had a tax rate of 40%. What was NMP’s residual income? 3. In 2020, Smithson–Williams Industries (SWI) achieved an operating profit after taxes of €10 million on total assets of €100 million. Half of its assets were financed with debt with a pretax cost of 9%. Its cost of equity capital is 12%, and its tax rate is 40%. Did SWI achieve a positive residual income? The following information relates to questions 4-6 Calculate the economic value added or residual income, as requested, for each of the following: 4. NOPAT = $100 Beginning book value of debt = $200 Beginning book value of equity = $300 Weighted average cost of capital (WACC) = 11% Calculate EVA. 5. Net income = €5.00 Dividends = €1.00 Beginning book value of equity = €30.00 Required rate of return on equity = 11% Calculate residual income. 6. Return on equity = 18% Required rate of return on equity = 12% Beginning book value of equity = €30.00 Calculate residual income. 257 258 Learning Module 3 Residual Income Valuation The following information relates to questions 7-8 Jim Martin is using economic value added and market value added to measure the performance of Sundanci. Martin uses the fiscal year 2020 information below for his analysis. ■ Adjusted net operating profit after taxes is $100 million. ■ Total capital is $700 million (no debt). ■ Closing stock price is $26. ■ Total shares outstanding is 84 million. ■ The cost of equity is 14%. Calculate the following for Sundanci. Show your work. 7. EVA for fiscal year 2020. 8. MVA as of fiscal year-end 2020. The following information relates to questions 9-16 Mangoba Nkomo, CFA, a senior equity analyst with Robertson-Butler Investments, South Africa, has been assigned a recent graduate, Manga Mahlangu, to assist in valuations. Mahlangu is interested in pursuing a career in equity analysis. In their first meeting, Nkomo and Mahlangu discuss the concept of residual income and its commercial applications. Nkomo asks Mahlangu to determine the market value added for a hypothetical South African firm using the data provided in Exhibit 1. Exhibit 1: Hypothetical Firm Data (amounts in South African rand) Current share price R25.43 Book value per share R20.00 Total shares outstanding 30 million Cost of equity 13% Market value of debt R55 million Accounting book value of total capital R650 million Intrinsic share value of equity derived from residual income model R22.00 Nkomo also shares his valuation report of the hypothetical firm with Mahlangu. Nkomo’s report concludes that the intrinsic value of the hypothetical firm, based on the residual income model, is R22.00 per share. To assess Mahlangu’s knowledge of residual income valuation, Nkomo asks Mahlangu two questions about the hypothetical firm: Question 1 What conclusion can we make about future residual earnings given the current book value per share and my estimate of intrinsic value per share? Practice Problems Question 2 259 Suppose you estimated the intrinsic value of a firm’s shares using a constant growth residual income model, and you found that your estimate of intrinsic value equaled the book value per share. What would that finding imply about that firm’s return on equity? Satisfied with Mahlangu’s response, Nkomo requests that Mahlangu use the single-stage residual income model to determine the intrinsic value of the equity of Jackson Breweries, a brewery and bottling company, using data provided in Exhibit 2. Exhibit 2: Jackson Breweries Data (amounts in South African rand) Constant long-term growth rate 9.5% Constant long-term ROE 13% Current market price per share R150.70 Book value per share R55.81 Cost of equity 11% Nkomo also wants to update an earlier valuation of Amersheen, a food retailer. The valuation report, completed at the end of 2020, concluded an intrinsic value per share of R11.00 for Amersheen. The share price at that time was R8.25. Nkomo points out to Mahlangu that in late 2020, Amersheen announced a significant restructuring charge, estimated at R2 million, that would be reported as part of operating earnings in Amersheen’s 2020 annual income statement. Nkomo asks Mahlangu the following question about the restructuring charge: Question 1 What was the correct way to treat the estimated R2 million restructuring charge in my 2020 valuation report? Satisfied with Mahlangu’s response, Nkomo mentions to Mahlangu that Amersheen recently (near the end of 2021) completed the acquisition of a chain of convenience stores. Nkomo requests that Mahlangu complete, as of the beginning of 2022, an updated valuation of Amersheen under two scenarios: Scenario 1 Estimate the value of Amersheen shares using a multistage residual income model with the data provided in Exhibit 3. Under Scenario 1, expected ROE in 2025 is 26%, but it is assumed that the firm’s ROE will slowly decline towards the cost of equity thereafter. Scenario 2 Estimate the value of Amersheen shares using a multistage residual income model with the data provided in Exhibit 3, but assume that at the end of 2024, share price is expected to equal book value per share. Scenario 3 Exhibit 3: Amersheen Data (amounts in South African rand) Long-term growth rate starting in 2025 9.0% Expected ROE in 2025 26% 260 Learning Module 3 Residual Income Valuation Current market price per share R16.55 Book value per share, beginning of 2022 R7.60 Cost of equity 10% Persistence factor 0.70 2022 2023 2024 Expected earnings per share R3.28 R3.15 R2.90 Expected dividend per share R2.46 R2.36 R2.06 9. Based on the information in Exhibit 1, the market value added of the hypothetical firm is closest to: A. R65 million. B. R113 million. C. R168 million. 10. The most appropriate response to Nkomo’s Question 1 would be that the present value of future residual earnings is expected to be: A. zero. B. positive. C. negative. 11. The most appropriate response to Nkomo’s Question 2 would be that the firm’s return on equity is: A. equal to the firm’s cost of equity. B. lower than the firm’s cost of equity. C. higher than the firm’s cost of equity. 12. Based on the information in Exhibit 2, the intrinsic value per share of the equity of Jackson Breweries is closest to: A. R97.67. B. R130.22. C. R186.03. 13. If Nkomo’s 2020 year-end estimate of Amersheen shares’ intrinsic value was accurate, then Amersheen’s shares were most likely: A. overvalued. B. undervalued. C. fairly valued. 14. The most appropriate treatment of the estimated restructuring charge, in re- Practice Problems 261 sponse to Nkomo’s Question 3, would be: A. an upward adjustment to book value. B. an upward adjustment to the cost of equity. C. to exclude it from the estimate of net income. 15. Under Scenario 1, the intrinsic value per share of the equity of Amersheen is closest to: A. R13.29. B. R15.57. C. R16.31. 16. Under Scenario 2, the intrinsic value per share of the equity of Amersheen is closest to: A. R13.29. B. R15.57. C. R16.31. The following information relates to questions 17-26 Elena Castovan is a junior analyst with Contralith Capital, a long-only equity investment manager. She has been asked to value three stocks on Contralith’s watch list: Portous, Inc. (PTU), SSX Financial (SSX), and Tantechi Ltd. (TTCI). During their weekly meeting, Castovan and her supervisor, Ariana Beckworth, discuss characteristics of residual income models. Castovan tells Beckworth the following. Statement 1 The present value of the terminal value in RI models is often a larger portion of the total intrinsic value than it is in other DCF valuation models. Statement 2 The RI model’s use of accounting income assumes that the cost of debt capital is appropriately reflected by interest expense. Statement 3 RI models cannot be readily applied to companies that do not have positive expected near-term free cash flows. Beckworth asks Castovan why an RI model may be more appropriate for valuing PTU than the dividend discount model or a free cash flow model. Castovan tells Beckworth that, over her five-year forecast horizon, she expects PTU to perform the following actions. Reason 1 Pay dividends that are unpredictable Reason 2 Generatepositiveand fairly predictable free cash flows Reason 3 Report significant amounts of other comprehensive income 262 Learning Module 3 Residual Income Valuation At the conclusion of their meeting, Beckworth asks Castovan to value SSX using RI models. Selected financial information on SSX is presented in Exhibit 1. Exhibit 1: SSX Financial (SSX) Selected Financial Data Total assets (millions) €4,000.00 Capital structure 60% debt/40% equity EBIT (millions) €700.00 Tax rate 35.00% Return on equity (ROE) Pretax cost of 23.37% debta 5.20% Cost of equity 15.00% Market price per share €48.80 Price-to-book ratio a 2.10 Interest expense is tax-deductible. Castovan’s final assignment is to determine the intrinsic value of TTCI using both a single-stage and a multistage RI model. Selected data and assumptions for TTCI are presented in Exhibit 2. Exhibit 2: Tantechi Ltd. (TTCI) Selected Financial Data and Assumptions Book value per share €45.25 Market price per share €126.05 Constant long-term ROE 12.00% Constant long-term earnings growth rate 4.50% Cost of equity 8.70% For the multistage model, Castovan forecasts TTCI’s ROE to be higher than its long-term ROE for the first three years. Forecasted earnings per share and dividends per share for TTCI are presented in Exhibit 3. Starting in Year 4, Castovan forecasts TTCI’s ROE to revert to the constant long-term ROE of 12% annually. The terminal value is based on an assumption that residual income per share will be constant from Year 3 into perpetuity. Exhibit 3: Tantechi Ltd. (TTCI) Forecasts of Earnings and Dividends Year 1 Year 2 Year 3 Earnings per share (€) 7.82 8.17 8.54 Dividends per share (€) 1.46 1.53 1.59 Beckworth questions Castovan’s assumption regarding the implied persistence factor used in the multistage RI valuation. She tells Castovan that she believes that a persistence factor of 0.10 is appropriate for TTCI. Practice Problems 17. Which of Castovan’s statements regarding residual income models is correct? A. Statement 1 B. Statement 2 C. Statement 3 18. Which of Castovan’s reasons best justifies the use of a residual income model to value PTU? A. Reason 1 B. Reason 2 C. Reason 3 19. The forecasted item described in Reason 3 will most likely affect: A. earnings per share. B. dividends per share. C. book value per share. 20. Based on Exhibit 1, residual income for SSX is closest to: A. €40.9 million. B. €90.2 million. C. €133.9 million. 21. Based on Exhibit 1 and the single-stage residual income model, the implied growth rate of earnings for SSX is closest to: A. 5.8%. B. 7.4%. C. 11.0%. 22. Based on the single-stage RI model and Exhibit 2, Castovan should conclude that TTCI is: A. undervalued. B. fairly valued. C. overvalued. 23. Based on Exhibit 2, the justified price-to-book ratio for TTCI is closest to: A. 1.79. B. 2.27. C. 2.79. 24. Based on Exhibits 2 and 3 and the multistage RI model, Castovan should estimate 263 264 Learning Module 3 Residual Income Valuation the intrinsic value of TTCI to be closest to: A. €54.88. B. €83.01. C. €85.71. 25. The persistence factor suggested by Beckworth will lead to a multistage value estimate of TTCI’s shares that is: A. less than Castovan’s multistage value estimate. B. equal to Castovan’s multistage value estimate. C. greater than Castovan’s multistage value estimate. 26. The best justification for Castovan to use Beckworth’s suggested persistence factor is that TTCI has: A. a low dividend payout. B. extreme accounting rates of return. C. a strong market leadership position. 27. Use the following information to estimate the intrinsic value of VIM’s common stock using the residual income model: ■ VIM had total assets of $3,000,000, financed with twice as much debt capital as equity capital. ■ VIM’s pretax cost of debt is 6% and cost of equity capital is 10%. ■ VIM had EBIT of $300,000 and was taxed at a rate of 40%. EBIT is expected to continue at $300,000 indefinitely. ■ VIM’s book value per share is $20. ■ VIM has 50,000 shares of common stock outstanding. 28. Palmetto Steel, Inc. (PSI) maintains a dividend payout ratio of 80% because of its limited opportunities for expansion. Its return on equity is 15%. The required rate of return on PSI equity is 12%, and its long-term growth rate is 3%. Compute the justified P/B based on forecasted fundamentals, consistent with the residual income model and a constant growth rate assumption. The following information relates to questions 29-30 Protected Steel Corporation (PSC) has a book value of $6 per share. PSC is expected to earn $0.60 per share forever and pays out all of its earnings as dividends. The required rate of return on PSC’s equity is 12%. Calculate the value of the stock using the following: Practice Problems 29. Dividend discount model. 30. Residual income model. The following information relates to questions 31-32 Notable Books (NB) is a family controlled company that dominates the retail book market. NB has book value of $10 per share, is expected to earn $2.00 per share forever, and pays out all of its earnings as dividends. Its required return on equity is 12.5%. Value the stock of NB using the following: 31. Dividend discount model. 32. Residual income model. The following information relates to questions 33-35 Simonson Investment Trust International (SITI) is expected to earn $4.00, $5.00, and $8.00 per share for the next three years. SITI will pay annual dividends of $2.00, $2.50, and $20.50 in each of these years. The last dividend includes a liquidating payment to shareholders at the end of Year 3 when the trust terminates. SITI’s book value is $8 per share and its required return on equity is 10%. 33. What is the current value per share of SITI according to the dividend discount model? 34. Calculate per-share book value and residual income for SITI for each of the next three years and use those results to find the stock’s value using the residual income model. 35. Calculate return on equity and use it as an input to the residual income model to calculate SITI’s value. 36. Foodsco Incorporated (FI), a leading distributor of food products and materials to restaurants and other institutions, has a remarkably steady track record in terms of both return on equity and growth. At year-end 2017, FI had a book value of $30 per share. For the foreseeable future, the company is expected to achieve a ROE of 15% (on trailing book value) and to pay out one-third of its earnings in dividends. The required return is 12%. Forecast FI’s residual income for the year ending 31 December 2022. The following information relates to questions 37-39 Thales S.A. (Paris: HO.PA) has a current stock price of €98.73. It also has book value per share of €26.83. and a P/B of 3.68. Assume that the single-stage growth model is appropriate for valuing the company. Thales S.A.’s adjusted beta is 0.68, the risk-free rate is 4.46%, and the equity risk premium is 5.50%. 265 266 Learning Module 3 Residual Income Valuation 37. If the growth rate is 5.50% and the ROE is 20%, what is the justified P/B for Thales? 38. If the growth rate is 5.50%, what ROE is required to yield Thales S.A.’s current P/B? 39. If the ROE is 20%, what growth rate is required for Thales to have its current P/B? 40. Retail fund manager Seymour Simms is considering the purchase of shares in upstart retailer Hottest Topic Stores (HTR). The current book value of HTS is $20 per share, and its market price is $35. Simms expects long-term ROE to be 18%, long-term growth to be 10%, and cost of equity to be 14%. What conclusion would you expect Simms to arrive at if he uses a single-stage residual income model to value these shares? 41. Dayton Manufactured Homes (DMH) builds prefabricated homes and mobile homes. Favorable demographics and the likelihood of slow, steady increases in market share should enable DMH to maintain its ROE of 15% and growth rate of 10% through time. DMH has a book value of $30 per share and the required rate of return on its equity is 12%. Compute the value of its equity using the single-stage residual income model. 42. Use the following inputs and the finite horizon form of the residual income model to compute the value of Southern Trust Bank (STB) shares as of 31 December 2020: ■ ROE will continue at 15% for the next five years (and 10% thereafter) with all earnings reinvested (no dividends paid). ■ Cost of equity equals 10%. ■ B0 = $10 per share (at year-end 2020). ■ Premium over book value at the end of five years will be 20%. The following information relates to questions 43-46 Shunichi Kobayashi is valuing Procter & Gamble Company (NYSE: PG). Kobayashi has made the following assumptions: ■ Book value per share is estimated at $21.30 on 31 March 2019. ■ EPS will be 18% of the beginning book value per share for the next eight years. ■ Cash dividends paid will be 70% of EPS. ■ At the end of the eight-year period, the market price per share will be four times the book value per share. ■ The beta for PG is 0.50, the risk-free rate is 2.0%, and the equity risk premium is 6.2%. The current market price of PG is $107.50, which indicates a current P/B of 5.05. 43. Prepare a table that shows the beginning and ending book values, net income, and cash dividends annually for the eight-year period. 44. Estimate the residual income and the present value of residual income for the Practice Problems eight years. 45. Estimate the value per share of PG stock using the residual income model. 46. Estimate the value per share of PG stock using the dividend discount model. How does this value compare with the estimate from the residual income model? 47. Consider the following information about Industrias Gómez. ■ Current book value per share is €20.00. ■ Expected earnings per share for the next five years are €1.50, €2.50, €3.50, €4.50, and €5.50. ■ Dividends per share are projected to be €1.00 for the first three years and €2.00 for the last two years. ■ The terminal share price (at the end of Year 5) is expected to be 14× trailing earnings. ■ The required rate of return on equity is 9%. ■ Estimate the residual income each year, the terminal residual value, and the value per share of Industrias Gómez shares using the residual income model. ■ Estimate the value per share of Industrias Gómez shares using the dividend discount model. 48. Lendex Electronics (LE) had a great deal of turnover of top management for several years and was not followed by analysts during this period of turmoil. Because the company’s performance has been improving steadily for the past three years, technology analyst Stephanie Kent recently reinitiated coverage of LE. A meeting with management confirms Kent’s positive impression of LE’s operations and strategic plan. Kent decides LE merits further analysis. Careful examination of LE’s financial statements revealed that the company had negative other comprehensive income from changes in the value of available-for-sale securities in each of the past five years. How, if at all, should this observation about LE’s other comprehensive income affect the figures that Kent uses for the company’s ROE and book value for those years? 267 268 Learning Module 3 Residual Income Valuation SOLUTIONS 1. Yes, VIM earned a positive residual income of $8,000. EBIT $300,000 Interest 120,000 Pretax income $180,000 Tax expense 72,000 Net income $108,000 ($2,000,000 × 6%) Equity charge = Equity capital × Required return on equity = (1/3)($3,000,000) × 0.10 = $1,000,000 × 0.10 = $100,000 Residual income = Net income – Equity charge = $108,000 – $100,000 = $8,000 2. In this problem (unlike Problems 1 and 2), interest expense has already been deducted in arriving at NMP’s pretax income of $5.1 million. Therefore, Net income = Pretax income × (1 – Tax rate) = $5.1 million × (1 – 0.4) = $5.1 × 0.6 = $3.06 million Equity charge = Total equity × Cost of equity capital = (0.1 × $450 million) × 12% = $45 million × 0.12 = $5,400,000 Residual income = Net income – Equity charge = $3,0600,000 - $5,400,000 = –$2,340,000 NMP had negative residual income of −$2,340,000. 3. To achieve a positive residual income, a company’s net operating profit after taxes as a percentage of its total assets can be compared with its weighted average cost of capital. For SWI, NOPAT/Assets = €10 million/€100 million = 10% WACC = Percent of debt × After-tax cost of debt + Percent of equity × Cost of equity = (0.5)(0.09)(0.6) + (0.5)(0.12) = (0.5)(0.054) + (0.5)(0.12) = 0.027 + 0.06 = 0.087 = 8.7% Therefore, SWI’s residual income was positive. Specifically, residual income equals €1.3 million [(0.10 − 0.087) × €100 million]. 4. EVA = NOPAT − WACC × Beginning book value of assets ​     ​ = $100 − ​(​ ​11%​)​​× ​​(​$200 + $300​)​​ = $100 − ​(​ ​11%​)​​​​(​$500​)​​ = $45 Solutions 269 5. ​ t​​− r ​Bt−1 ​ ​ I​t​ = E R ​ ​   = €5.00 − ​(​ ​11%​)​​​(​ ​€30.00​)​​ = 6. ​ ​ I​t​ = ​(​ ​ROE​t​− r​)​​× ​Bt−1 R ​  ​ = ​​(​18 %   − 12%​)​​× ​​(​€30​)​​ = €5.00 − €3.30 = €1.70 ​ €1.80 7. Economic value added = Net operating profit after taxes − (Cost of capital × Total capital) = $100 million − (14% × $700 million) = $2 million. In the absence of information that would be required to calculate the weighted average cost of debt and equity, and given that Sundanci has no long-term debt, the only capital cost used is the required rate of return on equity of 14%. 8. Market value added = Market value of capital − Total capital = $26 stock price × 84 million shares − $700 million = $1,484,000,000. Market value added per share = $1,484,000,000 / 84 million shares= $17.67 per share. 9. C is correct. Market value added equals the market value of firm minus total accounting book value of total capital. Market value added = Market value of company – Accounting book value of total capital Market value of firm = Market value of debt + Market value of equity Market value of firm = R55 million + (30,000,000 × R25.43) Market value of firm = R55 million + R762.9 million = R817.9 million Market value added = R817.9 million – R650 million = R167.9 million, or approximately R168 million. 10. B is correct. The intrinsic value of R22.00 is greater than the current book value of R20.00. The residual income model states that the intrinsic value of a stock is its book value per share plus the present value of expected (future) per share residual income. The higher intrinsic value per share, relative to book value per share, indicates that the present value of expected per share residual income is positive. 11. A is correct because the intrinsic value is the book value per share, B0,plus the expected residual income stream, or B0 + [(ROE – r)B0/(r – g)]. If ROE equals the cost of equity (r), then V0 = B0. This implies that ROE is equal to the cost of the equity, and therefore there is no residual income contribution to the intrinsic value. As a result, intrinsic value would be equal to book value. 12. B is correct. With a single-stage residual income (RI) model, the intrinsic value, V0, is calculated assuming a constant return on equity (ROE) and a constant earnings growth (g). ​(​ROE − r​)​ ​V0​ ​ = ​B0​ ​+ B ​ 0​ ​_ ​ ​ ​(​r − g​)​ ​​ ​ ​ = R55.81 + R55.81​_        ​​(​0.13 − 0.11​)​ ​V 0 ​V0​ ​ = R130.22 (​ ​0.11 − 0.095​)​ 13. B is correct. The share price of R8.25 was lower than the intrinsic value of R11.00. Shares are considered undervalued when the current share price is less than intrinsic value per share. 270 Learning Module 3 Residual Income Valuation 14. C is correct. The restructuring charge is a non-recurring item and not indicative of future earnings. In applying a residual income model, it is important to develop a forecast of future residual income based on recurring items. Using the net income reported in Amersheen’s 2020 net income statement to model subsequent future earnings, without adjustment for the restructuring charge, would understate the firm’s future earnings. By upward adjusting the firm’s net income, by adding back the R2 million restructuring charge to reflect the fact that the charge is non-recurring, future earnings will be more accurately forecasted. 15. C is correct. The multistage residual income model results in an intrinsic value of R16.31. This variation of the multistage residual income model, in which residual income fades over time, is: T−1 ​(​Et​​− r ​Bt−1 ​ ​) ​(​E​ ​− r B ​ ​ ​)​ ​​ ​1 + r − ω​ ​​ ​1 + r​ ​ T T−1 ​ ​(​1 + r​)​t​ + ________________       ​( ​ ​ 0​ ​+ ​ ∑​​​_ ​​V0​ ​ = B )( ) T−1 t=1 ​ where ω is the persistence factor. The first step is to calculate residual income per share for years 2022–2025: 2022 2023 2024 2025 Beginning book value per share R7.60 (given) R7.60 + R3.28 – R2.46 = R8.42 R8.42 + R3.15 – R2.36 = R9.21 R9.21 + 2.90 – R2.06 = R10.05 ROE R3.28/R7.60 = 0.4316 R3.15/R8.42 = 0.3741 R2.90/R9.21 = 0.3149 26% (given) Retention rate 1 – (R2.46/R3.28) = 0.25 1 – (R2.36/R3.15) = 0.2508 1 – (R2.06/R2.90) = 0.2897 N/A Growth rate 0.4316 × 0.25 =0.1079 0.3741 × 0.2508 = 0.0938 0.3149 × 0.2897 = 0.0912 9% (given) Equity charge per share R7.60 × 0.10 = R0.76 R8.42 × 0.10 = R0.842 R9.21 × 0.10 = R0.921 R10.05 × 0.10 = R1.005 Residual income per share R3.28 – R0.76 = R2.52 R3.15 – R0.842 = R2.31 R2.90 – 0.921 = R1.98 [0.26 × R10.05] – R1.005 = R1.608 ROE = Earnings/Book value Growth rate = ROE × Retention rate Retention rate = 1 – (Dividends/Earnings) Book valuet = Book valuet–1 + Earningst–1 – Dividendst–1 Residual income per share = EPS – Equity charge per share Equity charge per share = Book value per sharet × Cost of equity Using the residual income per share for 2015 of R1.608, the second step is to calculate the present value of the terminal value: R1.608    ) ( ​ = R3.0203​ ​PV of Terminal Value = __________________ ​   ( )3 ​​ ​1 + 0.10 − 0.70​ ​​ ​1.10​ ​ ​ Then, intrinsic value per share is: R2.52 R2.31 ​ ​1.10​ ​ R1.98 ​ ​1.10​ ​ ​​(​1.10​)​​+ _ ​( +_ + R3.0203 = R16.31 ​​V0​ ​ = R7.60 + _ )2 ( )3 16. A is correct. The multistage residual income model results in an intrinsic value of R13.29. The multistage residual income model, is: Solutions 271 T ​(​Et​​− r ​Bt−1 ​ ​) ​(​P​ ​− ​B​ ​) ​ ​1 + r​ ​ ​ ​ (​ ​1 + r​)​t​ + _ ​ ( T ) TT ​​V0​ ​ = ​B0​ ​+ ​∑​​_ t=1 The first step is to calculate residual income per share for years 2022–2024: 2022 2023 2024 Beginning book value per share R7.60 (given) R7.60 + R3.28 – R2.46 = R8.42 R8.42 + R3.15 – R2.36 = R9.21 ROE R3.28/R7.60 = 0.4316 R3.15/R8.42 = 0.3741 R2.90/R9.21 = 0.3149 Retention rate 1 – (R2.46/R3.28) = 0.25 1 – (R2.36/R3.15) = 0.2508 1 – (R2.06/R2.90) = 0.2897 Growth rate 0.4316 × 0.25=0.1079 0.3741 × 0.2508 = 0.0938 0.3149 × 0.2897= 0.0912 Equity charge per share R7.60 × 0.10 = R0.76 R8.42 × 0.10 = R0.842 R9.21 × 0.10 = R0.921 Residual income per share R3.28 – R0.76 = R2.52 R3.15 – R0.842 = R2.31 R2.90 – 0.921= R1.98 ROE = Earnings/Book value Growth rate = ROE × Retention rate Retention rate = 1 – (Dividends/Earnings) Book valuet = Book valuet–1 + Earningst–1 – Dividendst–1 Residual income per share = EPS – Equity charge per share Equity charge per share = Book value per sharet × Cost of equity Under Scenario 2, at the end of 2024, it is assumed that share price will be equal to book value per share. This results in the second term in the equation above, the present value of the terminal value, being equal to zero. Then, intrinsic value per share is: R2.52 R2.31 ​ ​1.10​ ​ R1.98 ​ ​1.10​ ​ ​​(​1.10​)​​+ _ ​( +_ = R13.29​ ​​V0​ ​ = R7.60 + _ )2 ( )3 17. B is correct. The residual income model’s use of accounting income assumes that the cost of debt capital is reflected appropriately by interest expense. 18. A is correct. Dividend payments are forecasted to be unpredictable over Castovan’s five-year forecast horizon. A residual income model is appropriate when a company does not pay dividends or when its dividends are not predictable, which is the case for PTU. 19. C is correct. Other comprehensive income bypasses the income statement and goes directly to the statement of stockholders’ equity (which is a violation of the clean surplus relationship). Therefore, book value per share for PTU will be affected by forecasted OCI. 20. C is correct. The residual income can be calculated using net income and the equity charge or using net operating profit after taxes and the total capital charge. Residual income = Net income – Equity charge Calculation of Net Income (values in millions): EBIT €700.0 Less Interest expense €124.8 Pretax income €575.2 Less Income tax expense €201.3 Net income €373.9 (= €4,000 × 0.60 × 0.052) (= €575.20 × 0.35) Equity charge = Total assets × Equity weighting × Cost of equity 272 Learning Module 3 Residual Income Valuation Equity charge = €4,000 million × 0.40 × 0.15 = €240 million Therefore, residual income = €373.9 million – €240 million = €133.9 million. Alternatively, residual income can be calculated from NOPAT as follows. Residual income = NOPAT – Total capital charge NOPAT = EBIT × (1 – Tax rate) NOPAT = €700 million × (1 – 0.35) = €455 million The total capital charge is as follows. Equity charge = Total assets × Equity weighting × Cost of equity = €4,000 million × 0.40 × 0.15 = €240 million Debt charge = Total assets × Debt weighting × Pretax cost of debt × (1 – Tax rate) = €4,000 million × 0.60 × 0.052(1 – 0.35) = €81.1 million Total capital charge = €240 million + €81.1 million = €321.1 million Therefore, residual income = €455 million – €321.1 million = €133.9 million. 21. B is correct. The implied growth rate of earnings from the single-stage RI model is calculated by solving for g in the following equation: ​​V0​ ​ = B ​ 0​ ​+ ​( ​_ ​ r−g ) ​ ​​B0​ ROE − r Book value per share can be calculated using the given price-to-book ratio and market price per share as follows. Book value per share (B0) = Market price per share/Price-to-book ratio = €48.80/2.10 = €23.24 Then, solve for the implied growth rate. ​€48.80 = €23.24 + ​( ​_ ​ 0.15 − g ) ​ ​​€23.24 0.2337 − 0.15 g = 7.4% 22. C is correct. Using the single-stage RI model, the intrinsic value of TTCI is calculated as ​V0​ ​ = B ​ 0​ ​+ ​( ​_ ​ r−g ) ​ ​​B0​ ROE − r 0.12 − 0.087 ​         = ​​ €45.25 + ​( ​_ ​0.087 − 0.045 ) ​ ​​€45.25 = €80.80 The intrinsic value of €80.80 is less than the market price of €126.05, so Castovan should conclude that the stock is overvalued. 23. A is correct. The justified price-to-book ratio is calculated as Solutions 273 ROE − r _ ​_ ​ r−g ) ​ ​P B ​ = 1 + ​( ​    ​​ 0.12 − 0.087 = 1 + ​( ​_ ​0.087 − 0.045 ​)​​ = 1.79 24. C is correct. Residual income per share for the next three years is calculated as follows. Year 1 Year 2 Year 3 Beginning book value per share 45.25 51.61 58.25 Earnings per share 7.82 8.17 8.54 Less dividends per share 1.46 1.53 1.59 Change in retained earnings 6.36 6.64 6.95 Ending book value per share 51.61 58.25 65.20 Earnings per share 7.82 8.17 8.54 Less per share equity charge* 3.94 4.49 5.07 Residual income 3.88 3.68 3.47 * Per share equity charge = Beginning book value per share × Cost of equity Year 1 per share equity charge = 45.25 × 0.087 = 3.94 Year 2 per share equity charge = 51.61 × 0.087 = 4.49 Year 3 per share equity charge = 58.25 × 0.087 = 5.07 Because Castovan forecasts that residual income per share will be constant into perpetuity, equal to Year 3 residual income per share, the present value of the terminal value is calculated using a persistence factor of 1. 8.54 − ​(​ ​0.087 × 58.25​)​​ ​​ ​1 + 0.087 − 1​ ​​ ​1 + 0.087​ ​ ​ Present value of terminal value = ____________________       ​( ​ )( )2 3.47 ​​ ​0.087​ ​​ ​1.087​ ​ ​ = ____________ ​   ​ ( )( )2 = 33.78 So, the intrinsic value of TTCI is then calculated as follows. 3.88 3.68 ​1.087​​ ​ ​1.087 ​+ _ ​ ​​V0​ ​ = €45.25 + _ 2 ​+ 33.78 = €85.71 25. A is correct. In Castovan’s multistage valuation, she assumes that TTCI’s residual income will remain constant in perpetuity after Year 3. This perpetuity assumption implies a persistence factor of 1 in the calculation of the terminal value. A persistence factor of 0.10 indicates that TTCI’s residual income is forecasted to decline at an average rate of 90% per year. This assumption would lead to a lower valuation than Castovan’s multistage value estimate, which assumes that residual income will remain constant in perpetuity after Year 3. 26. B is correct. Beckworth’s suggested persistence factor for TTCI is 0.10, which is quite low. Companies with extreme accounting rates of return typically have low persistence factors. Companies with strong market leadership positions and low dividend payouts are likely to have high persistence factors. 27. According to the residual income model, the intrinsic value of a share of common stock equals book value per share plus the present value of expected future per-share residual income. Book value per share was given as $20. If we note that debt is $2,000,000 [(2/3)($3,000,000)] so that interest is $120,000 ($2,000,000 × 6%), VIM’s residual income is $8,000, which is calculated (as in Problem 1) as 274 Learning Module 3 Residual Income Valuation follows: Residual income = Net income – Equity charge = [(EBIT – Interest)(1 – Tax rate)] – [(Equity capital)(Required return on equity)] = [($300,000 – $120,000)(1 – 0.40)] – [($1,000,000)(0.10)] = $108,000 – $100,000 = $8,000 Therefore, residual income per share is $0.16 per share ($8,000/50,000 shares). Because EBIT is expected to continue at the current level indefinitely, the expected per-share residual income of $0.16 is treated as a perpetuity. The present value of $0.16 is discounted at the required return on equity of 10%, so the present value of the residual income is $1.60 ($0.16/0.10). Intrinsic value = Book value per share + PV of expected future income per-share residual income = $20 + $1.60 = $21.60 28. With g = b × ROE = (1 − 0.80) (0.15) = (0.20) (0.15) = 0.03, P/B = (ROE – g)/(r – g) = (0.15 – 0.03)/(0.12 – 0.03) = 0.12/0.09 = 1.33 or P/B = 1 + (ROE – r)/(r – g) = 1 + (0.15 – 0.12)/(0.12 – 0.03) = 1.33 29. Because the dividend is a perpetuity, the no-growth form of the DDM is applied as follows: V0 = D/r = $0.60/0.12 = $5 per share 30. According to the residual income model, V0 = Book value per share + Present value of expected future per-share residual income. Residual income is calculated as: RIt = E – rBt–1 = $0.60 – (0.12)($6) = –$0.12 Present value of perpetual stream of residual income is calculated as: RIt/r = –$0.12/0.12 = –$1.00 The value is calculated as: V0 = $6.00 – $1.00 = $5.00 per share 31. According to the DDM, V0 = D/r for a no-growth company. Solutions 275 V0 = $2.00/0.125 = $16 per share 32. Under the residual income model, V0 = B0 + Present value of expected future per-share residual income. Residual income is calculated as: RIt = E – rBt–1 = $2 – (0.125)($10) = $0.75 Present value of stream of residual income is calculated as: RIt/r = 0.75/0.125 = $6 The value is calculated as: V0 = $10 + $6 = $16 per share ​ 0​ ​ = Present value of the future dividends V           33. ​= $2 / 1.10 + $2.50 / (​ ​1.1​)​2​+ $20.50 / (​ ​1.1​)​3 = $1.818 + $2.066 + $15.402 = $19.286 34. The book values and residual incomes for the next three years are as follows: Year Beginning book value 1 2 3 $ 8.00 $10.00 $12.50 Retained earnings (Net income − Dividends) 2.00 2.50 (12.50) Ending book value $10.00 $12.50 $ 0.00 Net income $ 4.00 $ 5.00 $ 8.00 0.80 1.00 1.25 $ 3.20 $ 4.00 $ 6.75 Less equity charge (r × Book value) Residual income Under the residual income model, V0 = B0 + Present value of expected future per-share residual income V0 = $8.00 + $3.20/1.1 + $4.00/(1.1)2 + $6.75/(1.1)3 V0 = 8.00 + $2.909 + $3.306 + $5.071 = $19.286 35. Year 1 2 3 Net income (NI) $4.00 $5.00 $8.00 Beginning book value (BV) 8.00 10.00 12.50 Return on equity (ROE) = NI/BV 50% 50% 64% ROE − r 40% 40% 54% 276 Learning Module 3 Residual Income Valuation Year Residual income (ROE − r) × BV 1 2 3 $3.20 $4.00 $6.75 Under the residual income model, V0 = B0 + Present value of expected future per-share residual income V0 = $8.00 + $3.20/1.1 + $4.00/(1.1)2 + $6.75/(1.1)3 V0 = 8.00 + $2.909 + $3.306 + $5.071 = $19.286 36. Year 2018 2019 2022 Beginning book value $30.00 $33.00 $43.92 Net income = ROE × Book value 4.50 4.95 6.59 Dividends = payout × Net income 1.50 1.65 2.20 Equity charge (r × Book value) 3.60 3.96 5.27 Residual income = Net income − Equity charge Ending book value 0.90 0.99 1.32 $33.00 $36.30 $48.32 The table shows that residual income in Year 2018 is $0.90, which equals Beginning book value × (ROE − r) = $30 × (0.15 − 0.12). The Year 2019 column shows that residual income grew by 10% to $0.99, which follows from the fact that growth in residual income relates directly to the growth in net income as this example is configured. When both net income and dividends are a function of book value and return on equity is constant, then growth, g, can be predicted from (ROE)(1 − Dividend payout ratio). In this case, g = 0.15 × (1 − 0.333) = 0.10 or 10%. Net income and residual income will grow by 10% annually. Therefore, residual income in Year 2022 = (Residual income in Year 2018) × (1.1)4 = 0.90 × 1.4641 = $1.32. 37. The justified P/B can be found with the following formula: ​P0​ ​ ROE − r _ ​ r−g ​​​B​ ​​ = 1 + _ 0 ROE is 20%, g is 5.5%, and r is 8.2% [RF + βi[E(RM) − RF] = 4.46% + (0.68)(5.5%)]. Substituting in the values gives a justified P/B of ​P0​ ​ 0.20 − 0.082 _ ​0.082 − 0.055 ​ = 4.37 ​​​B​ ​​ = 1 + _ 0 The assumed parameters give a justified P/B of 4.37, slightly above the current P/B of 3.68. 38. To find the ROE that would result in a P/B of 3.68, we substitute 3.68, r, and g into the following equation: ​P0​ ​ ROE − r _ ​ r−g ​​​B​ ​​ = 1 + _ 0 This yields ROE − 0.082 ​3.68 = 1 + _ ​0.082 − 0.055 Solving for ROE requires several steps to finally derive a ROE of 0.15435 or 15.4%. This value of ROE is consistent with a P/B of 3.68. Solutions 277 39. To find the growth rate that would result with a P/B of 3.68, use the expression given in Part B, but solve for g instead of ROE: ​P0​ ​ ROE − r _ ​ r−g ​​​B​ ​​ = 1 + _ 0 Substituting in the values gives: 0.20 − 0.082 ​3.68 = 1 + _ ​ 0.082 − g The growth rate g is 0.03797, or 3.8%. If we assume that the single-stage growth model is applicable to Thales, the current P/B and current market price can be justified with values for ROE or g that are quite a bit lower than the starting values of 20% and 5.5%, respectively. 40. V ​ 0​  ​​  = ​B​ 0​​  + ​ ​(​​ROE − r​)​​ ​​B0​  ​​  / ​ ​(​​r − g​)​​ ​ ​ − 0.10​)​​ ​          ​​= $20 + ​ ​(​​0.18 − 0.14​ ​​ )​​ ​​ ​(​​$20​)​​ ​  / ​ ​(​​0.14 = $20 + $20 = $40 Given that the current market price is $35 and the estimated value is $40, Simms will probably conclude that the shares are somewhat undervalued. ​ 0​ ​ = ​B0​ ​+ ​​(​ROE − r​)​​B0​ ​/ ​(​ ​r − g​) 41. V      ​    ​ − 0.10​) = $30 + ​(​ ​0.15 − 0.12​)​​(​ ​$30​)​​/ ​(​ ​0.12 = $30 + $45 = $75 per share 42. Net Income (Projected) Year Ending Book Value 2020 ROE (%) Equity Charge (in Currency) Residual Income PV of RI $10.00 2021 $1.50 11.50 15 $1.00 $0.50 $0.45 2022 1.73 13.23 15 1.15 0.58 0.48 2023 1.99 15.22 15 1.32 0.67 0.50 2024 2.29 17.51 15 1.52 0.77 0.53 2025 2.63 20.14 15 1.75 0.88 0.55 $2.51 Using the finite horizon form of residual income valuation, V0 = B0 + Sum of discounted RIs + Premium (also discounted to present) = $10 + $2.51 + (0.20)(20.14)/(1.10)5 = $10 + $2.51 + $2.50 = $15.01 43. Columns (a) through (d) in the table show calculations for beginning book value, net income, dividends, and ending book value. (a) (b) (c) (d) (e) (f) Beginning Book Value Net Income Dividends Ending Book Value Residual Income PV of RI 1 $21.300 $3.834 $2.684 $22.450 $2.748 $2.614 2 22.450 4.041 2.829 23.663 2.896 2.622 3 23.663 4.259 2.981 24.940 3.052 2.629 Year 278 Learning Module 3 Residual Income Valuation (a) (b) (c) (d) (e) (f) Beginning Book Value Net Income Dividends Ending Book Value Residual Income PV of RI 4 24.940 4.489 3.142 26.287 3.217 2.637 5 26.287 4.732 3.312 27.707 3.391 2.644 6 27.707 4.987 3.491 29.203 3.574 2.652 7 29.203 5.256 3.680 30.780 3.767 2.659 8 30.780 5.540 3.878 32.442 3.971 Year Total 2.667 $21.125 For each year, net income is 18% of beginning book value. Dividends are 70% of net income. The ending book value equals the beginning book value plus net income minus dividends. 44. Column (e) of the table in Part A shows Residual income, which equals Net income – Cost of equity (%) × Beginning book value. To find the cost of equity, use the CAPM: r = RF + βi[E(RM) – RF] = 2% + (0.50)(6.2%) = 5.1% For Year 1 in the table, Residual income = RIt = E – rBt–1 = 3.834 – (5.1%)(21.30) = 3.834 – 1.086 = $2.748 This same calculation is repeated for Years 2 through 8. Column (f ) of the table gives the present value of the calculated residual income, discounted at 5.1%. 45. To find the stock value with the residual income method, use this equation: T ​(​E​​− r ​B​ ​)​ ​P​ ​− ​B​ ​ T T _ t−1 ​​V0​ ​ = B ​ 0​ ​+ ​∑_ ​ (t ​+ ​( )T )t t=1 ■ ​ ​1 + r​ ​​ ​ ​1 + r​ ​ ​ In this equation, B0 is the current book value per share of $21.30. ■ The second term, the sum of the present values of the eight years’ residual income is shown in the table, $21.125. ■ To estimate the final term, the present value of the excess of the terminal stock price over the terminal book value, use the assumption that the terminal stock price is assumed to be 4.0× the terminal book value. So, by assumption, the terminal stock price is $129.767 [PT = 4.0(32.442)]. PT − BT is $97.325 (129.767 − 32.442), and the present value of this amount discounted at 5.1% for eight years is $65.374. ■ Summing the relevant terms gives a stock price of $107.799 (V0 = 21.30 + 21.125 + 65.374). 46. The appropriate DDM expression expresses the value of the stock as the sum of the present value of the dividends plus the present value of the terminal value: Solutions 279 T ​P​ ​ ​D​​ ​ 1 ​ + r​ ​ ​ ​ 1 ​ + r​ ​ ​ t=1 ​( T ) T ​ ​​V0​ ​ = ​∑ _ ​( t ) t ​+_ Discounting the dividends from the table shown in the solution to Part A above at 8.30% gives: Year Dividend PV of Dividend 1 $2.684 2.554 2 2.829 2.561 3 2.981 2.568 4 3.142 2.575 5 3.312 2.583 6 3.491 2.590 7 3.680 2.598 8 3.878 2.605 All $20.634 ■ The present value of the eight dividends is $20.634. The estimated terminal stock price, calculated in the solution to Part C above is $125.767, which equals $87.165 discounted at 5.1% for eight years. ■ The value for the stock, the present value of the dividends plus the present value of the terminal stock price, is V0 = 20.634 + 87.165 = $107.799. ■ The stock values estimated with the residual income model and the dividend discount model are identical. Because they are based on similar financial assumptions, this equivalency is expected. Even though the two models differ in their timing of the recognition of value, their final results are the same. 47. A. The value found with the residual income model is: Year Beginning BV Net Income Dividends Ending BV Residual Income PV of Residual Income 1 20.00 1.50 1.00 20.50 –0.300 –0.275 2 20.50 2.50 1.00 22.00 0.655 0.551 3 22.00 3.50 1.00 24.50 1.520 1.174 4 24.50 4.50 2.00 27.00 2.295 1.626 5 27.00 5.50 2.00 30.50 3.070 1.995 Sum PVRI Terminal PT – BT PV of PT – BT B0 Total value: 5.071 46.500 30.222 20.000 €55.293 Residual income each year is Net income – 0.09 × (Beginning BV). The PV of residual income is found by discounting at 9%. The terminal price is 14 × EPS in Year 5, or 14 × 5.50 = €77.00. The terminal residual value is PT – BT 280 Learning Module 3 Residual Income Valuation = 77.00 – 30.50 = €46.50. Discounted at 9%, the PV of €46.50 is €30.222. The value per share is B0 + PV of residual income + PV of terminal residual value, which is €55.293. B. The value found with the dividend discount model is as follows: Year Dividend or Price PV of Dividend or Price 1 1.00 0.917 2 1.00 0.842 3 1.00 0.772 4 2.00 1.417 5 2.00 1.300 5 77.00 50.045 Total PV €55.293 The values per share found with the DDM and the residual income model are an identical €55.293. 48. When such items as changes in the value of available-for-sale securities bypass the income statement, they are generally assumed to be nonoperating items that will fluctuate from year to year, although averaging to zero in a period of years. The evidence suggests, however, that changes in the value of available-for-sale securities are not averaging to zero but are persistently negative. Furthermore, these losses are bypassing the income statement. It appears that the company is either making an inaccurate assumption or misleading investors in one way or another. Accordingly, Kent might adjust LE’s income downward by the amount of loss for other comprehensive income for each of those years. ROE would then decline commensurately. LE’s book value would not be misstated because the decline in the value of these securities was already recognized and appears in the shareholders’ equity account “Accumulated Other Comprehensive Income.” Practice Problems PRACTICE PROBLEMS The following information relates to questions 1-5 Alan Chin, CEO of Thunder Corporation, has asked his chief financial officer, Constance Ebinosa, to prepare a valuation of Thunder for the purpose of selling the company to a private investment partnership. Thunder is a profitable US-domiciled manufacturer of generic household products with $200 million in annual sales. Customers consist of several grocery store chains in the United States. Competitors include large companies such as Procter & Gamble, The Clorox Company, and Unilever. Thunder has been in business for 15 years and is privately owned by the original shareholders, none of whom are employed by the company. Thunder’s senior management has been in charge of the company’s operations for most of the past 15 years and expects to remain in that capacity after any sale. The partnership has expectations about Thunder similar to the current shareholders and management of Thunder. These investors expect to hold Thunder for an intermediate period and then bring the company public when market conditions are more favorable than currently. Chin is concerned about what definition of value to use in analyzing Thunder. He notes that the stock market has been very volatile recently. He also wonders whether fair market value can be realistically estimated when the most similar recent private market transactions may not have been at arm’s length. Chin asks Ebinosa whether there will be differences in the process of valuing a private company like Thunder compared with a public company. Ebinosa replies that differences do exist and mentions several factors an analyst must consider. Ebinosa also explains that several approaches are available for valuing private companies. She mentions that one possibility is to use an asset-based approach because Thunder has a relatively large and efficient factory and warehouse for its products. A real estate appraiser can readily determine the value of these facilities. A second method would be the market approach and using an average of the price-to-earnings multiples for Procter & Gamble and Clorox. A third possibility is a discounted free cash flow approach. The latter would focus on a continuation of Thunder’s trend of slow profitable growth during the past 10 years. The private investment partnership has mentioned that it is likely to use an income approach as one of its valuation methods. Ebinosa decides to validate the estimates they make. She assumes that for the next 12 months, Thunder’s revenues will increase by the long-term annual growth rate of 3%. She also makes the following assumptions to calculate the free cash flow to the firm for the next 12 months: ■ Gross profit margin is 45%. ■ Depreciation is 2% of revenues. ■ Selling, general, and administrative expenses are 24% of revenues. ■ Capital expenditures equal 125% of depreciation to support the current level of revenues. ■ Additional capital expenditures of 15% of incremental revenues are needed to fund future growth. ■ Working capital investment equals 8% of incremental revenues. 321 322 Learning Module 4 Private Company Valuation ■ Marginal tax rate on EBIT is 35%. Chin knows that if an income approach is used, the choice of discount rate may have a large influence on the estimated value. He makes two statements regarding discount rate estimates: 1. If the CAPM method is used to estimate the discount rate with a beta estimate based on public companies with operations and revenues similar to Thunder, then a small stock premium should be added to the estimate. 2. The weighted average cost of capital of the private investment partnership should be used to value Thunder. Ebinosa decides to calculate a value of Thunder’s equity using the capitalized cash flow method and decides to use the build-up method to estimate Thunder’s required return on equity. She makes the following assumptions: ■ Growth of FCFE is at a constant annual rate of 3%. ■ Free cash flow to equity for the year ahead is $2.5 million. ■ Risk-free rate is 4.5%. ■ Equity risk premium is 5.0%. ■ Size premium is 2.0%. 1. The least likely factor that would be a source of differences in valuing Thunder compared with valuing a publicly traded company is: A. access to public debt markets. B. agency problems. C. the size of the company. 2. Ebinosa can best value Thunder using the: A. excess earnings approach. B. asset-based approach. C. discounted free cash flow approach. 3. The free cash flow to the firm is closest to: A. $23,031,000. B. $25,441,000. C. $36,091,000. 4. Regarding the two statements about discount rate estimates, Chin is: A. correct with respect to adding the small stock premium and correct with respect to the weighted average cost of capital. B. correct with respect to adding the small stock premium and incorrect with respect to the weighted average cost of capital. C. incorrect with respect to adding the small stock premium and incorrect with respect to the weighted average cost of capital. Practice Problems 323 5. The indicated value of Thunder’s equity using the build-up method and the capitalized cash flow method (CCM) based on free cash flow to equity is closest to: A. $29.41 million. B. $38.46 million. C. $125.00 million. 6. Two companies are considering the acquisition of Target Company. Buyer A is a strategic buyer and Buyer B is a financial buyer. The following information pertains to Target Company: Sales = £28,000,000 Reported EBITDA = £4,500,000 Reported executive compensation = £1,000,000 Normalized executive compensation = £500,000 Reduced SG&A from eliminating duplicate general and administrative functions = £600,000 Calculate the pro forma EBITDA estimates that the strategic and financial buyers would each develop in an acquisitions analysis of Target Company. The following information relates to questions 7-12 The senior vice president of acquisitions for Northland Industries, Angela Lanton, and her head analyst, Michael Powell, are evaluating several potential investments. Northland is a diversified holding company for numerous businesses. One of Northland’s divisions is a manufacturer of fine papers, and that division has alerted Lanton about Oakstar Timber, a supplier that may be available for purchase. Oakstar’s sole owner, Felix Tanteromo, has expressed interest in exchanging his ownership of Oakstar for a combination of cash and Northland Industries securities. Oakstar’s main asset is 10,000 hectares of timberland in western Canada. The land is a combination of new and old growth Douglas fir trees. The value of this timberland has been steadily increasing since Oakstar acquired it. Oakstar manages the land on a sustained yield basis (i.e., so it continues to produce timber indefinitely) and contracts with outside forestry companies to evaluate, harvest, and sell the timber. Oakstar’s income is in the form of royalties (fees paid to Oakstar based on the number of cubic meters harvested). Oakstar’s balance sheet as of 31 December 20X0, in Canadian dollars, is as follows. Oakstar Timber Balance Sheet Year Ended 31 December 20X0 Assets Cash C$500,000 Inventory 25,000 Accounts receivable 50,000 Plant and equipment (cost less depreciation) 750,000 324 Learning Module 4 Private Company Valuation Oakstar Timber Balance Sheet Year Ended 31 December 20X0 Assets Land 10,000,000 Total assets C$11,325,000 Liabilities and Equity Accounts payables C$75,000 Long-term bank loan 1,500,000 Common stock 9,750,000 Total liabilities and equity C$11,325,000 In addition to the balance sheet, Powell is gathering other data to assist in valuing Oakstar and has found information on recent sales of timberland in western Canada. Douglas fir properties have averaged C$6,178 per hectare for tracts that are not contiguous and do not have a developed road system for harvesting the timber. For tracts with these features, as possessed by Oakstar, the average price is C$8,750 per hectare. Properties near urban areas and having potential for residential and recreational second home development command up to C$20,000 per hectare. Oakstar’s land lacks this potential. Lanton believes these values would form the basis of an asset-based valuation for Oakstar, with the additional assumption that other assets and liabilities on the balance sheet are assumed to be worth their stated values. The second company under evaluation, FAMCO, Inc., is a family-owned electronic manufacturing company with annual sales of US$120 million. The family wants to monetize the value of its ownership in FAMCO with a view to later investing part of the proceeds in a diversified stock portfolio. Lanton has asked Powell to obtain data for both an income-based and market-based valuation. Powell has obtained the recent annual income statement and additional data needed to calculate normalized earnings as follows. FAMCO, Inc. Income Statement Year Ending 31 December 20XO Revenues US$120,000,000 Gross profit 85,000,000 Selling, general, and administrative expenses 23,000,000 Pro forma EBITDA US$62,000,000 Depreciation and amortization 3,500,000 Pro forma earnings before interest and taxes US$58,500,000 Less: Interest 1,000,000 Earnings before taxes (EBT) Pro forma taxes on EBT Operating income after tax US$57,500,000 40% 23,000,000 US$34,500,000 Additional data for FAMCO is provided in the following table. Included are estimates by Powell of the compensation paid to family members and the smaller amount of salary expense for replacement employees if Northland acquires the company (reflecting perceived above-market compensation of the family group executives). He believes the current debt of FAMCO can be replaced with a more Practice Problems 325 optimal level of debt at a lower interest rate. The additional data will be reflected in a normalized income statement. FAMCO, Inc. Current debt level US$10,000,000 Current interest rate 10% Salaries of employed family members US$7,000,000 Salaries of replacement employees US$5,400,000 New debt level US$25,000,000 New interest rate 8% Powell also recognizes that a value needs to be assigned to FAMCO’s intangibles consisting of patents and other intangible assets. Powell prepares an additional estimate of excess earnings and intangibles value using the capitalized cash flow method. He projects the following data for 20X1: FAMCO, Inc.—Intangibles Valuation Data Working capital balance US$10,000,000 Fair value of fixed assets US$45,000,000 Normalized income to the company US$35,000,000 Required return on working capital 8% Required return on fixed assets 12% Required return on intangible assets 20% Weighted average cost of capital 14.5% Future growth rate 6% Lanton asks Powell to also use the market approach to valuation with a focus on the guideline transactions method. Powell prepares a table showing relevant information regarding three recent guideline transactions and market conditions at the time of the transactions. Powell’s assumptions about FAMCO include its expected fast growth and moderate level of risk. Target Firm Target’s Risk Target’s Growth Consideration Market Conditions Firm 1 High Slow Cash Normal, rising trend Firm 2 Moderate Fast Stock Prices near peak Firm 3 Moderate Fast Cash Normal, rising trend Although Northland is interested in acquiring all of the stock of FAMCO, the acquisition of a 15% equity interest in FAMCO is also an option. Lanton asks Powell about the valuation of small equity interests in private entities and notes that control and marketability are important factors that lead to adjustments in value estimates for small equity interests. Powell mentions that the control premium paid for the most similar guideline firm used in the analysis suggests a discount for lack of control of 20%. The discount for lack of marketability was estimated at 15%. 7. Which of the following statements concerning asset-based valuation as applied to Oakstar is most accurate? The approach is applicable: A. only when a guideline public company is unavailable for the valuation. 326 Learning Module 4 Private Company Valuation B. because natural resources with determinable market values constitute the majority of Oakstar’s total value. C. because as a passive collector of royalties, Oakstar has no meaningful capital expenditures and free cash flow is irrelevant. 8. Using an asset-based approach, the value (net of debt) of Oakstar is closest to: A. C$62,250,000. B. C$87,250,000. C. C$199,750,000. 9. The normalized earnings after tax for FAMCO is closest to: A. US$32,940,000. B. US$34,260,000. C. US$34,860,000. 10. Using the excess earnings method, the value of FAMCO’s intangibles is closest to: A. US$144.0 million. B. US$205.7 million. C. US$338.8 million. 11. The guideline transaction that is most likely applicable to FAMCO is: A. Firm 1. B. Firm 2. C. Firm 3. 12. The total discount for both control and marketability is closest to: A. 15%. B. 32%. C. 35%. 13. Using the build-up method and assuming that no adjustment for industry risk is required, calculate an equity discount rate for a small company, given the following information: Equity risk premium = 5.0% Midcap equity risk premium = 3.5% Small stock risk premium = 4.2% Income return on long-term bonds = 5.1% Total return on intermediate-term bonds = 5.3% Company-specific risk premium = 3.0% Practice Problems 327 20-Year Treasury bond yield as of the valuation date = 4.5% The following information relates to questions 14-20 Donald Schmidt is a portfolio manager at FutureTech, an information technology company that is expanding into the food industry. Schmidt meets with Tim Beckett, a FutureTech analyst, to discuss several privately owned companies being investigated for possible acquisition. Schmidt and Beckett first analyze Dairy Foody, a large, privately owned online food retailer.Beckett observes that the free cash flow growth of Dairy Foody has been quite unstable over time. Schmidt and Beckett discuss the valuation method that is most appropriate to value 100% of Dairy Foody. Schmidt prefers the valuation to reflect market pricing from recent acquisition activity in the food industry, which has been high in recent months. Schmidt and Beckett then discuss FastDelivery, a family-owned food delivery service. Pro forma financial data for FastDelivery, representing performance mid-economic cycle, are presented in Exhibit 1. Exhibit 1: FastDelivery: Pro Forma Operating Income after Taxes Revenues $50,000,000 COGS 30,000,000 Gross Profit 20,000,000 SG&A 10,000,000 EBITDA 10,000,000 Depreciation and amortization 3,000,000 EBIT 7,000,000 Pro forma taxes (at 25%) 1,750,000 Operating income after taxes $5,250,000 Beckett notices two items in FastDelivery’s financials that could detract from the reliability of forecasting normalized earnings. Specifically, Beckett notes that in Exhibit 1, SG&A includes the following: ■ Management salaries of $5,000,000 ■ The family’s personal expenses of $550,000 Beckett estimates the actual market rate of the management salaries to be $3,500,000. Schmidt also asks Beckett to calculate the required rate of return on equity for FastDelivery using the build-up method. Beckett collects the following information: ■ The risk-free rate is 1%. ■ The equity risk premium is 6%. ■ The industry risk premium is 2%. ■ The small-stock risk premium is 3%. 328 Learning Module 4 Private Company Valuation Schmidt transitions the conversation to Spice World, a privately owned grocery retailer. Schmidt estimates a required rate of return from a selection of guideline public companies. He tells Beckett that the required rate of return needs additional adjustments so that it can be applied to Spice World. First, Schmidt points out that Spice World faces operational and customer concentration risks that do not affect the selected guideline public companies. Second, Schmidt notes that Spice World has fewer employees and a lower level of total assets relative to the guideline public companies. Having estimated a required rate of return on equity for Spice World, Beckett calculates the company’s weighted average cost of capital (WACC) using its existing capital structure, as Schmidt believes that the current debt level is optimal. Beckett utilizes the following information to calculate the WACC for Spice World: ■ The cost of equity is 18%. ■ The corporate tax rate is 25%. ■ The pre-tax cost of debt is 9%. ■ The ratio of debt to total capital is 20%. Schmidt also requests that Beckett value a 15% interest in Food Garden, another grocery retailer. Based on a recent transaction involving a change in control, Beckett estimates the total equity value of Food Garden to be $100,000,000. In arriving at a value for the 15% interest, Beckett concludes that a discount for lack of control (DLOC) of 20% and a discount for lack of marketability (DLOM) of 35% are applicable. Finally, Beckett and Schmidt discuss LiveLong Foods, a privately owned food preservation company. Schmidt asks Beckett to employ the guideline public company method to value 100% of LiveLong Foods. Beckett observes that a group of comparable public companies has a median market value of invested capital (MVIC) to EBITDA multiple of 10. Beckett notes the following: ■ LiveLong Foods’ EBITDA is $12,000,000. ■ The market value of LiveLong Foods’ debt is $6,000,000. ■ A combined downward adjustment of 12% for relative risk and growth characteristics should be applied to the median multiple. ■ An 18% control premium should also be applied to the median multiple. 14. Given Schmidt’s preference, which of the following valuation methods is most appropriate to value Dairy Foody? A. Excess earnings method B. Capitalized cash flow method C. Guideline transactions method 15. The normalized operating income for FastDelivery is: A. $5,662,500. B. $6,375,000. C. $6,787,500. 16. Using the build-up method, the required rate of return on equity for FastDelivery Practice Problems is: A. 10%. B. 11%. C. 12%. 17. Given the observations about Spice World noted by Schmidt, the required rate of return for Spice World should be adjusted to reflect: A. only a size premium. B. only a company-specific risk premium. C. both a size premium and a company-specific risk premium. 18. The WACC for Spice World is closest to: A. 12.15%. B. 15.75%. C. 16.20%. 19. Beckett should estimate the value of the 15% equity interest in Food Garden to be: A. $6,750,000. B. $7,800,000. C. $12,750,000. 20. Using the guideline public company method, Beckett should estimate the value of 100% of LiveLong Foods to be: A. $118,608,000. B. $121,200,000. C. $124,608,000. 21. Using the capitalized cash flow method (CCM), calculate the fair market value of 100% of the equity of a hypothetical company, given the following information: Current year’s reported free cash flow to equity = $1,400,000 Current year’s normalized free cash flow to equity = $1,800,000 Long-term interest-bearing debt = $2,000,000 Weighted average cost of capital = 15% Equity discount rate = 18% Long-term growth rate of FCFE = 5.5% 329 330 Learning Module 4 Private Company Valuation The following information relates to questions 22-23 You have been asked to value Pacific Corporation, Inc., using an excess earnings method, given the following information: Working capital balance = $2,000,000 Fair value of fixed assets = $5,500,000 Book value of fixed assets = $4,000,000 Normalized earnings of firm = $1,000,000 Required return on working capital = 5.0% Required return on fixed assets = 8.0% Required return on intangible assets = 15.0% Weighted average cost of capital = 10.0% Long-term growth rate of residual income = 5.0% Based on this information: 22. What is the value of Pacific’s intangible assets? 23. What is the market value of invested capital? 24. An appraiser has been asked to determine the combined level of valuation discounts for a small equity interest in a private company. The appraiser concluded that an appropriate control premium is 15%. A discount for lack of marketability was estimated at 25%. Given these factors, what is the combined discount? Solutions 331 SOLUTIONS 1. B is correct. Thunder’s size and its probable lack of access to public debt markets are potential factors affecting its valuation compared with a public company. Given that the separation of ownership and control at Thunder is similar to that at public companies, however, agency problems are not a distinguishing factor in its valuation. 2. C is correct. The excess earnings method would rarely be applied to value a company’s equity, particularly when it is not needed to value intangibles. The asset-based approach is less appropriate because it is infrequently used to estimate the business enterprise value of operating companies. By contrast, the free cash flow method is broadly applicable and readily applied in this case. 3. A is correct. Using Ebinosa’s assumptions: Revenues ($200,000,000 × 1.03 = ) Gross profit Selling, general, and administrative expenses Pro forma EBITDA Depreciation Pro forma EBIT Pro forma taxes on EBIT 45%a 24%a 2%a 35%b Operating income after tax Plus: Depreciation Less: Capital expenditures on current sales Less: Capital expenditures to support future sales Less: Working capital requirement Free cash flow to the firm $206,000,000 92,700,000 49,440,000 43,260,000 4,120,000 39,140,000 13,699,000 25,441,000 125%c 15%d 8%d 4,120,000 5,150,000 900,000 480,000 $23,031,000 a Percentage of revenues Percentage of EBIT c Percentage of depreciation d Percentage of incremental revenues b 4. C is correct. Both statements by Chin are incorrect. If the CAPM is used with public companies with similar operations and similar revenue size, as stated, then the calculation likely captures the small stock premium and should not be added to the estimate. Small stock premiums are associated with build-up models and the expanded CAPM, rather than the CAPM per se. The correct weighted average cost of capital should reflect the risk of Thunder’s cash flows, not the risk of the acquirer’s cash flows. 5. A is correct. The return on equity is the sum of the risk-free rate, equity risk premium, and the size premium for a total of 4.5 + 5.0 + 2.0 = 11.5%. The value of the firm using the CCM is V = FCFE1/(r – g) = 2.5/(0.115 – 0.03) = $29.41 million. 6. A strategic buyer seeks to eliminate unnecessary expenses. The strategic buyer would adjust the reported EBITDA by the amount of the officers’ excess compensation. A strategic buyer could also eliminate redundant manufacturing costs estimated at £600,000. The pro forma EBITDA a strategic buyer might use in its acquisition analysis is the reported EBITDA of £4,500,000 plus the non-market compensation expense of £500,000 plus the operating synergies (cost savings) of 332 Learning Module 4 Private Company Valuation £600,000. The adjusted EBITDA for the strategic buyer is £4,500,000 + £500,000 + £600,000 = £5,600,000. The financial buyer would also make the adjustment to normalize officers’ compensation but would be unable to eliminate redundant manufacturing expenses. Thus, adjusted EBITDA for the financial buyer would be £4,500,000 + £500,000 = £5,000,000. 7. B is correct. Oakstar’s primary asset is timberland, the market value of which can be determined from comparable land sales. 8. B is correct. In the absence of market value data for assets and liabilities, the analyst usually must use book value data (the assumption is explicitly made that book values accurately reflect market values as well). Except for timberland, market values for assets are unavailable. Thus, all other assets are assumed to be valued by their book values, which sum to C$500,000 + C$25,000 + C$50,000 + C$750,000 = C$1,325,000. The land’s value is determined by the value of C$8,750 per hectare for properties comparable to Oakstar’s. Thus, the value of Oakstar’s land is C$8,750 × 10,000 = C$87,500,000. Liabilities are assumed to be worth the sum of their book value, or C$1,575,000. Thus, Estimated value = Total assets – Liabilities = C$1,325,000 + C$87,500,000 – C$1,575,000 = C$87,250,000. 9. C is correct. The new interest level is US$2,000,000 instead of US$1,000,000. SG&A expenses are reduced by US$1,600,000 (= US$5,400,000 – US$7,000,000) to US$21,400,000 by salary expense savings. Other than a calculation of a revised provision for taxes, no other changes to the income statement results in normalized earnings before tax of US$58,100,000 and normalized earnings after tax of US$34,860,000. 10. B is correct: Return on working capital = 0.08 × US$10,000,000 = US$800,000 Return on fixed assets = 0.12 × US$45,000,000 = US$5,400,000 Return on intangibles = US$35,000,000 – US$800,000 – US$5,400,000 = US$28,800,000 Value of intangibles using CCM = US$28,800,000/(0.20 – 0.06) = US$205.71 million. 11. C is correct. Firm 3 matches FAMCO in both risk and growth. Firm 1 fails on these factors. In addition, Firm 3 is a better match to FAMCO than Firm 2 because the offer for Firm 3 was a cash offer in normal market conditions, whereas Firm 2 was a stock offer in a boom market and the value does not reflect risk and growth in the immediate future. 12. B is correct. Both discounts apply, and they are multiplicative rather than additive: 1 – (1 – 0.20)(1 – 0.15) = 1 – 0.68 13. The build-up method is substantially similar to the extended CAPM except that beta is excluded from the calculation. The equity return requirement is calculated as risk-free rate plus equity risk premium for large-capitalization stocks plus small stock risk premium plus company-specific risk premium: 4.5 + 5.0 + 4.2 + 3.0 = 16.7%. Although practice may vary, in this case, there was no adjustment for industry risk. 14. C is correct. Schmidt and Beckett are valuing 100% of Dairy Foody as part of their acquisition analysis. The guideline transactions method (GTM) uses a mul- Solutions 333 tiple that specifically relates to the sale of entire companies and establishes the value estimate of a business based on pricing multiples derived from the acquisition of control of entire public or private companies. The GTM is appropriate given that Schmidt prefers the valuation to reflect market pricing from recent acquisition activity in the food industry, which has been high in recent months. A is incorrect because the excess earnings method (EEM) would not likely be used to value 100% of Dairy Foody. Generally, the EEM is used to value intangible assets and very small businesses when other such market approach methods are not feasible, and the EEM is rarely used in pricing entire private businesses. Additionally, Schmidt prefers the valuation to reflect market pricing from recent acquisition activity in the food industry, which has been high in recent months; therefore, the GTM would be more appropriate than the EEM. B is incorrect because the capitalized cash flow method (CCM) would not likely be used to value 100% of Dairy Foody. The CCM is essentially a stable, growth-free cash flow model that is not a preferable valuation approach when a company is not expected to grow at a constant rate as the method employs a sustainable growth rate to estimate firm value. Dairy Foody has a growth rate that has been quite unstable over time, so the CCM would not be a preferable method to estimate its value. Additionally, the CCM is rarely used for the valuation of public companies, larger private companies, or in the context of acquisitions, but it may be appropriate if market pricing evidence from public companies or transactions is limited. Thus, the CCM would not likely be used to value Dairy Foody because Dairy Foody is a large, privately owned company that is being valued in the context of an acquisition. Finally, Schmidt prefers the valuation to reflect market pricing from recent acquisition activity in the food industry, which has been high in recent months; therefore, the GTM would be more appropriate than the CCM. 15. C is correct. In computing normalized operating income for FastDelivery, SG&A expenses are first reduced by $1,500,000 (from the current $5 million to the market rate of $3.5 million) to reflect the change of management from family members to professional management at the market rate. Second, SG&A should be reduced by an additional $550,000 to remove the family’s personal expenses. Based on those adjustments, FastDelivery’s pro forma income statement is: Exhibit 2: FastDelivery: Pro Forma Operating Income after Taxes Revenues $50,000,000 COGS 30,000,000 Gross profit 20,000,000 SG&A 7,950,000 EBITDA 12,050,000 Depreciation and amortization 3,000,000 EBIT 9,050,000 Pro forma taxes (at 25%) 2,262,500 Operating income after taxes $6,787,500 A is incorrect because it fails to adjust SG&A for the change in management salaries from $5 million to the market rate of $3.5 million: 334 Learning Module 4 Private Company Valuation Exhibit 3: FastDelivery: Pro Forma Operating Income after Taxes Revenues $50,000,000 COGS 30,000,000 Gross profit 20,000,000 SG&A 9,450,000 EBITDA 10,550,000 Depreciation and amortization 3,000,000 EBIT 7,550,000 Pro forma taxes (at 25%) 1,887,500 Operating income after taxes $5,662,500 B is incorrect because it fails to remove the family’s personal expenses from SG&A: Exhibit 4: FastDelivery: Pro Forma Operating Income after Taxes Revenues $50,000,000 COGS 30,000,000 Gross profit 20,000,000 SG&A 8,500,000 EBITDA 11,500,000 Depreciation and amortization 3,000,000 EBIT 8,500,000 Pro forma taxes (at 25%) 2,125,000 Operating income after taxes $6,375,000 16. C is correct. The required rate of return on equity for FastDelivery is 12%: Risk-free rate 1% Plus: Small-stock risk premium 3% Plus: Equity risk premium 6% Plus: Industry risk premium 2% Indicated return on equity 12% A is incorrect because the industry risk premium is excluded from the calculation: Risk-free rate 1% Plus : Small-stock risk premium 3% Plus: Equity risk premium 6% Indicated return on equity 10% B is incorrect because the risk-free rate is incorrectly excluded from the calculation: Solutions 335 Equity risk premium 6% Plus: Small-stock risk premium 3% Plus: Industry risk premium 2% Indicated return on equity 11% 17. C is correct. The required return for Spice World should be adjusted to reflect both a size premium and a company-specific risk premium. Spice World is smaller than the guideline public companies, as it has fewer employees and a lower level of total assets. Therefore, the required rate of return should be adjusted to reflect a size premium so that it reflects the level of risk of Spice World. Further, Spice World faces operational and customer concentration risks that do not impact the selected guideline public companies—risks specific to Spice World. These company-specific risks increase the level of risk of Spice World relative to that of the guideline public companies. Consequently, the required return for Spice World should also reflect a company-specific risk premium. A is incorrect because the required return for Spice World should be adjusted to reflect both a size premium and a company-specific risk premium (not just a size premium). Spice World faces operational and customer concentration risks that do not impact the selected guideline public companies—risks specific to Spice World. These company-specific risks increase the level of risk of Spice World relative to that of the guideline public companies. Consequently, the required return for Spice World should also reflect a company-specific risk premium. B is incorrect because the required rate of return for Spice World should be adjusted to reflect both a size premium and a company-specific risk premium (not just a company-specific risk premium). Spice World is smaller than the guideline public companies, as it has fewer employees and a lower level of total assets. Therefore, the required return should be adjusted to reflect a size premium so that it reflects the level of risk of Spice World. 18. B is correct. Spice World’s WACC is calculated as: Pre-tax cost of debt 9.00% Tax rate complement (1 – tax rate) × 0.75 After-tax cost of debt 6.75% Weight × 0.20 Weighted cost of debt 1.35% Cost of equity 18.00% Weight × 0.80 Weighted cost of equity 14.40% Weighted average cost of capital 15.75% A is incorrect because Spice World’s cost of equity is incorrectly adjusted for taxes: Pre-tax cost of debt 9% Tax rate complement (1 – tax rate) × 0.75 After-tax cost of debt 6.75% Weight × 0.20 Weighted cost of debt 1.35% Cost of equity 18.0% Tax rate complement (1 – tax rate) × 0.75 336 Learning Module 4 Private Company Valuation After-tax cost of equity 13.50 Weight × 0.80 Weighted cost of equity 10.80% Weighted average cost of capital 12.15% C is incorrect because Spice World’s pre-tax cost of debt, instead of its after-tax cost of debt, is incorrectly used to calculate its weighted cost of debt: Pre-tax cost of debt Weight 9% × 0.20 Weighted cost of debt 1.8% Cost of equity 18.00% Weight × 0.80 Weighted cost of equity 14.40% Weighted average cost of capital 16.20% 19. B is correct. The lack of control and lack of marketability discounts are applied in sequence. The discounts are multiplicative because the valuation process involves discrete steps—first moving from a controlling to a non-controlling basis, and then moving from a marketable to a non-marketable basis. Therefore, the 15% equity interest in Food Garden is calculated as: Indicated value of equity in operations Interest appraised $100,000,000 ×15% Pro rata value of 15% equity interest 15,000,000 Less: Lack of control discount of 20% 3,000,000 Value assuming ready marketability 12,000,000 Less: Lack of marketability discount of 35% 4,200,000 Indicated value of 15% equity interest $7,800,000 A is incorrect because the lack of control and marketability discounts are incorrectly added together. Valuation discounts are multiplicative and applied sequentially, not added: Indicated value of equity in operations Interest appraised $100,000,000 ×15% Pro rata value of 15% equity interest 15,000,000 Less: 55%, the sum of the DLOC of 20% and the DLOM of 35% 8,250,000 Indicated value of 15% equity interest $6,750,000 C is incorrect because the DLOC is incorrectly subtracted from the lack of marketability discount. Valuation discounts are multiplicative and applied sequentially, not subtracted from one another: Indicated value of equity in operations Interest appraised $100,000,000 ×15% Pro rata value of 15% equity interest 15,000,000 Less: 15%, the difference between the DLOM of 35% and the DLOC of 20% 2,250,000 Solutions 337 Indicated value of 15% equity interest $12,750,000 20. A is correct. Using the guideline public company method, the value of LiveLong Foods is calculated in two steps: First, calculate the pricing multiple for LiveLong Foods: Initial MVIC to EBITDA from public companies Relative risk and growth adjustment for LiveLong Foods 10.0 −12% (1.2) 18% 1.584 Multiple before control adjustment Control premium adjustment Multiple after control adjustment 8.8 10.384 Second, calculate the valuation of LiveLong Foods: Normalized EBITDA Pricing multiple Indicated value of invested capital Less: Debt capital Indicated value of equity $12,000,000 × 10.384 $124,608,000 $6,000,000 $118,608,000 B is incorrect because the downward adjustment to LiveLong Foods’ pricing multiple and control premium adjustment were incorrectly added together. The valuation adjustments should be applied sequentially, not added: Initial MVIC to EBITDA from public companies 6%, the sum of the control premium adjustment and the relative risk and growth adjustment for LiveLong Foods 10.0 18% − 12% Multiple after adjustment Normalized EBITDA Pricing multiple Indicated value of invested capital Less: Debt capital Indicated value of equity 0.6 10.6 $12,000,000 × 10.6 $127,200,000 $6,000,000 $121,200,000 C is incorrect because the market value of LiveLong Foods’ debt is incorrectly not deducted from the indicated value of invested capital in arriving at the value of LiveLong Foods’ equity: Normalized EBITDA Pricing multiple Indicated value of invested capital $12,000,000 × 10.384 $124,608,000 21. There are FCFF and FCFE variations of the CCM. In this problem, the data permit the application of only the FCFE variation. According to that variation, the estimated value of equity equals the normalized free cash flow to equity estimate for next period divided by the capitalization rate for equity. The capitalization rate is the required rate of return for equity less the long-term growth rate in free 338 Learning Module 4 Private Company Valuation cash flow to equity. Using the current $1.8 million of free cash flow to equity, the 18% equity discount rate, and the long-term growth rate of 5.5% yields a value indication of [($1.8 million)(1.055)]/(0.18 – 0.055) = $1.899 million/0.125 = $15.19 million. 22. The residual income for intangible assets is $460,000 (the normalized earnings of $1,000,000 less the $540,000 required return for working capital and fixed assets). The value of intangible assets can then be calculated using the capitalized cash flow method. The intangibles value is $4,830,000 based on $483,000 of year-ahead residual income available to the intangibles capitalized at 10.0% (15.0% discount rate for intangibles less 5.0% long-term growth rate of residual income). 23. The market value of invested capital is the total of the values of working capital, fixed assets, and intangible assets. This value is $2,000,000 + $5,500,000 + $4,830,000 = $12,330,000. 24. The valuation of a small equity interest in a private company would typically be calculated on a basis that reflects the lack of control and lack of marketability of the interest. The control premium of 15% must first be used to provide an indication of a discount for lack of control (DLOC). A lack of control discount can be calculated using the formula Lack of control discount = 1 – [1/ (1 + Control premium)]. In this case, a lack of control discount of approximately 13% is calculated as 1 – [1/(1 + 15%)]. The discount for lack of marketability (DLOM) was specified. Valuation discounts are applied sequentially and are not added. The formula is (Pro rata control value) × (1 – DLOC) × (1 – DLOM). A combined discount of approximately 35% is calculated as 1 – (1 – 13%) × (1 – 25%) = 0.348, or 34.8%. Fixed Income Practice Problems PRACTICE PROBLEMS 1. Given spot rates for one-, two-, and three-year zero coupon bonds, how many forward rates can be calculated? 2. Give two interpretations for the following forward rate: The two-year forward rate one year from now is 2%. 3. Describe the relationship between forward rates and spot rates if the yield curve is flat. 4. Which forward rate cannot be computed from the one-, two-, three-, and four-year spot rates? The rate for a: A. one-year loan beginning in two years B. two-year loan beginning in two years C. three-year loan beginning in two years 5. Consider spot rates for three zero-coupon bonds: z(1) = 3%, z(2) = 4%, and z(3) = 5%. Which statement is correct? The forward rate for a one-year loan beginning in one year will be: A. less than the forward rate for a one-year loan beginning in two years. B. greater than the forward rate for a two-year loan beginning in one year. C. greater than the forward rate for a one-year loan beginning in two years. 6. If one-period forward rates are decreasing with maturity, the yield curve is most likely: A. flat. B. upward sloping. C. downward sloping. The following information relates to questions 7-17 A one-year zero-coupon bond yields 4.0%. The two- and three-year zero-coupon bonds yield 5.0% and 6.0%, respectively. 7. The rate for a one-year loan beginning in one year is closest to: A. 4.5%. B. 5.0%. C. 6.0%. 8. The forward rate for a two-year loan beginning in one year is closest to: A. 5.0%. 387 388 Learning Module 1 The Term Structure and Interest Rate Dynamics B. 6.0%. C. 7.0%. 9. The forward rate for a one-year loan beginning in two years is closest to: A. 6.0%. B. 7.0%. C. 8.0%. 10. The five-year spot rate is not provided here; however, the forward price for a two-year zero-coupon bond beginning in three years is known to be 0.8479. The price today of a five-year zero-coupon bond is closest to: A. 0.7119. B. 0.7835. C. 0.9524. 11. The one-year spot rate z1 is 4%, the forward rate for a one-year loan beginning in one year is 6%, and the forward rate for a one-year loan beginning in two years is 8%. Which of the following rates is closest to the three-year spot rate? A. 4.0% B. 6.0% C. 8.0% 12. The one-year spot rate z1 is 5%, and the forward price for a one-year zero-coupon bond beginning in one year is 0.9346. The spot price of a two-year zero-coupon bond is closest to: A. 0.87. B. 0.89. C. 0.93. 13. In a typical interest rate swap contract, the swap rate is best describedas the interest rate for the: A. fixed-rate leg of the swap. B. floating-rate leg of the swap. C. difference between the fixed and floating legs of the swap. 14. A two-year fixed-for-floating MRR swap is 1.00%, and the two-year US Treasury bond is yielding 0.63%. The swap spread is closest to: A. 37 bps. B. 100 bps. C. 163 bps. 15. The swap spread is quoted as 50 bps. If the five-year US Treasury bond is yielding Practice Problems 389 2%, the rate paid by the fixed payer in a five-year interest rate swap is closest to: A. 0.50%. B. 1.50%. C. 2.50%. 16. If the three-month T-bill rate drops and MRR remains the same, the relevant TED spread: A. increases. B. decreases. C. does not change. 17. Given the yield curve for US Treasury zero-coupon bonds, which spread is most helpful pricing a corporate bond? The: A. Z-spread. B. TED spread. C. MRR–OIS spread, formerly the Libor–OIS spread. The following information relates to questions 18-24 Jane Nguyen is a senior bond trader for an investment bank, and Chris Alexander is a junior bond trader at the bank. Nguyen is responsible for her own trading activities and also for providing assignments to Alexander that will develop his skills and create profitable trade ideas. Exhibit 1 presents the current par and spot rates. Exhibit 1: Current Par and Spot Rates Maturity Par Rate Spot Rate One year 2.50% 2.50% Two years 2.99% 3.00% Three years 3.48% 3.50% Four years 3.95% 4.00% Five years 4.37% Note: Par and spot rates are based on annual-coupon sovereign bonds. Nguyen gives Alexander two assignments that involve researching various questions: Assignment 1 What is the yield-to-maturity of the option-free, default-risk-free bond presented in Exhibit 2? Assume that the bond is held to maturity, and use the rates shown in Exhibit 1. 390 Learning Module 1 The Term Structure and Interest Rate Dynamics Exhibit 2: Selected Data for $1,000 Par Bond Bond Name Maturity (T) Coupon Bond Z Three years 6.00% Note: Terms are today for a T-year loan. Assignment 2 Assuming that the projected spot curve two years from today will be below the current forward curve, is Bond Z fairly valued, undervalued, or overvalued? After completing his assignments, Alexander asks about Nguyen’s current trading activities. Nguyen states that she has a two-year investment horizon and will purchase Bond Z as part of a strategy to ride the yield curve. Exhibit 1 shows Nguyen’s yield curve assumptions implied by the spot rates. 18. Based on Exhibit 1, the five-year spot rate is closest to: A. 4.40%. B. 4.45%. C. 4.50%. 19. Based on Exhibit 1, the market is most likely expecting: A. deflation. B. inflation. C. no risk premiums. 20. Based on Exhibit 1, the forward rate of a one-year loan beginning in three years is closest to: A. 4.17%. B. 4.50%. C. 5.51%. 21. Based on Exhibit 1, which of the following forward rates can be computed? A. A one-year loan beginning in five years B. A three-year loan beginning in three years C. A four-year loan beginning in one year 22. For Assignment 1, the yield-to-maturity for Bond Z is closest to the: A. one-year spot rate. B. two-year spot rate. C. three-year spot rate. 23. For Assignment 2, Alexander should conclude that Bond Z is currently: A. undervalued. Practice Problems 391 B. fairly valued. C. overvalued. 24. By choosing to buy Bond Z, Nguyen is most likely making which of the following assumptions? A. Bond Z will be held to maturity. B. The three-year forward curve is above the spot curve. C. Future spot rates do not accurately reflect future inflation. The following information relates to questions 25-29 Laura Mathews recently hired Robert Smith, an investment adviser at Shire Gate Advisers, to assist her in investing. Mathews states that her investment time horizon is short, approximately two years or less. Smith gathers information on spot rates for on-the-run annual-coupon government securities and swap spreads, as presented in Exhibit 1. Shire Gate Advisers recently published a report for its clients stating its belief that, based on the weakness in the financial markets, interest rates will remain stable, the yield curve will not change its level or shape for the next two years, and swap spreads will also remain unchanged. Exhibit 1: Government Spot Rates and Swap Spreads Maturity (years) 1 2 3 4 Government spot rate 2.25% 2.70% 3.30% 4.05% Swap spread 0.25% 0.30% 0.45% 0.70% Smith decides to examine the following three investment options for Mathews: Investment 1: Buy a government security that would have an annualized return that is nearly risk free. Smith is considering two possible implementations: a two-year investment or a combination of two one-year investments. Investment 2: Buy a four-year, zero-coupon corporate bond and then sell it after two years. Smith illustrates the returns from this strategy using the swap rate as a proxy for corporate yields. Investment 3: Buy a lower-quality, two-year corporate bond with a coupon rate of 4.15% and a Z-spread of 65 bps. When Smith meets with Mathews to present these choices, Mathews tells him that she is somewhat confused by the various spread measures. She is curious to know whether there is one spread measure that could be used as a good indicator of the risk and liquidity of money market securities during the recent past. 25. In his presentation of Investment 1, Smith could show that under the no-arbitrage principle, the forward price of a one-year government bond to be 392 Learning Module 1 The Term Structure and Interest Rate Dynamics issued in one year is closest to: A. 0.9662. B. 0.9694. C. 0.9780. 26. In presenting Investment 1, using Shire Gate Advisers’ interest rate outlook, Smith could show that riding the yield curve provides a total return that is most likely: A. lower than the return on a maturity-matching strategy. B. equal to the return on a maturity-matching strategy. C. higher than the return on a maturity-matching strategy. 27. In presenting Investment 2, Smith should show an annual return closest to: A. 4.31%. B. 5.42%. C. 6.53%. 28. The bond in Investment 3 is most likely trading at a price of: A. 100.97. B. 101.54. C. 104.09. 29. The most appropriate response to Mathews question regarding a spread measure is the: A. Z-spread. B. TED spread. C. MRR–OIS spread, formerly the Libor–OIS spread. The following information relates to questions 30-40 Liz Tyo is a fund manager for an actively managed global fixed-income fund that buys bonds issued in Countries A, B, and C. She and her assistant are preparing the quarterly markets update. Tyo begins the meeting by distributing the daily rates sheet, which includes the current government spot rates for Countries A, B, and C as shown in Exhibit 1. Practice Problems 393 Exhibit 1: Today’s Government Spot Rates Maturity Country A Country B Country C One year 0.40% –0.22% 14.00% Two years 0.70 –0.20 12.40 Three years 1.00 –0.12 11.80 Four years 1.30 –0.02 11.00 Five years 1.50 0.13 10.70 Tyo asks her assistant how these spot rates were obtained. The assistant replies, “Spot rates are determined through the process of bootstrapping. It entails backward substitution using par yields to solve for zero-coupon rates one by one, in order from latest to earliest maturities.” Tyo then provides a review of the fund’s performance during the last year and comments, “The choice of an appropriate benchmark depends on the country’s characteristics. For example, although Countries A and B have both an active government bond market and a swap market, Country C’s private sector is much bigger than its public sector, and its government bond market lacks liquidity.” Tyo further points out, “The fund’s results were mixed; returns did not benefit from taking on additional risk. We are especially monitoring the riskiness of the corporate bond holdings. For example, our largest holdings consist of three four-year corporate bonds (Bonds 1, 2, and 3) with identical maturities, coupon rates, and other contract terms. These bonds have Z-spreads of 0.55%, 1.52%, and 1.76%, respectively.” Tyo continues, “We also look at risk in terms of the swap spread. We considered historical three-year swap spreads for Country B, which reflect that market’s credit and liquidity risks, at three different points in time.” Tyo provides the information in Exhibit 2. Exhibit 2: Selected Historical Three-Year Rates for Country B Period 1 month ago Government Bond Yield (%) Fixed-for-Floating MRR Swap (%) –0.10 0.16 6 months ago –0.08 0.01 12 months ago –0.07 0.71 Tyo then suggests that the firm was able to add return by riding the yield curve. The fund plans to continue to use this strategy but only in markets with an attractive yield curve for this strategy. She moves on to present her market views on the respective yield curves for a five-year investment horizon. Country A: “The government yield curve has changed little in terms of its level and shape during the last few years, and I expect this trend to continue. We assume that future spot rates reflect the current forward curve for all maturities.” Country B: “Because of recent economic trends, I expect a reversal in the slope of the current yield curve. We assume that future spot rates will be higher than current forward rates for all maturities.” 394 Learning Module 1 The Term Structure and Interest Rate Dynamics Country C: “To improve liquidity,Country C’s central bank is expected to intervene, leading to a reversal in the slope of the existing yield curve. We assume that future spot rates will be lower than today’s forward rates for all maturities.” Tyo’s assistant asks, “Assuming investors require liquidity premiums, how can a yield curve slope downward? What does this imply about forward rates?” Tyo answers, “Even if investors require compensation for holding longer-term bonds, the yield curve can slope downward—for example, if there is an expectation of severe deflation. Regarding forward rates, it can be helpful to understand yield curve dynamics by calculating implied forward rates. To see what I mean, we can use Exhibit 1 to calculate the forward rate for a two-year Country C loan beginning in three years.” 30. Did Tyo’s assistant accurately describe the process of bootstrapping? A. Yes B. No, with respect to par yields C. No, with respect to backward substitution 31. The swap curve is a better benchmark than the government spot curve for: A. Country A. B. Country B. C. Country C. 32. Based on Exhibit 2, the implied credit and liquidity risks as indicated by the historical three-year swap spreads for Country B were the lowest: A. 1 month ago. B. 6 months ago. C. 12 months ago. 33. Based on Exhibit 1 and Tyo’s expectations, which country’s term structure is currently best for traders seeking to ride the yield curve? A. Country A B. Country B C. Country C 34. Based on Exhibit 1 and assuming Tyo’s market views on yield curve changes are realized, the forward curve of which country will lie below its spot curve? A. Country A B. Country B C. Country C 35. Based on Exhibit 1 and Tyo’s expectations for the yield curves, Tyo most likely perceives the bonds of which country to be fairly valued? A. Country A Practice Problems B. Country B C. Country C 36. With respect to their discussion of yield curves, Tyo and her assistant are most likely discussing which term structure theory? A. Pure expectations theory B. Local expectations theory C. Liquidity preference theory 37. Tyo’s assistant should calculate a forward rate closest to: A. 9.07%. B. 9.58%. C. 9.97%. 38. During economic expansions, monetary authorities raise benchmark rates to help control inflation. This action is most often consistent with: A. bearish flattening. B. bullish steepening. C. bearish steepening. 39. When government budget deficits fall, fiscal supply-side effects are most likely to result in: A. higher bond yields. B. a steeper yield curve. C. lower bond yields. 40. A flight to quality is most often associated with: A. a general rise in the level of interest rates. B. bullish flattening. C. bearish flattening. The following information relates to questions 41-42 41. Define the yield-to-maturity for a coupon bond. 42. Is it possible for a coupon bond to earn less than the yield-to-maturity if held to maturity? 43. If a bond trader believes that current forward rates overstate future spot rates, 395 396 Learning Module 1 The Term Structure and Interest Rate Dynamics how might she profit from that conclusion? 44. Explain the strategy of rolling down the yield curve. 45. What are the advantages of using the swap curve as a benchmark of interest rates relative to a government bond yield curve? 46. What is the TED spread, and what type of risk does it measure? 47. What is the SOFR rate, and which market conditions does it reflect? The following information relates to questions 48-51 Rowan Madison is a junior analyst at Cardinal Capital. Sage Winter, a senior portfolio manager and Madison’s supervisor, meets with Madison to discuss interest rates and review two bond positions in the firm’s fixed-income portfolio. Winter begins the meeting by asking Madison to state her views on the term structure of interest rates. Madison responds: “Yields are a reflection of expected spot rates and risk premiums. Investors demand risk premiums for holding long-term bonds, and these risk premiums increase with maturity.” Winter tells Madison that, based on recent changes in spreads, she is concerned about a perceived increase in counterparty risk in the economy and its effect on the portfolio. Madison asks Winter: “Which spread measure should we use to assess changes in counterparty risk in the economy?” Winter is also worried about the effect of yield volatility on the portfolio. She asks Madison to identify the economic factors that affect short-term and long-term rate volatility. Madison responds: “Short-term rate volatility is mostly linked to uncertainty regarding monetary policy, whereas long-term rate volatility is mostly linked to uncertainty regarding the real economy and inflation.” Finally, Winter asks Madison to analyze the interest rate risk portfolio positions in a 5-year and a 20-year bond. Winter requests that the analysis be based on level, slope, and curvature as term structure factors. Madison presents her analysis in Exhibit 1. Exhibit 1: Three-Factor Model of Term Structure Time to Maturity (years) Factor 5 20 Level –0.4352% –0.5128% Steepness –0.0515% –0.3015% Curvature 0.3963% 0.5227% Note: Entries indicate how yields would change for a one standard deviation increase in a factor. Winter asks Madison to perform two analyses: Practice Problems Analysis 1: Calculate the expected change in yield on the 20-year bond resulting from a two-standard-deviation increase in the steepness factor. Analysis 2: Calculate the expected change in yield on the five-year bond resulting from a one-standard-deviation decrease in the level factor and a one-standard-deviation decrease in the curvature factor. 48. Madison’s views on the term structure of interest rates are most consistent with the: A. local expectations theory. B. segmented markets theory. C. liquidity preference theory. 49. Is Madison’s response regarding the factors that affect short-term and long-term rate volatility correct? A. Yes B. No, she is incorrect regarding factors linked to long-term rate volatility C. No, she is incorrect regarding factors linked to short-term rate volatility 50. Based on Exhibit 1, the results of Analysis 1 should show the yield on the 20-year bond decreasing by: A. 0.3015%. B. 0.6030%. C. 0.8946%. 51. Based on Exhibit 1, the results of Analysis 2 should show the yield on the five-year bond: A. decreasing by 0.8315%. B. decreasing by 0.0389%. C. increasing by 0.0389%. 52. According to the local expectations theory, what would be the difference in the one-month total return if an investor purchased a five-year zero-coupon bond versus a two-year zero-coupon bond? 53. Compare the segmented market and the preferred habitat term structure theories. The following information relates to questions 54-56 54. List the three factors that have empirically been observed to affect Treasury 397 398 Learning Module 1 The Term Structure and Interest Rate Dynamics security returns and explain how each of these factors affects returns on Treasury securities. 55. What has been observed to be the most important factor in affecting Treasury returns? 56. Which measures of yield curve risk can measure shaping risk? Solutions 399 SOLUTIONS 1. Three forward rates can be calculated from the one-, two- and three-year spot rates. The rate on a one-year loan that begins at the end of Year 1 can be calculated using the one- and two-year spot rates; in the following equation, one would solve for f1,1: [1 + z2]2 = [1 + z1]1[1 + f1,1]1 The rate on a one-year loan that starts at the end of Year 2 can be calculated from the two- and three-year spot rates. In the following equation, one would solve for f2,1: [1 + z3]3 = [1 + z2]2[1 + f2,1]1 Additionally, the rate on a two-year loan that begins at the end of Year 1 can be computed from the one- and three-year spot rates. In the following equation, one would solve for f1,2: [1 + z3]3 = [1 + z1]1[1 + f1,2]2 2. For the two-year forward rate one year from now of 2%, the two interpretations are as follows: ■ 2% is the rate that will make an investor indifferent between buying a three-year zero-coupon bond or investing in a one-year zero-coupon bond and, when it matures, reinvesting in a zero-coupon bond that matures in two years. ■ 2% is the rate that can be locked in today by buying a three-year zero-coupon bond rather than investing in a one-year zero-coupon bond and, when it matures, reinvesting in a zero-coupon bond that matures in two years. 3. A flat yield curve implies that all spot interest rates are the same. When the spot rate is the same for every maturity, successive applications of the forward rate model will show that all the forward rates will also be the same and equal to the spot rate. 4. C is correct. There is no spot rate information to provide rates for a loan that terminates in five years. That is f2,3 is calculated as follows: √ 3 _ 5 ​[​1 + ​z​5​]​ ​ _ ​​f2,3 ​ ​= ​ ​ 2 ​− ​[​1 + ​z​2​]​ ​ 1 This equation indicates that in order to calculate the rate for a three-year loan beginning at the end of two years, one needs the five-year spot rate, z5, and the two-year spot rate, z2. However, z5 is not provided. 5. A is correct. The forward rate for a one-year loan beginning in one year, f1,1, is 1.042/1.03 – 1 = 5%. The rate for a one-year loan beginning in two years, f2,1, is 1.053/1.042 – 1 = 7%. This confirms that an upward-sloping yield curve is consistent with an upward-sloping forward curve. 6. C is correct. If one-period forward rates are decreasing with maturity, then the forward curve is downward sloping. This turn implies a downward-sloping yield curve where longer-term spot rates zB–A are less than shorter-term spot rates zA. 400 Learning Module 1 The Term Structure and Interest Rate Dynamics 7. C is correct. From the forward rate model, we have [1 + z2]2 = [1 + z1]1[1 + f1,1]1 Using the one- and two-year spot rates, we have ​(​1 + 0.05​)​2​ ​ ​1 + 0.04​ ​ ​ ​ ​− 1​ = f1,1 = 6.010%. (1 + 0.05)2 = (1 + 0.04)1[1 + f1,1]1, so _ ( )1 8. C is correct. From the forward rate model, [1 + z3]3 = [1 + z1]1[1 + f1,2]2 Using the one- and three-year spot rates, we find (1 + 0.06)3 = (1 + 0.04)1[1 + f1,2 ]2, √ _ (​ ​1 + 0.06​)​3​ so ​ _ ​ ​ − 1​ = f1,2 = 7.014%. (​ ​1 + 0.04​)​1​ 9. C is correct. From the forward rate model, [1 + z3]3 = [1 + z2]2[1 + f2,1]1 Using the two- and three-year spot rates, we find (​ ​1 + 0.06​)​3 ​ ​1 + 0.05​ ​ ​ ​ ​− 1​ = f2,1 = 8.029%. (1 + 0.06)3 = (1 + 0.05)2[1 + f2,1]1, so _ ( )2 10. A is correct. We can convert spot rates to spot prices to find DF3 = _ ​ 1 3 ​​= ​(​1.06​)​ ​ 0.8396. The forward pricing model can be used to find the price of the five-year zero as DFB = DFA ×FA,B–A,so DF5 = DF3F3,2 = 0.8396 × 0.8479 = 0.7119. 11. B is correct. Applying the forward rate model, we find [1 + z3]3 = [1 + z1]1[1 + f1,1]1[1 + f2,1]1 3 _ So [1 + z3]3 = (1 + 0.04)1(1 + 0.06)1(1 + 0.08)1, ​√1.1906 ​− 1​ = z3 = 5.987%. 12. B is correct. We can convert spot rates to spot prices and use the forward pricing model, so we have DF1 = _ ​ 1 1 ​= 0.9524. ​(​1.05​)​ ​ The forward pricing model is DFB = DFA ×FA,B–A,so DF2 = DF1F1,1 = 0.9524 × 0.9346 = 0.8901. 13. A is correct. The swap rate is the interest rate for the fixed-rate leg of an interest rate swap. 14. A is correct. The swap spread = 1.00% − 0.63% = 0.37%, or 37 bps. 15. C is correct. The fixed leg of the five-year fixed-for-floating swap will be equal to the five-year Treasury rate plus the swap spread: 2.0% + 0.5% = 2.5%. 16. A is correct. The TED spread is the difference between the three-month MRR and the three-month Treasury bill rate. If the T-bill rate falls and MRR does not change, the TED spread will increase. 17. A is correct. The Z spread is the single rate that, when added to the rates of the spot yield curve, will provide the correct discount rates to price a particular risky bond. 18. B is correct. The five-year spot rate is determined by using forward substitution and using the known values of the one-year, two-year, three-year, and four-year spot rates, as follows: Solutions 401 0.0437 0.0437 0.0437 0.0437 1 + 0.0437 ​1 = _ ​1.025 ​+ _ ​( +_ ​ +_ ​ +_ ​ 5 ​ ​1.03​)​2 (​ ​1.035​)​3 (​ ​1.04​)​4 ​ ​1 + ​z​ ​ ​ ( 5) _ √ 1.0437 ​0.8394 ​ − 1 = 4.453% ​​z​5​ = ​ _ 5 19. B is correct. The spot rates imply an upward-sloping yield curve, z3 > z2 > z1. Because nominal yields incorporate a premium for expected inflation, an upward-sloping yield curve is generally interpreted as reflecting a market expectation of increasing, or at least level, future inflation (associated with relatively strong economic growth). 20. C is correct. A one-year loan beginning in three years, or f3,1, is calculated as follows: 3+1 ​​[​1 + ​z​3+1​]​ ​ = [​ ​1 + ​z​3​]​ ​​[​1 + ​f3,1 ​ ​]​ 1 3 ​​[​1.04​]​4​ = [​ ​1.035​]​3​​[​1 + ​f3,1 ​ ​] ​(​1.04​)​4​ ​( ​− 1 = 5.514% ​​f3,1 ​ ​= _ )3 ​ ​1.035​ ​ ​ 21. C is correct. Exhibit 1 provides five years of par rates, from which the spot rates for z1, z2, z3, z4, and z5 can be derived. Thus the forward rate f1,4 can be calculated as follows: √ 4 _ 5 ​[​1 + ​z​5​]​ ​ _ ​​f1,4 ​ ​= ​ ​ 1 ​− ​[​1 + ​z​1​]​ ​ 1 22. C is correct. The yield-to-maturity, y3, of Bond Z should be a weighted average of the spot rates used in the valuation of the bond. Because the bond’s largest cash flow occurs in Year 3, z3 will have a greater weight than z1 and z2 in determining y3. Using the spot rates: $1, 060 $60 $60 ​+ _ ​( ​+ _ ​( ​ = $1, 071.16 ​Price = ​_ ( )1 )2 )3 ​ ​1.025​ ​ ​ ​ ​1.030​ ​ ​ ​ ​1.035​ ​ ​ Using the yield-to-maturity: $1, 060 $60 $60 _ _ ​Price = _ ​ 1 ​+ ​ 2 ​+ ​ 3 ​ = $1, 071.16 ​[​1 + y​(​ ​3​)​]​ ​ ​ ​[​1 + y​(​ ​3​)​]​ ​ ​[​1 + y​(​ ​3​)​​]​ The computed result is y3 = 3.46%, which is closest to the three-year spot rate of 3.50%. 23. A is correct. Alexander projects that the spot curve two years from today will be below the current forward curve, which implies that her expected future spot rates beyond two years will be lower than the quoted forward rates. Alexander would perceive Bond Z to be undervalued in the sense that the market is effectively discounting the bond’s payments at a higher rate than she would, and the bond’s market price is below her estimate of intrinsic value. 24. B is correct. Nguyen’s strategy is to ride the yield curve, which is appropriate when the yield curve is upward sloping. The yield curve implied by Exhibit 1 is upward sloping, which implies that the three-year forward curve is above the current spot curve. When the yield curve slopes upward, as a bond approaches maturity or “rolls down the yield curve,” the bond is valued at successively lower 402 Learning Module 1 The Term Structure and Interest Rate Dynamics yields and higher prices. 25. B is correct. The forward pricing model is based on the no-arbitrage principle and is used to calculate a bond’s forward price based on the spot yield curve. The spot curve is constructed by using annualized rates from option-free and default-risk-free zero-coupon bonds. Equation 2: DFB = DFA ×FA,B-A; we need to solve for F1,1. DF1 = 1/(1 + 0.0225)1 and DF2 = 1/(1 + 0.0270)2, F1,1 = DF2/DF1 = 0.9481/0.9780 = 0.9694. 26. C is correct. When the spot curve is upward sloping and its level and shape are expected to remain constant over an investment horizon (Shire Gate Advisers’ view), buying bonds with a maturity longer than the investment horizon (i.e., riding the yield curve) will provide a total return greater than the return on a maturity-matching strategy. 27. C is correct. The swap spread is a common way to indicate credit spreads in a market. The four-year swap rate (fixed leg of an interest rate swap) can be used as an indication of the four-year corporate yield. Riding the yield curve by purchasing a four-year zero-coupon bond with a yield of 4.75% {i.e., 4.05% + 0.70%, [P4 = 100/(1 + 0.0475)4 = 83.058]} and then selling it when it becomes a two-year zero-coupon bond with a yield of 3.00% {i.e., 2.70% + 0.30%, [P2 = 100/(1 + 0.0300)2 = 94.260]} produces an annual return of 6.53%: (94.260/83.058)0.5 – 1.0 = 0.0653. 28. B is correct. The Z-spread is the constant basis point spread that is added to the default-free spot curve to price a risky bond. A Z-spread of 65 bps for a particular bond would imply adding a fixed spread of 65 bps to maturities along the spot curve to correctly price the bond. Therefore, for the two-year bond, z1 = 2.90% (i.e., 2.25% + 0.65%), z2 = 3.35% (i.e., 2.70% + 0.65%), and the price of the bond with an annual coupon of 4.15% is as follows: P = 4.15/(1 +0. 029)1 + 4.15/(1 + 0.0335)2 + 100/(1 + 0.0335)2, P = 101.54. 29. C is correct. The MRR–OIS spread is considered an indicator of the risk and liquidity of money market securities. This spread measures the difference between MRR and the OIS rate. 30. C is correct. The assistant states that bootstrapping entails backward substitution using par yields to solve for zero-coupon rates one by one, in order from latest to earliest maturities. Bootstrapping entails forward substitution, however, using par yields to solve for zero-coupon rates one by one, in order from earliest to latest maturities. 31. C is correct. Country C’s private sector is much bigger than the public sector, and the government bond market in Country C currently lacks liquidity. Under such circumstances, the swap curve is a more relevant benchmark for interest rates. 32. B is correct. The historical three-year swap spread for Country B was the lowest six months ago. Swap spread is defined as the spread paid by the fixed-rate payer of an interest rate swap over the rate of the “on the run” (most recently issued) government bond security with the same maturity as the swap. The lower (higher) the swap spread, the lower (higher) the return that investors require for credit Solutions and/or liquidity risks. The fixed rate of the three-year fixed-for-floating Libor swap was 0.01% six months ago, and the three-year government bond yield was –0.08% six months ago. Thus the swap spread six months ago was 0.01% – (–0.08%) = 0.09%. One month ago, the fixed rate of the three-year fixed-for-floating Libor swap was 0.16%, and the three-year government bond yield was –0.10%. Thus the swap spread one month ago was 0.16% – (–0.10%) = 0.26%. Twelve months ago, the fixed rate of the three-year fixed-for-floating Libor swap was 0.71%, and the three-year government bond yield was –0.07%. Thus, the swap spread 12 months ago was 0.71% – (–0.07%) = 0.78%. 33. A is correct. Country A’s yield curve is upward sloping—a condition for the strategy—and more so than Country B’s. 34. B is correct. The yield curve for Country B is currently upward sloping, but Tyo expects a reversal in the slope of the current yield curve. This means she expects the resulting yield curve for Country B to slope downward, which implies that the resulting forward curve would lie below the spot yield curve. The forward curve lies below the spot curve in scenarios in which the spot curve is downward sloping; the forward curve lies above the spot curve in scenarios in which the spot curve is upward sloping. A is incorrect because the yield curve for Country A is currently upward sloping and Tyo expects that the yield curve will maintain its shape and level. That expectation implies that the resulting forward curve would be above the spot yield curve. C is incorrect because the yield curve for Country C is currently downward sloping and Tyo expects a reversal in the slope of the current yield curve. She thus expects the resulting yield curve for Country C to slope upward, which implies that the resulting forward curve would be above the spot yield curve. 35. A is correct. Tyo’s projected spot curve assumes that future spot rates reflect, or will be equal to, the current forward rates for all respective maturities. This assumption implies that the bonds for Country A are fairly valued because the market is effectively discounting the bond’s payments at spot rates that match those projected by Tyo. B and C are incorrect because Tyo’s projected spot curves for the two countries do not match the current forward rates for all respective maturities. In the case of Country B, she expects future spot rates to be higher (than the current forward rates that the market is using to discount the bond’s payments). For Country C, she expects future spot rates to be lower (than the current forward rates). Hence, she perceives the Country B bond to be currently overvalued and the Country C bond to be undervalued. 36. C is correct. Liquidity preference theory suggests that liquidity premiums exist to compensate investors for the added interest rate risk that they face when lending long term and that these premiums increase with maturity. Tyo and her assistant are assuming that liquidity premiums exist. 37. A is correct. From the forward rate model, f3,2, is found as follows: [1 + z5]5 = [1 + z3]3[1 + f3,2]2 Using the three-year and five-year spot rates, we find (1 + 0.107)5 = (1 + 0.118)3[1 + f3,2]2, so √ _ (​ ​1 + 0.107​)​5​ ​( ​ − 1 = ​f3,2 ​ ​ = 9.07% ​ ​ _ ​ ​1 + 0.118​)​3​ 403 404 Learning Module 1 The Term Structure and Interest Rate Dynamics 38. A is correct. This action is most often consistent with bearish flattening, or short-term bond yields rising more than long-term bond yields resulting in a flatter yield curve. 39. C is correct. When government budget deficits fall, fiscal supply-side effects are most likely to result in lower bond yields. 40. B is correct. A flight to quality is most often associated with bullish flattening, in which the yield curve flattens as long term rates fall by more than short-term rates. 41. The yield-to-maturity of a coupon bond is the expected rate of return on a bond if the bond is held to maturity, there is no default, and the bond and all coupons are reinvested at the original yield-to-maturity. 42. Yes, it is possible. For example, if reinvestment rates for the future coupons are lower than the initial yield-to-maturity, a bondholder may experience lower realized returns. 43. If forward rates are higher than expected future spot rates, the market price of the bond will be lower than the intrinsic value. This dynamic occurs because, everything else held constant, the market is currently discounting the bonds cash flows at a higher rate than the investor’s expected future spot rates. The investor can capitalize on this scenario by purchasing the undervalued bond. If expected future spot rates are realized, then bond prices should rise, thus generating gains for the investor. 44. The strategy of rolling down the yield curve is one in which a bond trader attempts to generate a total return over a given investment horizon that exceeds the return to bond with maturity matched to the horizon. The strategy involves buying a bond with maturity more distant than the investment horizon. Assuming an upward-sloping yield curve, if the yield curve does not change level or shape, as the bond approaches maturity (or rolls down the yield curve) it will be priced at successively lower yields. So as long as the bond is held for a period less than maturity, it should generate higher returns because of price gains. 45. Some countries do not have active government bond markets with trading at all maturities. For those countries without a liquid government bond market but with an active swap market, there are typically more points available to construct a swap curve than a government bond yield curve. For those markets, the swap curve may be a superior benchmark. 46. The TED spread is the difference between MRR and the US T-bill rate of matching maturity. It is an indicator of perceived credit and liquidity risk. In particular, because sovereign debt instruments are typically the benchmark for the lowest default risk instruments in a given market, and loans between banks (often at MRR) have some counterparty risk, the TED spread is considered to at least in part reflect default (or counterparty) risk in the banking sector. 47. The secured overnight financing rate (SOFR), or overnight cash borrowing rate collateralized by US Treasuries, is a barometer of the US Treasury repurchase (or repo) market. SOFR is a volume-weighted index of all qualified repo market transactions on a given day and is influenced by supply and demand conditions in secured funding markets. 48. C is correct. Liquidity preference theory asserts that investors demand a risk premium, in the form of a liquidity premium, to compensate them for the added interest rate risk they face when buying long-maturity bonds. The theory also Solutions states that the liquidity premium increases with maturity. 49. A is correct. Madison’s response is correct; research indicates that short-term rate volatility is mostly linked to uncertainty regarding monetary policy, whereas long-term rate volatility is mostly linked to uncertainty regarding the real economy and inflation. 50. B is correct. Because the factors in Exhibit 1 have been standardized to have unit standard deviations, a two-standard-deviation increase in the steepness factor will lead to the yield on the 20-year bond decreasing by 0.6030%, calculated as follows: Change in 20-year bond yield = –0.3015% × 2 = –0.6030% 51. C is correct. Because the factors in Exhibit 1 have been standardized to have unit standard deviations, a one-standard-deviation decrease in both the level factor and the curvature factor will lead to the yield on the five-year bond increasing by 0.0389%, calculated as follows: Change in five-year bond yield = 0.4352% – 0.3963% = 0.0389% 52. The local expectations theory asserts that the total return over a one-month horizon for a five-year zero-coupon bond would be the same as for a two-year zero-coupon bond. 53. Both theories attempt to explain the shape of any yield curve in terms of supply and demand for bonds. In segmented market theory, bond market participants are limited to purchase of maturities that match the timing of their liabilities. In the preferred habitat theory, participants have a preferred maturity for asset purchases, but they may deviate from it if they feel returns in other maturities offer sufficient compensation for leaving their preferred maturity segment. 54. Studies have shown that three factors affect Treasury returns: (1) changes in the level of the yield curve, (2) changes in the slope of the yield curve, and (3) changes in the curvature of the yield curve. Changes in the level refer to upward or downward shifts in the yield curve. For example, an upward shift in the yield curve is likely to result in lower returns across all maturities. Changes in the slope of the yield curve relate to the steepness of the yield curve. Thus, if the yield curve steepens, higher returns for short-maturity bonds and lower returns for long-maturity bonds will likely occur. An example of a change in the curvature of the yield curve is a situation where rates fall at the short and long end of the yield curve while rising for intermediate maturities. In this situation, returns on short and long maturities are likely to rise while declining for intermediate-maturity bonds. 55. Empirically, the most important factor is the change in the level of interest rates. 56. Key rate durations and a measure based on sensitivities to level, slope, and curvature movements can address shaping risk, but effective duration cannot. 405 520 Learning Module 3 Valuation and Analysis of Bonds with Embedded Options PRACTICE PROBLEMS The following information relates to questions 1-10 Samuel & Sons is a fixed-income specialty firm that offers advisory services to investment management companies. On 1 October 20X0, Steele Ferguson, a senior analyst at Samuel, is reviewing three fixed-rate bonds issued by a local firm, Pro Star, Inc. The three bonds, whose characteristics are given in Exhibit 30, carry the highest credit rating. Exhibit 1: Fixed-Rate Bonds Issued by Pro Star, Inc. Bond Maturity Coupon Type of Bond Bond #1 1 October 20X3 4.40% annual Option-free Bond #2 1 October 20X3 4.40% annual Callable at par on 1 October 20X1 and on 1 October 20X2 Bond #3 1 October 20X3 4.40% annual Putable at par on 1 October 20X1 and on 1 October 20X2 The one-year, two-year, and three-year par rates are 2.250%, 2.750%, and 3.100%, respectively. Based on an estimated interest rate volatility of 10%, Ferguson constructs the binomial interest rate tree shown in Exhibit 31. Exhibit 2: Binomial Interest Rate Tree Year 0 Year 1 Year 2 4.6470% 3.5930% 2.2500% 3.8046% 2.9417% 3.1150% On 19 October 20X0, Ferguson analyzes the convertible bond issued by Pro Star given in Exhibit 32. That day, the option-free value of Pro Star’s convertible bond is $1,060 and its stock price $37.50. Practice Problems 521 Exhibit 3: Convertible Bond Issued by Pro Star, Inc. Issue Date: 6 December 20X0 Maturity Date: 6 December 20X4 Coupon Rate: Issue Price: Conversion Ratio: 2% $1,000 31 1. The call feature of Bond #2 is best described as: A. European style. B. American style. C. Bermudan style. 2. The bond that would most likely protect investors against a significant increase in interest rates is: A. Bond #1. B. Bond #2. C. Bond #3. 3. A fall in interest rates would most likely result in: A. a decrease in the effective duration of Bond #3. B. Bond #3 having more upside potential than Bond #2. C. a change in the effective convexity of Bond #3 from positive to negative. 4. The value of Bond #2 is closest to: A. 102.103% of par. B. 103.121% of par. C. 103.744% of par. 5. The value of Bond #3 is closest to: A. 102.103% of par. B. 103.688% of par. C. 103.744% of par. 6. All else being equal, a rise in interest rates will most likely result in the value of the option embedded in Bond #3: A. decreasing. B. remaining unchanged. C. increasing. 522 Learning Module 3 Valuation and Analysis of Bonds with Embedded Options 7. All else being equal, if Ferguson assumes an interest rate volatility of 15% instead of 10%, the bond that would most likely increase in value is: A. Bond #1. B. Bond #2. C. Bond #3. 8. All else being equal, if the shape of the yield curve changes from upward sloping to flattening, the value of the option embedded in Bond #2 will most likely: A. decrease. B. remain unchanged. C. increase. 9. The conversion price of the bond in Exhibit 32 is closest to: A. $26.67. B. $32.26. C. $34.19. 10. If the market price of Pro Star’s common stock falls from its level on 19 October 20X0, the price of the convertible bond will most likely: A. fall at the same rate as Pro Star’s stock price. B. fall but at a slightly lower rate than Pro Star’s stock price. C. be unaffected until Pro Star’s stock price reaches the conversion price. The following information relates to questions 11-18 John Smith, an investment adviser, meets with Lydia Carter to discuss her pending retirement and potential changes to her investment portfolio. Domestic economic activity has been weakening recently, and Smith’s outlook is that equity market values will be lower during the next year. He would like Carter to consider reducing her equity exposure in favor of adding more fixed-income securities to the portfolio. Government yields have remained low for an extended period, and Smith suggests considering investment-grade corporate bonds to provide additional yield above government debt issues. In light of recent poor employment figures and two consecutive quarters of negative GDP growth, the consensus forecast among economists is that the central bank, at its next meeting this month, will take actions that will lead to lower interest rates. Smith and Carter review par, spot, and one-year forward rates (Exhibit 1) and four fixed-rate investment-grade bonds issued by Alpha Corporation that are being considered for investment (Exhibit 2). Practice Problems 523 Exhibit 1: Par, Spot, and One-Year Forward Rates (annual coupon payments) Maturity (Years) Par Rate (%) Spot Rate (%) One-Year Forward (%) 1 1.0000 1.0000 1.0000 2 1.2000 1.2012 1.4028 3 1.2500 1.2515 1.3522 Exhibit 2: Selected Fixed-Rate Bonds of Alpha Corporation Bond Annual Coupon Type of Bond Bond 1 1.5500% Straight bond Bond 2 1.5500% Convertible bond: currently trading out of the money Bond 3 1.5500% Putable bond: putable at par one year and two years from now Bond 4 1.5500% Callable bond: callable at par without any protection periods Note: All bonds in Exhibit 2 have remaining maturities of exactly three years. Carter tells Smith that the local news media have been reporting that housing starts, exports, and demand for consumer credit are all relatively strong, even in light of other poor macroeconomic indicators. Smith explains that the divergence in economic data leads him to believe that volatility in interest rates will increase. Smith also states that he recently read a report issued by Brown and Company forecasting that the yield curve could invert within the next six months. Smith develops a binomial interest rate tree with a 15% interest rate volatility assumption to assess the value of Alpha Corporation’s bonds. Exhibit 3 presents the interest rate tree. Exhibit 3: Binomial Interest Rate Tree for Alpha Corporation with 15% Interest Rate Volatility Year 0 Year 1 Year 2 1.7862% 1.6121% 1.0000% 1.3233% 1.1943% 0.9803% Carter asks Smith about the possibility of analyzing bonds that have lower credit ratings than the investment-grade Alpha bonds. Smith discusses four other corporate bonds with Carter. Exhibit 4 presents selected data on the four bonds. 524 Learning Module 3 Valuation and Analysis of Bonds with Embedded Options Exhibit 4: Selected Information on Fixed-Rate Bonds for Beta, Gamma, Delta, and Rho Corporations Bond Issuer Bond Features Credit Rating Bond 5 Beta Corporation Coupon 1.70% Callable in Year 2 OAS of 45 bps B Bond 6 Gamma Corporation Coupon 1.70% Callable in Year 2 OAS of 65 bps B Bond 7 Delta Corporation Coupon 1.70% Callable in Year 2 OAS of 85 bps B Bond 8 Rho Corporation Coupon 1.70% Callable in Year 2 OAS of 105 bps CCC Notes: All bonds have remaining maturities of three years. OAS stands for option-adjusted spread. 11. Based on Exhibit 2, and assuming that the forecast for interest rates and Smith’s outlook for equity returns are validated, which bond’s option is most likely to be exercised? A. Bond 2 B. Bond 3 C. Bond 4 12. Based on Exhibit 2, the current price of Bond 1 is most likely greater than the current price of: A. Bond 2. B. Bond 3. C. Bond 4. 13. Assuming the forecast for interest rates is proven accurate, which bond in Exhibit 2 will likely experience the smallest price increase? A. Bond 1 B. Bond 3 C. Bond 4 14. Based on the information in Exhibit 1 and Exhibit 2, the value of the embedded option in Bond 4 is closest to: A. nil. B. 0.1906. C. 0.8789. 15. If Smith’s interest rate volatility forecast turns out to be true, which bond in Ex- Practice Problems 525 hibit 2 is likely to experience the greatest price increase? A. Bond 2 B. Bond 3 C. Bond 4 16. If the Brown and Company forecast comes true, which of the following is most likely to occur? The value of the embedded option in: A. Bond 3 decreases. B. Bond 4 decreases. C. both Bond 3 and Bond 4 increases. 17. Based on Exhibit 2 and Exhibit 3, the market price of Bond 4 is closest to: A. 100.0000. B. 100.5123. C. 100.8790. 18. Which of the following conclusions regarding the bonds in Exhibit 4 is correct? A. Bond 5 is relatively cheaper than Bond 6. B. Bond 7 is relatively cheaper than Bond 6. C. Bond 8 is relatively cheaper than Bond 7. The following information relates to questions 19-27 Rayes Investment Advisers specializes in fixed-income portfolio management. Meg Rayes, the owner of the firm, would like to add bonds with embedded options to the firm’s bond portfolio. Rayes has asked Mingfang Hsu, one of the firm’s analysts, to assist her in selecting and analyzing bonds for possible inclusion in the firm’s bond portfolio. Hsu first selects two corporate bonds that are callable at par and have the same characteristics in terms of maturity, credit quality, and call dates. Hsu uses the option adjusted spread (OAS) approach to analyze the bonds, assuming an interest rate volatility of 10%. The results of his analysis are presented in Exhibit 1. Exhibit 1: Summary Results of Hsu’s Analysis Using the OAS Approach Bond OAS (in bps) Bond #1 25.5 Bond #2 30.3 526 Learning Module 3 Valuation and Analysis of Bonds with Embedded Options Hsu then selects the four bonds issued by RW, Inc., given in Exhibit 2. These bonds all have a maturity of three years and the same credit rating. Bonds #4 and #5 are identical to Bond #3, an option-free bond, except that they each include an embedded option. Exhibit 2: Bonds Issued by RW, Inc. Bond Coupon Special Provision Bond #3 4.00% annual Bond #4 4.00% annual Callable at par at the end of years 1 and 2 Bond #5 4.00% annual Putable at par at the end of years 1 and 2 Bond #6 One-year reference rate annually, set in arrears To value and analyze RW’s bonds, Hsu uses an estimated interest rate volatility of 15% and constructs the binomial interest rate tree provided in Exhibit 3. Exhibit 3: Binomial Interest Rate Tree Used to Value RW’s Bonds Year 0 Year 1 Year 2 5.3340% 4.6343% 2.5000% 3.9515% 3.4331% 2.9274% Rayes asks Hsu to determine the sensitivity of Bond #4’s price to a 20 bps parallel shift of the benchmark yield curve. The results of Hsu’s calculations are shown in Exhibit 40. Exhibit 4: Summary Results of Hsu’s Analysis about the Sensitivity of Bond #4’s Price to a Parallel Shift of the Benchmark Yield Curve Magnitude of the Parallel Shift in the Benchmark Yield Curve +20 bps –20 bps Full Price of Bond #4 (% of par) 100.478 101.238 Hsu also selects the two floating-rate bonds issued by Varlep, plc, given in Exhibit 41. These bonds have a maturity of three years and the same credit rating. Practice Problems 527 Exhibit 5: Floating-Rate Bonds Issued by Varlep, plc Bond Coupon Bond #7 One-year reference rate annually, set in arrears, capped at 5.00% Bond #8 One-year reference rate annually, set in arrears, floored at 3.50% To value Varlep’s bonds, Hsu constructs the binomial interest rate tree provided in Exhibit 42. Exhibit 6: Binomial Interest Rate Tree Used to Value Varlep’s Bonds Year 0 Year 1 Year 2 6.3679% 4.5027% 3.0000% 5.0092% 3.5419% 3.9404% Last, Hsu selects the two bonds issued by Whorton, Inc., given in Exhibit 43. These bonds are close to their maturity date and are identical, except that Bond #9 includes a conversion option. Whorton’s common stock is currently trading at $30 per share. Exhibit 7: Bonds Issued by Whorton, Inc. Bond Type of Bond Bond #9 Convertible bond with a conversion price of $50 Bond #10 Identical to Bond #9 except that it does not include a conversion option 19. Based on Exhibit 1, Rayes would most likely conclude that relative to Bond #1, Bond #2 is: A. overpriced. B. fairly priced. C. underpriced. 20. The effective duration of Bond #6 is: A. close to 1. B. higher than 1 but lower than 3. C. higher than 3. 21. In Exhibit 2, the bond whose effective duration might lengthen if interest rates 528 Learning Module 3 Valuation and Analysis of Bonds with Embedded Options rise is: A. Bond #3. B. Bond #4. C. Bond #5. 22. The effective duration of Bond #4 is closest to: A. 0.76. B. 1.88. C. 3.77. 23. The value of Bond #7 is closest to: A. 99.697% of par. B. 99.936% of par. C. 101.153% of par. 24. The value of Bond #8 is closest to: A. 98.116% of par. B. 100.000% of par. C. 100.485% of par. 25. The value of Bond #9 is equal to the value of Bond #10: A. plus the value of a put option on Whorton’s common stock. B. plus the value of a call option on Whorton’s common stock. C. minus the value of a call option on Whorton’s common stock. 26. The minimum value of Bond #9 is equal to the greater of: A. the conversion value of Bond #9 and the current value of Bond #10. B. the current value of Bond #10 and a call option on Whorton’s common stock. C. the conversion value of Bond #9 and a call option on Whorton’s common stock. 27. The factor that is currently least likely to affect the risk–return characteristics of Bond #9 is: A. interest rate movements. B. Whorton’s credit spreads. C. Whorton’s common stock price movements. Practice Problems 529 The following information relates to questions 28-36 Jules Bianchi is a bond analyst for Maneval Investments, Inc. Bianchi gathers data on three corporate bonds, as shown in Exhibit 1. Exhibit 1: Selected Bond Data Coupon Rate Price Ayrault, Inc. (AI) 5.25% 100.200 Callable at par in one year and two years from today Blum, Inc. (BI) 5.25% 101.300 Option-free Cresson Enterprises (CE) 5.25% 102.100 Putable at par in one year from today Issuer Bond Description Note: Each bond has a remaining maturity of three years, annual coupon payments, and a credit rating of BBB. To assess the interest rate risk of the three bonds, Bianchi constructs two binomial interest rate trees based on a 10% interest rate volatility assumption and a current one-year rate of 4%. Panel A of Exhibit 2 provides an interest rate tree assuming the benchmark yield curve shifts down by 30 bps, and Panel B provides an interest rate tree assuming the benchmark yield curve shifts up by 30 bps. Bianchi determines that the AI bond is currently trading at an option-adjusted spread (OAS) of 13.95 bps relative to the benchmark yield curve. Exhibit 2: Binomial Interest Rate Trees Interest Rates Shift Down by 30 bps Year 0 Year 1 Year 2 7.0037% 5.1968% 3.7000% 5.7342% 4.2548% 4.6947% Interest Rates Shift Up by 30 bps Year 0 Year 1 Year 2 7.7432% 5.8605% 4.3000% 6.3396% 4.7982% 5.1904% Armand Gillette, a convertible bond analyst, stops by Bianchi’s office to discuss two convertible bonds. One is issued by DeLille Enterprises (DE), and the other is issued by Raffarin Incorporated (RI). Selected data for the two bonds are present- 530 Learning Module 3 Valuation and Analysis of Bonds with Embedded Options ed in Exhibits 3 and 4. Exhibit 3: Selected Data for DE Convertible Bond Issue price Conversion period €1,000 at par 13 September 20X5 to 12 September 20X8 Initial conversion price €10.00 per share Threshold dividend €0.50 per share Change of control conversion price €8.00 per share Common stock share price on issue date €8.70 Share price on 17 September 20X5 €9.10 Convertible bond price on 17 September 20X5 €1,123 Exhibit 4: Selected Data for RI Convertible Bond Straight bond value €978 Value of embedded issuer call option €43 Value of embedded investor put option €26 Value of embedded call option on issuer’s stock €147 Conversion price €12.50 Current common stock share price €11.75 Gillette makes the following comments to Bianchi: ■ “The DE bond does not contain any call or put options, but the RI bond contains both an embedded call option and put option. I expect that DeLille Enterprises will soon announce a common stock dividend of €0.70 per share.” ■ “My belief is that, over the next year, Raffarin’s share price will appreciate toward the conversion price but not exceed it.” 28. Based on Exhibits 1 and 2, the effective duration for the AI bond is closest to: A. 1.98. B. 2.15. C. 2.73. 29. If benchmark yields were to fall, which bond in Exhibit 1 would most likely experience a decline in effective duration? A. AI bond B. BI bond C. CE bond Practice Problems 30. Based on Exhibit 1, for the BI bond, one-sided: A. up-duration will be greater than one-sided down-duration. B. down-duration will be greater than one-sided up-duration. C. up-duration and one-sided down-duration will be about equal. 31. Based on Exhibit 1, which key rate duration is the largest for the BI bond? A. One-year key rate duration B. Two-year key rate duration C. Three-year key rate duration 32. Which bond in Exhibit 1most likely has the lowest effective convexity? A. AI bond B. BI bond C. CE bond 33. Based on Exhibit 3, if DeLille Enterprises pays the dividend expected by Gillette, the conversion price of the DE bond will: A. be adjusted downward. B. not be adjusted. C. be adjusted upward. 34. Based on Exhibit 3, the market conversion premium per share for the DE bond on 17 September 20X5 is closest to: A. €0.90. B. €2.13. C. €2.53. 35. Based on Exhibit 4, the arbitrage-free value of the RI bond is closest to: A. €814. B. €1,056. C. €1,108. 36. Based on Exhibit 4 and Gillette’s forecast regarding Raffarin’s share price, the return on the RI bond over the next year is most likely to be: A. lower than the return on Raffarin’s common shares. B. the same as the return on Raffarin’s common shares. C. higher than the return on Raffarin’s common shares. 531 532 Learning Module 3 Valuation and Analysis of Bonds with Embedded Options SOLUTIONS 1. C is correct. The call option embedded in Bond #2 can be exercised only at two predetermined dates: 1 October 20X1 and 1 October 20X2. Thus, the call feature is Bermudan style. 2. C is correct. The bond that would most likely protect investors against a significant increase in interest rates is the putable bond (i.e., Bond #3). When interest rates have risen and higher-yield bonds are available, a put option allows the bondholders to put back the bonds to the issuer prior to maturity and to reinvest the proceeds of the retired bonds in higher-yielding bonds. 3. B is correct. A fall in interest rates results in a rise in bond values. For a callable bond, such as Bond #2, the upside potential is capped because the issuer is more likely to call the bond. In contrast, the upside potential for a putable bond, such as Bond #3, is uncapped. Thus, a fall in interest rates would result in a putable bond having more upside potential than an otherwise identical callable bond. Note that A is incorrect because the effective duration of a putable bond increases, not decreases, with a fall in interest rates; the bond is less likely to be put and thus behaves more like an option-free bond. C is also incorrect because the effective convexity of a putable bond is always positive. It is the effective convexity of a callable bond that will change from positive to negative if interest rates fall and the call option is near the money. 4. A is correct: Year 0 Year 1 100 100.655 3.5930% 102.103 2.2500% 4.400 100 101.417 2.9417% Year 2 Year 3 99.764 4.6470% 104.400 4.400 100 100.574 3.8046% 4.400 100 101.246 3.1150% 5. C is correct: 104.400 104.400 Solutions Year 0 533 Year 1 101.056 3.5930% 103.744 2.2500% 4.400 102.301 2.9417% Year 2 100 99.764 4.6470% Year 3 104.400 4.400 100.574 3.8046% 104.400 4.400 101.246 3.1150% 104.400 6. C is correct. Bond #3 is a putable bond, and the value of a put option increases as interest rates rise. At higher interest rates, the value of the underlying option-free bond (straight bond) declines, but the decline is offset partially by the increase in the value of the embedded put option, which is more likely to be exercised. 7. C is correct. Regardless of the type of option, an increase in interest rate volatility results in an increase in option value. Because the value of a putable bond is equal to the value of the straight bond plus the value of the embedded put option, Bond #3 will increase in value if interest rate volatility increases. Put another way, an increase in interest rate volatility will most likely result in more scenarios where the put option is exercised, which increases the values calculated in the interest rate tree and, thus, the value of the putable bond. 8. C is correct. Bond #2 is a callable bond, and the value of the embedded call option increases as the yield curve flattens. When the yield curve is upward sloping, the one-period forward rates on the interest rate tree are high and opportunities for the issuer to call the bond are fewer. When the yield curve flattens or inverts, many nodes on the tree have lower forward rates, which increase the opportunities to call and, thus, the value of the embedded call option. 9. B is correct. The conversion price of a convertible bond is equal to the par value divided by the conversion ratio—that is, $1,000/31 = $32.26 per share. 10. B is correct. The conversion value of the bond is 31 × $37.50 or $1,162.50, which represents its minimum value. Thus, the convertible bond exhibits mostly stock risk–return characteristics; a fall in the stock price will result in a fall in the convertible bond price. However, the change in the convertible bond price is less than the change in the stock price because the convertible bond has a floor. That floor is the value of the straight (option-free) bond. 11. C is correct. If the central bank takes actions that lead to lower interest rates, the yields on Alpha’s bonds are likely to decrease. If the yield to maturity on Bond 4 (callable) falls below the 1.55% coupon rate, the call option will become valuable and Alpha may call the bond because it is in the money. A is incorrect because if the equity market declines, the market value of Alpha stock will also likely decrease. Therefore, Bond 2 (convertible) would have a lower conversion value; hence, the conversion option likely would not be exercised. Because Bond 2 is currently trading out of the money, it will likely trade further out of the money once the price of Alpha stock decreases. 534 Learning Module 3 Valuation and Analysis of Bonds with Embedded Options B is incorrect because Bond 3 (putable) is more likely to be exercised in an increasing rather than a decreasing interest rate environment. 12. C is correct. All four bonds in Exhibit 2 issued by Alpha Corporation offer the same coupon rate and have the same remaining term to maturity. Bond 4 (callable) most likely has a current price that is less than Bond 1 (straight or option free) because investors are short the call option and must be compensated for bearing call risk. Bond 2 (convertible) most likely has a current price that is greater than Bond 1 because investors are paying for the conversion option embedded in Bond 2 and the option has time value associated with it, even though the option is trading out of the money. Similarly, Bond 3 (putable) most likely has a current price that is greater than Bond 1 because investors are paying for the put option. 13. C is correct. The consensus economic forecast is for interest rates to decrease. In an environment of decreasing interest rates, all bond prices should rise, ignoring any price impact resulting from any embedded options. When interest rates fall, the value of the embedded call option in Bond 4 (callable) increases, causing an opposing effect on price. The put option of putable bonds, by contrast, increases in value when interest rates rise rather than decline. 14. C is correct. Bond 4 is a callable bond. Value of an issuer call option = Value of straight bond – Value of callable bond. The value of the straight bond may be calculated using the spot rates or the one-year forward rates. Value of an option-free (straight) bond with a 1.55% coupon using spot rates: 1.55/(1.0100)1 + 1.55/(1.012012)2 + 101.55/(1.012515)3 = 100.8789. The value of a callable bond (at par) with no call protection period cannot exceed 100, as at that price or higher the bond would be called. The value of the call option = 100.8789 – 100 = 0.8789. 15. B is correct. An increase in interest rate volatility will cause the value of the put and call options embedded in Bond 3 and Bond 4 to increase. Bond 3 (putable) would experience an increase in price because the increased value of the put option increases the bond’s value. In contrast, Bond 4 (callable) will experience a price decrease because the increased value of the call option reduces the callable bond’s value. Bond 2, an out-of-the-money convertible, will resemble the risk– return characteristics of a straight bond and will thus be unaffected by interest rate volatility. 16. A is correct. All else being equal, the value of a put option decreases as the yield curve moves from being upward sloping to flat to downward sloping (inverted). Alternatively, a call option’s value increases as the yield curve flattens and increases further if the yield curve inverts. Therefore, if the yield curve became inverted, the value of the embedded option in Bond 3 (putable) would decrease and the value of the embedded option in Bond 4 (callable) would increase. 17. A is correct. The market price of callable Bond 4 with no protection period cannot exceed 100. 18. B is correct. A bond with a larger option-adjusted spread (OAS) than that of a bond with similar characteristics and credit quality means that the bond is likely underpriced (cheap). Bond 7 (OAS 85 bps) is relatively cheaper than Bond 6 (OAS 65 bps). C is incorrect because Bond 8 (CCC) has a lower credit rating than Bond 7 (B) and the OAS alone cannot be used for the relative value comparison. The larger OAS (105 bps) incorporates compensation for the difference between the B and Solutions 535 CCC bond credit ratings. Therefore, there is not enough information to draw a conclusion about relative value. 19. C is correct. The option-adjusted spread (OAS) is the constant spread added to all the one-period forward rates that makes the arbitrage-free value of a risky bond equal to its market price. The OAS approach is often used to assess bond relative values. If two bonds have the same characteristics and credit quality, they should have the same OAS. If this is not the case, the bond with the largest OAS (i.e., Bond #2) is likely to be underpriced (cheap) relative to the bond with the smallest OAS (i.e., Bond #1). 20. A is correct. The effective duration of a floating-rate bond is close to the time to next reset. As the reset for Bond #6 is annual, the effective duration of this bond is close to 1. 21. B is correct. Effective duration indicates the sensitivity of a bond’s price to a 100 bps parallel shift of the benchmark yield curve assuming no change in the bond’s credit spread. The effective duration of an option-free bond, such as Bond #3, goes down as interest rates rise. As interest rates rise, a call option moves out of the money, which increases the value of the callable bond and lengthens its effective duration. In contrast, as interest rates rise, a put option moves into the money, which limits the price depreciation of the putable bond and shortens its effective duration. Thus, the bond whose effective duration might lengthen if interest rates rise is the callable bond (i.e., Bond #4). 22. B is correct. The effective duration of Bond #4 can be calculated using Equation 3, where ΔCurve is 20 bps, PV– is 101.238, and PV+ is 100.478. PV0, the current full price of the bond (i.e., with no shift), is not given but can be calculated using Exhibit 3 as follows: Year 0 Year 1 98.789 4.6343% 100.873 2.5000% 4.000 100 100.548 3.4331% Year 2 Year 3 98.734 5.3340% 104.000 4.000 100 100.047 3.9515% 4.000 100 101.042 2.9274% Thus, the effective duration of Bond #4 is: 101.238 − 100.478 ___________________ ​EffDur =    ​   2 × ​(​ ​0.0020​)​​× ​(​ ​100.873​)​​​ = 1.88. 23. A is correct: 104.000 104.000 536 Learning Module 3 Valuation and Analysis of Bonds with Embedded Options Year 0 Year 1 Year 2 98.714 6.3679% 99.381 4.5027% 99.697 3.0000% 3.0000 99.996 3.5419% Year 3 105.0000 106.3679 4.5027 99.991 5.0092% 105.0000 105.0092 3.5419 100.000 3.9404% 103.9404 Year 2 Year 3 100.000 6.3679% 106.3679 24. C is correct: Year 0 Year 1 100.000 4.5057% 100.485 3.0000% 3.5000 3.0000 100.000 3.5419% 4.5027 100.000 5.0092% 105.0092 3.5419 100.000 3.9404% 103.9404 25. B is correct. A convertible bond includes a conversion option, which is a call option on the issuer’s common stock. This conversion option gives the bondholders the right to convert their debt into equity. Thus, the value of Bond #9, the convertible bond, is equal to the value of Bond #10, the underlying option-free bond (straight bond), plus the value of a call option on Whorton’s common stock. 26. A is correct. The minimum value of a convertible bond is equal to the greater of the conversion value of the convertible bond (i.e., Bond #9) and the current value of the straight bond (i.e., Bond #10). 27. C is correct. The risk–return characteristics of a convertible bond depend on the market price of the issuer’s common stock (underlying share price) relative to the bond’s conversion price. When the underlying share price is well below the conversion price, the convertible bond exhibits mostly bond risk–return characteristics. In this case, the price of the convertible bond is mainly affected by interest rate movements and the issuer’s credit spreads. In contrast, when the underlying share price is above the conversion price, the convertible bond exhibits mostly stock risk–return characteristics. In this case, the price of the convertible bond is mainly affected by the issuer’s common stock price movements. The underlying share price ($30) is lower than the conversion price of Bond #9 ($50). Thus, Bond #9 exhibits mostly bond risk–return characteristics and is least affected by Whor- Solutions 537 ton’s common stock price movements. 28. B is correct. The AI bond’s value if interest rates shift down by 30 bps (PV–) is 100.78: Year 0 Year 1 98.800 5.3363% 100.780 3.8395% 5.25 100 100.538 4.3943% Year 2 Year 3 98.233 7.1432% 105.25 5.25 99.411 5.8737% 105.25 5.25 100 101.397 4.8342% 105.25 The AI bond’s value if interest rates shift up by 30 bps (PV+) is 99.487: ​​(​PV​ ​)​​− ​(​ ​PV​+​)​​ 100.780 − 99.487 _________________    ​ = _______________    ​   ​EffDur =    ​ ( − ) 2 × 0.003 × 100.200 ​ = 2.15. 2 × ​​ ​ΔCurve​ ​​× ​(​ ​PV​0​)​​ 29. A is correct. The AI bond is a callable bond, and the effective duration of a callable bond decreases when interest rates fall. The reason is because a decline in interest rates may result in the call option moving into the money, which limits the price appreciation of the callable bond. Exhibit 1 also shows that the price of the AI bond is 100.200 and that it is callable at par in one year and two years. Thus, the call option is already in the money and would likely be exercised in response to increases in the AI bond’s price. 30. C is correct. The BI bond is an option-free bond, and one-sided up-duration and one-sided down-duration will be about equal for option-free bonds. 31. C is correct. The BI bond is an option-free bond. Its longest key rate duration will be in the year of its maturity because the largest cash flow (payment of both coupon and principal) occurs in that year. 32. A is correct. All else being equal, a callable bond will have lower effective convexity than an option-free bond when the call option is in the money. Similarly, when the call option is in the money, a callable bond will also have lower effective convexity than a putable bond if the put option is out of the money. Exhibit 1 shows that the callable AI bond is currently priced slightly higher than its call price of par value, which means the embedded call option is in the money. The put option embedded in the CE bond is not in the money; the bond is currently priced 2.1% above par value. Thus, at the current price, the putable CE bond is more likely to behave like the option-free BI bond. Consequently, the effective convexity of the AI bond will likely be lower than the option-free BI bond and the putable CE bond. 33. A is correct. The conversion price would be adjusted downward because Gillette’s expected dividend payment of €0.70 is greater than the threshold dividend of €0.50. 538 Learning Module 3 Valuation and Analysis of Bonds with Embedded Options 34. B is correct. The market conversion premium per share is equal to the market conversion price minus the underlying share price. The market conversion price is calculated as follows: Convertible bond price ________________ Market conversion price =    ​ Conversion    ratio €1, 123 = ________________ ​€1, 000 / €10 per share ​ = €11.23 per share. The market conversion premium per share is then calculated as follows: Market conversion premium per share = Market conversion price – Underlying share price. = €11.23 – €9.10 = €2.13. 35. C is correct. The value of a convertible bond with both an embedded call option and a put option can be determined using the following formula: Value of callable putable convertible bond = Value of straight bond + Value of call option on the issuer’s stock – Value of issuer call option + Value of investor put option. Value of callable putable bond = €978 + €147 – €43 + €26 = €1,108. 36. A is correct. Over the next year, Gillette believes that Raffarin’s share price will continue to increase toward the conversion price but not exceed it. If Gillette’s forecast becomes true, the return on the RI bond will increase but at a lower rate than the increase in Raffarin’s share price because the conversion price is not expected to be reached. Practice Problems 593 PRACTICE PROBLEMS The following information relates to questions 1-8 Lena Liecken is a senior bond analyst at Taurus Investment Management. Kristel Kreming, a junior analyst, works for Liecken in helping conduct fixed-income research for the firm’s portfolio managers. Liecken and Kreming meet to discuss several bond positions held in the firm’s portfolios. Bonds I and II both have a maturity of one year, an annual coupon rate of 5%, and a market price equal to par value. The risk-free rate is 3%. Historical default experiences of bonds comparable to Bonds I and II are presented in Exhibit 1. Exhibit 1: Credit Risk Information for Comparable Bonds Bond Recovery Rate Percentage of Bonds That Survive and Make Full Payment I 40% 98% II 35% 99% Bond III is a zero-coupon bond with three years to maturity. Liecken evaluates similar bonds and estimates a recovery rate of 38% and a risk-neutral default probability of 2%, assuming conditional probabilities of default. Kreming creates Exhibit 2 to compute Bond III’s credit valuation adjustment. She assumes a flat yield curve at 3%, with exposure, recovery, and loss given default values expressed per 100 of par value. Exhibit 2: Analysis of Bond III Probability of Default Probability of Survival Expected Loss Present Value of Expected Loss Exposure Recovery Loss Given Default 1 94.2596 35.8186 58.4410 2.0000% 98.0000% 1.1688 1.1348 2 97.0874 36.8932 60.1942 1.9600% 96.0400% 1.1798 1.1121 3 100.0000 38.0000 62.0000 1.9208% 94.1192% 1.1909 1.0898 3.5395 3.3367 Date 0 Sum 5.8808% Bond IV is an AA rated bond that matures in five years, has a coupon rate of 6%, and a modified duration of 4.2. Liecken is concerned about whether this bond will be downgraded to an A rating, but she does not expect the bond to default during the next year. Kreming constructs a partial transition matrix, which is presented in Exhibit 3, and suggests using a model to predict the rating change of Bond IV using leverage ratios, return on assets, and macroeconomic variables. 594 Learning Module 4 Credit Analysis Models Exhibit 3: Partial One-Year Corporate Transition Matrix (entries in %) From/To AAA AAA 92.00 6.00 1.00 2.00 89.00 8.00 0.05 1.00 85.00 0.50 1.00 1.75 AA A Credit Spread (%) AA A Kreming calculates the risk-neutral probabilities, compares them with the actual default probabilities of bonds evaluated over the past 10 years, and observes that the actual and risk-neutral probabilities differ. She makes two observations regarding the comparison of these probabilities: Observation 1 Actual default probabilities include the default risk premium associated with the uncertainty in the timing of the possible default loss. Observation 2 The observed spread over the yield on a risk-free bond in practice includes liquidity and tax considerations, in addition to credit risk. 1. The expected exposure to default loss for Bond I is: A. less than the expected exposure for Bond II. B. the same as the expected exposure for Bond II. C. greater than the expected exposure for Bond II. 2. Based on Exhibit 1, the loss given default for Bond II is: A. less than that for Bond I. B. the same as that for Bond I. C. greater than that for Bond I. 3. Based on Exhibit 1, the expected future value of Bond I at maturity is closest to: A. 98.80. B. 103.74. C. 105.00. 4. Based on Exhibit 1, the risk-neutral default probability for Bond I is closest to: A. 2.000%. B. 3.175%. C. 4.762%. 5. Based on Exhibit 2, the credit valuation adjustment for Bond III is closest to: A. 3.3367. Practice Problems 595 B. 3.5395. C. 5.8808. 6. Based on Exhibit 3, if Bond IV’s credit rating changes during the next year to an A rating, its expected price change would be closest to: A. –8.00%. B. –7.35%. C. –3.15%. 7. Kreming’s suggested model for Bond IV is a: A. structural model. B. reduced-form model. C. term structure model. 8. Which of Kreming’s observations regarding actual and risk-neutral default probabilities is correct? A. Only Observation 1 B. Only Observation 2 C. Both Observation 1 and Observation 2 The following information relates to questions 9-23 Daniela Ibarra is a senior analyst in the fixed-income department of a large wealth management firm. Marten Koning is a junior analyst in the same department, and David Lok is a member of the credit research team. The firm invests in a variety of bonds. Ibarra is presently analyzing a set of bonds with some similar characteristics, such as four years until maturity and a par value of €1,000. Exhibit 1 includes details of these bonds. Exhibit 1: A Brief Description of the Bonds Being Analyzed Bond Description B1 A zero-coupon, four-year corporate bond with a par value of €1,000. The wealth management firm’s research team has estimated that the risk-neutral probability of default for each date for the bond is 1.50%, and the recovery rate is 30%. B2 A bond similar to B1, except that it has a fixed annual coupon rate of 6% paid annually. B3 A bond similar to B2 but rated AA. B4 A bond similar to B2 but the coupon rate is the one-year benchmark rate plus 4%. Ibarra asks Koning to assist her with analyzing the bonds. She wants him to perform the analysis with the assumptions that there is no interest rate volatility and 596 Learning Module 4 Credit Analysis Models that the government bond yield curve is flat at 3%. Ibarra performs the analysis assuming an upward-sloping yield curve and volatile interest rates. Exhibit 2 provides the data on annual payment benchmark government bonds. She uses these data to construct a binomial interest rate tree based on an assumption of future interest rate volatility of 20%. Exhibit 2: Par Curve for Annual Payment Benchmark Government Bonds Maturity Coupon Rate Price Discount Factor Spot Rate Forward Rate 1 –0.25% €100 1.002506 –0.2500% 2 0.75% €100 0.985093 0.7538% 1.7677% 3 1.50% €100 0.955848 1.5166% 3.0596% 4 2.25% €100 0.913225 2.2953% 4.6674% Answer the first five questions based on the assumptions made by Marten Koning, the junior analyst. Answer the subsequent questions based on the assumptions made by Daniela Ibarra, the senior analyst. Note: All calculations in this problem set are carried out on spreadsheets to preserve precision. The rounded results are reported in the solutions. 9. The market price of Bond B1 is €875. The bond is: A. fairly valued. B. overvalued. C. undervalued. 10. Koning realizes that an increase in the recovery rate would lead to an increase in the bond’s fair value, whereas an increase in the probability of default would lead to a decrease in the bond’s fair value. He is not sure, however, which effect would be greater. So, he increases both the recovery rate and the probability of default by 25% of their existing estimates and recomputes the bond’s fair value. The recomputed fair value is closest to: A. €843.14. B. €848.00. C. €855.91. 11. The fair value of Bond B2 is closest to: A. €1,069.34. B. €1,111.51. C. €1,153.68. 12. The market price of Bond B2 is €1,090. If the bond is purchased at this price and there is a default on Date 3, the rate of return to the bond buyer would be closest to: A. –28.38%. Practice Problems 597 B. –41.72%. C. –69.49%. 13. Bond B3 will have a modified duration of 2.75 at the end of the year. Based on the representative one-year corporate transition matrix in Exhibit 3 and assuming no default, how should the analyst adjust the bond’s yield to maturity to assess the expected return on the bond over the next year? Exhibit 3: Representative One-Year Corporate Transition Matrix (entries are in %) From/To AAA AA A BBB BB B CCC, CC, C D AAA 90.00 9.00 0.60 0.15 0.10 0.10 0.05 0.00 AA 1.50 88.00 9.50 0.75 0.15 0.05 0.03 0.02 A 0.05 2.50 87.50 8.40 0.75 0.60 0.12 0.08 BBB 0.02 0.30 4.80 85.50 6.95 1.75 0.45 0.23 BB 0.01 0.06 0.30 7.75 79.50 8.75 2.38 1.25 B 0.00 0.05 0.15 1.40 9.15 76.60 8.45 4.20 CCC, CC, C 0.00 0.01 0.12 0.87 1.65 18.50 49.25 29.60 0.60% 0.90% 1.10% 1.50% 3.40% 6.50% 9.50% Credit Spread A. Add 7.7 bps to YTM. B. Subtract 7.7 bps from YTM. C. Subtract 9.0 bps from YTM. 14. David Lok has estimated the probability of default of Bond B1 to be 1.50%. He is presenting the approach the research team used to estimate the probability of default. Which of the following statements is Lok likely to make in his presentation if the team used a reduced-form credit model? A. Option pricing methodologies were used, with the volatility of the underlying asset estimated based on historical data on the firm’s stock price. B. Regression analysis was used, with the independent variables including both firm-specific variables, such as the debt ratio and return on assets, and macroeconomic variables, such as the rate of inflation and the unemployment rate. C. The default barrier was first estimated, followed by the estimation of the probability of default as the portion of the probability distribution that lies below the default barrier. 15. In the presentation, Lok is asked why the research team chose to use a reduced-form credit model instead of a structural model. Which statement is he likely to make in reply? A. Structural models are outdated, having been developed in the 1970s; reduced-form models are more modern, having been developed in the 1990s. 598 Learning Module 4 Credit Analysis Models B. Structural models are overly complex because they require the use of option pricing models, whereas reduced-form models use regression analysis. C. Structural models require “inside” information known to company management, whereas reduced-form models can use publicly available data on the firm. 16. As previously mentioned, Ibarra is considering a future interest rate volatility of 20% and an upward-sloping yield curve, as shown in Exhibit 2. Based on her analysis, the fair value of Bond B2 is closest to: A. €1,101.24. B. €1,141.76. C. €1,144.63. 17. Ibarra wants to know the credit spread of Bond B2 over a theoretical comparable-maturity government bond with the same coupon rate as this bond. The foregoing credit spread is closest to: A. 108 bps. B. 101 bps. C. 225 bps. 18. Ibarra is interested in analyzing how a simultaneous decrease in the recovery rate and the probability of default would affect the fair value of Bond B2. She decreases both the recovery rate and the probability of default by 25% of their existing estimates and recomputes the bond’s fair value. The recomputed fair value is closest to: A. €1,096.59. B. €1,108.40. C. €1,111.91. 19. The wealth management firm has an existing position in Bond B4. The market price of B4, a floating-rate note, is €1,070. Senior management has asked Ibarra to make a recommendation regarding the existing position. Based on the assumptions used to calculate the estimated fair value only, her recommendation should be to: A. add to the existing position. B. hold the existing position. C. reduce the existing position. 20. The issuer of the floating-rate note, B4, is in the energy industry. Ibarra believes that oil prices are likely to increase significantly in the next year, which will lead to an improvement in the firm’s financial health and a decline in the probability of default from 1.50% in Year 1 to 0.50% in Years 2, 3, and 4. Based on these expectations, which of the following statements is correct? A. The CVA will decrease to €22.99. B. The note’s fair value will increase to €1,177.26. Practice Problems C. The value of the FRN, assuming no default, will increase to €1,173.55. 21. The floating-rate note, B4, is currently rated BBB by Standard & Poor’s and Fitch Ratings (and Baa by Moody’s Investors Service). Based on the research department assumption about the probability of default in Question 10 and her own assumption in Question 11, which action does Ibarra most likely expect from the credit rating agencies? A. Downgrade from BBB to BB. B. Upgrade from BBB to AAA. C. Place the issuer on watch with a positive outlook. 22. During the presentation about how the research team estimates the probability of default for a particular bond issuer, Lok is asked for his thoughts on the shape of the term structure of credit spreads. Which statement is he most likely to include in his response? A. The term structure of credit spreads typically is flat or slightly upward sloping for high-quality investment-grade bonds. High-yield bonds are more sensitive to the credit cycle, however, and can have a more upwardly sloped term structure of credit spreads than investment-grade bonds or even an inverted curve. B. The term structure of credit spreads for corporate bonds is always upward sloping—more so the weaker the credit quality because probabilities of default are positively correlated with the time to maturity. C. There is no consistent pattern for the term structure of credit spreads. The shape of the credit term structure depends entirely on industry factors. 23. The final question for Lok is about covered bonds. The person asking says, “I’ve heard about them but don’t know what they are.” Which statement is Lok most likely to make to describe a covered bond? A. A covered bond is issued in a non-domestic currency. The currency risk is then fully hedged using a currency swap or a package of foreign exchange forward contracts. B. A covered bond is issued with an attached credit default swap. It essentially is a “risk-free” government bond. C. A covered bond is a senior debt obligation giving recourse to the issuer as well as a predetermined underlying collateral pool, often commercial or residential mortgages. The following information relates to questions 24-30 Anna Lebedeva is a fixed-income portfolio manager. Paulina Kowalski, a junior analyst, and Lebedeva meet to review several positions in Lebedeva’s portfolio. Lebedeva begins the meeting by discussing credit rating migration. Kowalski asks Lebedeva about the typical impact of credit rating migration on the expected return on a bond. Lebedeva asks Kowalski to estimate the expected return over 599 600 Learning Module 4 Credit Analysis Models the next year on a bond issued by Entre Corp. The BBB rated bond has a yield to maturity of 5.50% and a modified duration of 7.54. Kowalski calculates the expected return on the bond over the next year given the partial credit transition and credit spread data in Exhibit 1. She assumes that market spreads and yields will remain stable over the year. Exhibit 1: One-Year Transition Matrix for BBB Rated Bonds and Credit Spreads AAA AA A BBB BB B CCC, CC, C Probability (%) 0.02 0.30 4.80 85.73 6.95 1.75 0.45 Credit spread 0.60% 0.90% 1.10% 1.50% 3.40% 6.50% 9.50% Lebedeva next asks Kowalski to analyze a three-year bond, issued by VraiRive S.A., using an arbitrage-free framework. The bond’s coupon rate is 5%, with interest paid annually and a par value of 100. In her analysis, she makes the following three assumptions: ■ The annual interest rate volatility is 10%. ■ The recovery rate is one-third of the exposure each period. ■ The annual probability of default each year is 2.00%. Selected information on benchmark government bonds for the VraiRive bond is presented in Exhibit 2, and the relevant binomial interest rate tree is presented in Exhibit 3. Exhibit 2: Par Curve Rates for Annual Payment Benchmark Government Bonds Coupon Rate Price Discount Factor Spot Rate Forward Rate 1 3.00% 100 0.970874 3.0000% 3.0000% 2 4.20% 100 0.920560 4.2255% 5.4656% 3 5.00% 100 0.862314 5.0618% 6.7547% Maturity Practice Problems 601 Exhibit 3: One-Year Binomial Interest Rate Tree for 10% Volatility (risk-neutral probabilities in parentheses) Date 0 Date 1 Date 2 8.1823% (0.2500) 6.0139% (0.5000) 3.0000% (1.0000) 6.6991% (0.5000) 4.9238% (0.5000) 5.4848% (0.2500) Kowalski estimates the value of the VraiRive bond assuming no default(VND) as well as the fair value of the bond. She then estimates the bond’s yield to maturity and the bond’s credit spread over the benchmark in Exhibit 2. Kowalski asks Lebedeva, “What might cause the bond’s credit spread to decrease?” Lebedeva and Kowalski next discuss the drivers of the term structure of credit spreads. Kowalski tells Lebedeva the following: Statement 1 The credit term structure for the most highly rated securities tends to be either flat or slightly upward sloping. Statement 2 The credit term structure for lower-rated securities is often steeper, and credit spreads widen with expectations of strong economic growth. Next, Kowalski analyzes the outstanding bonds of DLL Corporation, a high-quality issuer with a strong, competitive position. Her focus is to determine the rationale for a positive-sloped credit spread term structure. Lebedeva ends the meeting by asking Kowalski to recommend a credit analysis approach for a securitized asset-backed security (ABS) held in the portfolio. This non-static asset pool is made up of many medium-term auto loans that are homogeneous, and each loan is small relative to the total value of the pool. 24. The most appropriate response to Kowalski’s question regarding credit rating migration is that it has: A. a negative impact. B. no impact. C. a positive impact. 25. Based on Exhibit 1, the one-year expected return on the Entre Corp. bond is closest to: A. 3.73%. B. 5.50%. C. 7.27%. 602 Learning Module 4 Credit Analysis Models 26. Based on Kowalski’s assumptions and Exhibits 2 and 3, the credit spread on the VraiRive bond is closest to: A. 0.6949%. B. 0.9388%. C. 1.4082%. 27. The most appropriate response to Kowalski’s question relating to the credit spread is: A. an increase in the probability of default. B. an increase in the loss given default. C. a decrease in the risk-neutral probability of default. 28. Which of Kowalski’s statements regarding the term structure of credit spreads is correct? A. Only Statement 1 B. Only Statement 2 C. Both Statement 1 and Statement 2 29. DLL’s credit spread term structure is most consistent with the firm having: A. low leverage. B. weak cash flow. C. a low profit margin. 30. Given the description of the asset pool of the ABS, Kowalski should recommend a: A. loan-by-loan approach. B. portfolio-based approach. C. statistics-based approach. Solutions 603 SOLUTIONS 1. B is correct. The expected exposure is the projected amount of money that an investor could lose if an event of default occurs, before factoring in possible recovery. The expected exposure for both Bond I and Bond II is 100 + 5 = 105. 2. C is correct. The loss given default is a positive function of the expected exposure to default loss and a negative function of the recovery rate. Because Bond II has a lower recovery rate than Bond I and the same expected exposure to default loss (100 + 5 = 105), it will have a higher loss given default than Bond I will have. The loss given default for Bond I is 105 × (1 – 0.40) = 63.00. The loss given default for Bond II is 105 × (1 – 0.35) = 68.25. 3. B is correct. In the event of no default, the investor is expected to receive 105. In the event of a default, the investor is expected to receive 105 – [105 × (1 – 0.40)] = 42. The expected future value of the bond is, therefore, the weighted average of the no-default and default amounts, or (105 × 0.98) + (42 × 0.02) = 103.74. 4. B is correct. The risk-neutral default probability, P*, is calculated using the current price, the expected receipt at maturity with no default (that is, 100 + 5 = 105), the expected receipt at maturity in the event of a default (that is, 0.40 × 105 = 42), and the risk-free rate of interest (0.03): ​[​ ​105 × ​(​ ​1 − ​P​*​)​]​ ​​+ ​(​ ​42 × ​P​*​)​​        .​​ ​100 = ______________________ ​ 1.03 Solving for P* gives 0.031746, or 3.1746%. 5. A is correct. The CVA is the sum of the present value of expected losses on the bond, which from Exhibit 2 is 3.3367. 6. C is correct. The expected percentage price change is the product of the negative of the modified duration and the difference between the credit spread in the new rating and the old rating: Expected percentage price change = –4.2 × (0.0175 – 0.01) = –0.0315, or –3.15%. 7. B is correct. A reduced-form model in credit risk analysis uses historical variables, such as financial ratios and macroeconomic variables, to estimate the default intensity. A structural model for credit risk analysis, in contrast, uses option pricing and relies on a traded market for the issuer’s equity. 8. B is correct. Observation 1 is incorrect, but Observation 2 is correct. The actual default probabilities do not include the default risk premium associated with the uncertainty in the timing of the possible default loss. The observed spread over the yield on a risk-free bond in practice does include liquidity and tax considerations, in addition to credit risk. 9. B is correct. The following table shows that the credit valuation adjustment (CVA) for the bond is €36.49, the sum of the present values of expected loss. The steps taken to complete the table are as follows. Step 1 €1, 000 Exposure at date T is _ ​ ​,​ where r is 3%. That is, ​(​1 + r​)​4−T​ exposure is computed by discounting the face value of the bond using the risk-free rate and the number of years until maturity. 604 Learning Module 4 Credit Analysis Models Step 2 Recovery = Exposure × Recovery rate. Step 3 Loss given default (LGD) = Exposure – Recovery. Step 4 Probability of default (POD) on Date 1 is 1.50%. The probability of survival (POS) on Date 1 is 98.50%. For subsequent dates, POD is calculated as the annual default probability multiplied by the previous date’s POS. For example, to determine the Date 2 POD (1.4775%), the annual default probability (1.50%) is multiplied by the Date 1 POS (98.50%). Step 1 POS in Dates 2–4 = POS in the previous year – POD. That is, POS in year T = POS in year (T – 1) – POD in year T. POS can also be determined by subtracting the annual default probability from 100% and raising it to the power of the number of years: (100% – 1.5000%)1 = 98.5000%. (100% – 1.5000%)2 = 97.0225%. (100% – 1.5000%)3 = 95.5672%. (100% – 1.5000%)4 = 94.1337%. Step 2 Expected loss = LGD × POD. Step 3 Discount factor (DF) for date T is _ ​ 1 T ​​, where r is 3%. Step 4 PV of expected loss = Expected loss × DF. ​(​1 + r​)​ ​ Exposure Recovery LGD POD POS Expected Loss DF PV of Expected Loss 1 €915.14 €274.54 €640.60 1.5000% 98.5000% €9.61 0.970874 €9.33 2 €942.60 €282.78 €659.82 1.4775% 97.0225% €9.75 0.942596 €9.19 3 €970.87 €291.26 €679.61 1.4553% 95.5672% €9.89 0.915142 €9.05 4 €1,000.00 €300.00 €700.00 1.4335% 94.1337% €10.03 0.888487 €8.92 CVA = €36.49 Date 0 The value of the bond if it were default free would be 1,000 × DF for Date 4 = €888.49. Fair value of the bond considering CVA = €888.49 – CVA = €888.49 – €36.49 = €852.00. Because the market price of the bond (€875) is greater than the fair value of €852, B is correct. A is incorrect because the market price of the bond differs from its fair value. C is incorrect because although the bond’s value if the bond were default free is greater than the market price, the bond has a risk of default, and CVA lowers its fair value to below the market price. 10. B is correct. The recovery rate to be used now in the computation of fair value is 30% × 1.25 = 37.5%, whereas the default probability to be used is 1.50% × 1.25 = Solutions 605 1.875%. Using the steps outlined in the solution to Question 1, the following table is prepared, which shows that the bond’s CVA increases to 40.49. Thus, Koning concludes that a change in the probability of default has a greater effect on fair value than a similar change in the recovery rate. The steps taken to complete the table are the same as those in the previous problem. There are no changes in exposures and discount factors in this table. Exposure Recovery LGD POD POS Expected Loss DF PV of Expected Loss 1 €915.14 €343.18 €571.96 1.8750% 98.1250% €10.72 0.970874 €10.41 2 €942.60 €353.47 €589.12 1.8398% 96.2852% €10.84 0.942596 €10.22 3 €970.87 €364.08 €606.80 1.8053% 94.4798% €10.95 0.915142 €10.03 4 €1,000.00 €375.00 €625.00 1.7715% 92.7083% €11.07 0.888487 €9.84 CVA = €40.49 DF PV of Expected Loss Date 0 Changes in the default probability and recovery rates do not affect the value of the default-free bond. So, it is the same as in the previous question: €888.49. Fair value of the bond considering CVA = €888.49 – CVA = €888.49 – €40.49 = €848.00 11. A is correct. The following table shows that the CVA for the bond is €42.17, the sum of the present values of expected loss. The steps taken to complete the table are as follows. Step 1 Exposure at Date 4 is €1,000 + Coupon amount = €1,000 + €60 = €1,060. Exposure at a date T prior to that is the coupon on date T + PV at date T of subsequent coupons + PV of €1,000 to be received at Date 4. For example, exposure at Date 2 is €60 €60 €1, 000 €60 €1, 060 €60 + _ ​1 + 0.03 ​+ _ ​( ​+ _ ​ ​ = €60 + _ ​1 + 0.03 ​+ _ ​( ​ ​ ​1 + 0.03​)​2​ ​(​1 + 0.03​)​2​ ​ ​1 + 0.03​)​2​​. ​​        = €1, 117.40. Steps 2 through 8 are the same as those in the solution to Question 1. Date Exposure Recovery LGD POD POS Expected Loss 0 1 €1,144.86 €343.46 €801.40 1.5000% 98.5000% €12.02 0.970874 €11.67 2 €1,117.40 €335.22 €782.18 1.4775% 97.0225% €11.56 0.942596 €10.89 3 €1,089.13 €326.74 €762.39 1.4553% 95.5672% €11.10 0.915142 €10.15 4 €1,060.00 €318.00 €742.00 1.4335% 94.1337% €10.64 0.888487 €9.45 CVA = €42.17 The value of the bond if it were default free would be €60 × DF1 + €60 × DF2 + €60 × DF3 + €1,060 × DF4 = €1,111.51. Fair value of the bond considering CVA = €1,111.51 – €42.17 = €1,069.34. 12. A is correct. If default occurs on Date 3, the rate of return can be obtained by solving the following equation for internal rate of return (IRR): 606 Learning Module 4 Credit Analysis Models €60 €60 €326.74 ​€1, 090 = _ ​1 + IRR ​+ _ ​( +_ ​ ​.​ ​ ​1 + IRR​)​2​ (​ ​1 + IRR​)​3​ In this equation, €60 is the amount of coupon received at Dates 1 and 2 prior to default at Date 3. The amount €326.74 is the recovery at Time 3 (from the CVA table in the solution to the previous question). The solution to the foregoing equation can be obtained using the cash flow IRR function on your calculator. 13. B is correct. For each possible transition, the expected percentage price change, computed as the product of the modified duration and the change in the spread as shown in Exhibit 3 (relating to question 5), is calculated as follows: From AA to AAA: –2.75 × (0.60% – 0.90%) = +0.83%. From AA to A: –2.75 × (1.10% – 0.90%) = –0.55%. From AA to BBB: –2.75 × (1.50% – 0.90%) = –1.65%. From AA to BB: –2.75 × (3.40% – 0.90%) = –6.88%. From AA to B: –2.75 × (6.50% – 0.90%) = –15.40%. From AA to C: –2.75 × (9.50% – 0.90%) = –23.65%. The expected percentage change in the value of the AA rated bond is computed by multiplying each expected percentage price change for a possible credit transition by its respective transition probability given in Exhibit 3 and summing the products: (0.0150 × 0.83%) + (0.8800 × 0%) + (0.0950 × –0.55%) + (0.0075 × –1.65%) + (0.0015 × –6.88%) + (0.0005 × –15.40%) + (0.0003 × –23.65%) = –0.0774%. Therefore, the expected return on the bond over the next year is its YTM minus 0.0774%, assuming no default. 14. B is correct. Statement B is correct because a reduced-form credit model involves regression analysis using information generally available in the financial markets, such as the measures mentioned in the statement. Statement A is incorrect because it is consistent with the use of a structural model and not a reduced-form model. It is a structural model that is based on the premise that a firm defaults on its debt if the value of its assets falls below its liabilities and that the probability of that event has the characteristics of an option. Statement C is incorrect because it is consistent with the use of a structural model and not a reduced-form model. A structural model involves the estimation of a default barrier, and default occurs if the value of firm's assets falls below the default barrier. 15. C is correct. Structural models require information best known to the managers of the company. Reduced-form models require information only generally available in financial markets. A is incorrect because although it is literally true, when the models were developed is immaterial. Structural models are currently used in practice by commercial banks and credit rating agencies. B is incorrect because computer technology facilitates valuation using option pricing models as well as regression analysis. 16. A is correct. The following tree shows the valuation assuming no default of Bond B2, which pays a 6% annual coupon. Solutions Date 0 607 Date 1 Date 2 1,008.76 4.3999% 1,063.57 2.1180% 1,144.63 –0.2500% 60 Date 3 Date 4 980.75 8.0804% 1,060 60 1,005.54 5.4164% 60 1,043.43 2.9493% 60 1,099.96 1.4197% 60 1,022.86 3.6307% 1,067.73 1.9770% 60 1,034.81 2.4338% 1,060 1,060 1,060 The scheduled year-end coupon and principal payments are placed to the right of each forward rate in the tree. For example, the Date 4 values are the principal plus the coupon of 60. The following are the four Date 3 values for the bond, shown above the interest rate at each node: €1,060/1.080804 = €980.75. €1,060/1.054164 = €1,005.54. €1,060/1.036307 = €1,022.86. €1,060/1.024338 = €1,034.81. These are the three Date 2 values: ​(​ ​0.5 × €980.75​)​​+ ​(​ ​0.5 × €1, 005.54​)​​+ €60 _______________________________        ​ = €1, 008 . 76. 1.043999 ​(​ ​0.5 × €1, 005.54​)​​+ ​(​ ​0.5 × €1, 022.86​)​​+ €60 ________________________________         ​ = €1, 043.43. 1.029493 ​(​ ​0.5 × €1, 022.86​)​​+ ​(​ ​0.5 × €1, 034.81​)​​+ €60 ________________________________        ​ = €1, 067.73. 1.019770 These are the two Date 1 values: ​(​ ​0.5 × €1, 008.76​)​​+ ​(​ ​0.5 × €1, 043.43​)​​+ €60 ________________________________         ​ = €1, 063.57. 1.021180 ​(​ ​0.5 × €1, 043.43​)​​+ ​(​ ​0.5 × €1, 067.73​)​​+ €60 ________________________________        ​ = €1, 099.96. 1.014197 This is the Date 0 value: ​(​ ​0.5 × €1, 063.57​)​​+ ​(​ ​0.5 × €1, 099.96​)​​+ €60 ________________________________        ​ = €1, 144.63. 0.997500 So, the value of the bond assuming no default is 1,144.63. This value could also have been obtained more directly using the benchmark discount factors from Exhibit 2: 608 Learning Module 4 Credit Analysis Models €60 × 1.002506 + €60 × 0.985093 + €60 × 0.955848 + €1,060 × 0.913225 = €1,144.63. The benefit of using the binomial interest rate tree to obtain the VND is that the same tree is used to calculate the expected exposure to default loss. The credit valuation adjustment table is now prepared following these steps: Step 1 Compute the expected exposures as described in the following, using the binomial interest rate tree prepared earlier. The expected exposure for Date 4 is €1,060. The expected exposure for Date 3 is (0.1250 × €980.75) + (0.3750 × €1,005.54) + (0.3750 × €1,022.86) + (0.1250 × €1,034.81) + 60 = €1,072.60. The expected exposure for Date 2 is (0.25 × €1,008.76) + (0.50 × €1,043.43) + (0.25 × €1,067.73) + €60 = €1,100.84. The expected exposure for Date 1 is (0.50 × €1,063.57) + (0.50 × €1,099.96) + 60 = €1,141.76. Step 2 LGD = Exposure × (1 – Recovery rate). Step 3 The initial default probability is 1.50%. For subsequent dates, POD is calculated as the default probability multiplied by the previous date’s POS. For example, to determine the Date 2 POD (1.4775%), the default probability (1.5000%) is multiplied by the Date 1 POS (98.5000%). Step 4 POS is determined by subtracting the default probability from 100% and raising it to the power of the number of years: (100% – 1.5000%)1 = 98.5000%. (100% – 1.5000%)2 = 97.0225%. (100% – 1.5000%)3 = 95.5672%. (100% – 1.5000%)4 = 94.1337%. Step 5 Expected loss = LGD × POD. Step 6 Discount factors in year T are obtained from Exhibit 2. Step 7 PV of expected loss = Expected loss × DF. Exposure LGD POD POS Expected Loss DF PV of Expected Loss 1 €1,141.76 €799.23 1.5000% 98.5000% €11.99 1.002506 €12.02 2 €1,100.84 €770.58 1.4775% 97.0225% €11.39 0.985093 €11.22 Date 0 Solutions 609 Date Exposure LGD POD POS Expected Loss DF PV of Expected Loss 3 €1,072.60 €750.82 1.4553% 95.5672% €10.93 0.955848 €10.44 4 €1,060.00 €742.00 1.4335% 94.1337% €10.64 0.913225 €9.71 CVA = €43.39 DF PV of Expected Loss Fair value of the bond considering CVA = €1,144.63 – CVA = €1,144.63 – €43.39 = €1,101.24. 17. A is correct. The corporate bond’s fair value is computed in the solution to Question 8 as €1,101.24. The YTM can be obtained by solving the following equation for IRR: €60 €60 €60 €1, 060 ​( ​+ _ ​ ​+ _ ​ ​. ​€1, 101.24 = _ ​1 + IRR ​+ _ ​ ​1 + IRR​)​2​ (​ ​1 + IRR​)​3​ (​ ​1 + IRR​)​4​ The solution to this equation is 3.26%. Valuation of a four-year, 6% coupon bond under no default is computed in the solution to Question 8 as 1,144.63. So, the YTM of a theoretical comparable-maturity government bond with the same coupon rate as the corporate bond, B2, can be obtained by solving the following equation for IRR: €60 €60 €60 €1, 060 ​€1, 144.63 = _ ​1 + IRR ​+ _ ​( ​+ _ ​ ​+ _ ​ ​. ​ ​1 + IRR​)​2​ ​(​1 + IRR​)​3​ ​(​1 + IRR​)​4​ The solution to this equation is 2.18%. So, the credit spread that the analyst wants to compute is 3.26% – 2.18% = 1.08%, or 108 bps. B is incorrect because it is the spread over the four-year government par bond that has a YTM of 2.25% in Exhibit 2: 3.26% – 2.25% = 1.01%, or 101 bps. Although this spread is commonly used in practice, the analyst is interested in finding the spread over a theoretical 6% coupon government bond. C is incorrect because it is the YTM of the coupon four-year government bond in Exhibit 2. 18. B is correct. The recovery rate to be used now in the computation of fair value is 30% × 0.75 = 22.500%, whereas the default probability to be used is 1.50% × 0.75 = 1.125%. The tree that shows the valuation assuming no default of Bond B2 in the solution to Question 8 will not be affected by the foregoing changes. Accordingly, VND remains €1,144.63. Following the steps outlined in the solution to Question 8, the following table is prepared, which shows that the CVA for the bond decreases to €36.23. Thus, Ibarra concludes that a decrease in the probability of default has a greater effect on fair value than a similar decrease in the recovery rate. The steps taken to complete the table are the same as those in Question 8. There are no changes in exposures or discount factors in this table. Date Exposure LGD POD POS Expected Loss 0 1 €1,141.76 €884.87 1.1250% 98.8750% €9.95 1.002506 €9.98 2 €1,100.84 €853.15 1.1123% 97.7627% €9.49 0.985093 €9.35 3 €1,072.60 €831.26 1.0998% 96.6628% €9.14 0.955848 €8.74 4 €1,060.00 €821.50 1.0875% 95.5754% €8.93 0.913225 €8.16 CVA = €36.23 610 Learning Module 4 Credit Analysis Models Fair value of the bond considering CVA = €1,144.63 – CVA = €1,144.63 – €36.23 = €1,108.40. 19. A is correct. The following tree shows the valuation assuming no default of the floating-rate note (FRN), B4, which has a quoted margin of 4%. Date 0 Date 1 Date 2 Date 3 1,037.01 8.0804% 1,074.21 4.3999% 1,112.73 2.1180% 1,154.27 –0.2500% 61.18 Date 4 1,120.80 84.00 1,037.94 5.4164% 37.50 1,076.03 2.9493% 69.49 1,115.03 1.4197% 54.20 1,038.60 3.6307% 1,077.30 1.9770% 59.77 1,039.05 2.4338% 1,094.16 1,076.31 1,064.34 The scheduled year-end coupon and principal payments are placed to the right of each forward rate in the tree. For example, the four Date 4 values are the principal plus the coupon. €1,000 × (1 + 0.080804 + 0.04) = €1,120.80. €1,000 × (1 + 0.054164 + 0.04) = €1,094.16. €1,000 × (1 + 0.036307 + 0.04) = €1,076.31. €1,000 × (1 + 0.024338 + 0.04) = €1,064.34. The following are the four Date 3 bond values for the note, shown above the interest rate at each node: €1,120.80/1.080804 = €1,037.01. €1,094.16/1.054164 = €1,037.94. €1,076.31/1.036307 = €1,038.60. €1,064.34/1.024338 = €1,039.05. The three Date 3 coupon amounts are computed based on the interest rate at Date 2 plus the quoted margin of 4%: €1,000 × (0.043999 + 0.04) = €84.00. €1,000 × (0.029493 + 0.04) = €69.49. €1,000 × (0.019770 + 0.04) = €59.77. There are three Date 2 bond values: ​(​ ​0.5 × €1, 037.01​)​​+ ​(​ ​0.5 × €1, 037.94​)​​+ €84.00 __________________________________         ​ = €1, 074 . 21. 1.043999 Solutions 611 ​(​ ​0.5 × €1, 037.94​)​​+ ​(​ ​0.5 × €1, 038.60​)​​+ €69.49 __________________________________         ​ = €1, 076.03.​ 1.029493 ​(​ ​0.5 × €1, 038.60​)​​+ ​(​ ​0.5 × €1, 039.05​)​​+ €59.77 __________________________________         ​ = €1, 077.30. 1.019770 The two Date 2 coupon amounts are computed based on the interest rate at Date 1 plus the quoted margin of 4%: €1,000 × (0.021180 + 0.04) = €61.18. €1,000 × (0.014197 + 0.04) = €54.20. The Date 1 coupon amount is computed based on the interest rate at date 0 plus the quoted margin of 4%: €1,000 × (–0.0025 + 0.04) = €37.50. These are the calculations for the bond values for Date 1 and Date 0: ​(​ ​0.5 × €1, 074.21​)​​+ ​(​ ​0.5 × €1, 076.03​)​​+ €61 . 18 ___________________________________          ​ = €1, 112.73. 1.021180 ​(​ ​0.5 × €1, 076.06​)​​+ ​(​ ​0.5 × €1, 077.30​)​​+ €54.20 __________________________________         ​ = €1, 115.0. 1.014197 Then, the VND is calculated as follows: ​(​ ​0.5 × €1, 112.73​)​​+ ​(​ ​0.5 × €1, 115.03​)​​+ €37.50 __________________________________         ​ = €1, 154.27. 0.9975 The expected exposures are then computed using the binomial interest rate tree prepared earlier. For example, the expected exposure for Date 4 is computed as follows: (0.125 × €1,120.80) + (0.375 × €1,094.16) + (0.375 × €1,076.31) + (0.125 × €1,064.34) = €1,087.07. Similarly, the expected exposure for Date 3 is computed as follows: (0.125 × €1,037.01) + (0.375 × €1,037.94) + (0.375 × €1,038.60) + (0.125 × €1,039.05) + (0.250 × €84) + (0.500 × €69.49) + (0.250 × €59.77) = €1,108.90. The expected exposures for Dates 2 and 1 are computed similarly, and the credit valuation adjustment table is completed following Steps 2–7 outlined in the solution to Question 8. Exposure LGD POD POS Expected Loss DF PV of Expected Loss 1 €1,151.38 €805.97 1.5000% 98.5000% €12.09 1.002506 €12.12 2 €1,133.58 €793.51 1.4775% 97.0225% €11.72 0.985093 €11.55 3 €1,108.90 €776.23 1.4553% 95.5672% €11.30 0.955848 €10.80 4 €1,087.07 €760.95 1.4335% 94.1337% €10.91 0.913225 €9.96 CVA = €44.43 Date 0 Fair value of the FRN considering CVA = €1,154.27 – CVA = €1,154.27 – €44.43 = €1,109.84. Because the market price of €1,070 is less than the estimated fair value, the analyst should recommend adding to existing positions in the FRN. 612 Learning Module 4 Credit Analysis Models B and C are incorrect because the FRN is perceived to be undervalued in the market. 20. A is correct. The changing probability of default will not affect the binomial tree prepared in the solution to Question 11. The Date 1 value remains €1,154.27, which is also the VND. The expected exposures, loss given default, and discount factors are also unaffected by the changing probability of default. The following is the completed credit valuation adjustment table. Date Exposure LGD POD POS Expected Loss DF PV of Expected Loss 0 1 €1,151.38 €805.97 1.5000% 98.5000% €12.09 1.002506 €12.12 2 €1,133.58 €793.51 0.4925% 98.0075% €3.91 0.985093 €3.85 3 €1,108.90 €776.23 0.4900% 97.5175% €3.80 0.955848 €3.64 4 €1,087.07 €760.95 0.4876% 97.0299% €3.71 0.913225 €3.39 CVA = €22.99 Thus, CVA decreases to €22.99. 21. C is correct. The credit rating agencies typically make incremental changes, as seen in a transition matrix provided in Exhibit 3. Ibarra believes the bond is undervalued, because her assessment of the probability of default and the recovery rate is more optimistic than that of the agencies. Therefore, she most likely expects the credit rating agencies to put the issuer on a positive watch. A is incorrect because the bond is perceived to be undervalued, not overvalued. Ibarra is not expecting a credit downgrade. B is incorrect because it is not the most likely expectation. The rating agencies rarely change an issuer’s rating from BBB all the way to AAA. In Exhibit 3 (relating to question 5) the probability of a BBB rated issuer going from BBB to AAA is 0.02%, whereas to go from BBB to A it is 4.80%. 22. A is correct. B is incorrect because, although generally true for investment-grade bonds, the statement neglects the fact that high-yield issuers sometimes face a downward-sloping credit term structure. Credit term structures are not always upward sloping. C is incorrect because there is a consistent pattern for the term structure of credit spreads: Typically, it is upwardly sloped because greater time to maturity is associated with higher projected probabilities of default and lower recovery rates. 23. C is correct. A covered bond is a senior debt obligation of a financial institution that gives recourse to the originator/issuer as well as a predetermined underlying collateral pool. Each country or jurisdiction specifies the eligible collateral types as well as the specific structures permissible in the covered bond market. Covered bonds usually have either commercial or residential mortgages meeting specific criteria or public sector exposures as underlying collateral. A is incorrect. The term “covered” is used in foreign exchange analysis, for instance, “covered interest rate parity.” In the context of securitized debt, a covered bond is secured by specific assets in addition to the overall balance sheet of the issuer. B is incorrect because a covered bond does not involve a credit default swap. In addition, an issuer is not likely to sell a credit default swap on its own liability. Solutions 613 24. A is correct. Credit spread migration typically reduces the expected return for two reasons. First, the probabilities for rating changes are not symmetrically distributed around the current rating; they are skewed toward a downgrade rather than an upgrade. Second, the increase in the credit spread is much larger for downgrades than is the decrease in the spread for upgrades. 25. A is correct. The expected return on the Entre Corp. bond over the next year is its yield to maturity plus the expected percentage price change in the bond over the next year. In the following table, for each possible transition, the expected percentage price change is the product of the bond’s modified duration of 7.54, multiplied by –1, and the change in the spread, weighted by the given probability: Expected percentage price change = (0.0002 × 6.786%) + (0.0030 × 4.524%) + (0.0480 × 3.016%) + (0.8573 × 0.000%) + (0.0695 × –14.326%) + (0.0175 × –37.700%) + (0.0045 × –60.320%) = –1.76715%. So, the expected return on the Entre Corp. bond is its yield to maturity plus the expected percentage price change due to credit migration: Expected return = 5.50% – 1.77% = 3.73%. Expected % Price Change (1) Probability (2) Expected % Price Change × Probability (1 × 2) From BBB to AAA –7.54 × (0.60% – 1.50%) = 6.786% 0.0002 0.00136 From BBB to AA –7.54 × (0.90% – 1.50%) = 4.524% 0.0030 0.01357 From BBB to A –7.54 × (1.10% – 1.50%) = 3.016% 0.0480 0.14477 From BBB to BB –7.54 × (3.40% – 1.50%) = –14.326% 0.0695 –0.99566 From BBB to B –7.54 × (6.50% – 1.50%) = –37.700% 0.0175 –0.65975 From BBB to CCC, CC, or C –7.54 × (9.50% – 1.50%) = –60.320% 0.0045 –0.27144 Total: –1.76715 26. C is correct. The credit spread can be calculated in three steps: Step 1 Estimate the value of the three-year VraiRive bond assuming no default. Based on Kowalski’s assumptions and Exhibits 2 and 3, the value of the three-year VraiRive bond assuming no default is 100.0000. Date 0 100.0000 3.0000% Date 1 Date 2 Date 3 97.0584 8.1823% 105.00 96.9052 6.0139% 5.00 5.00 98.4076 6.6991% 99.0948 4.9238% 5.00 99.5404 5.4848% Supporting calculations: 105.00 105.00 614 Learning Module 4 Credit Analysis Models The bond value in each node is the value of next period’s cash flows discounted by the forward rate. For the three nodes on Date 2, the bond values are as follows: 105/1.081823 = 97.0584. 105/1.066991 = 98.4076. 105/1.054848 = 99.5404. For the two nodes on Date 1, the two bond values are as follows: [(0.5 × 97.0584) + (0.5 × 98.4076) + 5.00]/1.060139 = 96.9052. [(0.5 × 98.4076) + (0.5 × 99.5404) + 5.00]/1.049238 = 99.0948. Finally, for the node on Date 0, the bond value is [(0.5 × 96.9052) + (0.5 × 99.0948) + 5.00]/1.030000 = 100.0000. Therefore, the VND for the VraiRive bond is 100.0000. Step 2 Calculate the credit valuation adjustment, and then subtract the CVA from the VND from Step 1 to establish the fair value of the bond. The CVA equals the sum of the present values of each year’s expected loss and is calculated as follows: Date Expected Exposure Loss Given Default Probability of Default Discount Factor Present Value of Expected Loss 1 103.0000 68.6667 2.0000% 0.970874 1.3333 2 103.3535 68.9023 1.9600% 0.920560 1.2432 3 105.0000 70.0000 1.9208% 0.862314 1.1594 CVA = 3.7360 Supporting calculations: The expected exposures at each date are the bond values at each node, weighted by their risk-neutral probabilities, plus the coupon payment: Date 1: (0.5 × 96.9052) + (0.5 × 99.0948) + 5.00 = 103.0000. Date 2: (0.25 × 97.0584) + (0.5 × 98.4076) + (0.25 × 99.5404) + 5.00 = 103.3535. Date 3: 105.0000 The loss given default on each date is 2/3 of the expected exposure. The probability of default on each date is as follows: Date 1: 2% Date 2: 2% × (100% – 2%) = 1.96%. Date 3: 2% × (100% – 2%)2 = 1.9208%. Solutions 615 The discount factor on each date is 1/(1 + spot rate for the date) raised to the correct power. Finally, the credit valuation adjustment each year is the product of the LGD times the POD times the discount factor, as shown in the last column of the table. The sum of the three annual CVAs is 3.7360. So, the fair value of the VraiRive bond is the VND less the CVA, or VND – CVA = 100 – 3.7360 = 96.2640. Step 3 Based on the fair value from Step 2, calculate the yield to maturity of the bond, and solve for the credit spread by subtracting the yield to maturity on the benchmark bond from the yield to maturity on the VraiRive bond. The credit spread is equal to the yield to maturity on the VraiRive bond minus the yield to maturity on the three-year benchmark bond (which is 5.0000%). Based on its fair value of 96.2640, the VraiRive bond’s yield to maturity is 5 5 105 ​96.2640 = _ ​(​ ​1 + YTM​)​​+ _ ​( +_ ​ ​​. ​ ​1 + YTM​)​2​ (​ ​1 + YTM​)​3​ Solving for YTM, the yield to maturity is 6.4082%. Therefore, the credit spread on the VraiRive bond is 6.4082% – 5.0000% = 1.4082%. 27. C is correct. A decrease in the risk-neutral probability of default would decrease the credit valuation adjustment and decrease the credit spread. In contrast, increasing the bond’s loss-given-default assumption and increasing the probability-of-default assumption would increase the credit valuation adjustment and decrease the fair value of the bond (and increase the yield to maturity and the credit spread over its benchmark). 28. A is correct. For investment-grade bonds with the highest credit ratings, credit spreads are extremely low, and credit migration is possible only in one direction given the implied lower bound of zero on credit spreads. As a result, the credit term structure for the most highly rated securities tends to be either flat or slightly upward sloping. Securities with lower credit quality, however, face greater sensitivity to the credit cycle. Credit spreads would decrease, not increase, with the expectation of economic growth. There is a countercyclical relationship between credit spreads and benchmark rates over the business cycle. A strong economic climate is associated with higher benchmark yields but lower credit spreads because the probability of issuers defaulting declines in such good times. 29. A is correct. Positive-sloped credit spread curves may arise when a high-quality issuer with a strong competitive position in a stable industry has low leverage, strong cash flow, and a high profit margin. This type of issuer tends to exhibit very low short-term credit spreads that rise with increasing maturity given greater uncertainty due to the macroeconomic environment, potential adverse changes in the competitive landscape, technological change, or other factors that drive a higher implied probability of default over time. Empirical academic studies also tend to support the view that the credit spread term structure is upward sloping for investment-grade bond portfolios. 616 Learning Module 4 Credit Analysis Models 30. B is correct. The auto ABS is granular, with many small loans relative to the size of the total portfolio. The auto loans are also homogeneous. These characteristics support using the portfolio-based approach. A loan-by-loan approach would be inefficient because of the large number of basically similar loans; this approach is best for a portfolio of discrete, large loans that are heterogeneous. A statistics-based approach would work for a static book of loans, whereas the auto loan portfolio would be dynamic and would change over time. Practice Problems 643 PRACTICE PROBLEMS The following information relates to questions 1-6 UNAB Corporation On 1 January 20X2, Deem Advisors purchased a $10 million six-year senior unsecured bond issued by UNAB Corporation. Six months later (1 July 20X2), concerned about the portfolio’s credit exposure to UNAB, Doris Morrison, the chief investment officer at Deem Advisors, buys $10 million protection on UNAB with a standardized coupon rate of 5%. The reference obligation of the CDS is the UNAB bond owned by Deem Advisors. UNAB adheres to the ISDA CDS protocols. On 1 January 20X3, Morrison asks Bill Watt, a derivatives analyst, to assess the current credit quality of UNAB bonds and the value of Deem Advisors’ CDS on UNAB debt. Watt gathers the following information on UNAB’s debt issues currently trading in the market: Bond 1: A two-year senior unsecured bond trading at 40% of par Bond 2: A five-year senior unsecured bond trading at 50% of par Bond 3: A five-year subordinated unsecured bond trading at 20% of par With respect to the credit quality of UNAB, Watt makes the following statement: “There is severe near-term stress in the financial markets, andUNAB’s credit curve clearly reflects the difficult environment.” On 1 July 20X3, UNAB fails to make a scheduled interest payment on the outstanding subordinated unsecured obligation after a grace period; however, the company does not file for bankruptcy. Morrison asks Watt to determine if UNAB experienced a credit event and, if so, to recommend a settlement preference. Kand Corporation Morrison is considering purchasing protection on Kand Corporation debt to hedge the portfolio’s position in Kand. She instructs Watt to determine if an upfront payment would be required and, if so, the amount of the premium. Watt presents the information for the CDS in Exhibit 1. Exhibit 1: Summary Data for 10-year CDS on Kand Corporation Credit spread 700 bps Duration 7 years Coupon rate 5% Morrison purchases 10-year protection on Kand Corporation debt. Two months later the credit spread for Kand Corporation has increased by 200 bps. Morrison asks Watt to close out the firm’s CDS position on Kand Corporation by entering into a new, offsetting contract. 644 Learning Module 5 Credit Default Swaps Tollunt Corporation Deem Advisors’ chief credit analyst recently reported that Tollunt Corporation’s five-year bond is currently yielding 7% and a comparable CDS contract has a credit spread of 4.25%. Since the current market reference rate is 2.5%, Watt has recommended executing a basis trade to take advantage of the pricing of Tollunt’s bonds and CDS. The basis trade would consist of purchasing both the bond and the CDS contract. 1. If UNAB experienced a credit event on 1 July, Watt should recommend that Deem Advisors: A. prefer a cash settlement. B. prefer a physical settlement. C. be indifferent between a cash or a physical settlement. 2. According to Watt’s statement, the shape of UNAB’s credit curve is most likely: A. flat. B. upward-sloping. C. downward-sloping. 3. Should Watt conclude that UNAB experienced a credit event? A. Yes B. No, because UNAB did not file for bankruptcy C. No, because the failure to pay occurred on a subordinated unsecured bond 4. Based on Exhibit 1, the upfront premium as a percent of the notional for the CDS protection on Kand Corporation would be closest to: A. 2.0%. B. 9.8%. C. 14.0%. 5. If Deem Advisors enters into a new offsetting contract two months after purchasing protection on Kand Corporation, this action will most likely result in: A. a loss on the CDS position. B. a profit on the CDS position. C. neither a loss nor a profit on the CDS position. 6. If convergence occurs in the bond and CDS markets for Tollunt Corporation, a basis trade will capture a profit closest to: A. 0.25%. B. 1.75%. C. 2.75%. Practice Problems 645 The following information relates to questions 7-14 John Smith, a fixed-income portfolio manager at a €10 billion sovereign wealth fund (the Fund), meets with Sofia Chan, a derivatives strategist with Shire Gate Securities (SGS), to discuss investment opportunities for the Fund. Chan notes that SGS adheres to ISDA (International Swaps and Derivatives Association) protocols for credit default swap (CDS) transactions and that any contract must conform to ISDA specifications. Before the Fund can engage in trading CDS products with SGS, the Fund must satisfy compliance requirements. Smith explains to Chan that fixed-income derivatives strategies are being contemplated for both hedging and trading purposes. Given the size and diversified nature of the Fund, Smith asks Chan to recommend a type of CDS that would allow the Fund to simultaneously fully hedge multiple fixed-income exposures. Smith and Chan discuss opportunities to add trading profits to the Fund. Smith asks Chan to determine the probability of default associated with a five-year investment-grade bond issued by Orion Industrial. Selected data on the Orion Industrial bond are presented in Exhibit 1. Exhibit 1: Selected Data on Orion Industrial Five-Year Bond Year Hazard Rate 1 0.22% 2 0.35% 3 0.50% 4 0.65% 5 0.80% Chan explains that a single-name CDS can also be used to add profit to the Fund over time. Chan describes a hypothetical trade in which the Fund sells £6 million of five-year CDS protection on Orion, where the CDS contract has a duration of 3.9 years. Chan assumes that the Fund closes the position six months later, after Orion’s credit spread narrowed from 150 bps to 100 bps. Chan discusses the mechanics of a long/short trade. In order to structure a number of potential trades, Chan and Smith exchange their respective views on individual companies and global economies. Chan and Smith agree on the following outlooks. Outlook 1: The European economy will weaken. Outlook 2: The US economy will strengthen relative to that of Canada. Outlook 3: The credit quality of electric car manufacturers will improve relative to that of traditional car manufacturers. Chan believes US macroeconomic data are improving and that the general economy will strengthen in the short term. Chan suggests that a curve trade could be used by the Fund to capitalize on her short-term view of a steepening of the US credit curve. Another short-term trading opportunity that Smith and Chan discuss involves the merger and acquisition market. SGS believes that Delta Corporation may make an unsolicited bid at a premium to the market price for all of the publicly traded shares of Zega, Inc. Zega’s market capitalization and capital structure are 646 Learning Module 5 Credit Default Swaps comparable to Delta’s; both firms are highly levered. It is anticipated that Delta will issue new equity along with 5- and 10-year senior unsecured debt to fund the acquisition, which will significantly increase its debt ratio. 7. To satisfy the compliance requirements referenced by Chan, the Fund is most likely required to: A. set a notional amount. B. post an upfront payment. C. sign an ISDA master agreement. 8. Which type of CDS should Chan recommend to Smith? A. CDS index B. Tranche CDS C. Single-name CDS 9. Based on Exhibit 1, the probability of Orion defaulting on the bond during the first three years is closest to: A. 1.07%. B. 2.50%. C. 3.85%. 10. To close the position on the hypothetical Orion trade, the Fund: A. sells protection at a higher premium than it paid at the start of the trade. B. buys protection at a lower premium than it received at the start of the trade. C. buys protection at a higher premium than it received at the start of the trade. 11. The hypothetical Orion trade generated an approximate: A. loss of £117,000. B. gain of £117,000. C. gain of £234,000. 12. Based on the three economic outlook statements, a profitable long/short trade would be to: A. sell protection using a Canadian CDX IG and buy protection using a US CDX IG. B. buy protection using an iTraxx Crossover and sell protection using an iTraxx Main. C. buy protection using an electric car CDS and sell protection using a traditional car CDS. 13. The curve trade that would best capitalize on Chan’s view of the US credit curve Practice Problems is to: A. buy protection using a 20-year CDX and buy protection using a 2-year CDX. B. buy protection using a 20-year CDX and sell protection using a 2-year CDX. C. sell protection using a 20-year CDX and buy protection using a 2-year CDX. 14. A profitable equity-versus-credit trade involving Delta and Zega is to: A. short Zega shares and buy protection on Delta using the 10-year CDS. B. go long Zega shares and buy protection on Delta using 5-year CDS. C. go long Delta shares and buy protection on Delta using 5-year CDS. 647 648 Learning Module 5 Credit Default Swaps SOLUTIONS 1. A is correct. Deem Advisors would prefer a cash settlement. Deem Advisors owns Bond 2 (trading at 50% of par), which is worth more than the cheapest-to-deliver obligation (Bond 1, also a senior secured bond, trading at 40% of par). Based on the price of this cheapest-to-deliver security, the estimated recovery rate is 40%. Thus, Deem Advisors can cash settle for $6 million [= (1 – 40%) × $10 million] on its CDS contract and sell the bond it owns, Bond 2, for $5 million, for total proceeds of $11 million. If Deem Advisors were to physically settle the contract, only $10 million would be received, the face amount of the bonds, and it would deliver Bond 2. B is incorrect because if Deem Advisors were to physically settle the contract, it would receive only $10 million, which is less than the $11 million that could be obtained from a cash settlement. C is incorrect because Deem Advisors would not be indifferent between settlement protocols as the firm would receive $1 million more with a cash settlement in comparison to a physical settlement. 2. C is correct. A downward-sloping credit curve implies a greater probability of default in the earlier years than in the later years. Downward-sloping curves are less common and often are the result of severe near-term stress in the financial markets. A is incorrect because a flat credit curve implies a constant hazard rate (conditional probability of default). B is incorrect because an upward-sloping credit curve implies a greater probability of default in later years. 3. A is correct. UNAB experienced a credit event when it failed to make the scheduled coupon payment on the outstanding subordinated unsecured obligation. Failure to pay, a credit event, occurs when a borrower does not make a scheduled payment of principal or interest on outstanding obligations after a grace period, even without a formal bankruptcy filing. B is incorrect because a credit event can occur without filing for bankruptcy. The three most common credit events are bankruptcy, failure to pay, and restructuring. C is incorrect because a credit event (failure to pay) occurs when a borrower does not make a scheduled payment of principal or interest on any outstanding obligations after a grace period, even without a formal bankruptcy filing. 4. C is correct. An approximation for the upfront premium is (Credit spread – Fixed coupon rate) × Duration of the CDS. To buy 10-year CDS protection, Deem Advisors would have to pay an approximate upfront premium of 1,400 bps [(700 – 500) × 7], or 14% of the notional. A is incorrect because 200 bps, or 2%, is derived by taking the simple difference between the credit spread and the fixed coupon rate (700 – 500), ignoring the duration component of the calculation. B is incorrect because 980 bps, or 9.8%, is the result of dividing the credit spread by the fixed coupon rate and multiplying by the duration of the CDS [(700/500) × 7]. 5. B is correct. Deem Advisors purchased protection and therefore is economically short and benefits from an increase in the company’s spread. Since putting on the protection, the credit spread increased by 200 bps, and Deem Advisors realizes the profit by entering into a new, offsetting contract (sells protection to another party at a higher premium). A is incorrect because a decrease (not increase) in the spread would result in a loss for the credit protection buyer. C is incorrect because Deem Advisors, the Solutions credit protection buyer, would profit from an increase in the company’s credit spread, not break even. 6. A is correct. A difference in credit spreads in the bond market and CDS market is the foundation of the basis trade strategy. If the spread is higher in the bond market than in the CDS market, it is said to be a negative basis. In this case, the bond credit spread is currently 4.50% (bond yield minus MRR) and the comparable CDS contract has a credit spread of 4.25%. The credit risk is cheap in the CDS market relative to the bond market. Since the protection and the bond were both purchased, if convergence occurs, the trade will capture the 0.25% differential in the two markets (4.50% – 4.25%). B is incorrect because the bond market implies a 4.50% credit risk premium (bond yield minus the market reference rate) and the CDS market implies a 4.25% credit risk premium. Convergence of the bond market credit risk premium and the CDS credit risk premium would result in capturing the differential, 0.25%. The 1.75% is derived by incorrectly subtracting MRR from the credit spread on the CDS (= 4.25% – 2.50%). C is incorrect because convergence of the bond market credit risk premium and the CDS credit risk premium would result in capturing the differential, 0.25%. The 2.75% is derived incorrectly by subtracting the credit spread on the CDS from the current bond yield (= 7.00% – 4.25%). 7. C is correct. Parties to CDS contracts generally agree that their contracts will conform to ISDA specifications. These terms are specified in the ISDA master agreement, which the parties to a CDS sign before any transactions are made. Therefore, to satisfy the compliance requirements referenced by Chan, the sovereign wealth fund must sign an ISDA master agreement with SGS. 8. A is correct. A CDS index (e.g., CDX and iTraxx) would allow the Fund to simultaneously fully hedge multiple fixed-income exposures. A tranche CDS will also hedge multiple exposures, but it would only partially hedge those exposures. 9. A is correct. Based on Exhibit 1, the probability of survival for the first year is 99.78% (100% minus the 0.22% hazard rate). Similarly, the probability of survival for the second and third years is 99.65% (100% minus the 0.35% hazard rate) and 99.50% (100% minus the 0.50% hazard rate), respectively. Therefore, the probability of survival of the Orion bond through the first three years is equal to 0.9978 × 0.9965 × 0.9950 = 0.9893, and the probability of default sometime during the first three years is 1 – 0.9893, or 1.07%. 10. B is correct. The trade assumes that £6 million of five-year CDS protection on Orion is initially sold, so the Fund received the premium. Because the credit spread of the Orion CDS narrowed from 150 bps to 100 bps, the CDS position will realize a financial gain. This financial gain is equal to the difference between the upfront premium received on the original CDS position and the upfront premium to be paid on a new, offsetting CDS position. To close the position and monetize this gain, the Fund should unwind the position by buying protection for a lower premium (relative to the original premium collected). 649 650 Learning Module 5 Credit Default Swaps 11. B is correct. The gain on the hypothetical Orion trade is £117,000, calculated as follows. Approximate profit = Change in credit spread (in bps) × Duration × Notional amount. Approximate profit = (150 bps – 100 bps) × 3.9 × £6 million. Approximate profit = 0.005 × 3.9 × £6 million. = £117,000. The Fund gains because it sold protection at a spread of 150 bps and closed out the position by buying protection at a lower spread of 100 bps. 12. B is correct. Based on Outlook 1, Chan and Smith anticipate that Europe’s economy will weaken. In order to profit from this forecast, one would buy protection using a high-yield CDS index (e.g., iTraxx Crossover) and sell protection using an investment-grade CDS index (e.g., iTraxx Main). 13. B is correct. To take advantage of Chan’s view of the US credit curve steepening in the short term, a curve trade will entail shorting (buying protection using) a long-term (20-year) CDX and going long (selling protection using) a short-term (2-year) CDX. A steeper curve means that long-term credit risk increases relative to short-term credit risk. 14. B is correct. The shares of Zega can be sold at a higher price as a result of the unsolicited bid in the market. If Delta Corporation issues significantly more debt, there is a higher probability that it may default. If the Fund sells protection on Delta now, the trade will realize a profit as credit spreads widen. An equity-versus-credit trade would be to go long (buy) the Zega shares and buy protection on Delta.