Analysts use a wide range of models in practice, ranging from the simple to the sophisticated. These models often make very different assumptions about pricing, but they do share some common characteristics and can be classified in broader terms. There are several advantages to such a classification  it is easier to understand where individual models fit in to the big picture, why they provide different results, and when they have fundamental errors in logic.
Question 1  DCF Valuation Fundamentals
Discounted cash flow valuation is based upon the notion that the value of an asset is the present value of the expected cash flows on that asset, discounted at a rate that reflects the riskiness of those cash flows. Specify whether the following statements about discounted cash flow valuation are true or false, assuming that all variables are constant except for the variable discussed below:
A. As the discount rate increases, the value of an asset increases.
B. As the expected growth rate in cash flows increases, the value of an asset increases.
C. As the life of an asset is lengthened, the value of that asset increases.
D. As the uncertainty about the expected cash flows increases, the value of an asset increases.
E. An asset with an infinite life (i.e., it is expected to last forever) will have an infinite value.
Question 2  Approaches to DCF Valuation
There are two approaches to valuation. The first approach is to value the equity in the firm. The second approach is to value the entire firm. What is the distinction? Why does it matter?
Question 3  Mismatching Cash flows and Discount Rates
The following are the projected cash flows to equity and to the firm over the next five years:



























(The terminal value is the value of the equity or firm at the end of year 5.)
The firm has a cost of equity of 12% and a cost of capital of 9.94%. Answer the following questions:
A. What is the value of the equity in this firm?
B. What is the value of the firm?
Question 4  Problems in DCF Valuation
Why might discounted cash flow valuation be difficult to do for the following types of firms?
A. A private firm, where the owner is planning to sell the firm.
B. A biotechnology firm, with no current products or sales, but with several promising product patents in the pipeline.
C. A cyclical firm, during a recession.
D. A troubled firm, which has made significant losses and is not expected to get out of trouble for a few years.
E. A firm, which is in the process of restructuring, where it is selling some of its assets and changing its financial mix.
F. A firm, which owns a lot of valuable land that is currently unutilized.
Question 5  Relative Valuation: Fundamentals
An analyst tells you that he uses price/earnings multiples, rather than discounted cash flow valuation, to value stocks, because he does not like making assumptions about fundamentals  growth, risk, and payout ratios. Is his reasoning correct?
Question 6  Industry Average P/E Ratios
You are estimating the price/earnings multiple to use to value Paramount Corporation, by looking at the average price/earnings multiple of comparable firms. The following are the price/earnings ratios of firms in the entertainment business.
Firm  P/E Ratio 
Disney (Walt)  22.09 
Time Warner  36.00 
King World Productions  14.10 
New Line Cinema  26.70 
CCL  19.12 
PLG  23.33 
CIR  22.91 
GET  97.60 
GTK  26.00 
A. What is the average P/E ratio?
B. Would you use all the comparable firms in calculating the average? Why or why not?
C. What assumptions are you making when you use the industryaverage P/E ratio to value Paramount Communications?
ESTIMATION OF DISCOUNT RATES
The discount rate is a critical ingredient in discounted cash flow valuation. Errors in estimating the discount rate or mismatching cash flows and discount rates can lead to serious errors in valuation. At an intuitive level, the discount rate used should be consistent with both the riskiness and the type of cash flow being discounted. Though there is no consensus among practitioners on the right model to use for measuring risk, there is agreement that higherrisk cash flows should be discounted at a higher rate.
This chapter examines the process of estimating discount rates  the cost of equity to be used in discounting cash flows to equity, and the cost of capital to be used in discounting cash flows to the firm.
Question 1  CAPM: Historical Risk Premiums
The following table summarizes risk premiums for stocks in the United States, relative to treasury bills and bonds, for different time periods:
Risk Premium for Equity
Stocks  T.Bills  Stocks  T.Bonds  
Arithmetic  Geometric  Arithmetic  Geometric  
19261990  8.41%  6.41%  7.24%  5.50% 
19621990  4.10%  2.95%  3.92%  3.25% 
19811990  6.05%  5.38%  0.13%  0.19% 
A. What does this premium measure?
B. Why is the geometric mean lower than the arithmetic mean for both bonds and bills?
C. If you had to use a risk premium, would you use the most recent data (19811990), or would you use the longer periods? Explain your reasoning.
Question 2  CAPM: Premiums in Emerging Markets
You are an investor who is interested in the emerging markets of Asia. You are trying to value some stocks in Malaysia, which does not have a long history of financial markets. During the last two years, the stock market has gone up 60% a year, while the government borrowing rate has been 15%, yielding an historical premium of 45%. Would you use this as your risk premium, looking into the future? If not, what would you base your estimate of the premium on?
Question 3  Using the CAPM
The beta for Eastman Kodak is 1.10. The current sixmonth treasury bill rate is 3.25%, while the thirtyyear bond rate is 6.25%. Estimate the cost of equity for Eastman Kodak, based upon
(a) using the treasury bill rate as your riskfree rate.
(b) using the treasury bond rate as your riskfree rate.
(Use the premiums in the table in question 1, if necessary.)
Which one of these estimates would you use in valuation? Why?
Question 4  Estimating CAPM parameters
You are trying to estimate the cost of equity to use in valuing Daimler Benz, and a data service reports a beta estimate of 0.90. However, the beta is estimated relative to the Frankfurt Stock Exchange (DAX). If you were an international portfolio manager with holdings across many markets, would you use this beta estimate? How would you estimate beta to meet your needs?
Question 5  CAPM: Divisional and Corporate Betas
You have been asked to estimate the beta of a hightechnology firm which has three divisions with the following characteristics
Division  Beta  Market Value 
Personal Computers  1.60  $100 million 
Software  2.00  $150 million 
Computer Mainframes  1.20  $250 million 
A. What is the beta of the equity of the firm?
B. What would happen to the beta of equity if the firm divested itself of its software business?
C. If you were asked to value the software business for the divestiture, which beta would you use in your valuation?
Question 6  CAPM: Betas and Financial Leverage
The following are the betas of the equity of four forestry/paper product companies, and their debt/equity ratios.
Company  Beta  Debt/Equity Ratio  
Weyerhauser  1.15  33.91%  
Champion International  1.18  54.14%  
Intenational Paper  1.05  45.50%  
KimberlyClark  0.91  11.29% 
(All the firms face a corporate tax rate of 40%)
A. Estimate the unlevered beta of each firm. What do the unlevered betas tell you about these firms?
B. Assume now that Kimberly Clark is planning to increase its debt/equity ratio to 30%. What will its new beta be?
C. If you were valuing an initial public offering in the paper products area, what beta would you use in the valuation? (Assume that the firm going public plans to have a debt/equity ratio of 40%.)
Question 7  Betas and Operating Leverage
The following is a description of the cost structure and betas of five firms in the food production industry:
Company  Fixed Costs  Variable Costs  Beta  D/(D+E) 
CPC International  62%  38%  1.23  18.83% 
Ralston Purina  47%  53%  0.81  38.32% 
Quaker Oats  45%  55%  0.75  13.28% 
Chiquita  50%  50%  0.88  75.35% 
Kellogg's  40%  60%  0.76  5.57% 
(Assume that all firms have a tax rate of 40%.)
A. Based upon just the operating leverage, which firms would you expect to have the highest and lowest betas (assuming that they are in the same business)?
B. Chiquita's beta is believed to be misleading because its financial leverage has increased dramatically since the period when the beta was estimated. If the average D/(D+E) ratio during the period of the regression (to estimate the betas) was only 30%, what would your new estimate of Chiquita's beta be?
Question 8  CAPM: Betas and Private Firms
You are attempting to estimate the beta of a private firm that has no comparable firms. You decide to estimate an "accounting" beta using past earnings. You have six years of accounting data on the private firm, and the comparable information on earnings changes for the average S&P 500 company during the same period.
Year  Net Income of Private Company  Earnings ChangeAverage Firm in S&P  
1988  $10.00 million  + 7%  
1989  $15.00 million  +10%  
1990  $18.00 million  + 5%  
1991  $18.50 million   10%  
1992  $19.00 million   8%  
1993  $22.00 million  + 6% 
A. Estimate the accounting beta for the private company.
B. What are the limitations of an "accounting" beta?
Question 9  CAPM: Betas and Mergers
The following are the betas of three companies involved in a merger battle. The target firm is Paramount Communications, and the competing bidders are QVC and Viacom:
Company  Beta  Market Value of Equity  Debt  
Paramount  1.05  $6,500 million  $817 million  
QVC  1.70  $2,000 million  $100 million  
Viacom  1.15  $7,500 million  $2,500 million 
(Assume that all firms have a tax rate of 35%.)
A. If QVC acquires Paramount, using a mix of debt and equity comparable to its current debt/equity ratio, what would the beta of the combined firm be?
B. If QVC acquires Paramount, using only debt, what would the beta of the comparable firm be?
C. If Viacom acquires Paramount, using a mix of debt and equity comparable to its current debt/equity ratio, what would the beta of the combined firm be?
D. If Viacom acquires Paramount, using only equity, what would the beta of the comparable firm be?
Question 10  Arbitrage Pricing Model
Consider the following derivation of the arbitrage pricing model, where the expected return on a stock is written as the function of four variables:
E(Rj) = 0.062  1.855 b1 + 1.4450 b2  0.124 b3  2.744 b4 Assume that you have estimated the betas relative to each of the four factors for Paramount to be:
b1 = 0.07 b2 = 0.01 b3 = 0.02 b4 = 0.01
A. What does the intercept of this regression measure?
B. What economic and statistical significance do the four factors have?
C. What do the factor coefficients and betas measure?
D. What is the expected return on Paramount, using the arbitrage pricing model?
E. The beta for Paramount is 1.05. Assuming a riskfree rate of 6.25%, what would your estimate of expected return be under the CAPM? Why is it different from your answer in part (D)?
Question 11  Dividend Growth Model
The following is a list of companies, with prices, dividends per share and expected growth rates in dividends (from analyst projections) for each company:
Company  Market Price  Current DPS  Growth Rate in DPS  Beta 
Merck 




Ogden Co. 




Honda (ADR) 




Microsoft 




(Microsoft has an expected growth rate in earnings of 24% for the next five years.)
A. Estimate the cost of equity using the dividend growth model. Which, if any, of these firms may be reasonable candidates for using this model? Why?
B. Estimate the cost of equity using the CAPM. (The thirtyyear bond rate is 6.25%.)
C. Which estimate will you use in valuation and why?
Question 12  WACC Calculation
Merck & Company has 1.13 billion shares traded at a market value of $32 per share, and $1.918 billion in book value of outstanding debt (with an estimated market value of $2 billion). The equity has a book value of $5.5 billion, and the stock has a beta of 1.10. The firm paid interest expenses of $160 million in the most recent financial year, is rated AAA and paid 35% of its income as taxes. The thirtyyear government bond rate is 6.25%, and AAA bonds trade at a spread of twenty basis points (0.2%) over the treasury bond rate.
A. What are the market value and book value weights on debt and equity?
B. What is the cost of equity?
C. What is the aftertax cost of debt?
D. What is the cost of capital?
Question 13  WACC: Another exercise
General Motors has 710 million shares trading at $55 per share and $69 billion in debt outstanding (with a market value of $65 billion), on which it incurred an interest expense of $5 billion in the most recent year. It also has $4 billion in preferred stock outstanding, trading at par, on which it paid a dividend of $365 million. The stock has a beta of 1.10 and is rated A (which commands a spread of 1.25% over the treasury bond rate of 6.25%). The company faced a corporate tax rate of 40%.
A. What is the cost of equity for GM?
B. What is the aftertax cost of debt for GM?
C. What is the cost of preferred stock?
D. What is the cost of capital?
ESTIMATION OF CASH FLOWS
Much of the tedium in valuation is associated with estimating cash flows, a necessary element of discounted cash flow valuation. This chapter examines the process of estimating cash flows and establishes some general principles which should be adhered to in all valuation models. The one overriding principle governing cash flow estimation is the need to match cash flows to discount rates: equity cash flows to cost of equity; firm cash flows to cost of capital; pretax cash flows to pretax rates; posttax cash flows to posttax rates; nominal cash flows to nominal rates; and real cash flows to real rates. The process of estimating these cash flows is explained in detail in the pages that follow.
Question 1  Cash Flows to Equity: Concepts
Which of the following is the best description of the free cash flow to equity?
A. It is the cash that equity investors can take out of the firm.
B. It is the dividend that is paid to stockholders.
C. It is the cash that equity investors can take out of the firm after financing investment needed to sustain future growth.
D. It is the cash left over after meeting debt payments and paying taxes.
E. None of the above.
Question 2  Cash Flows: Concepts
Answer true or false to the following statements.
A. The free cash flow to equity will always be higher than the net income of the firm, because depreciation is added back.
B. The free cash flow to equity will always be higher than the dividend.
C. The free cash flow to equity will always be higher than cash flow to the firm, because the latter is a predebt cash flow.
D. The entire free cash flow to equity cannot be paid out as a dividend because some of it has to be invested in new projects.
Question 3  The Effects of Inflation
Answer true or false to the following statements relating to the effect of inflation on cash flows and value.
A. Discounting nominal cash flows at the real discount rate will result in too low an estimate of value.
B. Dicounting real cash flows at the nominal discount rate will result in too low an estimate of value.
C. If done right, the value estimated should be the same if either real cash flows are discounted at the real discount rate or nominal cash flows are discounted at the nominal discount rate.
D. If companies can raise prices at the same rate as inflation, their value should not be affected by changes in the inflation rate.
E. Inflation should increase the value of stocks because it increases expected future cash flows.
Question 4  Estimating Cash Flows: Diebold Incorporated
Diebold Incorporated manufactures, markets, and services automated teller machines in the United States. The following are selected numbers from the financial statements for 1992 and 1993 (in millions):
1992  1993  
Revenues  $544.0  $620.0 
(Less) Operating Expenses  ($465.1)  ($528.5) 
(Less) Depreciation  ($12.5)  ($14.0) 
= Earnings before Interest and Taxes  $66.4  $77.5 
(Less) Interest Expenses  ($0.0)  ($0.0) 
(Less) Taxes  ($25.3)  ($29.5) 
= Net Income  $41.1  $48.0 
Working Capital  $175.0  $240.0 
The firm had capital expenditures of $15 million in 1992 and $18 million in 1993. The working capital in 1991 was $180 million.
A. Estimate the cash flows to equity in 1992 and 1993.
B. What would the cash flows to equity in 1993 have been if working capital had remained at the same percentage of revenue it was in 1992.
Question 5  Estimating Cash Flows: Ryder System
Ryder System is a fullservice truck leasing, maintenance, and rental firm with operations in North America and Europe. The following are selected numbers from the financial statements for 1992 and 1993 (in millions).



Revenues  $5,192.0  $5,400.0 
(Less) Operating Expenses  ($3,678.5)  ($3848.0) 
(Less) Depreciation  ($573.5)  ($580.0) 
= EBIT  $940.0  $972.0 
(Less) Interest Expenses  ($170.0)  ($172.0) 
(Less) Taxes  ($652.1)  ($670.0) 
= Net Income  $117.9  $130.0 
Working Capital  $92.0  <$370.0> 
Total Debt  $2,000 mil  $2,200 mil 
The firm had capital expenditures of $800 million in 1992 and $850 million in 1993. The working capital in 1991 was $34.8 million, and the total debt outstanding in 1991 was $1.75 billion. There were 77 million shares outstanding, trading at $29 per share.
A. Estimate the cash flows to equity in 1992 and 1993.
B. Estimate the cash flows to the firm in 1992 and 1993.
C. Assuming that revenues and all expenses (including depreciation and capital expenditures) increase 6%, and that working capital remains unchanged in 1994, estimate the projected cash flows to equity and the firm in 1994. (The firm is assumed to be at its optimal financial leverage.)
D. How would your answer in (c) change if the firm planned to increase its debt ratio in 1994 by financing 75% of its capital expenditures (net of depreciation) with new debt issues?
Question 6  Estimating Cash Flows: Occidental Petroleum
Occidental Petroleum produces and markets crude oil. The following are selected numbers from the financial statements for 1992 and 1993 (in millions).



Revenues 

$9,000.0 
(Less) Operating Expenses 

($6,970.0) 
(Less) Depreciation 

($860.0) 
= EBIT 

$1,170.0 
(Less) Interest Expenses 

($515.0) 
(Less) Taxes 

($420.0) 
= Net Income 

$235.0 
Working Capital 

($50.0) 
Total Debt 


The firm had capital expenditures of $950 million in 1992 and $1 billion in 1993. The working capital in 1991 was $190 million, and the total debt outstanding in 1991 was $5.75 billion. There were 305 million shares outstanding, trading at $21 per share.
A. Estimate the cash flows to equity in 1992 and 1993.
B. Estimate the cash flows to the firm in 1992 and 1993.
C. Assuming that revenues and all expenses (including depreciation and capital expenditures) increase 4%, and that working capital remains unchanged in 1994, estimate the projected cash flows to equity and the firm in 1994. (The firm is assumed to be at its optimal financial leverage.)
D. How would your answer in (c) change if the firm planned to reduce its debt ratio in 1994 by financing 100% of its capital expenditures (net of depreciation) with new equity issues?
Question 7  Inflation and Value
Watts Industries, a manufacturer of valves for industrial and residential use, had the following projected free cash flows to equity per share for the next five years , in nominal terms.
Year  FCFE/sh  Terminal Value 
1  $1.12  
2  $1.25  
3  $1.40  
4  $1.57  
5  $1.76  $23.32 
The terminal price is based upon a stable nominal growth rate of 6% a year after year 5. The discount rate, based upon financial market rates, is 14%, and the expected inflation rate is 3%.
A. Estimate the value per share, using nominal cash flows and the nominal discount rate.
B. Estimate the value per share, using real cash flows and the real discount rate.
Question 8  Aftertax and Pretax Valuation
Consider the example of Polaroid Corporation. The stock is trading at $37.00 per share currently. The expected dividends, prior to personal taxes, as well as the expected terminal price, are given below:


















The expected return, prior to personal taxes, on Polaroid is 13%, of which 1.81% is expected to come from dividends. An investor facing a tax rate of 36% on dividends and 25% on capital gains is considering investing in the stock.
A. What is the expected return, after personal taxes, to this investor?
B. What are the expected dividends and terminal price, after personal taxes, to this investor?
C. What is the value of the stock, using these afterpersonaltax cash flows and discount rates?
D. What is the value of the stock, using the prepersonaltax cash flows and discount rates?
Question 9  Terminal Values for Cash Flow Calculation
The terminal value in a capital budgeting project is generally much lower than the initial investment. The terminal price in a stock valuation is generally much higher than the initial investment. How would you explain the difference?
ESTIMATION OF GROWTH RATES
The value of a firm is ultimately determined not by current cash flows but by expected future cash flows. The estimation of growth rates in earnings and cash flows is therefore central to doing a reasonable valuation. Growth rates can be obtained in many ways: they can be based upon past growth; drawn from estimates made by other analysts who follow the firm; or related to the firm's fundamentals. Since each of these approaches yields some valuable information, it makes sense to blend them to arrive at one composite growth rate to use in the valuation. This chapter examines different approaches to estimating future growth, and discusses the determinants of growth.
Question 1  Arithmetic and Geometric Means
The following are the earnings per share of Thermo Electron, a company that designs cogeneration and resource recovery plants, from 1987 to 1992:














A. Estimate the arithmetic average growth rate in earnings per share from 1987 to 1992.
B. Estimate the geometric average growth rate in earnings per share from 1987 to 1992.
C. Why are the growth rates different?
Question 2  Linear and Loglinear Models of Earnings Growth
Consider again the example of Thermo Electron, described in the prior example, using the historical data from 1987 to 1992.
A. Estimate the growth rate from a linear regression model.
B. Estimate the growth rate from a loglinear regression model.
C. Project the earnings per share in 1993 using both models.
Question 3  Dealing with Negative Earnings
The earnings per share from 1987 to 1993 are reported below for McDonnell Douglas, an aircraft manufacturer with extensive defense contracts:
















A. Estimate the geometric average growth rate in earnings from 1987 to 1993.
B. Estimate the arithmetic average growth rate in earnings from 1987 to 1993, using a correction for the negative earnings.
C. Estimate the growth rate in earnings, using the linear regression model.
Question 4  Earnings Growth and ROE
Johnson and Johnson, a leading manufacturer of healthcare products, had a return on equity in 1992 of 31.4%, and paid out 36% of its earnings as dividends. It earned a net income of $1,625 million on a book value of equity of $5,171 million. As a consequence of healthcare reform, it is expected that the return on equity will drop to 25% in 1993 and that the dividend payout ratio will remain unchanged.
A. Estimate the growth rate in earnings based upon 1992 numbers.
B. Estimate the growth rate in 1993, when the ROE drops from 31.4% to 25%.
C. Estimate the growth rate after 1993, assuming that 1993 numbers can be sustained.
Question 5  Earnings Growth, Leverage, and Risk
Eastman Kodak was, in the view of many observers, in serious need of restructuring in 1994. In 1993, the firm reported the following:
Net Income = $1,080 million
Interest Expense = $ 550 million
The firm also had the following estimates of debt and equity in the balance sheet:
Equity (Book Value) = $6,000 million
Debt (Book Value) = $6,880 million
The firm also paid out total dividends of $660 million in 1993. The stock was trading at $63, and there were 330 million shares outstanding. (It faced a corporate tax rate of 40%.) Eastman Kodak had a beta of 1.10.
Analysts believe that Kodak could take the following restructuring actions to improve its financial strength:
ï It could sell its chemical division, which has a total book value of assets of $2,500 million and has only $100 million in earnings before interest and taxes.
ï It could use the cash to pay down debt and improve its bond rating (leading to a decline in the interest rate to 7%).
ï It could reduce the dividend payout ratio to 50% and reinvest more back into the business.
A. What is the expected growth rate in earnings, assuming that 1993 numbers remain unchanged?
B. What is the expected growth rate in earnings, if the restructuring plan described above is put into effect?
C. What will the beta of the stock be, if the restructuring plan is put into effect?
Question 6  Corporate Strategy and Expected Growth
Philip Morris, a leading consumer products company, was forced to cut prices on its Marlboro brand of cigarettes in early 1993 to combat loss of sales to generic competitors. You are attempting to assess the effects on expected growth as a consequence.
In 1992, Philip Morris had earnings before interest and taxes of $10 billion on sales of $60 billion. The firm also had total assets of $30 billion in that year. As a consequence of its price cuts in 1993, the preinterest profit margin is expected to decline to 9%. The debt/equity ratio is expected to remain unchanged at 1.00, and the interest rate will remain at 6.5%. (The tax rate is 36%.) Philip Morris pays out 65% of its earnings as dividends.
A. Based upon 1992 numbers, what is the expected growth rate in earnings?
B. Assuming that the asset turnover ratio remains unchanged, what will the growth rate in earnings be after the price cuts in 1993?
C. How much will the asset turnover ratio have to increase for Philip Morris to return to the growth rate it had in 1992?
Question 7  Adjusting Inputs For Firm Type
Computer Associates makes software that enables computers to run more efficiently. It is still in its highgrowth phase and has the following financial characteristics:
Return on Assets = 25%
Dividend Payout Ratio = 7%
Debt/Equity Ratio = 10%
Interest rate on Debt = 8.5%
Corporate tax rate = 40%
It is expected to become a stable firm in ten years.
A. What is the expected growth rate for the highgrowth phase?
B. Would you expect the financial characteristics of the firm to change once it reaches a steady state? What form do you expect the change to take?
C. Assume now that the industry averages for larger, more stable firms in the industry are as follows:
Industry Average Return on Assets = 14%
Industry Average Debt/Equity Ratio = 40%
Industry Average Interest Rate on Debt = 7%
Industry Average Dividend Payout ratio = 50%
D. What would you expect the growth rate in the stable growth phase to be?
Question 8  Weighting Different Estimates of Growth Rate
The following are a number of valuation scenarios, where multiple estimates of growth are available. Specify how you weight the different growth rates and why.
A. A cyclical firm, whose earnings have dropped significantly (historical growth rate is negative) as a consequence of a recession, but which you believe has bottomed out and is in the process of recovering. The firm is heavily followed by analysts, who have a good track record in forecasting earnings growth.
B. A troubled firm, whose earnings have dropped significantly because of a combination of bad luck and bad management, but which is now restructuring. You have fairly good information on the form the restructuring will take and its expected impact. Analysts follow the firm, but their track record is spotty.
C. A healthy firm, where the estimates of growth from history, analysts, and fundamentals are fairly close.
D. A firm, which has a long and fairly reliable history of earnings growth, but which has just sold off three divisions (comprising almost half of the market value of the firm). Analysts follow the stock, but base forecasts primarily on historical growth.
CHAPTER 6
DIVIDEND DISCOUNT MODELS
The basic model for valuing equity is the dividend discount model: the value of a stock is the present value of its expected dividends. This chapter explores the general model and its permutations tailored for different assumptions about future growth. It also examines issues in using the dividend discount model and the results of studies that have looked at its efficacy.
Question 1  Uses of the Dividend Discount Model
Respond true or false to the following statements relating to the dividend discount model.
A. The dividend discount model cannot be used to value a high growth company that pays no dividends.
B. The dividend discount model will undervalue stocks, because it is too conservative.
C. The dividend discount model will find more undervalued stocks, when the overall stock market is depressed.
D. Stocks that are undervalued using the dividend discount model have generally made significant positive excess returns over long periods (five years or more).
E. Stocks which pay high dividends and have low price/earnings ratios are more likely to come out as undervalued using the dividend discount model.
Question 2  Gordon Growth Model : Concepts
An analyst complains that the Gordon Growth Model yields absurd results. He presents several problems that he has had with the model. Respond to each of these comments.
A. The model values stocks which do not pay dividends at zero.
B. The model sometimes yields negative values for stocks, when growth rates exceed the discount rate.
C. The model yields absurdly high values for other stocks, where the discount rate is very close to the growth rate.
D. No firm raises dividends by a fixed percent every year. The model's assumption is unrealistic and the values obtained from it will not hold.
E. Since cyclical firms have earnings which go up and down, based upon economic conditions, the model can never be used to value a cyclical firm.
Question 3  Gordon Growth Model
Ameritech Corporation paid dividends per share of $3.56 in 1992, and dividends are expected to grow 5.5% a year forever. The stock has a beta of 0.90, and the treasury bond rate is 6.25%.
A. What is the value per share, using the Gordon Growth Model?
B. The stock is trading for $80 per share. What would the growth rate in dividends have to be to justify this price?
Question 4  Growth Rate in the Gordon Growth Model
A key input for the Gordon Growth Model is the expected growth rate in dividends over the long term. How, if at all, would you factor in the following considerations in estimating this growth rate?
A. There is an increase in the inflation rate.
B. The economy in which the firm operates is growing very rapidly.
C. The growth potential of the industry in which the firm operates is very high.
D. The current management of the firm is of very high quality.
Question 5  TwoStage Dividend Discount Model: Basics
Newell Corporation, a manufacturer of doityourself hardware and housewares, reported earnings per share of $2.10 in 1993, on which it paid dividends per share of $0.69. Earnings are expected to grow 15% a year from 1994 to 1998, during which period the dividend payout ratio is expected to remain unchanged. After 1998, the earnings growth rate is expected to drop to a stable 6%, and the payout ratio is expected to increase to 65% of earnings. The firm has a beta of 1.40 currently, and it is expected to have a beta of 1.10 after 1998. The treasury bond rate is 6.25%.
A. What is the expected price of the stock at the end of 1998?
B. What is the value of the stock, using the twostage dividend discount model?
Question 6  TwoStage Dividend Discount Model: Estimating Terminal Payout Ratio
Church & Dwight, a large producer of sodium bicarbonate, reported earnings per share of $1.50 in 1993 and paid dividends per share of $0.42. In 1993, the firm also reported the following:
Net Income = $30 million
Interest Expense = $0.8 million
Book Value of Debt = $7.6 million
Book Value of Equity = $160 million
The firm faced a corporate tax rate of 38.5%. (The market value debtto equity ratio is 5%.) The treasury bond rate is 7%.
The firm expects to maintain these financial fundamentals from 1994 to 1998, after which its is expected to become a stable firm, with an earnings growth rate of 6%. The firm's financial characteristics will approach industry averages after 1998. The industry averages are as follows:
Return on Assets = 12.5%
Debt/Equity Ratio = 25%
Interest Rate on Debt = 7%
Church and Dwight had a beta of 0.85 in 1993, and the unlevered beta is not expected to change over time.
A. What is the expected growth rate in earnings, based upon fundamentals, for the highgrowth period (1994 to 1998)?
B. What is the expected payout ratio after 1998?
C. What is the expected beta after 1998?
D. What is the expected price at the end of 1998?
E. What is the value of the stock, using the twostage dividend discount model?
F. How much of this value can be attributed to extraordinary growth? to stable growth?
Question 7  The H Model
Oneida Inc. the world's largest producer of stainless steel and silver plated flatware, reported earnings per share of $0.80 in 1993, and paid dividends per share of $0.48 in that year. The firm is expected to report earnings growth of 25% in 1994, after which the growth rate is expected to decline linearly over the following six years to 7% in 1999. The stock is expected to have a beta of 0.85. (The treasury bond rate is 6.25%.)
A. Estimate the value of stable growth, using the H Model.
B. Estimate the value of extraordinary growth, using the H Model.
C. What are the assumptions about dividend payout in the H Model?
Question 8  The ThreeStage Dividend Discount Model
Medtronic Inc., the world's largest manufacturer of implantable biomedical devices, reported earnings per share in 1993 of $3.95, and paid dividends per share of $0.68. Its earnings are expected to grow 16% from 1994 to 1998, but the growth rate is expected to decline each year after that to a stable growth rate of 6% in 2003. The payout ratio is expected to remain unchanged from 1994 to 1998, after which it will increase each year to reach 60% in steady state. The stock is expected to have a beta of 1.25 from 1994 to 1998, after which the beta will decline each year to reach 1.00 by the time the firm becomes stable. (The treasury bond rate is 6.25%.)
A. Assuming that the growth rate declines linearly (and the payout ratio increases linearly) from 1999 to 2003, estimate the dividends per share each year from 1994 to 2003.
B. Estimate the expected price at the end of 2003.
C. Estimate the value per share, using the threestage dividend discount model.
FREE CASH FLOW TO EQUITY DISCOUNT MODELS
The dividend discount model is based upon the premise that the only cash flows received by stockholders are dividends. This chapter uses a more expansive definition of cash flows to equity as the cash flows left over after meeting all financial obligations, including debt payments, and after covering capital expenditure and working capital needs. It discusses the reasons for differences between dividends and free cash flows to equity, and presents the discounted free cash flow to equity model for valuation.
Question 1  FCFE Calculation: Concepts
Respond true or false to the following statements relating to the calculation and use of FCFE.
A. The free cash flow to equity will generally be more volatile than dividends.
B. The free cash flow to equity will always be higher than the dividends.
C. The free cash flow to equity will always be higher than net income.
D. The free cash flow to equity can never be negative.
Question 2  Constant Growth FCFE Model
KimberlyClark, a household product manufacturer, reported earnings per share of $3.20 in 1993, and paid dividends per share of $1.70 in that year. The firm reported depreciation of $315 million in 1993, and capital expenditures of $475 million. (There were 160 million shares outstanding, trading at $51 per share.) This ratio of capital expenditures to depreciation is expected to be maintained in the long term. The working capital needs are negligible. KimberlyClark had debt outstanding of $1.6 billion, and intends to maintain its current financing mix (of debt and equity) to finance future investment needs. The firm is in steady state and earnings are expected to grow 7% a year. The stock had a beta of 1.05. (The treasury bond rate is 6.25%.)
A. Estimate the value per share, using the Dividend Discount Model.
B. Estimate the value per share, using the FCFE Model.
C. How would you explain the difference between the two models, and which one would you use as your benchmark for comparison to the market price?
Question 3  TwoStage FCFE Model: Basics
Ecolab Inc. sells chemicals and systems for cleaning, sanitizing, and maintenance. It reported earnings per share of $2.35 in 1993, and expected earnings growth of 15.5% a year from 1994 to 1998, and 6% a year after that. The capital expenditure per share was $2.25, and depreciation was $1.125 per share in 1993. Both are expected to grow at the same rate as earnings from 1994 to 1998. Working capital is expected to remain at 5% of revenues, and revenues which were $1,000 million in 1993 are expected to increase 6% a year from 1994 to 1998, and 4% a year after that. The firm currently has a debt ratio (D/(D+E)) of 5%, but plans to finance future investment needs (including working capital investments) using a debt ratio of 20%. The stock is expected to have a beta of 1.00 for the period of the analysis, and the treasury bond rate is 6.50%. (There are 63 million shares outstanding.)
A. Assuming that capital expenditures and depreciation offset each other after 1998, estimate the value per share.
B. Assuming that capital expenditures continue to be 200% of depreciation even after 1998, estimate the value per share.
C. What would the value per share have been, if the firm had continued to finance new investments with its old financing mix (5%)? Is it fair to use the same beta for this analysis?
Question 4  TwoStage FCFE Model: An Extended Application
Dionex Corporation, a leader in the development and manufacture of ion chromography systems (used to identify contaminants in electronic devices), reported earnings per share of $2.02 in 1993, and paid no dividends. These earnings are expected to grow 14% a year for five years (1994 to 1998) and 7% a year after that. The firm reported depreciation of $2 million in 1993 and capital spending of $4.20 million, and had 7 million shares outstanding. The working capital is expected to remain at 50% of revenues, which were $106 million in 1993, and are expected to grow 6% a year from 1994 to 1998 and 4% a year after that. The firm is expected to finance 10% of its capital expenditures and working capital needs with debt. Dionex had a beta of 1.20 in 1993, and this beta is expected to drop to 1.10 after 1998. (The treasury bond rate is 7%.)
A. Estimate the expected free cash flow to equity from 1994 to 1998, assuming that capital expenditures and depreciation grow at the same rate as earnings.
B. Estimate the terminal price per share (at the end of 1998). Stable firms in this industry have capital expenditures which are 150% of depreciation, and maintain working capital at 25% of revenues.
C. Estimate the value per share today, based upon the FCFE model.
Question 5  ThreeStage FCFE Model: Manufacturing Firm
Biomet Inc., designs, manufactures and markets reconstructive and trauma devices, and reported earnings per share of $0.56 in 1993, on which it paid no dividends. (It had revenues per share in 1993 of $2.91). It had capital expenditures of $0.13 per share in 1993 and depreciation in the same year of $0.08 per share. The working capital was 60% of revenues in 1993 and will remain at that level from 1994 to 1998, while earnings and revenues are expected to grow 17% a year. The earnings growth rate is expected to decline linearly over the following five years to a rate of 5% in 2003. During the high growth and transition periods, capital spending and depreciation are expected to grow at the same rate as earnings, but are expected to offset each other when the firm reaches steady state. Working capital is expected to drop from 60% of revenues during the 19941998 period to 30% of revenues after 2003. The firm has no debt currently, but plans to finance 10% of its net capital investment and working capital requirements with debt.
The stock is expected to have a beta of 1.45 for the high growth period (19941998), and it is expected to decline to 1.10 by the time the firm goes into steady state (in 2003). The treasury bond rate is 7%.
A. Estimate the value per share, using the FCFE model.
B. Estimate the value per share, assuming that working capital stays at 60% of revenues forever.
C. Estimate the value per share, assuming that the beta remains unchanged at 1.45 forever.
Question 6  ThreeStage FCFE Model: Service Firm
Omnicare Inc., which provides pharmacy management and drug therapy to nursing homes, reported earnings per share of $0.85 in 1993 on revenues per share of $12.50. It had negligible capital expenditures, which were covered by depreciation, but had to maintain working capital at 40% of revenues. Revenues and earnings are expected to grow 20% a year from 1994 to 1998, after which the growth rate is expected to decline linearly over three years to 5% in 2001. The firm has a debt ratio of 15%, which it intends to maintain in the future. The stock has a beta of 1.10, which is expected to remain unchanged for the period of the analysis. The treasury bond rate is 7%.
A. Estimate the value per share, using the free cash flow to equity model.
B. Assume now that you find out that the way that Omnicare is going to create growth is by giving easier credit terms to their clients. How would that affect your estimate of value? (Will it increase or decrease?)
C. How sensitive is your estimate of value to changes in the working capital assumption?
Question 7 FCFE Model and Dividend Discount Model
Which of the following firms is likely to have a higher value from the dividend discount model, a higher value from the FCFE model or the same value from both models?
A. A firm that pays out less in dividends than it has available in FCFE, but which invests the balance in treasury bonds.
B. A firm which pays out more in dividends than it has available in FCFE, and then issues stock to cover the difference.
C. A firm which pays out, on average, its FCFE as dividends.
D. A firm which pays out less in dividends that it has available in FCFE, but which uses the cash at regular intervals to acquire other firms, with the intent of diversifying.
E. A firm which pays out more in dividends than it has available in FCFE, but borrows money to cover the difference. (The firm is already overlevered.)
VALUING A FIRM  THE FREE CASH FLOW TO FIRM (FCFF) APPROACH
There are two approaches to valuing the equity in the firm: the dividend discount model and the FCFE valuation model. This chapter develops another approach to valuation where the entire firm is valued, by discounting the cumulated cash flows to all claim holders in the firm by the weighted average cost of capital, and examines its limitations and applications.
Question 1  Free Cash Flow to the Firm: Concepts
Respond true or false to the following statements about the free cash flow to the firm.
A. The free cash flow to the firm is always higher than the free cash flow to equity.
B. The free cash flow to the firm is the cumulated cash flow to all investors in the firm, though the form of their claims may be different.
C. The free cash flow to the firm is a predebt, pretax cash flow.
D. The free cash flow to the firm is an afterdebt, aftertax cash flow.
E. The free cash flow to the firm cannot be estimated without knowing interest and principal payments, for a firm with debt.
Question 2  Free Cash Flow to Firm and Other Definitions of FCFF
Lay out how you would get to the free cash flow to the firm (what would you add and/or subtract to the base number?) from the following measures of cash flow.
A. Net Income
B. Earnings before taxes
C. EBIT (Earnings before interest and taxes)
D. EBITDA (Earnings before interest, taxes, and depreciation)
E. Net Operating Income
F. Free Cash Flow to Equity
Question 3  FCFF Steady State Model
Union Pacific Railroad reported net income of $770 million in 1993, after interest expenses of $320 million. (The corporate tax rate was 36%.) It reported depreciation of $960 million in that year, and capital spending was $1.2 billion. The firm also had $4 billion in debt outstanding on the books, rated AA (carrying a yield to maturity of 8%), trading at par (up from $3.8 billion at the end of 1992). The beta of the stock is 1.05, and there were 200 million shares outstanding (trading at $60 per share), with a book value of $5 billion. Union Pacific paid 40% of its earnings as dividends and working capital requirements are negligible. (The treasury bond rate is 7%.)
A. Estimate the free cash flow to the firm in 1993.
B. Estimate the value of the firm at the end of 1993.
C. Estimate the value of equity at the end of 1993, and the value per share, using the FCFF approach.
Question 4  TwoStage FCFF Model: Lockheed Corporation
Lockheed Corporation, one of the largest defense contractors in the U.S., reported EBITDA of $1290 million in 1993, prior to interest expenses of $215 million and depreciation charges of $400 million. Capital Expenditures in 1993 amounted to $450 million, and working capital was 7% of revenues (which were $13,500 million). The firm had debt outstanding of $3.068 billion (in book value terms), trading at a market value of $3.2 billion, and yielding a pretax interest rate of 8%. There were 62 million shares outstanding, trading at $64 per share, and the most recent beta is 1.10. The tax rate for the firm is 40%. (The treasury bond rate is 7%.)
The firm expects revenues, earnings, capital expenditures and depreciation to grow at 9.5% a year from 1994 to 1998, after which the growth rate is expected to drop to 4%. (Capital spending will offset depreciation in the steady state period.) The company also plans to lower its debt/equity ratio to 50% for the steady state (which will result in the pretax interest rate dropping to 7.5%.)
A. Estimate the value of the firm.
B. Estimate the value of the equity in the firm and the value per share.
Question 5  Valuing a Division
In the face of disappointing earnings results and increasingly assertive institutional stockholders, Eastman Kodak was considering a major restructuring in 1993. As part of this restructuring, it was considering the sale of its health division, which earned $560 million in earnings before interest and taxes in 1993, on revenues of $5.285 billion. The expected growth in earnings was expected to moderate to 6% between 1994 and 1998, and to 4% after that. Capital expenditures in the health division amounted to $420 million in 1993, while depreciation was $350 million. Both are expected to grow 4% a year in the long term. Working capital requirements are negligible.
The average beta of firms competing with Eastman Kodak's health division is 1.15. While Eastman Kodak has a debt ratio (D/(D+E)) of 50%, the health division can sustain a debt ratio (D/(D+E)) of only 20%, which is similar to the average debt ratio of firms competing in the health sector. At this level of debt, the health division can expect to pay 7.5% on its debt, before taxes. (The tax rate is 40%, and the treasury bond rate is 7%.)
A. Estimate the cost of capital for the division.
B. Estimate the value of the division.
C. Why might an acquirer pay more than this estimated value?
Question 6 Choosing the Optimal Leverage
Santa Fe Pacific, a major rail operator with diversified operations, had earnings before interest, taxes and depreciation, of $637 million in 1993, with depreciation amounting to $235 million (offset by capital expenditure of an equivalent amount). The firm is in steady state and expected to grow 6% a year in perpetuity. Santa Fe Pacific had a beta of 1.25 in 1993 and debt outstanding of $1.34 billion. The stock price was $18.25 at the end of 1993, and there were 183.1 million shares outstanding. The expected ratings and the costs of debt at different levels of debt for Santa Fe are shown in the following table (the treasury bond rate is 7%, and the firm faced a tax rate of 40%):

































The earnings before interest and taxes are expected to grow 3% a year in perpetuity, with capital expenditures offset by depreciation. (The tax rate is 40% and the treasury bond rate is 7%.)
A. Estimate the cost of capital at the current debt ratio.
B. Estimate the costs of capital at debt ratios ranging from 0% to 90%.
C. Estimate the value of the firm at debt ratios ranging from 0% to 90%.
Question 7  Choosing the Optimal Leverage and Moving There
Bally's Manufacturing, a large leisuretime company, that owns three casinos in Las Vegas and over 300 fitness centers had debt outstanding of $1.180 billion in 1993, and 45.99 million shares outstanding, trading at $9 per share. The debt is rated B, and commands a pretax interest rate of 10.31%. The company had $236 million in earnings before interest, taxes and depreciation in 1993, and depreciation of $109 million. (Capital expenditures amounted to $125 million in 1993.) The stock had a beta of 2.20.
Bally's is planning to pay down debt and reduce its debt ratio (D/(D+E)) to 50%, which should raise its debt rating to A (and lower the pretax rate to 7.51%). The tax rate for the firm is 40%. The treasury bond rate is 7%.
A. What is Ballys' current cost of capital?
B. What will the effect of the debt reduction be on the cost of capital?
C. The firm value is expected to increase by $100 million as a consequence of the debt reduction. Assuming that the firm is in steady state, what is the expected growth rate in cash flows to the firm that will yield this value increase?
SPECIAL CASES IN VALUATION
The standard discounted cash flow valuation models have to be modified in special cases  for cyclical firms, for troubled firms, for firms with special product options and for private firms. This chapter examines the problems associated with valuing these firms and suggests possible solutions.
Question 1  Cyclical Firm: Normalized Earnings Per Share
Intermet Corporation, the largest independent iron foundry organization in the country, reported a deficit per share of $0.15 in 1993. The earnings per share from 1984 to 1992, were as follows:
Year  EPS 
1984  $0.69 
1985  $0.71 
1986  $0.90 
1987  $1.00 
1988  $0.76 
1989  $0.68 
1990  $0.09 
1991  $0.16 
1992  <$0.07> 
The firm had capital expenditures of $1.60 per share, and depreciation per share of $1.20 in 1993. Working capital was expected to increase $0.10 per share in 1994. The stock has a beta of 1.2, which is expected to remain unchanged, and finances its capital expenditure and working capital requirements with 40% debt. (D/(D+E)). The firm is expected, in the long term, to grow at the same rate as the economy (6%).
A. Estimate the normalized earnings per share in 1994, using the average earnings approach.
B. Estimate the normalized free cash flow to equity per share in 1994, using the average earnings approach.
C. The firm is expected, in the long term, to grow at the same rate as the economy (6%).
Question 2  Valuing a Cyclical Firm: Normalized Earnings (ROC)
General Motors Corporation reported a deficit per share in 1993 of $4.85, following losses in the two earlier years (the average earnings per share is negative). The company had assets with a book value of $25 billion, and spent almost $7 billion on capital expenditures in 1993, which was partially offset by a depreciation charge of $6 billion. The firm had $19 billion in debt outstanding, on which it paid interest expenses of $1.4 billion. It intends to maintain a debt ratio (D/(D+E)) of 50%. The working capital requirements of the firm are negligible, and the stock has a beta of 1.10. In the last normal period of operations for the firm between 1986 and 1989, the firm earned an average return on assets of 12%. (The tax rate was 40%.) The treasury bond rate is 7%.
Once earnings are normalized, GM expects them to grow 5% a year forever, and capital expenditures and depreciation to keep track.
A. Estimate the value per share for GM, assuming earnings are normalized instantaneously.
B. How would your valuation be affected if GM is not going to reach its normalized earnings until 1995 (in two years)?
Question 3  Valuing a Cyclical Firm: Adjusted Growth Rate
Chrysler Corporation reported a significant loss of $2.74 per share in 1991, but reported positive earnings per share of $1.38 in 1992, as sales improved and profit margins increased. The improving economy is expected to quadruple earnings in 1993, after which earnings growth is expected to stabilize at 5% in the long term. Chrysler also reported capital spending per share of $5.50 and depreciation per share of $4.50 in 1992, and both items are expected to grow 5% a year in the long term. The working capital for the firm amounted to $2.50 per share in 1992, and was expected to grow 3% a year in the long term. The beta for the stock is 1.25, but is expected to stabilize at 1.10 after 1993. The firm expects to maintain a debt ratio of 40%. The treasury bond rate is 7%.
A. Estimate the value per share.
B. How sensitive is this estimate to assumptions about growth in 1993?
Question 4  Valuing a Troubled Firm: Using Bond Rating
Toro Corporation, which manufactures lawn mowers and tractors, had revenues of $635 million in 1992, on which it reported a loss of $7 million (largely as a consequence of the recession). It had interest expenses of $17 million in 1992, and its bonds were rated BBB. [A typical BBB rated company had an interest coverage ratio (EBIT/Interest Expenses) of 3.10.] The company faced a 40% tax rate. The stock had a beta of 1.10. The treasury bond rate is 7%.
Toro spent $25 million on capital expenditures in 1992, and had depreciation of $20 million. Working capital amounted to 25% of sales. The company expects to maintain a debt ratio of 25%. In the long term, growth in revenues and profits is expected to be 4%, once earnings return to normal levels.
A. Assuming that the bond rating reflects normalized earnings, estimate the normalized earnings for Toro Corporation.
B. Allowing for the longterm growth rate on normalized earnings, estimate the value of equity for Toro Corporation.
Question 5  Valuing a Troubled Firm: Normalized Earnings
Kollmorgen Corporation, a diversified technology company, reported sales of $194.9 million in 1992, and had a net loss of $1.9 million in that year. Its net income had traced a fairly volatile course over the previous five years:
Year  Net Income 
1987  $0.3 million 
1988  $11.5 million 
1989  $2.4 million 
1990  $7.2 million 
1991  $4.6 million 
The stock had a beta of 1.20, and the normalized net income is expected to increase 6% a year until 1996, after which the growth rate is expected to stabilize at 5% a year (the beta will drop to 1.00). The depreciation amounted to $8 million in 1992, and capital spending amounted to $10 million in that year. Both items are expected to grow 5% a year in the long term. The firm expects to maintain a debt ratio of 35%. (The treasury bond rate is 7%.)
A. Assuming that the average earnings from 1987 to 1992 represents the normalized earnings, estimate the normalized earnings and free cash flow to equity.
B. Estimate the value per share.
Question 6  Valuing a Troubled Firm: Operating Margin
Delta Airlines, the third ranking domestic airline, had revenues of $12 billion in 1993 and reported a loss of $415 million in that year. Between 1988 and 1990, which was the last period of significant profitability for the firm, the firm had a pretax operating margin of 12% (Pretax Operating Margin = EBIT/Sales). Delta Airlines had interest expenses of $340 million in 1993, and its capital expenditures were offset by depreciation. The company faces a tax rate of 40%. The stock had a beta of 1.15, and the treasury bond rate is 7%. Working capital requirements are negligible.
The expected growth rate in revenues/net incomes is 6% in the long term.
A. Assuming that the firm returns to 198890 levels of profitability by 1994, estimate the value of equity.
B. Estimate the value of equity, if the firm does not return to 198890 levels of profitability until 1995. (The firm continues to lose money in 1994.)
Question 7  Valuing a Troubled Firm: FCFF Approach
OHM Corporation, an environmental service provider, had revenues of $209 million in 1992 and reported losses of $3.1 million. It had earnings before interest and taxes of $12.5 million in 1992, and had debt outstanding of $109 million (in market value terms). There are 15.9 million shares outstanding, trading at $11 per share. The pretax interest rate on debt owed by the firm is 8.5%, and the stock has a beta of 1.15. The firm's EBIT is expected to increase 10% a year from 1993 to 1996, after which the growth rate is expected to drop to 4% in the long term. Capital expenditures will be offset by depreciation, and working capital needs are negligible. (The corporate tax rate is 40%, and the treasury bond rate is 7%.)
A. Estimate the cost of capital for OHM.
B. Estimate the value of the firm.
C. Estimate the value of equity (both total and on a per share basis).
Question 8  Valuing a Private Firm
You have been asked by the owner of a small firm that produces and sells computer software to estimate the value of his firm. The firm had revenues of $20 million in the most recent year, on which it made earnings before interest and taxes of $2 million. The firm had debt outstanding of $10 million, on which pretax interest expenses amounted to $1 million. The book value of equity is $10 million. The average beta of publicly traded firms that are in the same business is 1.30, and the average debtequity ratio is 0.2 (based upon the market value of equity). The market value of equity of these firms is, on average, three times the book value of equity. All firms face a 40% tax rate. Capital expenditures amounted to $1 million in the most recent year, and were twice the depreciation charge in that year. Both items are expected to grow at the same rate as revenues for the next five years, and to offset each other in steady state.
The revenues of this firm are expected to grow 20% a year for the next five years, and 5% after that. Net income is expected to increase 25% a year for the next five years, and 8% after that. The treasury bond rate is 7%.
A. Estimate the cost of equity for this private firm.
B. Estimate the cost of capital for this private firm.
C. Estimate the value of the owner's stake in this private firm, using both the firm approach and the equity approach.
Question 9  Estimating Value: Initial Public Offering
Boston Chicken, a company selling roasted chickens and accompaniments in outlets through the country, went public in 1993. In the year prior to going public, it had revenues of $40 million, on which it reported earnings before interest and taxes of $12 million. The firm had no debt outstanding, and expected revenues to grow 35% a year from 1993 to 1997, 15% a year from 1998 to 2000, and 5% a year after that, while pretax operating margins (EBIT/Revenues) were expected to remain stable. Capital expenditures ñ which exceeded depreciation by $5 million in the year prior to going public ñ were expected to grow 20% a year from 1993 to 1997, as is depreciation. After 1998, capital expenditures are expected to offset depreciation. Working capital requirements are negligible.
The average beta of publicly traded fastfood chains with which Boston Chicken will be competing is 1.15, and their average debtequity ratio is 25%. Boston Chicken plans to maintain its policy of no debt until 1997, and to move to the industry average debt ratio after that (the pretax cost of debt is expected to be 8%). The treasury bond rate is 7%. All firms face a tax rate of 40%.
A. Estimate the cost of equity for Boston Chicken.
B. Estimate the value of equity for Boston Chicken.
SOLUTIONS
APPROACHES TO VALUATION
Question 1
A. False. The reverse is generally true.
B. True. The value of an asset is an increasing function of its cash flows.
C. True. The value of an asset is an increasing function of its life.
D. False. Generally, the greater the uncertainty, the lower is the value of an asset.
E. False. The present value effect will translate the value of an asset from infinite to finite terms.
Question 2
When equity is valued, the cash flows to equity investors are discounted at their cost (the cost of equity) to arrive at a present value, which is the value of the equity stake in the business.
When the firm is valued, the cash flows to all investors in the firm (including equity investors, lenders and preferred stockholders) are discounted at the weighted average cost of capital to arrive at a present value, which equals the value of the entire firm (generally much higher than the value of just the equity stake.)
The distinction matters for two reasons:
(1) Mismatching cash flows and discount rates can cause significant errors in valuation.
(2) Not recognizing what the present value of the cash flows measures can also lead to misinterpretations. For instance, if the present value of cash flows to the firm is treated as the value of equity, there is an obvious problem.
Question 3
A. PV of CF to Equity = 250/1.12 + 262.50/1.12^2 + 275.63/1.12^3 + 289.41/1.12^4 + (303.88+3946.50)/1.12^5 = $3224
B. PV of CF to Firm = 340/1.0994 + 357/1.0994^2 + 374.85/1.0994^3 + 393.59/1.0994^4 + (413.27+6000)/1.0994^5 = $5149
Question 4
A. It might be difficult to estimate how much of the success of the private firm is due to the owner's special skills and contacts.
B. Since the firm has no history of earnings and cash flow growth and, in fact, no potential for either in the near future, estimating near term cash flows may be impossible.
C. The firm's current earnings and cash flows may be depressed due to the recession. Other measures, such as debtequity ratios and return on assets may also be affected.
D. Since discounted cash flow valuation requires positive cash flows some time in the near term, valuing troubled firms, which are likely to have negative cash flows in the foreseeable future, is likely to be difficult.
E. Restructuring alters the asset and liability mix of the firm, making it difficult to use historical data on earnings growth and cash flows on the firm.
F. Unutilized assets do not produce cash flows and hence do not show up in discounted cash flow valuation, unless they are considered separately.
Question 5
No. Any time a multiple is used, there is implicit, in that multiple, assumptions about growth, risk and payout. In fact, any multiple can be stated as an explicit function of these variables.
Question 6
A. Average P/E Ratio = 31.98
B. No. Eliminate the outliers, because they are likely to skew the average. The average P/E ratio without GET and King World is 25.16.
C. You are assuming that
(1) Paramount is similar to the average firm in the industry in terms of growth and risk.
(2) The marker is valuing communications firms correctly, on average.
SOLUTIONS
ESTIMATION OF DISCOUNT RATES
Question 1
A. It measures, on average, the premium earned by stocks over government securities. It is used as a measure of the expected risk premium in the future.
B. The geometric mean allows for compounding, while the arithmetic mean does not. The compounding effect, in conjunction with the variability of returns, will lower the geometric mean relative to the arithmetic mean.
C. The longer time period is most appropriate, because it covers more of the possible outcomes  crashes, booms, bull markets, bear markets. In contrast, a tenyear period can offer a slice of history that is not representative of all possible outcomes.
Question 2
Recent history is probably not an appropriate basis for estimating the premium, since this history can be skewed upward or downward by a couple of good or bad years. The premium should be based on the fundamentals driving the Malaysian market, relative to other emerging and developed markets, and estimate a premium accordingly. (Use 7.5% to 8.5% as the premium over the longterm bond rate.)
Question 3
CAPM: using T.Bill rate = 3.25% + 1.10 (8.41%) = 12.50%
CAPM: using T.Bond rate = 6.25% + 1.10 (5.50%) = 12.30%
The longterm bond rate should be used as the riskfree rate, because valuation is based upon a long time horizon.
* 8.41% is the arithmetic mean average premium earned by stocks over treasury bills between 1926 and 1990.
** 5.50% is the geometric mean average premium earned by stocks over treasury bonds between 1926 and 1990.
Question 4
An international portfolio manager would prefer a beta estimated relative to an international index. Daimler Benz returns would be regressed against returns on such an index to estimate its beta.
Question 5
A. Beta = 1.60 * 100/500 + 2.00 * 150/500 + 1.20 * 250/500 = 1.52
B. If they pay the cash out as a dividend: Beta = 1.60 * 100/350 + 1.20 * 250/350 = 1.31
If they keep the cash in the firm: Beta = 1.60*100/500 + 0*150/500 + 1.20*250/500 = 0.92
C. Use 2.00, the beta for the software division.
Question 6
A.




Weyerhauser 



Champion International 



Intenational Paper 



KimberlyClark 



The unlevered betas measure the business and operating leverage risk associated with each of these firms.
B. New beta for Kimberly Clark = 0.85 * (1 + (10.4) (0.30)) = 1.00
C. The average unlevered beta of these comparable firms should be relevered using the debt equity ratio of the initial public offering.
Average Unlevered Beta = 0.88
Beta of the Initial Public Offering = 0.88 (1 + (10.4) (0.40)) = 1.09
Question 7
A. CPC should have the highest beta (because of its high fixed costs) and Kellogg's should have the lowest beta (because of its low fixed costs).
B. Old Debt/Equity Ratio = D/(D+E)/( 1  D/(D+E)) = 0.3/ (10.3) = 0.4286
Unlevered Beta (using D/E ratio of 30%) = 0.88/(1 + (1  0.4) * 0.4286) = 0.70
New Debt/Equity Ratio = 0.7535/(1  0.7535) = 3.06
New Levered Beta = 0.70 (1 + (1  0.4) * (3.06)) = 1.985
Question 8
A.
Year  Earnings  _ Earnings  Market Earnings Change  





























Chg in Earnings =  0.1716 + 1.8273 Market Earnings Change 
The accounting beta is 1.8273.
B. The regression has only five observations, and earnings figures can be misleading.
Question 9
A. QVC/Paramount Beta = 1.05 * 6500/8500 + 1.70 * 2000/8500 = 1.20
When leverage will not change after the acquisition, the equity betas can be weighted by equity market values to get a approximate estimate of the beta after the acquisition.
B. QVC Unlevered Beta = 1.70/(1 + (1  0.35)(100/2000)) = 1.65
Paramount Unlevered Beta = 1.05/(1 + (1  0.35)(817/6500)) = 0.97
QVC/Paramount Unlevered Beta = 1.65 * 2100/(2100 + 7317) + 0.97* 7317/(2100 + 7317) =1.12
[Use values of the firm (debt + equity) as the weights for unlevered betas]
QVC/Paramount Levered Beta = 1.12 * (1+ (1  0.35) (7417/2000)) = 3.82
(New Debt = 817 + 100 + 6500 = 7417; New Equity = 2000)
C. Viacom/Paramount Beta = 1.05 * 6500/14000 + 1.15 * 7500/14000 = 1.10
D. Viacom Unlevered Beta = 1.15/(1 + (1  0.35)(2500/7500)) = 0.95
Paramount Unlevered Beta = 1.05/(1 + (1  0.35)(817/6500)) = 0.97
Viacom/Paramount Unlevered Beta = 0.95 * 10000/(10000 + 7317) + 0.97 * 7317/(10000 + 7317) = 0.958
Use values of the firm (debt + equity) as the weights for unlevered betas.
Viacom/Paramount Levered Beta = 0.958 * (1 + (1  0.35)(3317/14000)) = 1.11
(New Debt = 817 + 2500 = 3317)
Question 10
A. It measures the riskless rate during the period of the analysis.
B. There were four common economic factors driving stock returns over the estimation period.
C. The factor coefficients measure the risk premium relative to each factor, and the betas measure sensitivity to the factor.
D. Expected Return = 0.062  1.855 (0.07) + 1.4450 (0.01)  0.124 (0.02)  2.744 (0.01) = 0.17638 or 17.638%
E. Expected Return = 6.25% + 1.05 * 5.50% = 12.025%
The CAPM assumes that the market factor captures all systematic risk. The APM allows for multiple sources of systematic risk.
Question 11
A. See secondtolast column below.
B. See last column below.









Merck 






Ogden Co. 






Honda (ADR) 






Microsoft 






C. Use the CAPM estimate, because
(1) the DDM cost of equity cannot be calculated for many firms, with no dividends and/or no record of growth in the same; and
(2) the CAPM cost of equity has logical constraints. The DDM cost of equity does not.
Question 12
A.





Debt 




Equity 




B. Cost of Equity = 6.25% + 1.10 * 5.50% = 12.30%
C. Aftertax Cost of Debt = 6.45% (1  0.35) = 4.19%
D. Cost of Capital = 12.30% * (36160/38160) + 4.19% * (2000/38160) = 11.87%
Question 13
A. Cost of Equity = 6.25% + 1.10 * 5.50% = 12.30%
B. Aftertax Cost of Debt = 7.50% * (1  0.4) = 4.50%
C. Cost of Preferred Stock = 365/4000 = 9.125%
D. Cost of Capital = 12.30% * (710 * 55)/[(710 * 55)+ 65000 + 4000] + 4.50% * 65000/[(710 * 55) + 65000 + 4000] + 9.125% * 4000/[(710 * 55)+ 65000 + 4000] = 7.49%
SOLUTIONS
ESTIMATION OF CASH FLOWS
Question 1
C. It is the cash that equity investors can take out of the firm after financing investment needed to sustain future growth.
Question 2
A. False. Capital expenditures may be greater than depreciation.
B. False. The dividends can exceed the free cash flow to equity.
C. False. The FCFF is a predebt cash flow. In the long term, it can be equal to, but it cannot be lower than the FCFE. In any one year, however, the FCFE can exceed the FCFF is there are substantial new debt issues.
D. False. The free cash flow to equity is after capital expenditures.
Question 3
A. False. It will result in too high a value.
B. True.
C. True.
D. False. There might be loss of value due to loss of depreciation tax benefits.
E. False. The discount rate also goes up.
Question 4
A. FCFE in 1992 = $41.10 + $12.50  $15  (175  180) = $43.60 million
FCFE in 1993 = $48 + $14  $18  (240  175) =  $21 million
B. Working Capital as Proportion of Revenues: 1992 = 175/544 = 32.17%
Change in Revenues in 1993 = 620  544 = 76
FCFE in 1993 = $48 + $14  $18  (175/544) * (620  544)
= $19.55 million
Question 5
A. FCFE_{1992} = $117.9 + $573.5  $800  ($92  $34.8) + (20001750)
= $84.20 million
FCFE_{1993 }= $130 + $580  $850  (370  92) + (2200  2000)
= $522 million
B. FCFF1992 = $117.9 million + $170 (1  (652/770)) + $573.5  $800  ($92  $34.8)
=  $139.75 million
(The tax rate is extraordinarily high = 652/770; the taxable income is 770 million (940  170))
FCFF in 1993 = $130 million + $172 (1  (670/800)) + $580  $850 
(370  92) = $349.95 million
C. Debt Ratio = $2200 million/($2200 million + 77 * $29) = 49.63%
1994 projection  
Net Income =  $137.80 
 (1  0.4963) * (850  580) * 1.06 =  $144.16 
FCFE =  $6.36 
D. (Also in millions)
Net Income = 

 (1  0.75) * (850  580) * 1.06 = 

FCFE = 

Question 6








Revenues 





 Operating Expenses 





 Depreciation 





= EBIT 





 Interest Expenses 





 Taxes 





= Net Income 





+ Depreciation 





 Capital Expenditures 





 DWorking Capital 





+ Net Debt Issues 





=FCFE 





+ Interest Expenses (1t) 





 Net Debt Issues 





=FCFF 





Working Capital 





Total Debt 




Debt Ratio 


Tax Rate 




To get the new debt issues in 1994, take 43.84% of the net capital expenditures in that year (1040  894)
Question 7
























Real Cash Flow = Nominal Cash Flowt/(1.03)t
A. Present Value = 1.12/1.14 + 1.25/1.14^{2} + 1.40/1.14^{3} + 1.57/1.14^{4} + (1.76 + 23.32)/1.14^{5} = $16.84
B. Real Discount Rate = 1.14/1.03  1 = 10.68%
Present Value =1.09/1.1068 + 1.18/1.1068_{2} + 1.29/1.1068^{3} + 1.40/1.1068^{4} + (21.63)/1.1068^{5} = $16.84
(Use real discount rates on real cash flows.)
Question 8
A. Expected Return = Expected Dividend Yield * (1  0.36) + Expected Price Appreciation * (1  0.25) = 0.0181 * 0.64 + 0.1119 * 0.75 = 9.55%
B.



Year 

























Terminal price after taxes = $62.79  (62.79  37.00) * 0.25 = $56.34.
C. Present Value = $0.43/1.0955 + $0.48/1.0955^2 + $0.54/1.0955^3 + $0.60/1.0955^4 + ($0.68 + $56.34)/1.0955^5 = $37.76
D. Present Value = $0.67/1.13 + $0.75/1.13^2 + $0.84/1.13^3 + $0.94/1.13^4
+ ($1.06 + $62.79)/1.13^5 = $36.99
Question 9
A capital budgeting project generally has a finite life. Consequently it loses value over time. A stock has an infinite life. It generally increases in value over time, both as a consequence of inflation and real growth.
SOLUTIONS
ESTIMATION OF GROWTH RATES
Question 1





















A. Arithmetic Average =  17.68%  
B. Geometric Average =  17.65% 
C. The geometric average considers the compounded effects of growth. The arithmetic average does not.
Question 2
A. Linear Regression =  EPS = 0.4413 + 0.172 (t)  
Growth rate from this regression model equals  $0.172 a year.  
B. Loglinear Regression =  EPS = 0.5841 + 0.1674 t  
Growth rate from this regression model equals 16.74% a year.  
C. Expected EPS next year using linear regression:
= 0.4413 + 0.172 (7) = $1.65 

Expected EPS next year using loglinear regression:
= Exp(0.5841 + 0.1674 (t)) = $1.80 

Question 3







































A. Geometric Average = (10.75/7.27)(1/6) 1 =  6.74%  
B. Arithmetic Average Using Corrected Growth Rates =  33.20%  
C. Linear Regression =  EPS = 3.8271 + 0.1389 (t)  
Expected Growth Rate = 0.1389/Average EPS =  3.17% 
Question 4
A. Retention Ratio = 64%
Return on Equity = 1625/5171 = 31.4%
Expected Growth Rate = 0.64 * 31.4% = 20.10%
B. Growth Rate in 1993 = ( $5,171 * (.25 .314)/ $1,625) + 0.64 * 0.25 = 4.37%
C. Growth Rate After 1993 = 0.64 * 0.25 = 16%
Question 5
A. Retention Ratio = 1  $660/$1080 = 0.3889
Return on Assets = ($1080 + $550 * (1  0.4))/($6000 + $6880) = 10.95%
Debt/Equity Ratio = (6880/6000) = 1.14
Expected Growth Rate: = 0.3889 (10.95% + 1.14 (10.95%  (550/6880) * (1  0.4)) = 7.00%
B. Retention Ratio = 50%
Total Assets = $6000 + $6880  $2500 = $10.380.
New Return on Assets = (1020 + 550 * (1  0.4))/10380 = 13.01%
(The earnings before interest and taxes goes down by $100. The earnings after taxes will drop by $60. Note that interest expenses will be lower after debt is paid off, but the net income will go up by an equivalent amount.)
New Debt Equity Ratio = (4380/6000) = 0.73
New Expected Growth Rate = 0.50 (13.01% + 0.73 (13.01%  7%*(1  0.4))) = 9.72%
(The growth rate next year will be much higher as a result of the shift in the return on equity, but the long term growth rate will now be 9.72%)
C. Beta Before Change = 1.10
Unlevered Beta = 1.10/(1 + (6880/(330 * $63)) * (1  0.4)) = 0.9178
(Use market value of equity for this calculation)
Beta After Change = 0.9178 * (1 + (4380/(330 * $63)) * (1  0.4)) = 1.04
Question 6
A. PreInterest, AfterTax Profit Margin = EBIT (1T)/Sales = 10 * (1  0.36)/ 60 = 10.67%
Asset Turnover = Sales/Total Assets = 60/30 = 2
Return On Assets = 0.1067 * 2 = 0.2134 Or 21.34%
Retention Ratio = 35%
Expected Growth Rate = 0.35 (0.2134 + 1 (0.2134  0.065 * 0.64)) = 13.48%
B. PreInterest, AfterTax Profit Margin = 9.00%
Return on Assets = 0.09 * 2 = 18%
New Growth Rate = 0.35 (0.18 + 1 (0.18  0.065 * 0.64))= 11.14%
C. BreakEven Asset Turnover = 0.2134/0.09 = 2.37
Question 7
A. Expected Growth Rate = 0.93 (25% + 0.10 (25%  8.50% * (1  0.4)) = 25.10%
B. The following would be the expected changes :
(1) ROC will decline as the firm gets larger and the marginal projects are no longer as lucrative.
(2) Dividend payout ratio will increase.
(3) Debt/Equity ratio will increase as the firm gets larger and safer.
(4) The interest rate on debt will decline for the same reasons.
C. Expected Growth Rate = 0.5 (0.14 + 0.4 (0.14  0.07 * (1  0.4)) = 8.96%
Question 8
A. Weight analysts' forecasts the most, and historical growth rates the least (or not at all). In estimating growth rates from fundamentals, use predicted values for the fundamentals, rather than current values.
B. Use growth rates from fundamentals, and reflect the expected changes from the restructuring in these fundamentals.
C. Weight all three growth rates equally.
D. Use fundamentals on the remaining divisions to predict growth.
SOLUTIONS
DIVIDEND DISCOUNT MODELS
Question 1
A. False. The dividend discount model can still be used to value the dividends that the company will pay after the high growth eases.
B. False. It depends upon the assumptions made about expected future growth and risk.
C. False. This will be true only if the stock market falls more than merited by changes in the fundamentals (such as growth and cash flows).
D. True. Portfolios of stocks that are undervalued using the dividend discount model seem to earn excess returns over long time periods.
E. True. The model is biased towards these stocks because of its emphasis on dividends.
Question 2
A. A stock that pays no dividends is not a stable stock. The Gordon Growth model is not designed to value such a stock. If a company with stable growth insists on not paying dividends, but retains the FCFE, this FCFE can be used in the Gordon Growth model as the dividend.
B. A stable stock cannot have a growth rate greater than the discount rate, because no company can grow much faster than the economy in which it operates in the Gordon Growth Model. This upper limit on how high growth rates can go operates as a constraint in the model.
C. This should not happen for a stable stock, for the same reasons stated above.
D. It is true that the model smooths out growth rates in dividends. In present value terms, though, this smoothing effect cannot have a large effect on the value estimate obtained from the model.
E. The model requires that, in the long term, the growth rate for a firm is stable (close to the growth rate in the economy). Thus, cyclical firms, which maintain an average growth rate close to a stable rate, cyclical ups and downs notwithstanding, can be valued using this model.
Question 3
A. Cost of Equity = 6.25% + 0.90 * 5.5% = 11.20%
Value Per Share = $3.56 * 1.055/(.1120  .055) = $65.89
B. $3.56 (1 + g)/(.1120  g) = $80
Solving for g,
g = (80 * .112  3.56)/(80 + 3.56) = 6.46%
Question 4
A. This should increase both the cost of equity (by raising interest rates) and the nominal growth rate. Whether the increase will be the same in both variables will depend in large part on whether an increase in inflation will adversely impact real economic growth.
B. This should affect the estimation of a stable growth rate. A much higher stable growth rate can be used for firms in economies which are growing rapidly.
C. An analyst has very limited flexibility when it comes to using the Gordon Growth model in estimating growth. If the growth potential of the industry in which the firm operates is very high, a growth rate slightly higher (1 to 2%) than the growth rate in the economy can be used as a stable growth rate. Alternatively, a twostage or threestage growth model can be used to value the stock.
D. Same as the answer to 3.
Question 5
A. Expected Earnings Per Share in 1999 = $2.10 * 1.155 * 1.06 = $4.48
Expected Dividends Per Share in 1999 = $4.48 * 0.65 = $2.91
Cost Of Equity After 1999 = 6.25% + 1.1 * 5.5% = 12.30%
Expected Price at the End of 1998 = Expected DPS in 1999/(ke, 1999  g)
= $2.91/(.1230  .06) = $46.19
B.
























Cost of Equity = 6.25% + 1.40 * 5.5% = 


PV of Dividends and Terminal Price (@ 13.95%) = 

Question 6
A. Retention Ratio = 1  Payout Ratio = 1  0.42/1.50 = 72%
Return on Assets
= (Net Income + Int Exp (1t))/(BV of Debt + BV of Equity)
= (30 + 0.8 * (1  0.385))/(7.6 + 160) = 18.19%
Debt/Equity Ratio = 7.6/160 = .0475
Interest Rate on Debt = 0.8/7.6 = 10.53%
Expected Growth Rate
= 0.72 [.1819 + .0475 (.1819  .1053 * (1  0.385))] = 13.5%
Alternatively, and much more simply,
Return on Equity = 30/160 = .1875
Expected Growth Rate = 0.72 * .1875 = 13.5%
B. Expected payout ratio after 1998:
= 1  g/[ROC + D/E (ROC  i (1t))]
= 1  .06/(.125+.25(.125  .07(1.385))
= 0.5876
C. Beta in 1993 = 0.85
Unlevered Beta = 0.85/(1 + (1  0.385) * 0.05) = 0.8246
Beta After 1998 = 0.8246 * (1 + (1  0.385) * 0.25) = 0.95
D. Cost of Equity in 1999 = 7% + 0.95 * 5.5% = 12.23%
Expected Dividend in 1999
= ( $1.50 * 1.1355 * 1.06) * 0.5876 = $1.76
Expected Price at End of 1998 = $1.76/(.1223  .06) = $28.25
E.
























Cost of Equity = 7% + 0.85 * 5.5% = 


PV of Dividends and Terminal Price (@ 11.68%) = 

F. Total Value per Share = $18.47
Value Per Share Using Gordon Growth Model
= $1.50 * 1.06 * 0.5876/(.1223  .06) = $15.00
Value Per Share With No Growth = $1.50 * 0.5876/.1223 = $7.21
Value of Extraordinary Growth = $18.47  $15.00 = $3.47
Value of Stable Growth = $15.00  $7.21 = $7.79
Question 7
A. Cost of Equity = 6.25% + 0.85 * 5.5% = 10.93%
Value of Stable Growth = $0.48 * 1.07/(.1093  .07) = $13.07
B. Value of Extraordinary Growth
= $0.48 * (6/2) * (.25  .07)/(.1093  .07) = $6.60
C. The payout ratio is assumed to remain unchanged as the growth rate changes. The payout ratio in this case is assumed to remain at 60% (0.48/0.80).
Question 8
A.

































B. Expected Price at End of 2003
= ($13.34 * 1.06 * 0.60)/(.1175  .06) = $147.54
(Cost of Equity = 6.25% = 5.5% = 11.75%)
C.
PV of Dividends  High Growth = 


PV of Dividends  Transition = 


PV of Terminal Price = 


Value Per Share = 

SOLUTIONS
FREE CASH FLOW TO EQUITY DISCOUNT MODELS
Question 1
A. True. Dividends are generally smoothed out. Free cash flows to equity reflect the variability of the underlying earnings as well as the variability in capital expenditures.
B. False. Firms can have negative free cash flows to equity. Dividends cannot be less than zero.
C. False. Firms with high capital expenditures, relative to depreciation, may have lower FCFE than net income.
D. False. The free cash flow to equity can be negative for companies, which either have negative net income and/or high capital expenditures, relative to depreciation. This implies that new stock has to be issued.
Question 2
A. Value Per Share = $1.70 * 1.07/(.1203  .07) = $36.20
(Cost of Equity = 6.25% + 1.05 * 5.50% = 12.03%)
B.
Current Earnings per share = 


 (1  Desired Debt Fraction) *  
(Capital Spending  Depreciation) = 83.61%* 



 (1  Desired Debt Fraction) *  
DWorking Capital  = 83.61% * 



Free Cash Flow to Equity = 

Cost of Equity = 6.25% + 1.05 * 5.5% = 12.03%
Value Per Share = $2.36 * 1.07/(.1203  .07) = $50.20
This is based upon the assumption that the current ratio of capital expenditures to depreciation is maintained in perpetuity.
C. The FCFE is greater than the dividends paid. The higher value from the model reflects the additional value from the cash accumulated in the firm. The FCFE value is more likely to reflect the true value.
Question 3
A.






Term Price 









































The net capital expenditures (Cap Ex  Depreciation) and working capital change is offset partially by debt (20%). The balance comes from equity. For instance, in year 1:
FCFE = $2.71  ($2.60  $1.30) * (1  0.20)  $0.05 * (1  0.20) = $1.64)
Cost of Equity = 6.5% + 1 * 5.5% = 12%
Terminal Value Per Share = $5.08/(.12  .06) = $84.74
Present Value Per Share = 1.64/1.12 + 1.89/1.12^2 + 2.19/1.12^3 + 2.54/1.12^4 + (2.93 + 84.74)/1.12^5 = $55.89
B.






Term Price 









































Terminal Value Per Share = $3.13/(.12  .06) = $52.09
Present Value Per Share = 1.64/1.12 + 1.89/1.12^2 + 2.19/1.12^3 + 2.54/1.12^4 + (2.93+52.09)/1.12^5 = $37.36
C.






Term Price 









































Terminal Value Per Share = $2.75/(.12  .06) = $45.85
Present Value Per Share = 1.43/1.12 + 1.66/1.12^2 + 1.92/1.12^3 + 2.23/1.12^4 + (2.58 + 45.85)/1.12^5 = $32.87
The beta will probably be lower because of lower leverage.
Question 4
A.
1 
$2.30 
$0.68 
$0.33 
$0.45 
$1.57 
Term. Price 


































The net capital expenditures (Cap Ex  Depreciation) and working capital change is offset partially by debt (10%). The balance comes from equity. For instance, in year 1 
FCFE = $2.30  ($0.68  $0.33) * (1  0.10)  $0.45 * (1  0.10) = $1.57)
B. Terminal Price = $3.71/ (.1305  .07) = $52.69
C. Present Value Per Share = 1.57/1.136 + 1.82/1.136^2 + 2.11/1.136^3 + 2.45/1.136^4 + (2.83 + 52.69)/1.136^5 = $35.05
Question 5
A.






Earnings 





(CapExDeprec'n) * (1_) 





DWorking Capital * (1_) 





FCFE 





Present Value 





Transition Period (up to ten years)
Year 






Growth Rate 






Cumulated Growth 






Earnings 






(CapExDeprec'n) * (1_) 






DWorking Capital * (1_) 






FCFE 






Beta 






Cost of Equity 






Present Value 






EndofLife Index 


Stable Growth Phase  
Growth Rate: Stable Phase = 


FCFE in Terminal Year = 


Cost of Equity in Stable Phase = 


Price at the End of Growth Phase = 


PV of FCFE in High Growth Phase = 


Present Value of FCFE in Transition Phase = 


Present Value of Terminal Price = 


Value of the Stock = 

B.






Earnings 





(CapExDeprec'n)* (1_) 





DWorking Capital * (1_) 





FCFE 





Present Value 





Transition Period (up to ten years)
Year 





Growth Rate 





Cumulated Growth 





Earnings 





(CapExDeprec'n)*(1_) 





DWorking Capital *(1_) 





FCFE 





Beta 





Cost of Equity 





Present Value 





EndofLife Index 

Stable Growth Phase
Growth Rate in Stable Phase = 


FCFE in Terminal Year = 


Cost of Equity in Stable Phase = 


Price at the End of Growth Phase = 


PV of FCFE in High Growth Phase = 


Present Value of FCFE in Transition Phase = 


Present Value of Terminal Price = 


Value of the Stock = 

C.






Earnings 





(CapExDeprec'n) * (1_) 





DWorking Capital * (1_) 





FCFE 





Present Value 





Transition Period (up to ten years)
Year 





Growth Rate 





Cumulated Growth 





Earnings 





(CapExDeprec'n) * (1_) 





DWorking Capital * (1_) 





FCFE 





Beta 





Cost of Equity 





Present Value 





EndofLife Index 

Stable Growth Phase
Growth Rate in Stable Phase = 


FCFE in Terminal Year = 


Cost Of Equity in Stable Phase = 


Price at End of Growth Phase = 


PV of FCFE In High Growth Phase = 


Present Value of FCFE in Transition Phase = 


Present Value of Terminal Price = 


Value of the Stock = 

Question 6
A.
Year 





Earnings 





(CapExDeprec'n)* (1_) 





DWorking Capital * (1_) 





FCFE 





Present Value 





Transition Period (up to ten years)
Year 




Growth Rate 




Cumulated Growth 




Earnings 




(CapExDeprec'n)*(1_) 




D Working Capital *(1_) 




FCFE 




Beta 




Cost of Equity 




Present Value 




EndofLife Index 


Stable Growth Phase  
Growth Rate in Stable Phase =  5.00%  
FCFE in Terminal Year = 


Cost of Equity in Stable Phase =  13.05%  
Price at the End of Growth Phase = 


PV of FCFE in High Growth Phase =  $0.85  
Present Value of FCFE in Transition Phase =  $1.83  
Present Value of Terminal Price =  $10.46  
Value of the Stock =  $13.14 
B. It is impossible to say. Easier credit terms will increase working capital as a percentage of revenues, and thus act as a drain on cash flows. On the other hand, the higher growth in revenues and earnings will create a positive effect. The net effect can be either positive or negative.
C.
Working Capital as % of Revenues  Value Per Share 
60%  $8.62 
50%  $10.88 
40  $13.14 
30%  $15.40 
20%  $17.66 
* This assumes that there is no change in expected growth, as a consequence.
Question 7
A. Both models should have the same value, as long as a higher growth rate in earnings is used in the dividend discount model to reflect the growth created by the interest earned, and a lower beta to reflect the reduction in risk. The reality, however, is that most analysts will not make this adjustment, and the dividend discount model value will be lower than the FCFE model value.
B. The dividend discount model will overstate the true value, because it will not reflect the dilution that is inherent in the issue of new stock.
C. Both models should provide the same value.
D. Since acquisition, with the intent of diversifying, implies that the firm is paying too much (i.e., negative net present value), the dividend discount model will provide a lower value than the FCFE model.
E. If the firm is overlevered to begin with, and borrows more money, there will be a loss of value from the overleverage. The FCFE model will reflect this lost value, and will thus provide a lower estimate of value than the dividend discount model.
SOLUTIONS
VALUING A FIRM  THE FCFF APPROACH
Question 1
A. False. It can be equal to the FCFE if the firm has no debt.
B. True.
C. False. It is predebt, but aftertax.
D. False. It is aftertax, but predebt.
E. False. The free cash flow to firm can be estimated directly from the earnings before interest and taxes.
Question 2
A. FCFF = Net Income + Interest (1t) + Depreciation  Capital Spending  DWorking Capital
B. FCFF = (Earnings before taxes + Interest Expenses) (1  tax rate) + Depreciation  Capital Spending  DWorking Capital
C. FCFF = EBIT (1 tax rate) + Depreciation  Capital Spending  DWorking Capital
D. FCFF = (EBITDA  Depreciation) (1 tax rate) + Depreciation  Capital Spending  DWorking Capital
E. FCFF = (NOI  Nonoperating Expenses) (1 tax rate) + Depreciation  Capital Spending  DWorking Capital
F. FCFF = FCFE + Interest Expenses (1  tax rate)  New Debt Issues + Principal Repayments
* Assumed no preferred stock is outstanding.
Question 3
A. FCFF in 1993 = Net Income + Depreciation  Capital Expenditures  DWorking Capital + Interest Expenses (1  tax rate)
= $770 + $960  $1200  0 + $320 (1  0.36) = $734.80 million
B. EBIT = Net Income/(1  tax rate) + Interest Expenses = 770/0.64 + 320 = $1523.125 million
Return on Assets = EBIT (1t)/ (BV of Debt + BV of Equity) = 974.80/9000 = 10.83%
Expected Growth Rate in FCFF = Retention Ratio * ROC = 0.6 * 10.83% = 6.50%
Cost of Equity = 7% + 1.05 * 5.5% = 12.775%
Cost of Capital = 8% (1  0.36) (4000/(4000 + 12000)) + 12.775% = (12000/(4000 + 12000)) = 10.86%
Value of the Firm = 734.80/(.1086  .065) = $16,853 millions
C. Value of Equity = Value of Firm  Market Value of Debt
= $16,853  $4,000 = $12,853 millions
Value Per Share = $12,853/200 = $64.27
Question 4
A.
Yr  EBITDA  Deprec'n  EBIT 

Cap  _ WC  FCFF  Term  

Exp.  Value  
0 




$450 



1 




$493 



2 




$540 



3 




$591 



4 




$647 



5 




$708 




'9397  After 1998  
Cost of Equity =  13.05%  12.30%  
AT Cost of Debt =  4.80%  4.50%  
Cost of Capital =  9.37%  9.69% 
Terminal Value
= {EBIT (1t)(1+g)  (Rev1998  Rev1997) * WC as % of Rev}/(WACCg)
= (841 * 1.04)  (13500 * 1.0955 * 1.04  13500 * 1.0955)
* 0.07 /(.0969.04) = $14,326
Value of the Firm
= 440/1.0937 + 482/1.0937^{2} + 528/1.0937^{3} + 578/1.0937^{4} + (633 + 14941)/1.0937^{5} = $11,172
B. Value of Equity in the Firm = ($11566  Market Value of Debt) = 11172  3200 = $7,972
Value Per Share = $7,972/62 = $128.57
Question 5
A. Beta for the Health Division = 1.15
Cost of Equity = 7% + 1.15 * 5.5% = 13.33%
Cost of Capital = 13.33% * 0.80 + (7.5% * 0.6) * 0.2 = 11.56%
B.
Year 

















































After 5 years  











Value of the Division = 283/1.1156 + 302/1.11562 + 321/1.11563 + 342/1.11564 + (364 + 5014)/1.11565 = $4,062 millions
C. There might be potential for synergy, with an acquirer with related businesses. The health division at Kodak might also be mismanaged, creating the potential for additional value from better management.
Question 6
A. Cost of Equity = 7% + 1.25 * 5.5% = 13.88%
Current Debt Ratio = 1340/(1340 + 18.25 * 183.1) = 28.63%
Aftertax Cost of Debt = 7.43% (1  0.4) = 4.46%
Cost of Capital = 13.88% (0.7137) + 4.46% (0.2863) = 11.18%
B. & C. See table below.



















































































Unlevered Beta = 1.25/(1 + 0.6 * (1340/(183.1 * 18.25)) = 1.01
Levered Beta at 10% D/(D+E) = 1.01 * (1 + 0.6 * (10/90)) = 1.07
FCFF to Firm Next Year = (637  235) * (1  0.4) * 1.03 = $248.43 million
Value of the Firm = 255.67 * 1.03/(WACC.03)
Question 7
A. Cost of Equity = 7% + 2.2 * 5.5% = 19.10%
Aftertax Cost of Debt = 10.31%(1  0.4) = 6.19%
Market Value of Equity = 45.99 * 9 = $413.91 million
Cost of Capital = 19.10% (413.91/(413.91 + 1180)) + 6.19% (1180/(413.91 + 1180)) = 9.54%
B. Unlevered Beta = 2.2/(1 + 0.6 * (1180/413.91)) = 0.81
New Beta = 0.81 (1 + 0.6 * 1) = 1.30
New Cost of Equity = 14.14%
Aftertax Cost of Debt = 7.51%(1  0.6) = 4.51%
Cost of Capital = 14.14% (0.5) + 4.51% (0.5) = 9.32%
C.
















The value of the firm is calculated as follows:
FCFF in Current Year = 236 * (1  0.4) + 109  125 = $125.6 million
Value of the Firm Before the Change = 125.6 (1+g)/(.0954g)
Value of the Firm After the Change = 125.6 (1+g)/(.0932g)
SOLUTIONS
SPECIAL CASES IN VALUATION
Question 1
A.
































Average Earnings Per Share = $0.48
Normalized Earnings Per Share in 1994 = $0.48 * 1.06 = $0.51
B.
Normalized Earnings Per Share = 


 (Cap Ex  Deprec'n) * (1  Debt ratio) = 


 D Working Capital * (1 Debt ratio) = 


Normalized FCFE Next Year = 


(Assume that capital expenditures and depreciation will grow 6%
in 1994.) 
Question 2
A.
Total Assets in 1993 = 

(in millions)  
Normalized Return on Assets = 


Normalized Return on Assets (pretax) = 


Normalized Income statement (based upon 12% ROC)  
Earnings Before Interest and Taxes = 


Interest Expenses = 


Earnings Before Taxes = 


Taxes (at 40%) = 


Net Income = 


 (Cap Ex  Deprec'n) * (1Debt ratio) = 


FCFE = 


Cost of Equity = 7% + 1.1 * 5.5% = 


Expected Growth Rate = 

Earnings before interest and taxes is calculated using the ROC:
ROC = EBIT (1 tax rate) / Total Assets = 12% (given in the problem)
Value of Equity = (1660 * 1.05)/(.1305  .05) = $21,652
B. Value of Equity = $21,652/1.1305^2 = $16,942
Question 3
A.
Earnings Per Share Next Year = 

 (Cap Ex  Deprec'n) *(1 Debt ratio) = 

 ( _ Working Capital) * (1  Debt ratio) = 

FCFE Next Year = 

Cost of Equity = 7% + 1.1 * 5.5% = 

Expected Growth Rate = 

Capital Expenditures  Depreciation = ($5.50 * 1.05  $4.50 * 1.05)
Debt Ratio = 40%
Value Per Share = $4.85/(.1305  .05) = $60.19
B. The value is very sensitive to assumptions about growth in 1993. If the earnings do not quadruple in 1993, the free cash flow to equity will be significantly below $4.85, and the value lower than $60.19.
Question 4
A.
Earnings Before Interest and Taxes = 


 Interest Expense = 


Earnings Before Taxes = 


 Taxes (40%) = 


Earnings After Taxes = 


 (Cap Ex  Deprec'n) * (1Debt Ratio) = 


 DWorking Capital * (1 Debt Ratio) = 


FCFE = 

EBIT = Interest Expense * Interest Coverage Rate = $17 * 3.10 = $ 52.70
The change in working capital is based upon revenues growing at 4%.
B. Cost of Equity = 7% + 1.1 * 5.5% = 13.05%
Expected Growth Rate = 4%
Value of Equity = 12.91 * 1.04/(.1305  .04) = $148.36 million
Question 5
A.





















Average = 


Net Income = 


 (Cap Ex  Deprec'n) * (1  Debt ratio) = 


= FCFE = 

B. Cost of Equity (until 1996) = 7% +1.2 * 5.5% = 13.6%
Cost of Equity (after 1996) = 7% + 5.5% = 12.5%
Year  Net Income 

FCFE  Terminal Value 


























Capital expenditures are offset by depreciation in the terminal year.
Terminal Value = $2.23/(.125  .05) = $29.73
Value of Equity
= 0.42/1.136 + 0.45/1.136^2 + 0.50/1.136^3 + (0.54 + 29.73)/1.136^4
= $19.24 million
Question 6
A.
EBIT (based upon operating margin of 12%) = 


Interest Expenses = 


Earnings Before Taxes = 


Taxes (at 40%) = 


Net Income (also FCFE)= 


Cost of Equity = 7% + 1.15 * 5.5% = 


Expected Growth Rate in FCFE = 


Value of Equity= (660 * 1.06)/ (.1333  .06) = 

B. Value of Equity (assuming two year delay in return to profitability)
= 9010/1.133252 = $7,016 million
Question 7
A.



Market Value Weight 


Cost of Component 


Cost of Capital = 13.33% (0.6161) + 5.1% (0.3839) = 10.17%
B.






EBIT (1t) 





 (Cap Ex  Deprec'n) 





 DWorking Capital 





= FCFF 





Terminal Value 

Terminal Value = $11.42/(.1017  .04) = $185.18
Present Value = $8.25/1.1017 + $9.08/1.1017^2 + $9.98/1.1017^3 + ($10.98 + $185.18)/1.1017^4 = $155.60 million
C. Value of Equity = Value of Firm  Market Value of Debt = $155.60  $109 = $46.60 million
Value of Equity Per Share = $46.60/15.9 = $2.93
Question 8
A. Unlevered Beta for Publicly Traded Firms in Same Business
= 1.30/(1 + 0.6 * 0.2) = 1.16
Debt/Equity Ratio for Private Firm
= Debt/Estimated Market Value of Equity = 10/30 = 33.33%
New Levered Beta For Private Firm = 1.16 * (1 + 0.6 * .3333) = 1.39
New Cost Of Equity = 7% + 1.39 * 5.5% = 14.66%
B. PreTax Cost of Debt = $1/$10 = 10%
AfterTax Cost of Debt = 10% (1  0.4) = 6%
Cost of Capital = 6% (0.25) + 14.66% (0.75) = 12.49%
C. Using the Firm Approach:









EBIT 






EBIT (1  tax rate) 






 (Cap Ex  Deprec'n) 






= FCFF 






Terminal Value 

Terminal Value = $3.14/(.1249  .05) = $41.85
Present Value (Value of Firm) (@ 12.49%) = $0.84/1.1249 + $1.01/1.1249^{2} + $1.21/1.1249^{3} + $1.45/1.1249^{4} + ($1.74 + $41.85)/1.1249^{5} = $27.50 million
Value of Equity = $27.50 million  $10 million = $17.50 million
Using the Equity approach:









Net Income 






 (Cap Ex  Deprec'n) * (1 Debt ratio) = 
$0.45 
$0.54 
$0.65 
$0.78 
$0.93 
$0.00 
= FCFE 






Terminal Value of Equity 

Terminal Value of Equity = $1.98/(.1466  .05) = $29.71
Present Value (using Cost of Equity of 14.66%) = $0.30/1.1466 + $0.40/1.1466^{2} + $0.52/1.1466^{3} + $0.69/1.1466^{4} + ($0.90 + $29.71)/1.1466^{5} = $16.76 million
Question 9
A. Unlevered Beta for Comparable Firms = 1.15/(1 + 0.6 * .25) = 1.00
Cost of Equity (until 1997) = 7% + 5.5% = 12.50%
Cost of Capital (until 1997) = 12.50%
Beta after 1997 for Boston Chicken = 1.15
Cost of Equity (after 1997) = 7% + 1.15 * 5.5% = 13.325%
Cost of Capital (after 1997)
= 13.325% (0.8) + 8% (1  0.4) (0.2) = 11.62%
B.






EBIT (1t) 





Cap Ex  Deprec'n 





FCFF 










EBIT (1t) 




Cap Ex  Deprec'n 




FCFF 




Terminal Value 

Terminal Value = $51.56 /(.1162  .05) = $778.80 million
Present Value (using 12.50% for the first five years, and 11.62% after that) =$3.72/1.125 + $5.92/1.125^{2} + $9.07/1.125^{3} + $13.55/1.125^{4} + $19.84/1.125^{5} + $37.13/(1.125^{5} * 1.1162) + $42.70/(1.125^{5} * 1.1162^{2}) + ($49.10 + 778.80)/(1.125^{5} * 1.1162^{3}) = $401.67 million
This is both the value of the firm and the value of equity.