How Much Is the Concatenator Business Worth per Share

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Suppose the concatenator division is spun off from its parent as an independent company, Concatco, with one million outstanding shares. What would each share sell for?

We have already calculated the value of Concatco's free cash flow as $18.8 million, using the constant-growth DCF formula to calculate horizon value. If this value is right, and there are one million shares, each share should be worth $18.80. This amount should also be the present value of Concatco's dividends per share— although here we must slow down and be careful. Note from Table 4.7 that free cash flow is negative from years 1 to 6. Dividends can't be negative, so Concatco will have to raise outside financing. Suppose it issues additional shares. Then Con-catco's one million existing shares will not receive all of Concatco's dividend payments when the company starts paying out cash in year 7.

11The PV of free cash flow before the horizon improves to -$2.0 million because inflows in years 7 and

8 are now included.

80 PART I Value

There are two approaches to valuing a company's existing shares when new shares will be issued. The first approach discounts the net cash flow to existing shareholders if they buy all the new shares issued. In this case the existing shareholders would pay out cash to Concatco in years 1 to 6, and then receive all subsequent dividends; they would pay for or receive all free cash flow from year 1 to year 8 and beyond. The value of a share therefore equals free cash flow for the company as a whole, taking account of negative as well as positive amounts, divided by the number of existing shares. We have already done this calculation: If the value of the company is $18.8 million, the value of each of the one million existing shares should be $18.80.

The second approach discounts the dividends that will be paid when free cash flow turns positive. But you must discount only the dividends paid on existing shares. The new shares issued to finance the negative free cash flows in years 1 to 6 will claim a portion of the dividends paid out later.

Let's check that the second method gives the same answer as the first. Note that the present value of Concatco's free cash flow from years 1 to 6 is —$3.6 million. Concatco decides to raise this amount now and put it in the bank to take care of the future cash outlays through year 6. To do this, the company has to issue 191,500 shares at a price of $18.80:

Cash raised = price per share X number of new shares = 18.80 X 191,500 = $3,600,000

If the existing stockholders buy none of the new issue, their ownership of the company shrinks to

Existing shares 1,000,000

Existing + new shares 1,191,500

The value of the existing shares should be 83.9 percent of the present value of each dividend paid after year 6. In other words, they are worth 83.9 percent of PV(horizon value), which we calculated as $22.4 million from the constant-growth DCF formula.

PV to existing stockholders = .839 X PV(horizon value)

Since there are one million existing shares, each is worth $18.80.

Finally, let's check whether the new stockholders are getting a fair deal. They end up with 100 — 83.9 = 16.1 percent of the shares in exchange for an investment of $3.6 million. The NPV of this investment is

NPV to new stockholders = —3.6 + .161 X PV(horizon value)

On reflection, you will see that our two valuation methods must give the same answer. The first assumes that the existing shareholders provide all the cash whenever the firm needs cash. If so, they will also receive every dollar the firm pays out. The second method assumes that new investors put up the cash, relieving existing shareholders of this burden. But the new investors then receive a share of future payouts. If investment by new investors is a zero-NPV transaction, then it doesn't make existing stockholders any better or worse off than if they had invested themselves. The key assumption, of course, is that new shares are issued on fair terms, that is, at zero NPV.12

12The same two methods work when the company will use free cash flow to repurchase and retire outstanding shares. We discuss share repurchases in Chapter 16.

In this chapter we have used our newfound knowledge of present values to examine the market price of common stocks. The value of a stock is equal to the stream of cash payments discounted at the rate of return that investors expect to receive on other securities with equivalent risks.

Common stocks do not have a fixed maturity; their cash payments consist of an indefinite stream of dividends. Therefore, the present value of a common stock is


However, we did not just assume that investors purchase common stocks solely for dividends. In fact, we began with the assumption that investors have relatively short horizons and invest for both dividends and capital gains. Our fundamental valuation formula is, therefore,

This is a condition of market equilibrium. If it did not hold, the share would be overpriced or underpriced, and investors would rush to sell or buy it. The flood of sellers or buyers would force the price to adjust so that the fundamental valuation formula holds.

This formula will hold in each future period as well as the present. That allowed us to express next year's forecasted price in terms of the subsequent stream of dividends DIV2, DIV3,

We also made use of the formula for a growing perpetuity presented in Chapter 3. If dividends are expected to grow forever at a constant rate of g, then

It is often helpful to twist this formula around and use it to estimate the market capitalization rate r, given P0 and estimates of DIVX and g:

DIVi Po g

Remember, however, that this formula rests on a very strict assumption: constant dividend growth in perpetuity. This may be an acceptable assumption for mature, low-risk firms, but for many firms, near-term growth is unsustainably high. In that case, you may wish to use a two-stage DCF formula, where near-term dividends are forecasted and valued, and the constant-growth DCF formula is used to forecast the value of the shares at the start of the long run. The near-term dividends and the future share value are then discounted to present value.

The general DCF formula can be transformed into a statement about earnings and growth opportunities:



The ratio EPS^ r is the capitalized value of the earnings per share that the firm would generate under a no-growth policy. PVGO is the net present value of the investments t=1

82 PART I Value that the firm will make in order to grow. A growth stock is one for which PVGO is large relative to the capitalized value of EPS. Most growth stocks are stocks of rapidly expanding firms, but expansion alone does not create a high PVGO. What matters is the profitability of the new investments.

The same formulas that are used to value a single share can also be applied to value the total package of shares that a company has issued. In other words, we can use them to value an entire business. In this case we discount the free cash flow thrown off by the business. Here again a two-stage DCF model is deployed. Free cash flows are forecasted and discounted year by year out to a horizon, at which point a horizon value is estimated and discounted.

Valuing a business by discounted cash flow is easy in principle but messy in practice. We concluded this chapter with a detailed numerical example to show you what practice is really like. We extended this example to show how to value a company's existing shares when new shares will be issued to finance growth.

In earlier chapters you should have acquired—we hope painlessly—a knowledge of the basic principles of valuing assets and a facility with the mechanics of discounting. Now you know something of how common stocks are valued and market capitalization rates estimated. In Chapter 5 we can begin to apply all this knowledge in a more specific analysis of capital budgeting decisions.


There are a number of discussions of the valuation of common stocks in investment texts. We suggest:

Z. Bodie, A. Kane, and A. J. Marcus: Investments, 5th ed., Irwin/McGraw-Hill, 2002.

W. F. Sharpe, G. J. Alexander, and J. V. Bailey: Investments, 6th ed., Prentice-Hall, Inc., En-glewood Cliffs, N.J., 1999.

J. B. Williams's original work remains very readable. See particularly Chapter V of:

J. B. Williams: The Theory of Investment Value, Harvard University Press, Cambridge, Mass., 1938.

The following articles provide important developments of Williams's early work. We suggest, however, that you leave the third article until you have read Chapter 16:

D. Durand: "Growth Stocks and the Petersburg Paradox," Journal of Finance, 12:348-363 (September 1957).

M. J. Gordon and E. Shapiro: "Capital Equipment Analysis: The Required Rate of Profit," Management Science, 3:102-110 (October 1956).

M. H. Miller and F. Modigliani: "Dividend Policy, Growth and the Valuation of Shares," Journal of Business, 34:411-433 (October 1961).

Leibowitz and Kogelman call PVGO the "franchise factor." They analyze it in detail in:

M. L. Leibowitz and S. Kogelman: "Inside the P/E Ratio: The Franchise Factor," Financial Analysts Journal, 46:17-35 (November-December 1990).

Myers and Borucki cover the practical problems encountered in estimating DCF costs of equity for regulated companies; Harris and Marston report DCF estimates of rates of return for the stock market as a whole:

S. C. Myers and L. S. Borucki: "Discounted Cash Flow Estimates of the Cost of Equity Capital—A Case Study," Financial Markets, Institutions and Instruments, 3:9-45 (August 1994).

R. S. Harris and F. C. Marston: "Estimating Shareholder Risk Premia Using Analysts' Growth Forecasts," Financial Management, 21:63-70 (Summer 1992).

The following book covers valuation of businesses in great detail:

T. Copeland, T. Koller, and J. Murrin: Valuation: Measuring and Managing the Value of Companies, John Wiley & Sons, Inc., New York, 1994.

1. True or false?

a. All stocks in an equivalent-risk class are priced to offer the same expected rate of return.

b. The value of a share equals the PV of future dividends per share.

2. Respond briefly to the following statement.

"You say stock price equals the present value of future dividends? That's crazy! All the investors I know are looking for capital gains."

3. Company X is expected to pay an end-of-year dividend of $10 a share. After the dividend its stock is expected to sell at $110. If the market capitalization rate is 10 percent, what is the current stock price?

4. Company Y does not plow back any earnings and is expected to produce a level dividend stream of $5 a share. If the current stock price is $40, what is the market capitalization rate?

5. Company Z's earnings and dividends per share are expected to grow indefinitely by 5 percent a year. If next year's dividend is $10 and the market capitalization rate is 8 percent, what is the current stock price?

6. Company Z-prime is like Z in all respects save one: Its growth will stop after year 4. In year 5 and afterward, it will pay out all earnings as dividends. What is Z-prime's stock price? Assume next year's EPS is $15.

7. If company Z (see question 5) were to distribute all its earnings, it could maintain a level dividend stream of $15 a share. How much is the market actually paying per share for growth opportunities?

8. Consider three investors:

a. Mr. Single invests for one year.

b. Ms. Double invests for two years.

c. Mrs. Triple invests for three years.

Assume each invests in company Z (see question 5). Show that each expects to earn an expected rate of return of 8 percent per year.

9. True or false?

a. The value of a share equals the discounted stream of future earnings per share.

b. The value of a share equals the PV of earnings per share assuming the firm does not grow, plus the NPV of future growth opportunities.

10. Under what conditions does r, a stock's market capitalization rate, equal its earnings-price ratio EPSj/P0?

11. What do financial managers mean by "free cash flow"? How is free cash flow related to dividends paid out? Briefly explain.

12. What is meant by a two-stage DCF valuation model? Briefly describe two cases where such a model could be used.

13. What is meant by the horizon value of a business? How is it estimated?

14. Suppose the horizon date is set at a time when the firm will run out of positive-NPV investment opportunities. How would you calculate the horizon value?


Brealey-Meyers: Principles of Corporate Finance, Seventh Edition

I. Value

4. The Value of Common Stocks

© The McGraw-H Companies, 2003

PART I Value


Look in a recent issue of The Wall Street Journal at "NYSE-Composite Transactions."

a. What is the latest price of IBM stock?

b. What are the annual dividend payment and the dividend yield on IBM stock?

c. What would the yield be if IBM changed its yearly dividend to $1.50?

e. Use the P/E to calculate IBM's earnings per share.

f. Is IBM's P/E higher or lower than that of Exxon Mobil?

g. What are the possible reasons for the difference in P/E?

The present value of investing in a stock should not depend on how long the investor plans to hold it. Explain why.

Define the market capitalization rate for a stock. Does it equal the opportunity cost of capital of investing in the stock?

Rework Table 4.1 under the assumption that the dividend on Fledgling Electronics is $10 next year and that it is expected to grow by 5 percent a year. The capitalization rate is 15 percent.

In March 2001, Fly Paper's stock sold for about $73. Security analysts were forecasting a long-term earnings growth rate of 8.5 percent. The company was paying dividends of $1.68 per share.

a. Assume dividends are expected to grow along with earnings at g = 8.5 percent per year in perpetuity. What rate of return r were investors expecting?

b. Fly Paper was expected to earn about 12 percent on book equity and to pay out about 50 percent of earnings as dividends. What do these forecasts imply for g? For r? Use the perpetual-growth DCF formula.

You believe that next year the Superannuation Company will pay a dividend of $2 on its common stock. Thereafter you expect dividends to grow at a rate of 4 percent a year in perpetuity. If you require a return of 12 percent on your investment, how much should you be prepared to pay for the stock? Consider the following three stocks:

a. Stock A is expected to provide a dividend of $10 a share forever.

b. Stock B is expected to pay a dividend of $5 next year. Thereafter, dividend growth is expected to be 4 percent a year forever.

c. Stock C is expected to pay a dividend of $5 next year. Thereafter, dividend growth is expected to be 20 percent a year for 5 years (i.e., until year 6) and zero thereafter.

If the market capitalization rate for each stock is 10 percent, which stock is the most valuable? What if the capitalization rate is 7 percent?

Crecimiento S.A. currently plows back 40 percent of its earnings and earns a return of 20 percent on this investment. The dividend yield on the stock is 4 percent.

a. Assuming that Crecimiento can continue to plow back this proportion of earnings and earn a 20 percent return on the investment, how rapidly will earnings and dividends grow? What is the expected return on Crecimiento stock?

b. Suppose that management suddenly announces that future investment opportunities have dried up. Now Crecimiento intends to pay out all its earnings. How will the stock price change?

c. Suppose that management simply announces that the expected return on new investment would in the future be the same as the market capitalization rate. Now what is Crecimiento's stock price?

Look up General Mills, Inc., and Kellogg Co. on the Standard & Poor's Market Insight website ( The companies' ticker symbols are GIS and K.

a. What are the current dividend yield and price-earnings ratio (P/E) for each company? How do the yields and P/Es compare to the average for the food

industry and for the stock market as a whole? (The stock market is represented by the S & P 500 index.)

b. What are the growth rates of earnings per share (EPS) and dividends for each company over the last five years? Do these growth rates appear to reflect a steady trend that could be projected for the long-run future?

c. Would you be confident in applying the constant-growth DCF valuation model to these companies' stocks? Why or why not?

Look up the following companies on the Standard & Poor's Market Insight website ( Citigroup (C), Dell Computer (DELL), Dow Chemical (DOW), Harley Davidson (HDI), and Pfizer, Inc. (PFE). Look at "Financial Highlights" and "Company Profile" for each company. You will note wide differences in these companies' price-earnings ratios. What are the possible explanations for these differences? Which would you classify as growth (high-PVGO) stocks and which as income stocks?

Vega Motor Corporation has pulled off a miraculous recovery. Four years ago, it was near bankruptcy. Now its charismatic leader, a corporate folk hero, may run for president.

Vega has just announced a $1 per share dividend, the first since the crisis hit. Analysts expect an increase to a "normal" $3 as the company completes its recovery over the next three years. After that, dividend growth is expected to settle down to a moderate long-term growth rate of 6 percent.

Vega stock is selling at $50 per share. What is the expected long-run rate of return from buying the stock at this price? Assume dividends of $1, $2, and $3 for years 1, 2, 3. A little trial and error will be necessary to find r.

P/E ratios reported in The Wall Street Journal use the latest closing prices and the last 12 months' reported earnings per share. Explain why the corresponding earnings-price ratios (the reciprocals of reported P/Es) are not accurate measures of the expected rates of return demanded by investors.

Each of the following formulas for determining shareholders' required rate of return can be right or wrong depending on the circumstances:

DIVl epsl

For each formula construct a simple numerical example showing that the formula can give wrong answers and explain why the error occurs. Then construct another simple numerical example for which the formula gives the right answer.

14. Alpha Corp's earnings and dividends are growing at 15 percent per year. Beta Corp's earnings and dividends are growing at 8 percent per year. The companies' assets, earnings, and dividends per share are now (at date 0) exactly the same. Yet PVGO accounts for a greater fraction of Beta Corp's stock price. How is this possible? Hint: There is more than one possible explanation.

15. Look again at the financial forecasts for Growth-Tech given in Table 4.3. This time assume you know that the opportunity cost of capital is r = .12 (discard the .099 figure calculated in the text). Assume you do not know Growth-Tech's stock value. Otherwise follow the assumptions given in the text.

a. Calculate the value of Growth-Tech stock.

b. What part of that value reflects the discounted value of P3, the price forecasted for year 3?

c. What part of P3 reflects the present value of growth opportunities (PVGO) after year 3?

PART I Value

d. Suppose that competition will catch up with Growth-Tech by year 4, so that it can earn only its cost of capital on any investments made in year 4 or subsequently. What is Growth-Tech stock worth now under this assumption? (Make additional assumptions if necessary.) Look up Hawaiian Electric Co. (HI) on the Standard & Poor's Market Insight website ( Hawaiian Electric was one of the companies in Table 4.2. That table was constructed in 2001.

a. What is the company's dividend yield? How has it changed since 2001?

b. Table 4.2 projected growth of 2.6 percent. How fast have the company's dividends and EPS actually grown since 2001?

c. Calculate a sustainable growth rate for the company based on its five-year average return on equity (ROE) and plowback ratio.

d. Given this updated information, would you modify the cost-of-equity estimate given in Table 4.2? Explain.

Browse through the companies in the Standard & Poor's Market Insight website ( Find three or four companies for which the earnings-price ratio reported on the website drastically understates the market capitalization rate r for the company. (Hint: you don't have to estimate r to answer this question. You know that r must be higher than current interest rates on U.S. government notes and bonds.)

The Standard & Poor's Market Insight website ( contains information all of the companies in Table 4.6 except for Chubb and Weyerhaeuser. Update the calculations of PVGO as a percentage of stock price. For simplicity use the costs of equity given in Table 4.6. You will need to track down an updated forecast of EPS, for example from MSN money ( of Yahoo (

Compost Science, Inc. (CSI), is in the business of converting Boston's sewage sludge into fertilizer. The business is not in itself very profitable. However, to induce CSI to remain in business, the Metropolitan District Commission (MDC) has agreed to pay whatever amount is necessary to yield CSI a 10 percent book return on equity. At the end of the year CSI is expected to pay a $4 dividend. It has been reinvesting 40 percent of earnings and growing at 4 percent a year.

a. Suppose CSI continues on this growth trend. What is the expected long-run rate of return from purchasing the stock at $100? What part of the $100 price is attributable to the present value of growth opportunities?

b. Now the MDC announces a plan for CSI to treat Cambridge sewage. CSI's plant will, therefore, be expanded gradually over five years. This means that CSI will have to reinvest 80 percent of its earnings for five years. Starting in year 6, however, it will again be able to pay out 60 percent of earnings. What will be CSI's stock price once this announcement is made and its consequences for CSI are known?

List at least four different formulas for calculating PV(horizon value) in a two-stage DCF valuation of a business. For each formula, describe a situation where that formula would be the best choice. Look again at Table 4.7.

a. How do free cash flow and present value change if asset growth rate is only 15 percent in years 1 to 5? If value declines, explain why.

b. Suppose the business is a publicly traded company with one million shares outstanding. Then the company issues new stock to cover the present value of negative free cash flow for years 1 to 6. How many shares will be issued and at what price?

c. Value the company's one million existing shares by the two methods described in Section 4.5.

Icarus Air has one million shares outstanding and expects to earn a constant $10 million per year on its existing assets. All earnings will be paid out as dividends. Suppose that next year Icarus plans to double in size by issuing an additional one million shares at $100 a share. Everything will be the same as before but twice as big. Thus from year 2 onward the company earns a constant $20 million, all of which is paid out as dividends on the 20 million shares. What is the value of the company? What is the value of each existing Icarus Air share?

23. Look one more time at Table 4.1, which applies the DCF stock valuation formula to Fledgling Electronics. The CEO, having just learned that stock value is the present value of future dividends, proposes that Fledgling pay a bumper dividend of $15 a share in period 1. The extra cash would have to be raised by an issue of new shares. Recalculate Table 4.1 assuming that profits and payout ratios in all subsequent years are unchanged. You should find that the total present value of dividends per existing share is unchanged at $100. Why?

1. Look again at Tables 4.3 (Growth-Tech) and 4.7 (Concatenator Manufacturing). Note the discontinuous increases in dividends and free cash flow when asset growth slows down. Now look at your answer to Practice Question 11: Dividends are expected to grow smoothly, although at a lower rate after year 3. Is there an error or hidden inconsistency in Practice Question 11? Write down a general rule or procedure for deciding how to forecast dividends or free cash flow.

2. The constant-growth DCF formula



is sometimes written as

where BVPS is book equity value per share, b is the plowback ratio, and ROE is the ratio of earnings per share to BVPS. Use this equation to show how the price-to-book ratio varies as ROE changes. What is price-to-book when ROE = r?

3. Portfolio managers are frequently paid a proportion of the funds under management. Suppose you manage a $100 million equity portfolio offering a dividend yield (DIVj / P0) of 5 percent. Dividends and portfolio value are expected to grow at a constant rate. Your annual fee for managing this portfolio is .5 percent of portfolio value and is calculated at the end of each year. Assuming that you will continue to manage the portfolio from now to eternity, what is the present value of the management contract?


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  • Kimi
    What does return on equity of 161x mean?
    8 years ago
  • fallon
    How many shares of stock does reeby sports have to issue to raise 4.3?
    8 years ago
  • hessu
    What part of p3 reflects the present value of growth opportunities (pvgo) after?
    8 years ago
    What part of p3 reflects the pvgo after year 3?
    8 years ago
  • caramella
    What part of that value reflects the discounted value of p3, the price forecastedfor year 3?
    8 years ago
  • keri
    Has pulled off a miraculous recovery. four years ago, it was near?
    8 years ago
  • kaleb
    Has just announced a $1 per share dividend, the first since the crisis hit?
    8 years ago
  • Italia
    What part of that value reflects the discounted value of p3, the price forecasted for year 3 .?
    8 years ago
  • marko
    What part of p3 reflects the present value of growth opportunities (pvgo) after year 3?
    8 years ago
  • liya
    When r equal to eps1/p0?
    7 years ago
  • Fernanda
    When does a stock's market capitalization rate equal its earningsprice ratio?
    7 years ago

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