Bonds
The price, or current value, of a bond is equal to the present value of interest payments (It) over the life of the bond plus the present value of the principal payment (Pn) at maturity. The discount rate used in the analytical process is the yield to maturity (Y). The yield to maturity (required rate of return) is determined in the marketplace by such factors as the real rate of return, an inflation premium, and a risk premium.
The equation for bond valuation was presented as Formula 10-1.
The actual terms in the equation are solved by the use of present value tables. We say the present value of interest payments is:
The present value of the principal payment at maturity is:
We add these two values together to determine the price of the bond. We use annual or semiannual analysis.
FINANCIAL CALCULATOR | |
Bond Price | |
Value | Function |
n | N |
Y | I/Y |
Pn | FV |
It | PMT |
Function | Solution |
CPT | |
PV | Pb |
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The value of the bond will be strongly influenced by the relationship of the yield to maturity in the market to the interest rate on the bond and also the length of time to maturity.
If you know the price of the bond, the size of the interest payments, and the maturity of the bond, you can solve for the yield to maturity through a trial and error approach, by an approximation approach, or by using financially oriented calculators (in Appendix 10B at the end of the chapter) or appropriate computer software.
Preferred Stock
In determining the value of preferred stock, we are taking the present value of an infinite stream of level dividend payments. This would be a tedious process if the mathematical calculations could not be compressed into a simple formula. The appropriate equation is Formula 10-3.
According to Formula 10-3, to find the preferred stock price (Pp) we take the constant annual dividend payment (Dp) and divide this value by the rate of return that preferred stockholders are demanding (Kp).
If, on the other hand, we know the price of the preferred stock and the constant annual dividend payment, we can solve for the required rate of return on preferred stock as:
Common Stock
The value of common stock is also based on the concept of the present value of an expected stream of future dividends. Unlike preferred stock, the dividends are not necessarily level. The firm and shareholders may experience:
1. No growth in dividends.
2. Constant growth in dividends.
3. Variable or supernormal growth in dividends.
It is the second circumstance that receives most of the attention in the financial literature. If a firm has constant growth (g) in dividends (D) and the required rate of return (Ke) exceeds the growth rate, Formula 10-8 can be utilized.
In using Formula 10-8, all we need to know is the value of the dividend at the end of the first year, the required rate of return, and the discount rate. Most of our valuation calculations with common stock utilize Formula 10-8.
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If we need to know the required rate of return (Ke) for common stock, Formula 10-9 can be employed.
The first term represents the dividend yield on the stock and the second term the growth rate. Together they provide the total return demanded by the investor.
LIST OF TERMS
required rate of return 296
yield to maturity 299
real rate of return 299
inflation premium 299
risk-free rate of return 300
risk premium 300
business risk 300
financial risk 300
perpetuity 307
dividend valuation model 310
dividend yield 314
price-earnings ratio 314
supernormal growth 315
DISCUSSION QUESTIONS
1. How is valuation of any financial asset related to future cash flows? (LO10-2)
2. Why might investors demand a lower rate of return for an investment in Microsoft as compared to United Airlines? (LO10-2)
3. What are the three factors that influence the required rate of return by investors? (LO10-2)
4. If inflationary expectations increase, what is likely to happen to the yield to maturity on bonds in the marketplace? What is also likely to happen to the price of bonds? (LO10-2)
5. Why is the remaining time to maturity an important factor in evaluating the impact of a change in yield to maturity on bond prices? (LO10-4)
6. What are the three adjustments that have to be made in going from annual to semiannual bond analysis? (LO10-4)
7. Why is a change in required yield for preferred stock likely to have a greater impact on price than a change in required yield for bonds? (LO10-4)
8. What type of dividend pattern for common stock is similar to the dividend payment for preferred stock? (LO10-1)
9. What two conditions must be met to go from Formula 10-7 to Formula 10-8 in using the dividend valuation model? (LO10-5)
10. What two components make up the required rate of return on common stock? (LO10-5)
11. What factors might influence a firm’s price-earnings ratio? (LO10-3)
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12. How is the supernormal growth pattern likely to vary from the normal, constant growth pattern? (LO10-5)
13. What approaches can be taken in valuing a firm’s stock when there is no cash dividend payment? (LO10-5)
PRACTICE PROBLEMS AND SOLUTIONS
Bond value
(LO10-3)
1. The Titan Corp. issued a $1,000 par value bond paying 8 percent interest with 15 years to maturity. Assume the current yield to maturity on such bonds is 10 percent. What is the price of the bond? Do annual analysis.
Common stock value
(LO10-5)
2. Host Corp. will pay a $2.40 dividend (D1) in the next 12 months. The required rate of return (Ke) is 13 percent and the constant growth rate (g) is 5 percent.
a. Compute the stock price (P0).
b. If Ke goes up to 15 percent, and all else remains the same, what will be the stock price (P0)?
c. Now assume in the next year, D1 = $2.70, Ke = 12 percent, and g is equal to 6 percent. What is the price of the stock?
Solutions
1. Present Value of Interest Payments
Present Value of the Principal Payment at Maturity
Total Present Value (Bond Price)
Present value of interest payments | $608.49 |
Present value of principal payment at maturity | 237.39 |
Bond price | $847.88 |
FINANCIAL CALCULATOR | |
Bond Price | |
Value | Function |
15 | N |
10% | I/V |
1000 | FV |
80 | PMT |
Function | Solution |
CPT | |
PV | –847.88 |
2.
a.
b.
c.
PROBLEMS
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Selected problems are available with Connect. Please see the preface for more information.
Basic Problems
For the first 20 bond problems, assume interest payments are on an annual basis.
Bond value
(LO10-3)
1. The Lone Star Company has $1,000 par value bonds outstanding at 10 percent interest. The bonds will mature in 20 years. Compute the current price of the bonds if the present yield to maturity is
a. 6 percent.
b. 9 percent.
c. 13 percent.
Bond value
(LO10-3)
2. Midland Oil has $1,000 par value bonds outstanding at 8 percent interest. The bonds will mature in 25 years. Compute the current price of the bonds if the present yield to maturity is
a. 7 percent.
b. 10 percent.
c. 13 percent.
Bond value
(LO10-3)
3. Exodus Limousine Company has $1,000 par value bonds outstanding at 10 percent interest. The bonds will mature in 50 years. Compute the current price of the bonds if the percent yield to maturity is
a. 5 percent.
b. 15 percent.
Bond value
(LO10-3)
4. Barry’s Steroids Company has $1,000 par value bonds outstanding at 16 percent interest. The bonds will mature in 40 years. If the percent yield to maturity is 13 percent, what percent of the total bond value does the repayment of principal represent?
Bond value
(LO10-3)
5. Essex Biochemical Co. has a $1,000 par value bond outstanding that pays 15 percent annual interest. The current yield to maturity on such bonds in the market is 17 percent. Compute the price of the bonds for these maturity dates:
a. 30 years.
b. 20 years.
c. 4 years.
Bond value
(LO10-3)
6. Kilgore Natural Gas has a $1,000 par value bond outstanding that pays 9 percent annual interest. The current yield to maturity on such bonds in the market is 12 percent. Compute the price of the bonds for these maturity dates:
a. 30 years.
b. 15 years.
c. 1 year.
Bond maturity effect
(LO10-3)
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7. Toxaway Telephone Company has a $1,000 par value bond outstanding that pays 6 percent annual interest. If the yield to maturity is 8 percent, and remains so over the remaining life of the bond, the bond will have the following values over time:
Remaining Maturity | Bond Price |
15 | $795.67 |
10 | $830.49 |
5 | $891.86 |
1 | $973.21 |
Graph the relationship in a manner similar to the bottom half of Figure 10-2. Also explain why the pattern of price change takes place.
Interest rate effect
(LO10-3)
8. Go to Table 10-1, which is based on bonds paying 10 percent interest for 20 years. Assume interest rates in the market (yield to maturity) decline from 11 percent to 8 percent:
a. What is the bond price at 11 percent?
b. What is the bond price at 8 percent?
c. What would be your percentage return on investment if you bought when rates were 11 percent and sold when rates were 8 percent?
Interest rate effect
(LO10-3)
9. Look at Table 10-1 again, and now assume interest rates in the market (yield to maturity) increase from 9 to 12 percent.
a. What is the bond price at 9 percent?
b. What is the bond price at 12 percent?
c. What would be your percentage return on the investment if you bought when rates were 9 percent and sold when rates were 12 percent?
Interest rate effect
(LO10-3)
10. Using Table 10-1, assume interest rates in the market (yield to maturity) are 14 percent for 20 years on a bond paying 10 percent.
a. What is the price of the bond?
b. Assume five years have passed and interest rates in the market have gone down to 12 percent. Now, using Table 10-2 for 15 years, what is the price of the bond?
c. What would your percentage return be if you bought the bonds when interest rates in the market were 14 percent for 20 years and sold them 5 years later when interest rates were 12 percent?
Effect of maturity on bond price
(LO10-3)
11. Using Table 10-2:
a. Assume the interest rate in the market (yield to maturity) goes down to 8 percent for the 10 percent bonds. Using column 2, indicate what the bond price will be with a 10-year, a 15-year, and a 20-year time period.
b. Assume the interest rate in the market (yield to maturity) goes up to 12 percent for the 10 percent bonds. Using column 3, indicate what the bond price will be with a 10-year, a 15-year, and a 20-year period.
c. Based on the information in part a, if you think interest rates in the market are going down, which bond would you choose to own?
d. Based on information in part b, if you think interest rates in the market are going up, which bond would you choose to own?
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Intermediate Problems
Bond value
(LO10-3)
12. Jim Busby calls his broker to inquire about purchasing a bond of Disk Storage Systems. His broker quotes a price of $1,180. Jim is concerned that the bond might be overpriced based on the facts involved. The $1,000 par value bond pays 14 percent interest, and it has 25 years remaining until maturity. The current yield to maturity on similar bonds is 12 percent. Compute the new price of the bond and comment on whether you think it is overpriced in the marketplace.
Effect of yield to maturity on bond price
(LO10-3)
13. Tom Cruise Lines Inc. issued bonds five years ago at $1,000 per bond. These bonds had a 25-year life when issued and the annual interest payment was then 15 percent. This return was in line with the required returns by bondholders at that point as described next:
Real rate of return | 4% |
Inflation premium | 6 |
Risk premium | 5 |
Total return | 15% |
Assume that five years later the inflation premium is only 3 percent and is appropriately reflected in the required return (or yield to maturity) of the bonds. The bonds have 20 years remaining until maturity. Compute the new price of the bond.
Analyzing bond price changes
(LO10-3)
14. Katie Pairy Fruits Inc. has a $1,000 20-year bond outstanding with a nominal yield of 15 percent (coupon equals 15% × $1,000 = $150 per year). Assume that the current market required interest rate on similar bonds is now only 12 percent.
a. Compute the current price of the bond.
b. Find the present value of 3 percent × $1,000 (or $30) for 20 years at 12 percent. The $30 is assumed to be an annual payment. Add this value to $1,000.
c. Explain why the answers in parts a and b are basically the same. (There is a slight difference due to rounding in the tables.)
Effect of yield to maturity on bond price
(LO10-2 & 10-3)
15. Media Bias Inc. issued bonds 10 years ago at $1,000 per bond. These bonds had a 40-year life when issued and the annual interest payment was then 12 percent. This return was in line with the required returns by bondholders at that point in time as described next:
Real rate of return | 2% |
Inflation premium | 5 |
Risk premium | 5 |
Total return | 12% |
Assume that 10 years later, due to good publicity, the risk premium is now 2 percent and is appropriately reflected in the required return (or yield to maturity) of the bonds. The bonds have 30 years remaining until maturity. Compute the new price of the bond.
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Effect of yield to maturity on bond price
(LO10-2 & 10-3)
16. Wilson Oil Company issued bonds five years ago at $1,000 per bond. These bonds had a 25-year life when issued and the annual interest payment was then 15 percent. This return was in line with the required returns by bondholders at that point in time as described next:
Real rate of return | 8% |
Inflation premium | 3 |
Risk premium | 4 |
Total return | 15% |
Assume that 10 years later, due to bad publicity, the risk premium is now 7 percent and is appropriately reflected in the required return (or yield to maturity) of the bonds. The bonds have 15 years remaining until maturity. Compute the new price of the bond.
Deep discount bonds
(LO10-3)
17. Lance Whittingham IV specializes in buying deep discount bonds. These represent bonds that are trading at well below par value. He has his eye on a bond issued by the Leisure Time Corporation. The $1,000 par value bond pays 4 percent annual interest and has 18 years remaining to maturity. The current yield to maturity on similar bonds is 14 percent.
a. What is the current price of the bonds?
b. By what percent will the price of the bonds increase between now and maturity?
c. What is the annual compound rate of growth in the value of the bonds? (An approximate answer is acceptable.)
Yield to maturity—calculator or Excel required
(LO10-3)
18. Bonds issued by the Coleman Manufacturing Company have a par value of $1,000, which of course is also the amount of principal to be paid at maturity. The bonds are currently selling for $690. They have 10 years remaining to maturity. The annual interest payment is 13 percent ($130). Compute the yield to maturity.
Yield to maturity—calculator or Excel required
(LO10-3)
19. Stilley Resources bonds have four years left to maturity. Interest is paid annually, and the bonds have a $1,000 par value and a coupon rate of 5 percent. If the price of the bond is $841.51, what is the yield to maturity?
Yield to maturity—calculator or Excel required
(LO10-3)
20. Evans Emergency Response bonds have six years to maturity. Interest is paid semiannually. The bonds have a $1,000 par value and a coupon rate of 8 percent. If the price of the bond is $1,073.55, what is the annual yield to maturity?
For the next two problems, assume interest payments are on a semiannual basis.
Bond value––semiannual analysis
(LO10-3)
21. Heather Smith is considering a bond investment in Locklear Airlines. The $1,000 par value bonds have a quoted annual interest rate of 11 percent and the interest is paid semiannually. The yield to maturity on the bonds is 14 percent annual interest. There are seven years to maturity. Compute the price of the bonds based on semiannual analysis.
Bond value––semiannual analysis
(LO10-3)
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22. You are called in as a financial analyst to appraise the bonds of Olsen’s Clothing Stores. The $1,000 par value bonds have a quoted annual interest rate of 10 percent, which is paid semiannually. The yield to maturity on the bonds is 10 percent annual interest. There are 15 years to maturity.
a. Compute the price of the bonds based on semiannual analysis.
b. With 10 years to maturity, if yield to maturity goes down substantially to 8 percent, what will be the new price of the bonds?
Preferred stock value
(LO10-4)
23. The preferred stock of Denver Savings and Loan pays an annual dividend of $5.70. It has a required rate of return of 6 percent. Compute the price of the preferred stock.
Preferred stock value
(LO10-4)
24. North Pole Cruise Lines issued preferred stock many years ago. It carries a fixed dividend of $6 per share. With the passage of time, yields have soared from the original 6 percent to 14 percent (yield is the same as required rate of return).
a. What was the original issue price?
b. What is the current value of this preferred stock?
c. If the yield on the Standard & Poor’s Preferred Stock Index declines, how will the price of the preferred stock be affected?
Preferred stock value
(LO10-4)
25. X-Tech Company issued preferred stock many years ago. It carries a fixed dividend of $12.00 per share. With the passage of time, yields have soared from the original 10 percent to 17 percent (yield is the same as required rate of return).
a. What was the original issue price?
b. What is the current value of this preferred stock?
c. If the yield on the Standard & Poor’s Preferred Stock Index declines, how will the price of the preferred stock be affected?
Preferred stock rate of return
(LO10-4)
26. Analogue Technology has preferred stock outstanding that pays a $9 annual dividend. It has a price of $76. What is the required rate of return (yield) on the preferred stock?
All of the following problems pertain to the common stock section of the chapter.
Common stock value
(LO10-5)
27. Stagnant Iron and Steel currently pays a $12.25 annual cash dividend (D0). The company plans to maintain the dividend at this level for the foreseeable future as no future growth is anticipated. If the required rate of return by common stockholders (Ke) is 18 percent, what is the price of the common stock?
Common stock value
(LO10-5)
28. BioScience Inc. will pay a common stock dividend of $3.20 at the end of the year (D1). The required return on common stock (Ke) is 14 percent. The firm has a constant growth rate (g) of 9 percent. Compute the current price of the stock (P0).
Advanced Problems
Common stock value under different market conditions
(LO10-5)
29. Ecology Labs Inc. will pay a dividend of $6.40 per share in the next 12 months (D1). The required rate of return (Ke) is 14 percent and the constant growth rate is 5 percent.
a. Compute P0.
(For parts b, c, and d in this problem, all variables remain the same except the one specifically changed. Each question is independent of the others.)
b. Assume Ke, the required rate of return, goes up to 18 percent. What will be the new value of P0?
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c. Assume the growth rate (g) goes up to 9 percent. What will be the new value of P0? Ke goes back to its original value of 14 percent.
d. Assume D1 is $7.00. What will be the new value of P0? Assume Ke is at its original value of 14 percent and g goes back to its original value of 5 percent.
Common stock value under different market conditions
(LO10-5)
30. Maxwell Communications paid a dividend of $3 last year. Over the next 12 months, the dividend is expected to grow at 8 percent, which is the constant growth rate for the firm (g). The new dividend after 12 months will represent D1. The required rate of return (Ke) is 14 percent. Compute the price of the stock (P0).
Common stock value based on determining growth rate
(LO10-5)
31. Justin Cement Company has had the following pattern of earnings per share over the last five years:
Year | Earnings per Share |
20X1 | $5.00 |
20X2 | 5.30 |
20X3 | 5.62 |
20X4 | 5.96 |
20X5 | 6.32 |
The earnings per share have grown at a constant rate (on a rounded basis) and will continue to do so in the future. Dividends represent 40 percent of earnings. Project earnings and dividends for the next year (20X6).
If the required rate of return (Ke) is 13 percent, what is the anticipated stock price (P0) at the beginning of 20X6?
Common stock required rate of return
(LO10-5)
32. A firm pays a $4.80 dividend at the end of year one (D1), has a stock price of $80, and a constant growth rate (g) of 5 percent. Compute the required rate of return (Ke).
Common stock required rate of return
(LO10-5)
33. A firm pays a $1.50 dividend at the end of year one (D1), has a stock price of $155 (P0), and a constant growth rate (g) of 10 percent.
a. Compute the required rate of return (Ke).
Indicate whether each of the following changes would make the required rate of return (Ke) go up or down. (Each question is separate from the others. That is, assume only one variable changes at a time.) No actual numbers are necessary.
b. The dividend payment increases.
c. The expected growth rate increases.
d. The stock price increases.
Common stock value based on PV calculations
(LO10-5)
34. Trump Office Supplies paid a $3 dividend last year. The dividend is expected to grow at a constant rate of 7 percent over the next four years. The required rate of return is 14 percent (this will also serve as the discount rate in this problem). Round all values to three places to the right of the decimal point where appropriate.
a. Compute the anticipated value of the dividends for the next four years. That is, compute D1, D2, D3, and D4—for example, D1 is $3.21 ($3.00 × 1.07).
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b. Discount each of these dividends back to present at a discount rate of 14 percent and then sum them.
c. Compute the price of the stock at the end of the fourth year (P4).
(D5 is equal to D4 times 1.07)
d. After you have computed P4, discount it back to the present at a discount rate of 14 percent for four years.
e. Add together the answers in part b and part d to get P0, the current value of the stock. This answer represents the present value of the four periods of dividends, plus the present value of the price of the stock after four periods (which, in turn, represents the value of all future dividends).
f. Use Formula 10-8 to show that it will provide approximately the same answer as part e.
For Formula 10-8, use D1 = $3.21, Ke = 14 percent, and g = 7 percent. (The slight difference between the answers to part e and part f is due to rounding.)
g. If current EPS were equal to $5.32 and the P/E ratio is 1.1 times higher than the industry average of 8, what would the stock price be?
h. By what dollar amount is the stock price in part g different from the stock price in part f?
i. In regard to the stock price in part f, indicate which direction it would move if (1) D1 increases, (2) Ke increases, and (3) g increases.
Common stock value based on PV calculations
(LO10-5)
35. Beasley Ball Bearings paid a $4 dividend last year. The dividend is expected to grow at a constant rate of 2 percent over the next four years. The required rate of return is 15 percent (this will also serve as the discount rate in this problem). Round all values to three places to the right of the decimal point where appropriate.
a. Compute the anticipated value of the dividends for the next four years. That is, compute D1, D2, D3, and D4; for example, D1 is $4.08 ($4 × 1.02).
b. Discount each of these dividends back to present at a discount rate of 15 percent and then sum them.
c. Compute the price of the stock at the end of the fourth year (P4).
(D5 is equal to D4 times 1.02)
d. After you have computed P4, discount it back to the present at a discount rate of 15 percent for four years.
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e. Add together the answers in part b and part d to get P0, the current value of the stock. This answer represents the present value of the four periods of dividends, plus the present value of the price of the stock after four periods (which in turn represents the value of all future dividends).
f. Use Formula 10-8 to show that it will provide approximately the same answer as part e.
For Formula 10-8, use D1 = $4.08, Ke = 15 percent, and g = 2 percent. (The slight difference between the answers to part e and part f is due to rounding.)
g. If current EPS were equal to $4.98 and the P/E ratio is 1.2 times higher than the industry average of 6, what would the stock price be?
h. By what dollar amount is the stock price in part g different from the stock price in part f?
i. In regard to the stock price in part f, indicate which direction it would move if (1) D1 increases, (2) Ke increases, and (3) g increases.
COMPREHENSIVE PROBLEM
Preston Products
(Dividend valuation model, P/E ratio)
(LO10-5)
Mel Thomas, the chief financial officer of Preston Resources, has been asked to do an evaluation of Dunning Chemical Company by the president and chair of the board, Sarah Reynolds. Preston Resources was planning a joint venture with Dunning (which was privately traded), and Sarah and Mel needed a better feel for what Dunning’s stock was worth because they might be interested in buying the firm in the future.
Dunning Chemical paid a dividend at the end of year one of $1.30, the anticipated growth rate was 10 percent, and the required rate of return was 14 percent.
a. What is the value of the stock based on the dividend valuation model (Formula 10-8)?
b. Indicate that the value you computed in part a is correct by showing the value of D1, D2, and D3 and by discounting each back to the present at 14 percent. D1 is $1.30, and it increases by 10 percent (g) each year. Also discount the anticipated stock price at the end of year three back to the present and add it to the present value of the three dividend payments.
The value of the stock at the end of year three is:
If you have done all these steps correctly, you should get an answer approximately equal to the answer in part a.
c. As an alternative measure, you also examine the value of the firm based on the price-earnings (P/E) ratio times earnings per share.Page 329
Since the company is privately traded (not in the public stock market), you will get your anticipated P/E ratio by taking the average value of five publicly traded chemical companies. The P/E ratios were as follows during the time period under analysis:
P/E Ratio | |
Dow Chemical | 15 |
DuPont | 18 |
Georgia Gulf | 7 |
3M | 19 |
Olin Corp | 21 |
Assume Dunning Chemical has earnings per share of $2.10. What is the stock value based on the P/E ratio approach? Multiply the average P/E ratio you computed times earnings per share. How does this value compare to the dividend valuation model values that you computed in parts a and b?
d. If in computing the industry average P/E, you decide to weight Olin Corp. by 40 percent and the other four firms by 15 percent, what would be the new weighted average industry P/E? (Note: You decided to weight Olin Corp. more heavily because it is similar to Dunning Chemical.) What will the new stock price be? Earnings per share will stay at $2.10.
e. By what percent will the stock price change as a result of using the weighted average industry P/E ratio in part d as opposed to that in part c?
WEB EXERCISE
1. ExxonMobil was referred to at the beginning of the chapter as a firm that had a low valuation in the marketplace. Go to finance.yahoo.com and type XOM into the “Get Quotes” box.
Click on “Profile” in the left margin of the home page and write a one–paragraph description of the company’s activities. Return to the summary page and write down the company’s P/E ratio. Is it still relatively low (under 15)? Click on “Competitors” and compare ExxonMobil to others in the industry based on the P/E ratio and the PEG ratio (the P/E ratio divided by annual growth).
2. Go back to the summary page. Is the stock up or down from the prior day? (See the number in parentheses next to the share price.)
3. What is its 52-week range?
4. Scroll down and click on “Analyst Opinion.” What are the Mean Target, the High Target, and the Low Target? How many brokers follow the firm?
Note: Occasionally a topic we have listed may have been deleted, updated, or moved into a different location on a website. If you click on the site map or site index, you will be introduced to a table of contents that should aid you in finding the topic you are looking for.
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APPENDIX | 10A
Valuation of a Supernormal Growth Firm
The equation for the valuation of a supernormal growth firm is:
The formula is not difficult to use. The first term calls for determining the present value of the dividends during the supernormal growth period. The second term calls for computing the present value of the future stock price as determined at the end of the supernormal growth period. If we add the two, we arrive at the current stock price. We are adding together the present value of the two benefits the stockholder will receive: a future stream of dividends during the supernormal growth period and the future stock price.
Let’s assume the firm paid a dividend over the last 12 months of $1.67; this represents the current dividend rate. Dividends are expected to grow by 20 percent per year over the supernormal growth period (n) of three years. They will then grow at a normal constant growth rate (g) of 5 percent. The required rate of return (discount rate) as represented by Ke is 9 percent. We first find the present value of the dividends during the supernormal growth period.
1. Present Value of Supernormal Dividends
D0 = | $1.67. We allow this value to grow at 20 percent per year over the three years of supernormal growth. |
D1 = | D0 (1 + 0.20) = $1.67(1.20) = $2.00 |
D2 = | D1 (1 + 0.20) = $2.00(1.20) = $2.40 |
D3 = | D2 (1 + 0.20) = $2.40(1.20) = $2.88 |
We then discount these values back at 9 percent to find the present value of dividends during the supernormal growth period.
The present value of the supernormal dividends is $6.07. We now turn to the future stock price.
2. Present Value of Future Stock Price
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We first find the future stock price at the end of the supernormal growth period. This is found by taking the present value of the dividends that will be growing at a normal, constant rate after the supernormal period. This will begin after the third (and last) period of supernormal growth.
Since after the supernormal growth period the firm is growing at a normal, constant rate (g = 5 percent) and Ke (the discount rate) of 9 percent exceeds the new, constant growth rate of 5 percent, we have fulfilled the two conditions for using the constant dividend growth model after three years. That is, we can apply Formula 10-8 (without subscripts for now).
In this case, however, D is really the dividend at the end of the fourth period because this phase of the analysis starts at the beginning of the fourth period and D is supposed to fall at the end of the first period of analysis in the formula. Also the price we are solving for now is the price at the beginning of the fourth period, which is the same concept as the price at the end of the third period (P3).
We thus say:
D4 is equal to the previously determined value for D3 of $2.88 compounded for one period at the constant growth rate of 5 percent.
D4 = $2.88(1.05) = $3.02
Also:
Ke = 0.09 discount rate (required rate of return)
g = 0.05 constant growth rate
This is the value of the stock at the end of the third period. We discount this value back to the present.
Stock Price after Three Years | Discount Rate * Ke = 9% | Present Value of Future Price |
$75.50 | 0.772 | $58.29 |
* Note: n is equal to 3. |
The present value of the future stock price (P3) of $75.50 is $58.29.
By adding together the answers in parts (1) and (2) of this appendix, we arrive at the total present value, or price, of the supernormal growth stock.
(1) Present value of dividends during the normal growth period | $ 6.07 |
(2) Present value of the future stock price | 58.29 |
Total present value, or price | $64.36 |
The process is also illustrated in Figure 10A-1.
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Figure 10A-1 Stock valuation under supernormal growth analysis
Problem
Valuation of supernormal growth firm
(LO10-5)
10A-1. Surgical Supplies Corporation paid a dividend of $1.12 per share over the last 12 months. The dividend is expected to grow at a rate of 25 percent over the next three years (supernormal growth). It will then grow at a normal, constant rate of 7 percent for the foreseeable future. The required rate of return is 12 percent (this will also serve as the discount rate).
a. Compute the anticipated value of the dividends for the next three years (D1, D2, and D3).
b. Discount each of these dividends back to the present at a discount rate of 12 percent and then sum them.
c. Compute the price of the stock at the end of the third year (P3).
d. After you have computed P3, discount it back to the present at a discount rate of 12 percent for three years.
e. Add together the answers in part b and part d to get the current value of the stock. (This answer represents the present value of the first three periods of dividends plus the present value of the price of the stock after three periods.)