Case study

10

Valuation and Rates of Return

LEARNING OBJECTIVES

LO 10-1	The valuation of a financial asset is based on the present value of future cash flows.
LO 10-2	The required rate of return in valuing an asset is based on the risk involved.
LO 10-3	Bond valuation is based on the process of determining the present value of interest payments plus the present value of the principal payment at maturity.
LO 10-4	Preferred stock valuation is based on the dividend paid and the market required return.
LO 10-5	Stock valuation is based on determining the present value of the future benefits of equity ownership.

Valuation appears to be a fickle process to stockholders of some corporations. For example, if you held The Coca-Cola Company common stock in February 2015, you would be pleased to see that stockholders were valuing your stock at 26 times earnings. Certainly there was some justification for such a high valuation. Coca-Cola is sold in more than 200 countries, and it is the best-known brand in the world.

But keep in mind that the company’s earnings were only 5 percent higher in 2014 than they had been in 2009, although dividends were up 50 percent from $0.88 to $1.32 per share.

If stockholders of Coca-Cola were happy with the firm’s strong P/E (price-earnings ratio) valuation in February 2015, those who invested in ExxonMobil were not. The corporate giant was trading at a P/E ratio of 11 even though both its earnings and dividends had grown more rapidly than Coke’s (69 percent and 64 percent, respectively). Keep in mind ExxonMobil is one of the largest companies in the world in terms of revenue (almost $400 billion per year). It is a dominant player among integrated oil companies (those that not only discover oil but also sell it at the retail level). Furthermore, the stock has outperformed the popular market averages over the last 10-, 20-, and 30-year time periods. Thus ExxonMobil stockholders probably were not pleased with a low P/E ratio of 11.

The question then becomes, why are P/E ratios so different and why do they change so much? Informed investors care about their money and vote with their dollars. The factors that influence valuation are many and varied, and you will be exposed to many of them in this chapter.

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In  Chapter 9 , we considered the basic principles of the time value of money. In this chapter, we will use many of those concepts to determine how financial assets (bonds, preferred stock, and common stock) are valued and how investors establish the rates of return they demand. In the next chapter, we will use material from this chapter to determine the overall cost of financing to the firm. We merely turn the coin over. Once we know how much bondholders and stockholders demand in rates of return, we will observe what the corporation is required to pay them to attract their funds. The cost of corporate financing (capital) is subsequently used in analyzing whether a project is acceptable for investment. These relationships are depicted in  Figure 10-1 .

Figure 10-1 The relationship between time value of money, required return, cost of financing, and investment decisions

Valuation Concepts

The valuation of a financial asset is based on determining the present value of future cash flows. Thus we need to know the value of future cash flows and the discount rate to be applied to the future cash flows to determine the current value.

The market-determined required rate of return, which is the discount rate, depends on the market’s perceived level of risk associated with the individual security. Also important is the idea that required rates of return are competitively determined among the many companies seeking financial capital. For example, Microsoft, due to its low financial risk, relatively high return, and strong market position, is likely to raise debt capital at a significantly lower cost than can United Airlines, a firm with high financial risk. This implies that investors are willing to accept low return for low risk, and vice versa. The market allocates capital to companies based on risk, efficiency, and expected returns—which are based to a large degree on past performance. The reward to the financial manager for efficient use of capital in the past is a lower required return for investors than that of competing companies that did not manage their financial resources as well.

Throughout the balance of this chapter, we apply concepts of valuation to corporate bonds, preferred stock, and common stock. Although we describe the basic characteristics of each form of security as part of the valuation process, extended discussion of each security is deferred until later chapters.

Valuation of Bonds

As previously stated, the value of a financial asset is based on the concept of the present value of future cash flows. Let’s apply this approach to bond valuation. A bond provides an annuity stream of interest payments and a $1,000 principal payment at maturity. 1  These cash flows are discounted at Y, the yield to maturity. The value of Y is determined in the bond market and represents the required rate of return for bonds of a given risk and maturity. More will be said about the concept of yield to maturity in the next section.

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The price of a bond is thus equal to the present value of regular interest payments discounted by the yield to maturity added to the present value of the principal (also discounted by the yield to maturity).

The following timeline depicts a bond’s cashflows:

This relationship can be expressed mathematically as follows:

where

Pb =	Price of the bond
It =	Interest payments
Pn =	Principal payment at maturity
t =	Number corresponding to a period; running from 1 to n
n =	Number of periods
Y =	Yield to maturity (or required rate of return)

The first term in the equation says to take the sum of the present values of the interest payments (It); the second term directs you to take the present value of the principal payment at maturity (Pn). The discount rate used throughout the analysis is the yield to maturity (Y). The answer derived is referred to as Pb (the price of the bond). The analysis is carried out for n periods.

Let’s look at an example:

In this timeline, each interest payment (It) equals $100; Pn (principal payment at maturity) equals $1,000; Y (yield to maturity) is 10 percent; and n (total number of periods) equals 20. We could say that Pb (the price of the bond) equals:

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We take the present value of the interest payments and then add this value to the present value of the principal payment at maturity.

Present Value of Interest Payments In this case, we determine the present value of a $100 annuity for 20 years. 2  The discount rate is 10 percent. We can use  Formula 9-6  to find the following:

Present Value of Principal Payment (Par Value) at Maturity This single value of $1,000 will be received after 20 years. Note the term principal payment at maturity is used interchangeably with par value or face value of the bond. We discount $1,000 back to the present at 10 percent.

We can use  Formula 9-2  to find the following:

The current price of the bond, based on the present value of interest payments and the present value of the principal payment at maturity, is $1,000.

Present value of interest payments	$   851.40
Present value of principal payment at maturity	148.64
Total present value, or price, of the bond	$1,000.00

The price of the bond in this case is essentially the same as its par, or stated, value to be received at maturity of $1,000. This is because the annual interest rate is 10 percent (the annual interest payment of $100 divided by $1,000) and the yield to maturity, or discount rate, is also 10 percent. When the interest rate on the bond and the yield to maturity are equal, the bond will trade at par value. Later we will examine the mathematical effects of varying the yield to maturity above or below the interest rate on the bond.

Bond Valuation Using a Financial Calculator

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Bond values can be found using the PV function on a financial calculator. The first calculator solution box in the margin shows the present value of the twenty $100 coupon payments. The calculator keystrokes are identical to those used to find the present value of an annuity. Notice that the PMT value is entered as a negative number. As we found earlier, the present value of the coupon payment annuity stream is $851.36.

The second calculator solution shows the present value of the $1,000 principal payment that will be received at the end of 20 years. The calculator keystrokes are identical to those used to find the present value of a single amount. If the FV value is entered as a negative number, the present value of the principal will be $148.64. This is the value found using the present value equation earlier. Of course, the value of the bond ($1,000) is the sum of the present values in Panel A and Panel B.

Finally, the third calculator solution demonstrates how we calculate the bond value when entering the principal and coupon payments simultaneously. Again, the coupon amounts are entered as a negative value using the PMT key, and the principal is entered as a negative value using the FV key. The value of the bond is $1,000.

FINANCIAL CALCULATOR
PV of Interest Payments
Value	Function
20	N
10	I/Y
0	FV
−100	PMT
Function	Solution
CPT
PV	851.36

Using Excel’s PV Function to Calculate a Bond Price

FINANCIAL CALCULATOR
PV of Principal
Value	Function
20	N
10	I/Y
−1000	FV
0	PMT
Function	Solution
CPT
PV	148.64

Excel’s PV function can calculate the price of a bond. In order to produce a positive bond price, the coupon payment annuity amount is input as a negative value for the pmt argument. The principal payment (−1000) is entered as the fv argument. The function in cell D1 references the arguments in cells B1 to B4. The function in cell D5 uses hardcoded numerical values. In both cases, the bond values produced by the PV function are identical to the calculator solution.

FINANCIAL CALCULATOR
Bond Price
Value	Function
20	N
10	I/V
−1000	FV
−100	PMT
Function	Solution
CPT
PV	1000.00

Concept of Yield to Maturity

In the previous example, the yield to maturity that was used as the discount rate was 10 percent. The yield to maturity, or discount rate, is the rate of return required by bondholders. The bondholder, or any investor for that matter, will allow three factors to influence his or her required rate of return:

1. The required real rate of return—This is the rate of return the investor demands for giving up the current use of the funds on a noninflation-adjusted basis. It is the financial “rent” the investor charges for using his or her funds for one year, five years, or any given period. Although it varies from time to time, historically the real rate of return demanded by investors has been about 2 to 3 percent.

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2. Inflation premium—In addition to the real rate of return discussed above, the investor requires a premium to compensate for the eroding effect of inflation on the value of the dollar. It would hardly satisfy an investor to have a 3 percent total rate of return in a 5 percent inflationary economy. Under such circumstances, the lender (investor) would be paying the borrower 2 percent for use of the funds, or in other words, losing 2 percent in purchasing power. This would represent an irrational action. No one wishes to pay another party to use his or her fund. The inflation premium added to the real rate of return ensures that this will not happen. The size of the inflation premium will be based on the investor’s expectations about future inflation. In the last two decades, the inflation premium has been 2 to 4 percent. In the late 1970s, it was in excess of 10 percent.

If one combines the real rate of return (part 1) and the inflation premium (part 2), the risk-free rate of return is determined. This is the rate that compensates the investor for the current use of his or her funds and for the loss in purchasing power due to inflation, but not for taking risks. As an example, if the real rate of return were 3 percent and the inflation premium were 4 percent, we would say the risk-free rate of return is 7 percent. 3

3. Risk premium—We must now add the risk premium to the risk-free rate of return. This is a premium associated with the special risks of a given investment. Of primary interest to us are two types of risk: business risk and financial risk. Business risk relates to the inability of the firm to hold its competitive position and maintain stability and growth in its earnings. Financial risk relates to the inability of the firm to meet its debt obligations as they come due. In addition to the two forms of risk mentioned above, the risk premium will be greater or less for different types of investments. For example, because bonds possess a contractual obligation for the firm to pay interest to bondholders, they are considered less risky than common stock where no such obligation exists. 4

The risk premium of an investment may range from as low as zero on a very-short-term U.S. government–backed security to 10 to 15 percent on a gold mining expedition. The typical risk premium is 2 to 6 percent. Just as the required real rate of return and the inflation premium change over time, so does the risk premium. For example, high-risk corporate bonds (sometimes referred to as junk bonds) normally require a risk premium of about 5 percentage points over the risk-free rate. However, in September 1989 the bottom fell out of the junk bond market as Campeau Corp., International Resources, and Resorts International began facing difficulties in making their payments. Risk premiums almost doubled. The same phenomenon took place in the fall of 2008 in reaction to the U.S. financial crisis and in the spring of 2010 in reaction to the debt crisis in Greece, Portugal, Ireland, Italy, and Spain. As is emphasized in many parts of the text, there is a strong correlation between the risk the investor is taking and the return the investor demands. Supposedly, in finance as in other parts of business, “There is no such thing as a free lunch.” As you take more risk hoping for higher returns, you also expose yourself to the possibility of lower or negative returns on the other end of the probability curve.

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We shall assume that in the investment we are examining the risk premium is 3 percent. If we add this risk premium to the two components of the risk-free rate of return developed in parts 1 and 2, we arrive at an overall required rate of return of 10 percent.

+ Real rate of return	3%
+ Inflation premium	4
= Risk-free rate	7%
+ Risk premium	3
= Required rate of return	10%

In this instance, we assume we are evaluating the required return on a bond issued by a firm. If the security had been the common stock of the same firm, the risk premium might be 5 to 6 percent and the required rate of return 12 to 13 percent.

Finally, in concluding this section, you should recall that the required rate of return on a bond is effectively the same concept as required yield to maturity.

Changing the Yield to Maturity and the Impact on Bond Valuation

In the earlier bond value calculation, we assumed the interest rate was 10 percent ($100 annual interest on a $1,000 par value bond) and the yield to maturity was also 10 percent. Under those circumstances, the price of the bond was basically equal to par value. Now let’s assume conditions in the market cause the yield to maturity to change.

Increase in Inflation Premium For example, assume the inflation premium goes up from 4 to 6 percent. All else remains constant. The required rate of return would now be 12 percent.

+ Real rate of return	3%
+ Inflation premium	6
= Risk-free rate	9%
+ Risk premium	3
= Required rate of return	12%

With the required rate of return, or yield to maturity, now at 12 percent, the price of the bond will change. 5  A bond that pays only 10 percent interest when the required rate of return (yield to maturity) is 12 percent will fall below its current value of approximately $1,000. The new price of the bond is $850.61.

We can calculate the bond price by using the calculator keystrokes shown in the margin or by using the time-value equations as follows:

FINANCIAL CALCULATOR
Bond Price
Value	Function
20	N
12	I/V
1000	FV
100	PMT
Function	Solution
CPT
PV	–850.61

Present Value of Interest Payments  We take the present value of a $100 annuity for 20 years. The discount rate is 12 percent.

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Present Value of Principal Payment at Maturity  We take the present value of $1,000 after 20 years. The discount rate is 12 percent.

Total Present Value

Present value of interest payments	$746.94
Present value of principal payment at maturity	103.67
Total present value, or price, of the bond	$850.61

In this example, we assumed increasing inflation caused the required rate of return (yield to maturity) to go up and the bond price to fall by approximately $150. The same effect would occur if the business risk increased or the demanded level for the real rate of return became higher.

Decrease in Inflation Premium The opposite effect would happen if the required rate of return went down because of lower inflation, less risk, or other factors. Let’s assume the inflation premium declines and the required rate of return (yield to maturity) goes down to 8 percent.

The 20-year bond with the 10 percent interest rate (coupon rate) would now sell for $1,196.36 as shown in the calculator keystrokes in the margin or using the following calculations:

FINANCIAL CALCULATOR
Bond Price
Value	Function
20	N
8	I/V
1000	FV
100	PMT
Function	Solution
CPT
PV	–1196.36

Present Value of Interest Payments

Present Value of Principal Payment at Maturity

Total Present Value

Present value of interest payments	$   981.81
Present value of principal payment at maturity	214.55
Total present value, or price, of the bond	$1,196.36

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The bond is now trading at $196.36 over par value. This is certainly the expected result because the bond is paying 10 percent interest when the yield required in the market is only 8 percent. The 2 percentage point differential on a $1,000 par value bond represents $20 per year. The investor will receive this differential for the next 20 years. The present value of $20 for the next 20 years at the current market rate of interest of 8 percent is approximately $196.36. This explains why the bond is trading at $196.36 over its stated, or par, value.

The further the yield to maturity on a bond changes from the stated interest rate on the bond, the greater the price change effect will be. This is illustrated in  Table 10-1  for the 10 percent coupon rate, 20-year bonds discussed in this chapter.

Table 10-1 Bond price table

We clearly see the impact that different yields to maturity have on the price of a bond.

Time to Maturity

The impact of a change in yield to maturity on valuation is also affected by the remaining time to maturity. The effect of a bond paying 2 percentage points more or less than the going rate of interest is quite different for a 20-year bond than it is for a 1-year bond. In the latter case, the investor will only be gaining or giving up $20 for one year. That is certainly not the same as having this $20 differential for an extended period. Let’s once again return to the 10 percent interest rate bond and show the impact of a 2 percentage point decrease or increase in yield to maturity for varying times to maturity. The values are shown in  Table 10-2  and graphed in  Figure 10-2 . The upper part of  Figure 10-2  shows how the amount (premium) above par value is reduced as the number of years to maturity becomes smaller and smaller.  Figure 10-2  should be read from left to right. The lower part of the figure shows how the amount (discount) below par value is reduced with progressively fewer years to maturity. Clearly, the longer the maturity, the greater the impact of changes in yield.

Determining Yield to Maturity from the Bond Price

Until now we have used yield to maturity as well as other factors, such as the interest rate on the bond and number of years to maturity, to compute the price of the bond. We shall now assume we know the price of the bond, the interest rate on the bond, and the years to maturity, and we wish to determine the yield to maturity. Once we have computed this value, we have determined the rate of return that investors are demanding in the marketplace to provide for inflation, risk, and other factors.

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Table 10-2 Impact of time to maturity on bond prices

(10% Interest Payment, Various Times to Maturity)
Time Period in Years to Maturity	Bond Price with 8% Yield to Maturity	Bond Price with 12% Yield to Maturity
0	$1,000.00	$1,000.00
1	1,018.52	982.14
5	1,079.85	927.90
10	1,134.20	887.00
15	1,171.19	863.78
20	1,196.36	850.61
25	1,213.50	843.14
30	1,225.16	838.90

Figure 10-2 Relationship between time to maturity and bond price*

Let’s once again present  Formula 10-1 :

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We now determine the value of Y, the yield to maturity, that will equate the interest payments (It) and the principal payment (Pn) to the price of the bond (Pb).

Consider the following timeline for payments on a 15-year bond that pays $110 per year (11 percent of the face amount) in interest and $1,000 in principal repayment after 15 years. The current price of the bond is $931.89.

We wish to compute the yield to maturity, or discount rate, that equates future flows with the current price. It turns out that there is no algebraic formula that allows us to solve for the yield to maturity directly. Once upon a time, this presented a difficult puzzle that required tedious trial-and-error estimations to be checked before a solution could be found.

Fortunately, our tools have improved. Both Excel and financial calculators are able to do these calculations so rapidly that the user is frequently left unaware that they are using the same trial-and-error process that was once done by hand.

Let us start by reorganizing the timeline in an Excel spreadsheet as shown in  Table 10-3 . The Excel function RATE(n, pmt, pv, fv) shown at the bottom of the spreadsheet can also be used to find the yield to maturity, but the full spreadsheet has the advantage of making all the steps transparent to the reader. The spreadsheet also introduces Excel’s very flexible “Goal Seek” feature, which has many uses in addition to finding yields to maturity.

In the spreadsheet, the time (n) of each payment is shown in column B, and each payment amount is shown in column C. The last two payments are at time n = 15 when both the last coupon payment and the principal are paid. In column D, we see a “PV factor” that is used to find the present value of each payment. The general equation for each factor is shown in the first comment box that points to cell D2. The comment box pointing to cell D4 shows the actual Excel equation and syntax for that cell. Each of the PV factor cells references the discount rate in cell D$1, which is also the yield to maturity. The dollar sign in the cell ensures that each row in the D column is referencing cell D1. Column E shows the present value of each payment, and the sum of the present value of all these payments is shown in cell E20. This is the bond price. Once you have created the spreadsheet and entered the data and appropriate equations, you are ready to use “Goal Seek.”

The yield to maturity of Y = 12.00% is shown in red in cell D1. This cell was calculated using the “Goal Seek” function in Excel. Goal Seek is used when you know the result that you want for a formula, but you are not sure what input value the formula needs to get the result. In the case of the yield to maturity, we know the bond price should be $931.89, but we do not know the discount rate that produces that price.

The Goal Seek function can be found in the most recent version of Excel on the Data tab, in the Data Tools group, under What-If Analysis. See  Figure 10-3  for a picture of the Excel Ribbon location. Earlier versions of Excel also include Goal Seek, but the feature may be in a menu or toolbar instead of on the Excel Ribbon. The financial calculator keystrokes function much like Excel’s RATE(nper, pmt, pv,(fv)) function. These keystrokes are shown in the margin near  Table 10-3 .

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FINANCIAL CALCULATOR
Bond Yield
Value	Function
15	N
−931.89	PV
110	PMT
1000	FV
Function	Solution
CPT
I/V	12.00

Table 10-3 Excel functions for YTM

Figure 10-3 Finding the Goal Seek function in Excel

These are the steps used to find the yield to maturity:

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1. Make sure that you have calculated an arbitrary bond price by putting an interest rate in cell D1. Any rate should work, but using 11% is a good place to start because you already know that, at 11%, the price of the bond should be $1,000 since the coupon payment of $110 is 11% of the principal.

2. Open the Goal Seek feature.

3. In the “Set cell” box, enter the reference for the cell containing the formula for the bond price.

4. In the “To value” box, type the value 931.89, which is the price of the bond.

5. In the “By changing cell” box, enter the reference for the cell that contains the discount rate that you wish to find. This is cell D1 for this example.

6. Click “OK.”

Goal Seek runs and produces the result in cell D1: Y = 12%. The RATE (nper, pmt, pv,(fv)) function in cell A22 also produces a value of 12%.

Semiannual Interest and Bond Prices

We have been assuming that interest was paid annually in our bond analysis. In actuality, most bonds pay interest semiannually. Thus a 10 percent interest rate bond may actually pay $50 twice a year instead of $100 annually. To make the conversion from an annual to semiannual analysis, we follow three steps:

1. Divide the annual interest rate by 2.

2. Multiply the number of years by 2.

3. Divide the annual yield to maturity by 2.

FINANCIAL CALCULATOR
Bond Price
Value	Function
40	N
6	I/Y
50	PMT
1000	FV
Function	Solution
CPT
PV	–849.54

Assume a 10 percent, $1,000 par value bond has a maturity of 20 years. The annual yield to maturity is 12 percent. In following the three steps above, we would show this:

1. 10%/2 = 5% semiannual interest rate; therefore, 5% × $1,000 = $50 semiannual interest.

2. 20 × 2 = 40 periods to maturity.

3. 12%/2 = 6% yield to maturity, expressed on a semiannual basis.

The calculator solution for this problem is shown in the margin.

The answer of $849.54 is slightly below what we found previously for the same bond, assuming an annual interest rate ($850.61). This value was initially shown on  page 301 . In terms of accuracy, the semiannual analysis is a more acceptable method and is the method used in bond tables. As is true in many finance texts, we present the annual interest rate approach first for ease of presentation, and then the semiannual basis is given. In the problems at the back of the chapter, you will be asked to do problems on both an annual and semiannual interest payment basis.

Valuation and Preferred Stock

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Preferred stock usually represents a perpetuity or, in other words, has no maturity date. It is valued in the market without any principal payment since it has no ending life. If preferred stock had a maturity date, the analysis would be similar to that of the preceding bond example. Preferred stock has a fixed dividend payment carrying a higher order of precedence than common stock dividends, but not the binding contractual obligation of interest on debt. Preferred stock, being a hybrid security, has neither the ownership privilege of common stock nor the legally enforceable provisions of debt. To value a perpetuity such as preferred stock, we first consider this formula:

where

Pp =	the price of preferred stock
Dp =	the annual dividend for preferred stock (a constant value)
Kp =	the required rate of return, or discount rate, applied to preferred stock dividends

Notice that, unlike a bond, the preferred stock never matures. Because the dividend payments are promised to continue forever, a preferred stock is valued as a perpetuity. A perpetuity is described by a timeline that stretches to infinity as shown here:

The preferred stock is easily valued as

Actually,  Formula 10-3  can be used to value any perpetuity, as long as the first payment occurs one year from the valuation date. All we have to do to find the price of preferred stock (Pp) is to divide the constant annual dividend payment (Dp) by the required rate of return that preferred stockholders are demanding (Kp). For example, if the annual dividend were $10 and the stockholder required a 10 percent rate of return, the price of preferred stock would be $100.

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As was true in our bond valuation analysis, if the rate of return required by security holders changes, the value of the financial asset (in this case, preferred stock) will change. You may also recall that the longer the life of an investment, the greater the impact of a change in required rate of return. It is one thing to be locked into a low-paying security for one year when the rate goes up; it is quite another to be locked in for 10 or 20 years. With preferred stock, you have a perpetual security, so the impact is at a maximum. Assume in the prior example that because of higher inflation or increased business risk, Kp (the required rate of return) increases to 12 percent. The new value for the preferred stock shares is:

If the required rate of return were reduced to 8 percent, the opposite effect would occur. The preferred stock price would be computed as:

It is not surprising that the preferred stock is now trading well above its original price of $100. It is still offering a $10 dividend (10 percent of the original offering price of $100), and the market is demanding only an 8 percent yield. To match the $10 dividend with the 8 percent rate of return, the market price will advance to $125.

Determining the Required Rate of Return (Yield) from the Market Price

In our analysis of preferred stock, we have used the value of the annual dividend (Dp) and the required rate of return (Kp) to solve for the price of preferred stock (Pp). We could change our analysis to solve for the required rate of return (Kp) as the unknown, given that we know the annual dividend (Dp) and the preferred stock price (Pp). We take  Formula 10-3  and rewrite it as  Formula 10-4 , where the unknown is the required rate of return (Kp).

Using  Formula 10-4 , if the annual preferred dividend (Dp) is $10 and the price of preferred stock (Pp) is $100, the required rate of return (yield) would be 10 percent as follows:

If the price goes up to $130, the yield will be only 7.69 percent:

We see the higher market price provides quite a decline in the yield.

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Valuation of Common Stock

The value of a share of common stock may be interpreted by the shareholder as the present value of an expected stream of future dividends. Although in the short run stockholders may be influenced by a change in earnings or other variables, the ultimate value of any holding rests with the distribution of earnings in the form of dividend payments. Though the stockholder may benefit from the retention and reinvestment of earnings by the corporation, at some point the earnings must be translated into cash flow for the stockholder. A stock valuation model based on future expected dividends, which is termed a dividend valuation model, can be stated as:

where

P0 =	Price of stock today
D =	Dividend for each year
Ke =	the required rate of return for common stock (discount rate)

This formula, with modification, is generally applied to three different circumstances:

1. No growth in dividends.

2. Constant growth in dividends.

3. Variable growth in dividends.

No Growth in Dividends

Under the no-growth circumstance, common stock is very similar to preferred stock. The common stock pays a constant dividend each year. For that reason, we merely translate the terms in  Formula 10-3 , which applies to preferred stock, to apply to common stock. This is shown as new  Formula 10-6 :

P0 =	Price of common stock today
D1 =	Current annual common stock dividend (a constant value)
Ke =	Required rate of return for common stock

Assume D1 = $1.87 and Ke = 12 percent; the price of the stock would be $15.58:

A no-growth policy for common stock dividends does not hold much appeal for investors and so is seen infrequently in the real world. 6

Constant Growth in Dividends

A firm that increases dividends at a constant rate is a more likely circumstance. Perhaps a firm decides to increase its dividends by 7 percent per year. The general valuation approach is shown in  Formula 10-7 :

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where

P0 =	Price of common stock today
D0(1 + g)1 =	Dividend in year 1, D1
D0(1 + g)2 =	Dividend in year 2, D2, and so on
g =	Constant growth rate in dividends
Ke =	Required rate of return for common stock (discount rate)

As shown in  Formula 10-7 , the current price of the stock is the present value of the future stream of dividends growing at a constant rate. If we can anticipate the growth pattern of future dividends and determine the discount rate, we can ascertain the price of the stock.

For example, assume the following information:

D0 =	Last 12-month’s dividend (assume $1.87)
D1 =	First year, $2.00 (growth rate, 7%)
D2 =	Second year, $2.14 (growth rate, 7%)
D3 =	Third year, $2.29 (growth rate, 7%) etc.
Ke =	Required rate of return (discount rate), 12%

Then

To find the price of the stock, we take the present value of each year’s dividend. This is no small task when the formula calls for us to take the present value of an infinite stream of growing dividends. Fortunately,  Formula 10-7  can be compressed into a much more usable form if two circumstances are satisfied:

1. The firm must have a constant dividend growth rate (g).

2. The discount rate (Ke) must be higher than the growth rate (g).

For most introductory courses in finance, these assumptions are usually made to reduce the complications in the analytical process. This allows us to reduce or rewrite  Formula 10-7  as  Formula 10-8 .  Formula 10-8  is the basic equation for finding the value of common stock and is referred to as the constant growth dividend valuation model:

This is an extremely easy formula to use in which:

P0 =	Price of the stock today
D1 =	Dividend at the end of the first year
Ke =	Required rate of return (discount rate)
g =	Constant growth rate in dividends

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In this formula, P0 is sometimes referred to as the value of a “growing perpetuity” because it is a perpetuity that grows at a constant rate. In order for  Formula 10-8  to work, it is critical that the first dividend come at the end of the first year. Based on the current example:

D1 =	$2.00
Ke =	0.12
g =	0.07

and P0 is computed as:

Thus, given that the stock has a $2 dividend at the end of the first year, a discount rate of 12 percent, and a constant growth rate of 7 percent, the current price of the stock is $40.

Let’s take a closer look at  Formula 10-8  shown earlier and the factors that influence valuation. For example, what is the anticipated effect on valuation if Ke (the required rate of return, or discount rate) increases as a result of inflation or increased risk? Intuitively, we would expect the stock price to decline if investors demand a higher return and the dividend and growth rate remain the same. This is precisely what happens.

If D1 remains at $2.00 and the growth rate (g) is 7 percent, but Ke increases from 12 percent to 14 percent, using  Formula 10-8 , the price of the common stock will now be $28.57 as shown below. This is considerably lower than its earlier value of $40:

Similarly, if the growth rate (g) increases while D1 and Ke remain constant, the stock price can be expected to increase. Assume D1 = $2.00, Ke is set at its earlier level of 12 percent, and g increases from 7 percent to 9 percent. Using  Formula 10-8  once again, the new price of the stock would be $66.67:

We should not be surprised to see that an increasing growth rate has enhanced the value of the stock.

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Stock Valuation Based on Future Stock Value The discussion of stock valuation to this point has related to the concept of the present value of future dividends. This is a valid concept, but suppose we wish to approach the issue from a slightly different viewpoint. Assume we are going to buy a stock and hold it for three years and then sell it. We wish to know the present value of our investment. This is somewhat like the bond valuation analysis. We will receive a dividend for three years (D1, D2, D3) and then a price (payment) for the stock at the end of three years (P3). What is the present value of the benefits? To solve this, we add the present value of three years of dividends and the present value of the stock price after three years. Assuming a constant growth dividend analysis, the stock price after three years is simply the present value of all future dividends after the third year (from the fourth year on). Thus the current price of the stock in this case is nothing other than the present value of the first three dividends, plus the present value of all future dividends (which is equivalent to the stock price after the third year). Saying the price of the stock is the present value of all future dividends is also the equivalent of saying it is the present value of a dividend stream for a number of years, plus the present value of the price of the stock after that time period. The appropriate formula would be  Formula 10-7 , where the fourth term would be replaced by P3 = D4/(Ke − g).

Determining the Required Rate of Return from the Market Price

In our analysis of common stock, we have used the first year’s dividend (D1), the required rate of return (Ke), and the growth rate (g) to solve for the stock price (P0) based on  Formula 10-8 .

We could change the analysis to solve for the required rate of return (Ke) as the unknown, given that we know the first year’s dividend (D1), the stock price (P0), and the growth rate (g). We take the preceding formula and algebraically change it to provide Formula 10-9.

Formula 10-9 allows us to compute the required return (Ke) for the investment. Returning to the basic data from the common stock example:

Ke =	Required rate of return (to be solved)
D1 =	Dividend at the end of the first year, $2.00
P0 =	Price of the stock today, $40
g =	Constant growth rate 0.07, or 7%

In this instance, we would say the stockholder demands a 12 percent return on the common stock investment. Of particular interest are the individual parts of the formula for Ke that we have been discussing. Let’s write out Formula 10-9 again.

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The first term represents the dividend yield the stockholder will receive, and the second term represents the anticipated growth in dividends, earnings, and stock price. While we have been describing the growth rate primarily in terms of dividends, it is assumed the earnings and stock price will also grow at that same rate over the long term if all else holds constant. You should also observe that the preceding formula represents a total-return concept. The stockholder is receiving a current dividend plus anticipated growth in the future. If the dividend yield is low, the growth rate must be high to provide the necessary return. Conversely, if the growth rate is low, a high dividend yield will be expected. The concepts of dividend yield and growth are clearly interrelated.

The Price-Earnings Ratio Concept and Valuation

In  Chapter 2 , we introduced the concept of the price-earnings ratio. The price-earnings ratio represents a multiplier applied to current earnings to determine the value of a share of stock in the market. It is considered a pragmatic, everyday approach to valuation. If a stock has earnings per share of $3 and a price-earnings (P/E) ratio of 15 times, it will carry a market value of $45. Another company with the same earnings but a P/E ratio of 20 times will enjoy a market price of $60.

The price-earnings ratio is influenced by the earnings and sales growth of the firm, the risk (or volatility in performance), the debt-equity structure of the firm, the dividend policy, the quality of management, and a number of other factors. Firms that have bright expectations for the future tend to trade at high P/E ratios while the opposite is true for low P/E firms.

For example, the average P/E for the S&P 500 Index firms was 19 in early 2015, but Facebook traded at a P/E of 72 because its earnings were expected to grow dramatically, and ExxonMobil traded at a P/E of 11 because oil prices had fallen and profits were expected to follow oil prices down over the next year.

P/E ratios can be looked up in Barron’s, at  finance.yahoo.com , and a number of other publications and Internet sites. Quotations from Barron’s are presented in  Table 10-4 . The first column after the company’s name shows the ticker symbol and is followed by volume. The third column indicates the yield (dividends per share divided by stock price). The fourth column is the item of primary interest and it indicates the current price-earnings (P/E) ratio. The remaining columns cover the stock price (last), the weekly price change, and earnings and dividend data.

For IBM, which is highlighted in white in  Table 10-4 , the P/E ratio is 13, indicating that the company’s stock price of $158.72 represents approximately 13 times earnings of $11.90 for the past 12 months. 7  Firms that are operating at a loss (deficit) have the symbol dd in the P/E ratio column.

The dividend valuation approach (based on the present value of dividends) that we have been using throughout the chapter is more theoretically sound than P/E ratios and more likely to be used by sophisticated financial analysts. To some extent, the two concepts of P/E ratios and dividend valuation models can be brought together. A stock that has a high required rate of return (Ke) because it’s risky will generally have a low P/E ratio. Similarly, a stock with a low required rate of return (Ke) because of the predictability of positive future performance will normally have a high P/E ratio. These are generalized relationships. There are, of course, exceptions to every rule.

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Table 10-4 Quotations from Barron’s

Source: Barron’s, February 9, 2015, p. M18.

Variable Growth in Dividends

In the discussion of common stock valuation, we have considered procedures for firms that had no growth in dividends and for firms that had a constant growth. Most of the discussion and literature in finance assumes a constant growth dividend model. However, there is also a third case, and that is one of variable growth in dividends. The most common variable growth model is one in which the firm experiences supernormal (very rapid) growth for a number of years and then levels off to more normal, constant growth. The supernormal growth pattern is often experienced by firms in emerging industries, such as in the early days of electronics or microcomputers.

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Finance in ACTION Managerial An Important Question—What’s a Small Business Really Worth?

The value of a small, privately held business takes on importance when the business is put up for sale, is part of a divorce settlement, or is being valued for estate purposes at the time of the owner’s death. The same basic principles that establish valuation for Fortune 500 companies apply to small businesses as well. However, there are important added considerations.

One factor is that private businesses often lack liquidity. Unlike a firm trading in the public securities market, there is no ready market for a local clothing goods store, a bowling alley, or even a doctor’s clinic. Therefore, after the standard value has been determined, it is usually reduced for lack of liquidity. Although circumstances vary, the normal reduction is in the 30 percent range. Thus a business that is valued at $100,000 on the basis of earnings or cash flow may be assigned a value of $70,000 for estate valuation purposes.

There are other factors that are important to small business valuation as well. For example, how important was a key person to the operation of a business? If the founder of the business was critical to its functioning, the firm may have little or no value in his or her absence. For example, a bridal consulting shop or a barber shop may have minimal value upon the death of the owner. On the other hand, a furniture company with established brand names or a small TV station with programming under contract may retain most of its value.

Another consideration that is important in valuing a small business is the nature of the company’s earnings. They are often lower than they would be in a publicly traded company. Why? First of all, the owners of many small businesses intermingle personal expenses with business expenses. Thus family cars, health insurance, travel, and so on may be charged as business expenses when, in fact, they have a personal element to them. While the IRS tries to restrict such practices, there are fine lines in distinguishing between personal and business uses. As a general rule, small, private businesses try to report earnings as low as possible to minimize taxes. Contrast this with public companies that report earnings quarterly with the intent of showing ever-growing profitability. For this reason, in valuing a small, privately held company, analysts often rework stated earnings in an attempt to demonstrate earning power that is based on income less necessary expenditures. The restated earnings are usually higher.

After these and many other factors are taken into consideration, the average small, private company normally sells at 5 to 10 times average adjusted earnings for the previous three years. It is also important to identify recent sale prices of comparable companies, and business brokers may be able to supply such information. When establishing final value, many people often look to their CPA or a business consultant to determine the true worth of a firm.

In evaluating a firm with an initial pattern of supernormal growth, we first take the present value of dividends during the exceptional growth period. We then determine the price of the stock at the end of the supernormal growth period by taking the present value of the normal, constant dividends that follow the supernormal growth period. We discount this price to the present and add it to the present value of the supernormal dividends. This gives us the current price of the stock.

A numerical example of a supernormal growth rate evaluation model is presented in  Appendix 10A  at the end of this chapter.

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Finally, in the discussion of common stock valuation models, readers may ask about the valuation of companies that currently pay no dividends. Since virtually all our discussion has been based on values associated with dividends, how can this “no dividend” circumstance be handled? One approach is to assume that even for the firm that pays no current dividends, at some point in the future, stockholders will be rewarded with cash dividends. We then take the present value of their deferred dividends.

A second approach to valuing a firm that pays no cash dividends is to take the present value of earnings per share for a number of periods and add that to the present value of a future anticipated stock price. The discount rate applied to future earnings is generally higher than the discount rate applied to future dividends.

SUMMARY AND REVIEW OF FORMULAS

The primary emphasis in this chapter is on valuation of financial assets: bonds, preferred stock, and common stock. Regardless of the security being analyzed, valuation is normally based on the concept of determining the present value of future cash flows. Thus we draw on many of the time-value-of-money techniques developed in  Chapter 9 . Inherent in the valuation process is a determination of the rate of return that investors demand. When we have computed this value, we have also identified what it will cost the corporation to raise new capital. Let’s specifically review the valuation techniques associated with bonds, preferred stock, and common stock.

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.)

APPENDIX | 10B

Using Calculators for Financial Analysis

This appendix is designed to help you use either an algebraic calculator (Texas Instruments BAII Plus Business Analyst) or the Hewlett-Packard 12C Financial Calculator. We realize that most calculators come with comprehensive instructions, and this appendix is meant only to provide basic instructions for commonly used financial calculations.

There are always two things to do before starting your calculations as indicated in the first table: Clear the calculator and set the decimal point. If you do not want to lose data stored in memory, do not perform steps 2 and 3 in the first box on the next page.

Each step is listed vertically as a number followed by a decimal point. After each step you will find either a number or a calculator function denoted by a box . Entering the number on your calculator is one step and entering the function is another. Notice that the HP 12C is color coded. When two boxes are found one after another, you may have an f or a g in the first box. An f is orange coded and refers to the orange functions above the keys. After typing the f function, you will automatically look for an orange-coded key to punch. For example, after f in the first Hewlett-Packard box (right-hand panel), you will punch in the orange-color-coded REG. If the f function is not followed by another box, you merely type in f and the value indicated.

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The g is coded blue and refers to the functions on the bottom of the function keys. After the g function key, you will automatically look for blue-coded keys. The TI BAII Plus is also color coded. The gold 2nd key, located near the top left corner of the calculator, refers to the gold functions above the keys. Upon pressing the 2nd key, the word “2nd” appears in the top left corner, indicating the gold function keys are active.

Familiarize yourself with the keyboard before you start. In the more complicated calculations, keystrokes will be combined into one step.

In the first four calculations, we solve for the future value (FV), present value (PV), future value of an ordinary annuity (FVA), and present value of an ordinary annuity (PVA), each for $100.

On the following pages, you can determine bond valuation, yield to maturity, net present value of an annuity, net present value of an uneven cash flow, internal rate of return for an annuity, and internal rate of return for an uneven cash flow.

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Bond Valuation Using Both the TI BAII Plus and the HP 12C

Solve for V = Price of the bond

Given:

Ct =	$80 annual coupon payments or 8% coupon ($40 semiannually)
Pn =	$1,000 principal (par value)
n =	10 years to maturity (20 periods semiannually)
i =	9.0% rate in the market (4.5% semiannually)

You may choose to refer to  Chapter 10  for a complete discussion of bond valuation.

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Yield to Maturity on Both the TI BAII Plus and HP 12C

Solve for Y = Yield to maturity

Given:

V =	$895.50 price of bond
Ct =	$80 annual coupon payments or 8% coupon ($40 semiannually)
Pn =	$1,000 principal (par value)
n =	10 years to maturity (20 periods semiannually)

You may choose to refer to  Chapter 10  for a complete discussion of yield to maturity.

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Net Present Value of an Annuity on Both the TI BAII Plus and the HP 12C

Solve for A = Present value of annuity

Given:

n =	10 years (number of years cash flow will continue)
PMT =	$5,000 per year (amount of the annuity)
i =	12% (cost of capital Ka)
Cost =	$20,000

You may choose to refer to  Chapter 12  for a complete discussion of net present value.

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Net Present Value of an Uneven Cash Flow on Both the TI BAII Plus and the HP 12C

Solve for NPV = Net present value

Given:

n =	5 years (number of years cash flow will continue)
PMT =	$5,000 (yr. 1); $6,000 (yr. 2); $7,000 (yr. 3); $8,000 (yr. 4); $9,000 (yr. 5)
i =	12% (cost of capital Ka)
Cost =	$25,000

You may choose to refer to  Chapter 12  for a complete discussion of net present value concepts.

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Internal Rate of Return for an Annuity on Both the TI BAII Plus and the HP 12C

Solve for IRR = Internal rate of return

Given:

n =	10 years (number of years cash flow will continue)
PMT =	$10,000 per year (amount of the annuity)
Cost =	$50,000 (this is the present value of the annuity)

You may choose to refer to  Chapter 12  for a complete discussion of internal rate of return.

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Internal Rate of Return with an Uneven Cash Flow on Both the TI BAII Plus and the HP 12C

Solve for IRR = Internal rate of return (return that causes present value of outflows to equal present value of the inflows)

Given:

n =	5 years (number of years cash flow will continue)
PMT =	$5,000 (yr. 1); $6,000 (yr. 2); $7,000 (yr. 3); $8,000 (yr. 4); $9,000 (yr. 5)
Cost =	$25,000

You may choose to refer to  Chapter 12  for a complete discussion of internal rate of return.

1 The assumption is that the bond has a $1,000 par value. If the par value is higher or lower, then this value would be discounted to the present from the maturity date.

2 For now, we are using annual interest payments for simplicity. Later in the discussion, we will shift to semiannual payments, and more appropriately determine the value of a bond.

3 Actually a slightly more accurate representation would be this: Risk-free rate = (1 + Real rate of return)(1 + Inflation premium) − 1. We would show: (1.03)(1.04) − 1 = 1.0712 − 1 = .0712 = 7.12 percent.

4 On the other hand, common stock carries the potential for very high returns when the corporation is quite profitable.

5 Of course the required rate of return on all other financial assets will also go up proportionally.

6 Since this is a no-growth stock, D1 equals D0.  Formula 10-6  uses D1 to emphasize that the first dividend payment comes at the end of year 1.

7 This EPS value for the past 12 months is different from the value in the table for the latest year, which represents a calendar year.