BUS 650 Week 2 Work
Chapter 4
Time Value Applications: Security Valuation and Expected Returns
Alan Schein Photography/Corbis
Learning Objectives
A�er studying this chapter, you should be able to:
Solve for the value of zero-coupon bonds using �me value of money mathema�cs. Determine the value of preferred stock using �me value of money mathema�cs. Solve for the value of constant-growth common stock using �me value of money mathema�cs. Determine the value of nonconstant-growth common stock using �me value of money mathema�cs. Explain how factors such as coupon rate, interest rates, and maturity affect bond values over �me. Solve for the expected rates of return for securi�es, given their market prices and cash flow characteris�cs.
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Ch. 4 Introduction
In Chapter 3, we began the study of money's �me value. In this chapter, we apply those basics to the valua�on of securi�es (stocks and bonds) and to solving for expected returns from inves�ng. Along the way, we present some of the terminology and features of corporate securi�es.
The ability to solve for the value of a share of common stock is a fundamental skill for a corporate manager to have. Recall from Chapter 1 that it is management's job to maximize shareholders' wealth, a task impossible to carry out without knowledge of what factors influence share prices and therefore determine the wealth of shareholders. Common and preferred stock valua�on as well as bond valua�on are also important topics for anyone who may wish to personally invest in such securi�es. The first part of this chapter introduces security valua�on.
Solving for expected returns is the topic that concludes Chapter 4. When price is known, it may be helpful for the manager (or the investor) to es�mate the return or yield that can reasonably be expected from a project or investment. Such an expected return can be compared to returns offered by compe�ng projects or investments. An investor, for example, would not want to invest in a corporate bond whose expected yield was below that of a less risky government bond.
Before beginning, let's quickly review value. Recall from earlier chapters that value is dependent on cash flows to investors, the �ming of those cash flows, and their riskiness. The cash flow that a security holder receives is the principal benefit of ownership. Without that benefit, the security would be nearly worthless. Cash flows from the firm to shareholders come in the form of dividends, and for bondholders the cash received comes in the form of coupon interest payment. As we will demonstrate, shareholders also receive cash flows from other investors when they sell their stock at a (hopefully!) higher price. This price apprecia�on is due to the expecta�on of higher future dividends, making the claim on future cash flows more valuable.
Now we apply the �me value of money techniques introduced in Chapter 3 to the valua�on of commonly encountered securi�es.
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What is the present value of a zero-coupon bond that pays $1,000 in 20 years with a required return of 8% annually?
One of the most common types of zero-coupon bonds is the U.S. savings bond. Can you think of any other examples of zero-coupon bonds?
Associated Press
4.1 Zero-Coupon Bonds: A Single Cash Flow
Corpora�ons and the government some�mes issue bonds known as zero-coupon bonds. These bonds differ from typical bonds in that they make no payments to the bondholders un�l maturity. Let's consider a bond that matures in 20 years, pays no coupon interest, and has a par value, or maturity value, of $1,000. That is, the investor will receive $1,000 on the bond's maturity date but no other cash payments during the life of the bond. If investors require an 8% annual return from this security, based on annual compounding, what should be the selling price of the bond? The problem is illustrated with a �meline in Figure 4.1, and prac�ced in the Applying Finance: Price of a Zero-Coupon Bond feature.
Figure 4.1: Determining the present value of a zero-coupon bond
Use Equa�on (3.7) from Chapter 3 to find the current value:
(3.7) PV0 = FVn(1 + r) -n or FVn /(1 + r)
n
PV0 = $1,000(1.08) -20 = $1,000(0.21455) + $214.55
Applying Finance: Price of a Zero-Coupon Bond
Present Value of a Zero-Coupon Bond: How much would an investor pay today for a zero-coupon bond that pays $1,000 in 20 years and earns 8% per year?
To Solve Using TI Business Analyst Calculator
1000 [FV]
8 [I/Y]
20 [N]
0 [PMT]
[CPT] [PV]
= $214.55
Note: Similar to Excel, the PV is displayed as a nega�ve. Also, you may enter the keystrokes in any order you wish so long as you enter CPT PV at the end.
To Solve Using Excel
Use the PV func�on. The inputs for this func�on are: = PV(Rate%,NPER,PMT,FV,Type)
= PV(8%,20,0,1000,0) = (214.55)
The answer displayed is nega�ve (in parentheses or red or signed nega�ve) because that is how much the investor will pay today (an ou�low or nega�ve cash flow) to receive $1,000 in 20 years. Remember that Excel requires an ou�low and an inflow (i.e., a cash flow signed posi�ve and a cash flow signed nega�ve). When we entered the posi�ve FV of $1,000 that meant that the PV had to be nega�ve. Cau�on: Remember that numbers cannot be entered with commas separa�ng thousands of dollars because commas separate inputs in Excel func�ons.
The secondary market for zero-coupon bonds is very ac�ve. Suppose one is selling for $425, maturing in 14 years, at which �me it will pay $1,000 to its holder. In this case, investors would be interested in the yield-to- maturity (YTM), or the return that the bond offers given its current market price and other characteris�cs. This problem is illustrated in Figure 4.2, and prac�ced in the Applying Finance: Yield-to-Maturity of a Zero-Coupon Bond feature.
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What is the YTM of a zero-coupon bond selling for $425 today, if it will pay $1,000 in 14 years?
Figure 4.2: Determining the YTM of a zero-coupon bond
To solve for r, Equa�on (3.4) from Chapter 3 could be used.
This bond is expected to yield 6.303% if held to maturity.
Applying Finance: Yield-to-Maturity of a Zero-Coupon Bond
Yield-to-Maturity of a Zero-Coupon Bond: What annual rate of return will an investor earn if she pays $425 today for a zero coupon that pays $1,000 when it matures in 14 years?
To Solve Using TI Business Analyst
14 [N], 0 [PMT], 425 [+/–] [PV], 1000 [FV], [CPT] [I/I]
Note that, like the Excel keystrokes, either the price (PV) or the par value (FV) has to be nega�ve in order to "tell" the calculator that one cash flow is going to the firm and one is going to the investor.
To Solve Using Excel
Use the Rate func�on with the format: RATE(NPER, PMT, PV, FV, TYPE, GUESS)
=RATE(14,0, –425,1000,0,10%) = 6.303%
Note: One of the cash flows is nega�ve (the $425) and the other is posi�ve ($1000). There is no comma or dollar sign in the $1000. The TYPE is zero because we assume interest accrues at the end of the period. GUESS can be le� out or enter something that seems reasonable. The display of the answer can be adjusted to show more or fewer decimal places by forma�ng the cells.
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Public u�lity companies like Pacific Gas & Electric are among the main issuers of preferred stocks. What do you think are the benefits of preferred stocks?
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4.2 Preferred Stock: A Perpetuity
The most common type of perpetuity is preferred stock. Preferred stock generally pays a fixed dividend. Thus, eight-dollar preferred refers to a share of preferred stock that promises to pay a dividend of $8 once per year into the foreseeable future. Preferred stock is known as a hybrid security in that it combines features of both fixed claims (bonds—i.e., debt) and residual claims (stocks—i.e., equity). Table 4.1 outlines the hybrid quali�es of preferred stock.
Table 4.1: Hybrid security features of preferred stock
Bond-like quali�es Stock-like quali�es
• Fixed dividend payments • Dividend payments not legally guaranteed
• Callable • Interest is not tax deduc�ble
• No maturity date
Preferred stock is fixed in the sense that the amount the issuing corpora�on is obligated to pay does not vary; it is fixed like the coupon rate on debt (in this case it is $8 once every year). Also, similar to some bond issues, preferred stock can be redeemable or callable (we will discuss this further in Sec�on 4.6).
On the other hand, preferred stock is residual because there is no legal obliga�on for the company to pay a dividend unless it has cash flows le� over a�er all other fixed claims (such as interest on bonds) have been paid. Preferred claims have a lower priority than other fixed claims, but a higher priority than common stock. Therefore, no dividends can be paid to common stockholders unless preferred dividends have been paid first. This contrasts to interest on debt, which must be paid or the company risks legal ac�on by bondholders. Interest on debt is tax deduc�ble, but dividends on preferred stock (and common stock) is not, making $1.00 of dividends more expensive for the company than $1.00 of interest. Also, like common stock, preferred stock has no maturity date.
The present value of a perpetuity formula is used to find the price of a share of eight-dollar preferred. The interest rate equals 16% in the example.
PV0 = CF/r is re-expressed as = D/r because today's price (P0) is equal to the present value of future cash flows (PV0) and
preferred's dividend (D) is the perpetuity's cash flow (CF). The price is $50 per share.
Most preferred stock in the United States is issued by public u�li�es, financial ins�tu�ons, or REITs (real estate investment trusts). For example, Pacific Gas & Electric (PG&E with �cker symbol PCG), a large California public u�lity, has eight different issues of preferred stock. One issue is the 6.0% nonredeemable preferred with a par value of $25.00. Each year the stock pays a dividend of 6% of $25 or $1.50. Dividends are paid quarterly so each quarter an investor receives $0.375 per share in dividends. The 5% preferred, also with a $25.00 par value, pays quarterly dividends of $0.3125 = 5% × $25/4.
On May 29, 2012, the 6% PG&E preferred stock sold for $29.35, down a bit from a price of $29.92 a few days before. At the $29.35 price, investors are earning a return of 5.1107%. See the Web Resources at the end of the chapter for a link to the trades in this preferred stock. Preferred stock issues, like PG&E, do not trade very o�en.
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Common stocks for companies such as Google are more familiar to us than preferred stocks. What are the benefits and drawbacks to common stocks?
Associated Press
4.3 Common Stock: A Growing Perpetuity
Unlike preferred stock, common stock does not pay dividends that are a constant amount through �me. On the other hand, common dividends are equally spaced in �me and do con�nue indefinitely. Common stock, therefore, sa�sfies all the criteria of a perpetuity except the changing amount of its dividend payment.
To find the price of common stock we might use the general formula for the present value of a stream of cash flows. Again, recognizing that PV0 = P0 and CF1 = D1, CF2 = D2, and so on, we may re-express the formula in
terms of the price (P0) and dividends (D1, D2, . . .) of the common stock:
Clearly, it is impossible to solve this equa�on explicitly because the cash flows (dividends) go on forever.
A number of models have been developed to allow this formula to be solved. The simplest model requires the assump�on that successive dividends grow at a constant rate. We call this a growing perpetuity.
Let that rate be termed gN, the long-run normal growth rate of dividends. The dividends may be expressed as (1 + gN) mul�plied by the preceding year's dividend payment:
D0 = the current dividend
D1 = D0(1 + gN)
D2 = D1(1 + gN) = D0(1 + gN) 2
D3 = D2(1 + gN) = D0(1 + gN) 3
Subs�tu�ng into Equa�on (4.1) yields a geometric series:
Mathema�cians have shown that as long as g is less than r, this series can be summed fairly easily.
A constant-growth stock may be valued using the constant-growth formula,
To illustrate the formula, let's assume a stock has just paid a $5.00-per-share dividend. We believe that future dividends will grow at a 6% rate forever, and investors require a 13% return on their investment in this stock. The stock's price should be
The growth rate plays a very important role in determining the value of a share of stock (or any asset). In this example, suppose the dividend growth rate had been 2% instead of 6%. Then the value of the stock today would be $46.36 = $5.00(1.02)/(0.13 – 0.02). Had the growth rate been zero (like a share of preferred stock), the value today would be just $5.00/0.13 = $38.46. You can see that the addi�onal growth has a large effect on the value of the stock.
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Even a company as successful as McDonald's experiences non-constant growth and drops in stock prices. There are many other examples of non- constant growth corpora�ons. How is non-constant growth significant to a financial manager?
4.4 Common Stock: Nonconstant Dividend Growth
The constant-growth valua�on model works well for securi�es whose forecasted financial behavior corresponds to the model's assump�on of dividends that grow at a constant rate. Some companies, such as electric u�li�es, compete in mature markets that offer few prospects for rapid growth. Demand for their product is pre�y stable, varying li�le with economic cycles. Such firms may be good candidates for valua�on using the constant-growth model.
For many corpora�ons, however, the constant-growth assump�on does not hold. O�en firms have new products that have compe��ve advantages over their compe�tors' products. Patent protec�on, new technology, low-cost produc�on methods, and brand name recogni�on may enable a firm to experience rapid growth for a period of �me. In the long run, though, this rapid growth is not sustainable as compe�tors' technology, manufacturing efficiency, and so on, catches up with the industry leader's, leveling the playing field in the marketplace. A constant-growth valua�on model is clearly inappropriate for firms that experience a period of nonconstant growth.
McDonald's Rise from Financial Downturn
Table 4.2 shows the annual dividends that Johnson & Johnson has paid since 2000. No�ce that the constant-growth model would probably not be appropriate for Johnson & Johnson since the growth rates have been somewhat erra�c. Also no�ce that the firm experienced double-digit growth rates between 2000 and 2008, but then dividend growth tapered off fairly drama�cally.
Table 4.2: Johnson & Johnson annual dividends since 2000
Year Dividend Annual % change
2000 $0.620 13.8%
2001 $0.700 12.9%
2002 $0.795 13.6%
2003 $0.925 16.4%
2004 $1.095 18.4%
2005 $1.275 16.4%
2006 $1.455 14.1%
2007 $1.620 11.3%
2008 $1.795 10.8%
2009 $1.930 7.5%
2010 $2.110 9.3%
2011 $2.250 6.6%
Note: Based on data from h�p://finance.yahoo.com (h�p://finance.yahoo.com)Processing math: 0%
There are many reasons why a company might have nonconstant dividend growth. What might account for abnormal growth periods?
Most corpora�ons, even large ones like Johnson & Johnson, experience nonconstant growth due to compe��on with other companies. How is the ability to calculate dividends helpful when buying stock?
Ge�y Images News/Ge�y Images
One method for valuing firms in a nonconstant-growth cycle is presented here. Let's assume that we are valuing a stock whose dividends are expected to grow at an 18% rate for each of the next three years. A�er this abnormal growth period, normal growth will con�nue at a 5% annual rate. The company's last annual dividend was $2.00 per share. The discount rate for the stock is 16%. The �meline in Figure 4.3 illustrates the growth assump�ons of this example.
Figure 4.3: Nonconstant dividend growth
As shown in Figure 4.3
gA = abnormal growth rate = 18%
A = length of abnormal growth period = 3
gN = normal or constant-growth rate = 5%
Because today's price should equal the present value of future dividends, the first step is to find the size of these dividends.
D0 = $2.00 last dividend paid
D1 = $2.00(1.18) = $2.36 In year 1 dividends grow at 18%
D2 = $2.36(1.18) = $2.78 In year 2 dividends grow at 18%
D3 = $2.78(1.18) = $3.28 In year 3 dividends grow at 18%
D4 = $3.28(1.05) = $3.44 In year 4 dividends grow at 5%
Dividends growth at 5% from D4. . .
It is impossible to solve explicitly for the value of all future dividends, and, thus, it is also impossible to find explicitly the present value of all future dividends. But, note that from point A forward, the growth rate is constant. This means that the assump�ons of the constant-growth valua�on model are met from period 3 onward. We can, therefore, solve for P3, the stock's price at �me 3, using the constant-growth model. This value, P3, incorporates the value of all the dividends from �me 3 onward. P3 includes the present value of D4, D5, D6, and so on. Recognizing this gives us a strategy for solving for P0, the current price.
But
So
Note that P3 is discounted for three periods because it is the price as of period 3 in Figure 4.3. We already know the value of D1, D2, D3, and r, so these values may be
subs�tuted into Equa�on (4.4).
To solve for P3, recall the constant-growth formula from the prior sec�on:
It solved for P0 using D1 because the constant-growth assump�on held from �me 0 onward. In this example, the constant growth holds from �me 3 onward, so we can
adjust the formula rela�ve to �me 3 and solve for P3.
We now have all the values we need to solve for P0, the current price of the stock.
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This price, $26.24, accounts for the present value of all future dividends. The present values of D0, D1 through D3 are solved for explicitly. The present values of D4, D5, D6,
and so on are solved for implicitly by finding the present value of P3. P3 is able to incorporate the values of all dividends a�er �me 3 because dividends grow at a constant
rate from �me 3 onward.
This method may be generalized in the following formula.
where
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Bonds have characteris�cs of annui�es and single cash flows.
A bond's price is equal to the present value of the coupon stream, plus the present value of par value.
4.5 Bonds: An Annuity and Single Cash Flow
Bonds can be thought of as a combina�on of an annuity and a single cash flow. Bond investors receive from the corpora�on both a stream of coupon interest payments over the life of the bond (annuity), and a payment of par value at maturity (single cash flow). Most bonds make coupon payments semiannually, and corporate bonds generally carry a $1,000 par value. The cash flows for a typical bond are illustrated in Figure 4.4.
Figure 4.4: Cash flows for a typical bond
In the figure, m is the number of coupon payment periods un�l the bond matures. For bonds paying coupons semiannually, m is twice the number of years un�l maturity. Every semiannual coupon payment equals one-half the coupon rate mul�plied by the bond's par value, so a bond with an 8% coupon rate would make two $40 interest payments every year (8% × $1,000)/2.
The price of a bond is the present value of the coupon stream plus the present value of par value. The coupon stream is an annuity and the repayment of par value is a single cash flow. The �meline in Figure 4.5 may be used to find the present value (the price) of a bond.
Figure 4.5: Determining the present value of a bond
The formula for solving for a bond's value is given here; keep in mind that r is the investors' required return for the bond (the discount rate per payment period).
A bond that carries an annual coupon rate of 6.5%, makes coupon payments semiannually, has a $1,000 par value, and matures in 10 years would have a value of $684.58 if the investors discount its cash flows at a 12% annual rate. Note that the 6.5% annual coupon rate is equal to 3.25% semiannually, yielding the $32.50 semiannual coupon payment. The 12% annual required return is re-expressed as 6% semiannually to agree with the semiannual payment period, and the number of periods is (10)(2) = 20.
This problem is also prac�ced in the Applying Finance: The Price of a Bond feature.
Applying Finance: The Price of a Bond
Finding the Price of a Corporate Bond: If an investor wants to earn a 12% annual return, how much would she pay today for a bond that carries an annual coupon rate of 6.5%, makes coupon payments semiannually, has a $1,000 par value, and matures in 10 years?
To Solve Using TI Business Analyst
6 [I/Y], 20 [N], 1000 [FV], 32.50 [PMT], [CPT] [PV]
Note that both the FV and the PMT are signed posi�ve because they are both cash inflows for investors, and therefore the answer for PV will be nega�ve because it will be the price investors are willing to pay. Of course, one could make both the FV and PMT nega�ve, and the answer would be posi�ve, taking the cash flows from the firm's perspec�ve.
To Solve Using Excel
Excel Solu�on: Use the PV func�ons with the following inputs:
RATE = 6% (This is the semiannual version of the 12% annual discount rate.)Processing math: 0%
Important informa�on can be obtained by examining the historical price changes of corporate bonds.
Many corpora�ons, like Caterpillar Incorporated, rely on various types of bond issues. What do you think are the benefits of bonds versus stocks?
Associated Press
NPER = 20 (There are 20 semiannual periods in 10 years.)
PMT = $32.50 (This is the semiannual interest payment to investors = $1,000x6.5%/2)
FV = $1000 (This is the par or face value of the bond that is repaid at maturity.)
TYPE = 0 (Interest payments are paid at the end of periods a�er interest has had �me to accrue.) = PV(6%,20,32.50,1000,0)
= PV(6%,20,32.50,1000,0)
Display shows ($684.577). This is signed nega�ve because this is the amount the investor will pay to purchase the future promised payments.
Note that this bond is selling below its par value ($684.58 < $1,000). It is said to be selling at a discount. Had the bond been valued at $1,000 so that the price was the par value, the bond would be selling at par. A bond whose price is above par is selling at a premium. These pricing differences can be a�ributed to the rela�onship between the bond's annual coupon rate and the investor's required return for the bond. In our example, the annual coupon rate was below the annual required return (6.5% < 12%). If investors paid full par value for the bond, it would only yield the coupon rate—below their requirements for making the investment. Bondholders could not sell this bond for $1,000 because there would be no demand. In order to market the bond, the bondholder must lower the price un�l the yield to the buyer equals the required return. Note that when purchasing a bond at a discount, investors will receive not only coupon payments but also a capital gain because they invest less than $1,000 yet receive the full par value when the bond matures. Buying a bond priced at a premium will lower the yield to investors because they will realize a capital loss over the life of the bond, offse�ng a por�on of their return from the coupon payments. A capital gain or loss becomes part of the bond's return to investors.
Caterpillar Incorporated (CAT) has a number of bonds outstanding. One bond issue has 9.375% coupon rate and matures in 2021. In Figure 4.6, we show how the price of the bond (the red line) changes as the yield on BAA- rated corporate bonds changes (the blue line) from 2003 through 2010. You can see the almost inverse rela�onship between market yields and the price of the bond. As yields fall, as they did in early 2008, the price of the bond rises. Since the bond has a high coupon rate (9.375%), investors place a high value on the bond when yields are low.
Figure 4.6: CAT bond price change in rela�on to BAA-rated corporate bonds
Based on data from www.bondsonline.com (h�p://www.bondsonline.com)
The CAT 9.375% bonds are not callable; that is, they cannot be redeemed before their maturity date in 2021. CAT has over a dozen different bonds issues. Some are callable, so the company can redeem them (buy them back for a fixed price) before the maturity date if it wants. To see the en�re list of CAT bonds (as well as bonds issued by its financing subsidiary), see the Web Resources sec�on at the end of the chapter.
Bonds are useful for illustra�ng the rela�onship between the �me value of cash flows and interest rates. Consider a 20-year bond that carries a 10% annual coupon rate, has a $1,000 par value, and makes coupon payments semiannually. If investors require a 10% return on the date the bond is ini�ally sold to the public, then the bond's price will be $1,000. It will sell at par. On the following day, let's assume that interest rates rise drama�cally and investors now require a 12% annual return on the bond. Those investors who bought the bond on the previous day own a security that pays a series of fixed payments that yield 10% on their $1,000 outlay. In order to sell the bond, they
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As interest rates rise, bond prices fall, and vice versa. Why is this?
must lower the price so the series of payments will yield a 12% return to the purchaser. Solving for the present value of the bond, given the new 12% annual discount rate, yields a price of $849.54. If rates had dropped to 8%, for example, the bond would sell for $1,197.93. Thus, the price of a bond moves in the opposite direc�on from interest rates. To see how to find the price for both these rates using Excel, see the Applying Finance: Solving for Price on a 20-year Bond feature.
Applying Finance: Solving for Price on a 20-year Bond
We can find the bond prices for the 20-year bond easily in Excel:
Excel formula: =PV(6%,40,50,1000,0) = $849.54
Excel formula: =PV(4%,40,50,1000,0) = $1197.93
Now let's consider a bond iden�cal to the one just described, except it matures in 30 years rather than 20 years. Again, when the appropriate discount rate is 10%, the bond will sell at par, $1,000. We solve for the price of the bond using a 12% discount rate and an 8% discount rate (see Applying Finance: Solving for Price on a 30-year Bond).
Applying Finance: Solving for Price on a 30-year Bond
We can find the bond prices for the 30-year bond in Excel:
Excel formula: =PV(12%,60,50,1000,0) = $838.39
Excel formula: =PV(8%,60,50,1000,0) = $1226.23
Compare the way that the 20-year and 30-year bond's prices responded to changes in the interest rate. Note that the longer the maturity of the bond, the more sensi�ve it is to interest rate changes. Investors, knowing this, generally require a higher return for longer maturity bonds because their prices will have greater responses to any changes in interest rates. For this reason, yields on longer-term bonds tend to be higher than short-term bond yields, assuming they are issued by equally risky borrowers. Table 4.3 shows how the prices of bonds vary with maturity and market interest rates. No�ce the longer the maturity is, the greater price change is due to shi�s in market interest rates.
Table 4.3: Varia�ons in price for a 10% coupon (semiannual payments)
Years to maturity 6.00% Market rate
8.00% Market rate
10.00% Market rate
12.00% Market rate
14.00% Market rate
5 $1,170.60 $1,081.11 $1,000.00 $926.40 $859.53
10 $1,297.55 $1,135.90 $1,000.00 $885.30 $788.12
15 $1,392.01 $1,172.92 $1,000.00 $862.35 $751.82
20 $1,462.30 $1,197.93 $1,000.00 $849.54 $733.37
25 $1,514.60 $1,214.82 $1,000.00 $842.38 $723.99
30 $1,553.51 $1,226.23 $1,000.00 $838.39 $719.22
There are two important lessons here. First, bond prices move in the opposite direc�on as movements in interest rates; and second, the longer the maturity of the bond is, the greater the change in its price is for a given change in rates. This rela�onship is illustrated in Figure 4.7 using a teeter to�er. When the interest rate side goes down, the price side goes up. The length of the right-hand side of the teeter-to�er may be thought of as the �me un�l the bond matures. The longer the right-hand side, the greater the movement in price for a given movement in interest rates. Therefore, the longer the maturity, the more risk there is of a large adverse price change. This risk is termed interest rate risk.
Figure 4.7: The interest rate-bond price teeter-to�er
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Sources like the Wall Street Journal are useful when calcula�ng expected returns. What are some other helpful resources?
Jerry Arcieri/Corbis
4.6 Solving for Expected Returns
In the preceding sec�ons, we solved for the value of preferred stock, common stock, and bonds. Many issues of these securi�es are ac�vely traded in financial markets. It is o�en more useful for investors to solve for the returns they might expect to realize from an investment in such securi�es than to solve for their value. A�er all, prices are generally known in the marketplace, so investors would be more interested in expected returns on compe�ng securi�es, rather than prices. Similarly, corporate managers can compare expected returns from prospec�ve LHS projects when deciding how to allocate the firm's investment dollars among assets. Solving for expected returns is analogous to finding value because the same formulas are used. Instead of knowing the discount rate and solving for price, however, now we know the price and are solving for the rate of return.
Let's consider a preferred stock with a price, as quoted in the Wall Street Journal, of $53.50. We note that this preferred stock pays a $4.50 dividend annually. Recognizing that this preferred stock is a perpetuity, we subs�tute the known quan��es into the perpetuity formula:
The return on this preferred stock is 8.41%. More precisely, 8.41% is the expected return because buyers cannot be certain that they will realize the 8.41% return (the firm could go bankrupt).
The expected return for common stock is found using Equa�on (4.3), if we assume the stock's dividends will grow at a constant rate.
Equa�on (4.3) is useful for two reasons. First, it may be used to find the expected return on a share of stock. For example, if a share is selling for $35, next year's dividend is expected to be $3 per share, and dividends are expected to grow at a 6% rate indefinitely, then the expected return on an investment in the stock is 14.57%:
The second use of Equa�on (4.3) is to illustrate the sources of the expected return. The first term to the right of the equal sign in Equa�on (4.3) is the dividend yield, D1/ P0.
The second term, gN, is equal to the capital gains rate. For our stock, investors expect an 8.57% return each year from dividends and a 6% return from price apprecia�on.
Let's now turn to bonds. Because of the complexity of the bond formula, expected returns from bond investments must be solved using either trial and error or a good financial calculator. To illustrate the trial-and-error method, let's solve for the expected return on a bond that sells for $800, pays coupons semiannually, matures in 10 years, carries a 9% coupon rate, and has a $1,000 par value.
Now we must take an educated guess at what r might be. We do have a clue about r: The bond is selling at a discount. Recall that a bond sells at a discount when its yield is greater than the coupon rate. Therefore, we know that r > 4.5% (expressing rates on a semiannual basis to conform to the coupon payment period). Say that our first guess for r is 5.5%:
Because $880 is above the actual price, we know we must raise the interest rate, lowering the value of the right-hand side of the equa�on. This �me let's try 6%. Using 6%, we get a value of $827.95, s�ll too high, but closer. Now let's try 6.25%. This �me the answer is $803.29, close enough using trial and error. The approximate expected return when buying this bond for $800 is 6.25% semiannually, or 12.5% per year. For a bond, the expected return is also called the bond's yield to maturity, or YTM. This problem is prac�ced in the Applying Finance: Corporate Bond YTM feature.
Applying Finance: Corporate Bond YTM
Solving for the rate that equates the price to the promised future cash flows: What is the YTM (yield to maturity) of a 10-year bond with a 9% coupon rate, $1,000 face value, if its price today is $800, and it pays interest semiannually?
To Solve Using TI Business Analyst
20 [N], 45 [PMT], 800 [+/–] [PV], 1000 [FV], [CPT][I/Y]
= 6.284%
Note: The answer will be given as a percentage and will be a semiannual rate so it must be doubled to find the YTM. 12.568% = 6.284 x 2
To Solve Using Excel
=RATE(20,45,–800,1000,0,5%) = 6.284%
This is the semiannual rate, so the annual return (yield to maturity) is 12.568%.
GUESS can be le� out as in : =RATE(20,45,–800,1000,0) Processing math: 0%
Many bonds are what is known as callable, meaning that the issuing corpora�on has the op�on to repurchase the bond at a fixed price above the bond's par value at some date prior to the bond's maturity. These features are a�rac�ve to corpora�ons because if the firm issues bonds at a �me when interest rates are high, the call allows the company to repurchase the bonds early and avoid high interest payments in the future. Corpora�ons that call a bond issue usually finance the repurchase by issuing new bonds that carry lower yields. This process is known as refunding debt. For an investor, a callable bond carries the risk that the corpora�on may repurchase the bond prior to maturity, and the bondholder, therefore, will not collect the high interest payment for the length of �me ini�ally an�cipated. Thus, the yield to maturity for a callable bond may be misleading, and most investors also calculate the yield to call to see what return on their bond investment they are likely to realize. The yield to call is calculated in the same fashion as the yield to maturity except the �me un�l call is subs�tuted for the �me un�l maturity (thus the number of coupon payments is reduced), and the call price is subs�tuted for the call premium (thus the ending cash flow is greater than par).
Let's demonstrate finding the yield to call by using the same bond we just used in the YTM example (see Applying Finance: Yield to Call). We assume that the call price is $1,100, and the call date is five years from now. This changes the future value to $1,100 rather than the $1,000 used in the earlier example and changes the number of semiannual coupon payment periods to 10 instead of 20. The discount rate that equates the $800 price with the future cash flows is 8.2% semiannually, for a yield to call of 16.4%. The yield to call is higher than the yield to maturity of 12.5% because of the higher ending cash flow that will be paid sooner if the bond is called. In this case, investors should not expect that the bond will be called. The bond is selling at a discount, meaning that market rates are currently above the coupon rate offered by the bond. Thus, the corpora�on would not choose to refund such an issue because it would have to issue bonds carrying a higher yield to replace these exis�ng bonds. It is cheaper for the corpora�on to let the bonds mature rather than call them. This is the case whenever the yield to maturity is below the yield to call.
Applying Finance: Yield to Call
To Solve Using TI Business Analyst
800 [+/–] [PV] Today's Price
1100 [FV] The Call Price
10 [N] Semiannual Periods Un�l the Call Date
45 [PMT] Semiannual Coupon
[CPT][ I/Y] = 8.19%, the Semiannual Yield
8.19% x 2 Yield to Call
Most callable bonds have a period during which they cannot be redeemed, usually the first three to seven years a�er issuance. This assures investors that they will receive some of the interest payments before the bond is redeemed. Another standard feature of callable bonds is a call premium. If the bond is called before it reaches half of its stated maturity, the company has to pay investors a bonus to repay the bond. Think of this as an early payment penalty. Usually the call premium is an extra year's interest.
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Ch. 4 Conclusion
Chapter 4 has applied the �me value skills from Chapter 3 to the valua�on of corporate securi�es. Pricing preferred stock was shown to be an applica�on of the formula for valuing perpetuity. Common stock, when dividends are expected to grow at a constant rate, was valued using the growing perpetuity formula. Bonds were priced using a combina�on of the present value of a single cash flow (to value the return of par value at maturity) and the formula for finding present value of an annuity (to value the coupon payments). Varia�ons of the formulas were also used to solve for the expected returns of traded securi�es.
The ability to express equivalent values of cash flows at different points in �me is a fundamental skill in finance. As with any skill, prac�ce increases proficiency and understanding in solving �me value problems.
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Ch. 4 Learning Resources
Key Ideas
Zero-coupon bonds differ from typical bonds in that they make no payments to the bondholders un�l maturity. Preferred stock is the most common type of perpetuity and is known as a hybrid security because it combines features of both fixed and residual claims. Common stock does not pay dividends that are a constant amount through �me; rather, common dividends are equally spaced in �me and con�nue indefinitely. The constant-growth assump�on does not hold for all companies, and these firms must use the nonconstant-growth dividend model. Bonds can be thought of as a combina�on of an annuity and a single cash flow. Callable bonds carry the risk that the corpora�on may repurchase the bond before maturity, meaning the bondholder would lose the high interest payment for the length of �me ini�ally an�cipated.
Key Equa�ons
Cri�cal Thinking Ques�ons
1. Corporate bonds can have lots of different features. To test your intui�on about how investors look at bonds consider how the presence of the following features would affect the price an investor would be willing to pay for a bond.
Sinking Fund: A sinking fund requires a company to set aside money over �me to re�re its bonds. For example, if a bond has a 20-year maturity, the company might begin se�ng aside funds in Year 11 so it will have a significant por�on of the principal accumulated before the bond matures. Some�mes the sinking fund amount is used to purchase bonds early, but for this ques�on assume that the fund is invested and the bonds are repaid at maturity. How would the existence of a sinking fund change the price investors are willing to pay for a bond? Call Feature: This allows a company to repurchase bonds before maturity. Usually callable bonds will have a no-call period (say the first five years) during which the company cannot repay the debt. For the next five years, the company may have to pay a premium to repay the bonds early. How would the existence of a call feature change the price investors are willing to pay for a bond? Think about when companies are most likely to want to repay bonds early. Collateral: Some loans are backed up with collateral; that is, the lenders have the first right to funds from specific corporate assets. We usually call such loans mortgages. How would the existence of collateral �ed to a bond change the price investors are willing to pay for the bond?
2. Suppose you purchase a bond with an 8% coupon rate for $1,000, which is the bond's face value. If you hold the bond to maturity, you will earn 8%. If you sell the bond before it matures, why might you not earn 8%?
3. Preferred stock has fixed dividend payments, though they are somewhat discre�onary. The dividends are not tax-deduc�ble like interest on debt. The dividend payout rate must be higher than the yield on a company's debt since the preferred stock has lower priority in bankruptcy. Why would companies issue such a security?
4. We developed a model to value stock based on dividends and dividend growth. If a stock doesn't pay any dividends does it have no value? Since there are many companies with valuable stock that don't pay dividends, we know the answer is "No." Explain why.
5. What type of investors would be a�racted to zero-coupon bonds? Why would they give up the interim coupon payments in favor of a larger payout at maturity? 6. Common stock is o�en modeled as a growing perpetuity in which the growth rate is constant. If the overall economy is expected to grow at 6% per year over the long run,
would it be reasonable to expect a firm's stock to grow at an average constant rate of 15% forever? Why or why not? (Hint: Imagine the economy as a pie that gets 6% bigger each year; then imagine the firm as a piece of that pie that gets 15% bigger every year.)
7. Consider the constant-growth model for valuing a share of stock:
Can you develop a rule regarding the maximize value of gN that can be used in the constant-growth model?
8. Nega�ve values for gN are allowable. Under what circumstances might a firm's growth be seen as nega�ve? How would stock price be affected as investors con�nue to lower
their expecta�ons about a company's growth prospects? Do the results from the constant-growth model match your intui�on about how the price would change? 9. If the constant-growth formula is applied to a stock whose growth rate is zero (gN = 0), what will the formula resemble? Will D1 differ from D0 from for a zero-growth stock?
Key Terms
Click on each key term to see the defini�on.
call premium (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
Amount in excess of par value that a company must pay when it calls a bond. It is the difference between the call price and the maturity value.
SLIDE 1 OF 7
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callable bond (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
A bond that gives the issuing corpora�on the op�on to repurchase the bond at a price above the bond's par value at some date prior to the bond's maturity.
constant-growth formula (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
A present value formula applied to stock valua�on where dividends are modeled as a growing perpetuity.
coupon rate (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
The fixed interest paid by a bond, stated as a percentage of par value.
dividend yield (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
The return due to dividends received equals the annual dividend divided by share price.
expected return (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
The probability weighted value of an investment, computed by assigning a probability of occurrence to the various possible future values.
growing perpetuity (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
An infinite cash flow stream that makes payments at regular intervals (e.g., monthly, annually, etc.), with each payment equaling its predecessor �mes a fixed growth factor.
hybrid security (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
Investments that combines features of both fixed claims and residual claims.
interest rate risk (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
The amount a bond will repay to the bondholder when it matures. Corporate bonds o�en have a face amount of $1.000.
preferred stock (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
The most common type of perpetuity and generally pays a fixed dividend.
refunding debt (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
Corpora�ons that call a bond issue usually finance the repurchase by issuing new bonds that carry lower yields.
selling at a discount (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
Selling the bond below par value.
selling at par (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
Selling the bond at the face amount.
selling at a premium (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
Selling the bond at a price above par.
yield to call (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
Calcula�on used to see what the return on bond investment is for callable bonds.
yield to maturity (YTM) (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
The yield of a debt security computed by considering its price and the �ming of all cash flows, iden�cal to an IRR (internal rate of return).
zero coupon bond (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro
Bonds issued by corpora�ons and the government which make no payments to bondholders un�l maturity. Processing math: 0%
Web Resources
PG&E 6% nonredeemable preferred price and volume data for May 2012: h�p://finance.yahoo.com/echarts?s=PCG-PA+Interac�ve#symbol=pcg- pa;range=5d;compare=;indicator=volume;char�ype=area;crosshair=on;ohlcvalues=0;logscale=off;source=undefined; (h�p://finance .yahoo.com/echarts?s=PCG- PA+Interac�ve#symbol5pcg-pa;range=5d;compare=;indicator=volume;char�ype=area;crosshair=on;ohlcvalues=0;logscale=off;source=undefined;)
List of CAT bonds: h�p://quicktake.morningstar.com/StockNet/bonds.aspx?Symbol=CAT&Country=USA (h�p://quicktake.morningstar.com/StockNet/bonds.aspx?Symbol=CAT&Country=USA)
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