Week 2 Journal
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chapter05.pdf
Learning Objectives
Upon completion of Chapter 5, you will be able to:
• Describe the significance of preferred stock perpetuities, and be able to determine their value using time value of money mathematics.
• Identify the significance of constant-growth common stock perpetuities, and be able to determine their value using time value of money mathematics.
• Describe the significance of non-constant-growth common stock perpetuities, and be able to determine their value using time value of money mathematics.
• Explain the significance of zero coupon bonds, and be able to determine their value using time value of money mathematics.
• Explain the significance of bonds and how factors such as coupon rate, interest rates, and maturity affect their value over time.
• Solve for the expected rates of return for these securities, given their market prices and cash flow characteristics.
5
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Security Valuation and Expected Returns
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CHAPTER 5Section 5.1 Stocks
In Chapter 4 we began the study of money’s time value. In this chapter we apply those basics to the valuation of securities (stocks and bonds) and to solving for expected returns from investing. Along the way, we present some of the terminology and fea- tures of corporate securities.
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 part of 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 valuation, as well as bond valuation, are also important top- ics for anyone who may wish to personally invest in such securities. The first part of this chapter introduces security valuation.
Solving for expected returns is the topic that concludes Chapter 5. When price is known, it may be helpful for the manager (or the investor) to estimate the return or yield that can reasonably be expected from a project or investment. Such an expected return can be com- pared to returns offered by competing projects or investments. An investor, for example, would never 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 timing of those cash flows, and their riskiness. The cash flow that a security holder receives is the principal benefit of ownership. With- out that benefit, the security would be nearly worthless. Cash flows from the firm to shareholders come in the form of dividends and the eventual sale of the security, and for bondholders the cash received comes in the form of coupon interest payment and the eventual maturity value of the bond. As we will demonstrate, shareholders also receive cash flows from other investors when they sell their stock at a (hopefully!) higher price. This price appre- ciation is due to the expectation of higher future dividends, making the claim on future cash flows more valuable.
Now we apply the time value of money techniques introduced in Chapter 4 to the valua- tion of commonly encountered securities.
5.1 Stocks
Stocks are securities that that give a purchaser an equity interest in a business. There are two types of stocks—preferred and common—and we will discuss these in this section. Shares of stock are issued by a corporation, each share representing an inter- est in the corporation and a claim on its profits and cash flows. When shares are first sold by the business to raise funds, the company engages in what is known as an initial public offering (an IPO). IPOs are said to be sold in the primary market. During the lives of these shares, they may be bought and sold hundreds of times in trading between investors. This type of trading is done on what are known as secondary markets. A great majority of stock trading takes place in secondary markets, and these include some of the most well known markets in the world, including the New York Stock Exchange (NYSE),
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CHAPTER 5Section 5.1 Stocks
the National Association of Securi- ties Dealers Automated Quotation system (NASDAQ), and the London Stock Exchange (LSE). Secondary markets are extremely important because the continuous trading of shares among investors ensures that prices are a good reflection of value, as the traders compete in try- ing to identify mispriced shares. Bonds, which are discussed later in the chapter, also are traded on sec- ondary markets. It is interesting to consider other familiar secondary markets in which assets are traded after their initial sales (eBay, Craig- slist, and classified newspaper advertisements, among others).
Preferred Stock
The most familiar type of perpetuity is preferred stock. Preferred stock generally pays a fixed dividend. Thus, “$8 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) and residual claims (stocks). It is fixed in the sense that the amount the issuing corporation is obli- gated to pay does not vary; in this case it is $8 once every year. Preferred stock is residual because the dividend need not be paid unless the corporation has cash flows left over once all other fixed claims (such as interest on bonds) have been paid. Preferred claims have a lower priority than do 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. The present value of a perpetuity formula is used to find the price of an $8 share preferred stock. The interest rate equals 16% in the example.
We start with the present value formula from Chapter 4,
(4.19) PV0 5 CF r
where CF is cash flow and r is the periodic interest rate. For a preferred stock we rewrite this as
(5.1) P0 5 D r
C. Covert Darbyshire/The New Yorker Collection/www.cartoonbank.com
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CHAPTER 5Section 5.1 Stocks
We have re-expressed PV 0 5 CF/r as P
0 5 D/r because today’s price (P
0 ) is equal to the
present value of future cash flows (PV 0 ), and the preferred stock’s dividend (D) is the per-
petuity’s cash flow (CF). Substituting in our numerical values, we get
P0 5 $8
0.16 5 $50.00
The price is $50 per share.
Common Stock: Constant-Growth Case
Common stock, unlike preferred, does not pay dividends that are a constant amount through time. On the other hand, common dividends are equally spaced in time and often continue indefinitely. Common stock, therefore, satisfies all the perpetuity criteria 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 PV
0 5 P
0 and CF
1 5 D
1 , CF
2 5 D
2 , and
so on, we may re-express the formula in terms of the price (P 0 ) and dividends (D
1 , D
2 , . . .)
of the common stock:
(5.2) P0 5 D1
11 1 r2 1 1 D2
11 1 r2 2 1 D3
11 1 r2 3 1 c5 a n 5 1
` Dn
11 1 r2 n
Clearly, it is impossible to solve this equation 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 assumption that successive dividends grow at a constant rate. We call this a growing perpetuity.
Let that rate be termed g N , the long-run normal growth rate of dividends. The dividends
may be expressed as (1 1 g N ) times the preceding year’s dividend payment. If D
0 is the last
dividend paid by the firm, then the dividend for the upcoming year is
D 1 5 D
0 (1 1 g
N )
A common source of error in valuing constant growth stocks is confusion between D 0 and
D 1 , which are the last divided (past tense) and the next dividend (future tense), respec-
tively. Pay careful attention to this distinction as you read problems and use the formula we are developing here to solve them.
The dividend for the second year is
D 2 5 D
1 (1 1 g
N ) 5 D
0 (1 1 g
N )2
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CHAPTER 5Section 5.1 Stocks
and the dividend for the third year is
D 3 5 D
2 (1 1 g
N ) 5 D
0 (1 1 g
N )3
Substituting into Equation (5.2) yields a geometric series:
(5.3) P0 5 D0 11 1 gN 2 11 1 r2 1
D0 11 1 gN 2 2 11 1 r2 2 1
D0 11 1 gN 2 3 11 1 r2 3 1
c5 a n 5 1 `
D0 11 1 gN 2 n 11 1 r2 n
Mathematicians 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,
(5.4) P0 5 D0 11 1 gN 2
r 2 gN 5
D1 r 2 gN
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
P0 5 $5.0011.062 0.13 2 0.06
5 $5.30 0.07
5 $75.71
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 $5.00(1.02)/(0.13 2 0.02). Had the growth rate been zero (like a share of preferred stock) the value today would be just $5.00/0.13 5 $38.46. You can see that the additional growth has a large effect on the value of the stock.
Common Stock: Non-Constant-Growth Case
The constant-growth valuation model works well for securities whose forecasted finan- cial behavior corresponds to the model’s assumption of dividends that grow at a con- stant rate. Some companies, such as electric utilities, compete in mature markets that offer few prospects for rapid growth. Demand for their product is pretty stable, varying little with economic cycles. Such firms may be good candidates for valuation using the constant-growth model.
For many corporations, however, the constant-growth assumption does not hold. Often firms have new products that have competitive advantages over their competitors’ prod- ucts. Patent protection, new technology, low-cost production methods, and brand name
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CHAPTER 5Section 5.1 Stocks
recognition may enable a firm to experience rapid growth for a period of time. In the long run, though, this rapid growth is not sustainable, as competitors’ technology, manufactur- ing efficiency, and so on, catch up with the industry leader’s, leveling the playing field in the marketplace. A constant-growth valuation model is clearly inappropriate for firms that experience a period of nonconstant growth.
One method for valuing firms in a non-constant-growth cycle is presented here. Let’s assume that we are valuing a stock whose dividends are expected to grow at an 18% rate each of the next 3 years. After this abnormal growth period, normal growth will continue at a 5% annual rate. The company’s last annual dividend was $2.00 per share. The dis- count rate for the stock is 16%. The timeline in Figure 5.1 illustrates the growth assump- tions of this example.
Figure 5.1
In Figure 5.1
g A 5 abnormal growth rate 5 18%
A 5 length of abnormal growth period 5 3 g
N 5 normal or constant growth rate 5 5%
Because today’s price should equal the present value of future dividends, the first step is to find the size of these dividends. If D
0 5 $2.00 last dividend paid, then
D 1 5 $2.00(1.18) 5 $2.36 (in year 1 dividends grow at 18%)
D 2 5 $2.36(1.18) 5 $2.78 (in year 2 dividends grow at 18%)
D 3 5 $2.78(1.18) 5 $3.28 (in year 3 dividends grow at 18%)
and
D 4 5 $3.28(1.05) 5 $3.44 (in year 4 dividends grow at 5%)
Dividends grow at 5% from D 4 onward.
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 (as we can easily see in Figure 5.2). This means that the assumptions of the constant-growth valuation model are met from period 3 onward. We can, therefore, solve for P
3 , the stock’s price at time 3, using the
constant-growth model. This value, P 3 , incorporates the value of all the dividends from
time 3 onward. P 3 includes the present value of D
4 , D
5 , D
6 , and so on. Recognizing this
gives us a strategy for solving for P 0 , the current price.
0 2 31 4
t = 0 A = 3
g A = 0.18 gN = 0.05
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CHAPTER 5Section 5.1 Stocks
Figure 5.2
We start with
P0 5 D1
11 1 r2 1 1 D2
11 1 r2 2 1 D3
11 1 r2 3 1 D4
11 1 r2 4 1 D5
11 1 r2 5 1 c
but
P3 5 D4
11 1 r2 4 1 D5
11 1 r2 5 1 c
so
(5.5) P0 5 D1
11 1 r2 1 1 D2
11 1 r2 2 1 D3
11 1 r2 3 1 P3
11 1 r2 3
Note that P 3 is discounted for three periods, because it is the price as of period 3 on our
timeline. We already know the value of D 1 , D
2 , D
3 , and r, so these values may be substi-
tuted into Equation (5.5):
P0 5 $2.36 1.16
1 $2.78 11.162 2 1
$3.28 11.162 3 1
P3 11.162 3
To solve for P 3 , recall the constant-growth formula from the prior section:
P0 5 D1
r 2 gN
0 1 2 3 4 5 6 7 8
D iv id e n d
Year
1
2
3
4
5
D 0
D 1
D 2
D 3
D 4
D 5
D 6
D 7
D 8
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CHAPTER 5Section 5.1 Stocks
This equation is solved for P 0 using D
1 because the constant-growth assumption held from
time 0 onward. In this example the constant growth holds from time 3 onward, so we can adjust the formula relative to time 3 and solve for P
3 , giving us
(5.6) P3 5 D4
r 2 gN
Substituting in numbers we get
P3 5 $3.44
0.16 2 0.05 5
$3.44 0.11
5 $31.27
We now have all the values we need to solve for P 0 , the current price of the stock. Therefore
P0 5 $2.36 1.16
1 $2.78 11.162 2 1
$3.28 11.162 3 1
$31.27 11.162 3
5 $2.03 1 $2.07 1 $2.10 1 $20.04 5 $26.24
This price, $26.24, accounts for the present value of all future dividends. The present val- ues of D
0 , D
1 through D
3 are solved for explicitly. The present values of D
4 , D
5 , D
6 , and so
on are solved for implicitly by finding the present value of P 3 . P
3 is able to incorporate the
values of all dividends after time 3 because dividends grow at a constant rate from time 3 onward.
This method may be generalized in the following formula:
(5.7) P0 5 a t 5 1 A Dt
11 1 r2 t 1 PA
11 1 r2 A
where
(5.8) PA 5 DA 1 1
r 2 gN
Here A 5 the number of years until constant growth begins, and
D A11
5 D A (1 1 g
N ) 5 D
0 (1 1 g
A )A (1 1 g
N )
In other words, D 1 , D
2 , …, D
A are the dividends during the non-constant-growth period.
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CHAPTER 5Section 5.2 Bonds
Demonstration Problem 5.1: Stock Valuation
Suppose a stock just paid a dividend of $1.75 per share. Next year the dividend is expected to grow at a 25% rate. After next year, dividends are expected to grow at a normal rate of 6.5%. If investors require a 16% return for this stock, estimate the stock’s price.
Solution
First prepare a timeline:
Because growth is normal from t 5 1 onward, the present value of all future dividends relative to time 1 (D
2 , D
3 , D
4 , etc.) can be found by solving for P
1 using the constant-growth model.
t = 0 t = 1 t = 2 t = 3
D 1
D 3
D 2
g A = 25%
g N = 6.5%
5.2 Bonds
Bonds are publicly traded long-term securities. The bond issuer (usually a corpora-tion or government agency) agrees to pay a fixed amount of interest, called coupon payments, over a specified time period. When the bond “matures” (reaches the end of its specified time period), the issuer repays the fixed amount of the principal invest- ment. Bonds can be categorized into three types: zero coupon bonds, bonds with coupons, and callable bonds. These are discussed in this section.
Zero Coupon Bonds
Corporations and the government sometimes issue bonds known as zero coupon bonds. These bonds differ from typical bonds in that they make no payments to the bondholders until 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 the selling price of the bond be? The problem is illustrated with the following timeline shown in Figure 5.3.
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CHAPTER 5Section 5.2 Bonds
Try It: Calculator Key Strokes and Excel Functions—Price of a Zero Coupon Bond
TI Business Analyst
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?
1000 [FV] This enters $1,000 as the future value.
8 [I/Y] The discount rate is entered as a percentage using the [I/ Y] key.
20 [N] The number of periods until maturity: 20.
0 [PMT] There are no payments; therefore “0” is entered.
[CPT] [PV] These keystrokes tell the calculator to compute the present value.
Answer: $214.55.
Note: Similar to what happens in Excel, the PV is displayed as a negative. Also, you may enter the keystrokes in any order you wish so long as you enter [CPT][PV] at the end.
Excel
Use the PV function. The inputs for this function are:
5PV(Rate%,NPER,PMT,FV,Type)
Solution: 5PV(8%,20,0,1000,0) 5 (214.55)
The answer displayed is negative (in parentheses or red or signed negative) because that is how much the investor will pay today (an outflow or negative cash flow) to receive $1,000 in 20 years. Remember that Excel requires an outflow and an inflow (that is, a cash flow signed positive and a cash flow signed negative). When we entered the positive FV of $1,000, that meant that the PV had to be negative.
Caution: Remember that numbers cannot be entered with commas separating thousands of dollars because commas separate inputs in Excel functions.
Figure 5.3
We use Equation (4.14) from Chapter 4 to find the current value:
(4.14) PV 0 5 FV
n (1
1 r)2n or FV
n /(1 1 r)n
5 $1,000(1.08)220 5 $1,000(0.21455) 5 $214.55
n = 0 n = 20
PV 0 = price = ? FV20 = $1,000
r = 8%
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CHAPTER 5Section 5.2 Bonds
The secondary market for zero coupon bonds is very active. Suppose one is selling a zero coupon bond for $425, maturing in 14 years, at which time 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 characteristics. The timeline is shown in Figure 5.4.
Figure 5.4
To solve for r, either Equation (4.11) or Equation (4.14) from Chapter 4 could be used. Using
(4.11) FV n 5 PV
0 (1 1 r)n
and substituting in for the future and present values, we have
$1,000 5 $425(1 1 r)14
Rearranging gives
11 1 r2 14 5 1,000 425
5 2.352941
so
1 1 r 5 (2.352941)1/14
or
r 5 (2.352941)1/14 2 1
Solving for r gives
r 5 0.06303
Therefore
YTM 5 6.303%
This bond is expected to yield 6.303% if held to maturity.
n = 0 n = 14
PV 0 = $425.00 FV14 = $1,000
r = YTM = ?
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CHAPTER 5Section 5.2 Bonds
Bonds with Coupons
Bond investors receive both a stream of coupon interest payments over the life of the bond and a payment of par value at maturity from the corporations in which they invest. 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 5.5.
Figure 5.5
In the figure, m is the number of coupon payment periods until the bond matures. For bonds paying coupons semiannually, m is twice the number of years until maturity. Every semiannual coupon payment equals 1/2 the coupon rate times the bond’s par value, so a bond with an 8% coupon rate would make two $40 interest payments every year (8% 3 $1,000)/2. The timeline may be used to find the present value (the price) of a bond, as illustrated in Figure 5.6.
n = 0
Coupon + Par Value
2
Coupon
m – 1
Coupon
1
Coupon
m
Try It: Calculator Key Strokes and Excel Functions—Yield to Maturity of a Zero Coupon Bond
The 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 bond that pays $1,000 when it matures in 14 years?
TI Business Analyst
14 [N]
0 [PMT]
425 [1/2] [PV]
1000 [FV]
[CPT] [I/I]
Note that, like the Excel keystrokes, either the price (PV) or the par value (FV) has to be negative in order to “tell” the calculator that one cash flow is going to the firm and one is going to the investor.
Excel
Use the Rate function with the format: RATE(NPER,PMT, PV, FV, TYPE, GUESS).
5RATE(14,0,2425,1000,0,10%) 5 6.303%
Note: One of the cash flows is negative (the $425) and the other is positive ($1,000). There is no comma or dollar sign in the $1,000 in Excel. The TYPE is zero because we assume interest accrues at the end of the period. GUESS can be left out, or enter something that seems reasonable. The display of the answer can be adjusted to show more or fewer decimal places by formatting the cells.
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CHAPTER 5Section 5.2 Bonds
Figure 5.6
Therefore, 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 formula for solving for a bond’s value is
(5.9) PV0 5 1coupon2 11 2 31/ 11 2 r2 m 4 2
r 1
par value 11 1 r2 m
Keep in mind that r is the investors’ required return for the bond (the discount rate per payment period).
Let’s look at an example. A bond that carries an annual coupon rate of 6.5%, makes cou- pon 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) 5 20.
PV0 5 132.502 11 2 31/ 1 1.06) 20 4 2
0.06 1
$1,000 11.062 20
5 $372.77 1 311.80 5 $684.57
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. The reason underlying such pricing differences is the relationship 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. To market the bond, the bondholder must lower the price until
n = 0
+ PV of par value
PV of coupons
Total present value = price of bond
n = 2
Coupon
n = 19
Coupon
n = 1
Coupon
n = 20
0
Par Value
2 19
Coupon
1 m = 20
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CHAPTER 5Section 5.2 Bonds
Try It: Calculator Key Strokes and Excel Functions—Price of a Bond
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?
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 positive because they are both cash inflows for inves- tors, and therefore the answer for PV will be negative because it will be the price investors are willing to pay. Of course, one could make both the FV and PMT negative and the answer would be positive, taking the cash flows from the firm’s perspective.
Excel
Excel Solution: Use the PV functions with the following inputs:
RATE 5 6% (This is the semiannual version of the 12% annual discount rate.)
NPER 5 20 (There are 20 semiannual periods in 10 years.)
PMT 5 $32.50 (This is the semiannual interest payment to investors.)
(5 $1,000 3 6.5%/2)
FV 5 $1000 (This is the par or face value of the bond that is repaid at maturity.)
TYPE 5 0 (Interest payments are paid at the end of periods after interest has had time to accrue.)
5 PV (6%,20,32.50,1000,0)
Display shows ($684.577). This is signed negative because this is the amount the investor will pay to purchase the future promised payments.
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, offsetting a portion of their return from the coupon payments. A capital gain or loss becomes part of the bond’s return to investors.
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CHAPTER 5Section 5.2 Bonds
Demonstration Problem 5.2: Bonds and YTM
A bond sells for $1,250, matures in 10 years, and has an annual coupon rate of 14%, payable semi- annually. What is the bond’s YTM?
Solution
We begin with a timeline and by setting up the problem:
Since
Bond Price 5 (PV of Annuity) 1 (PV of Single Cash Flow)
we have
$1,250 5 1$702 11 2 31/ 11 1 r2 20 4 2
r 1
1,000 11 1 r2 20
Table Solution
$1,250 5 $70(PVIFA n 5 20
) 1 $1,000 (PVIF n 5 20
) r 5 ? r 5 ?
Solving this equation requires using trial and error. If 6% were the initial guess, then the problem would proceed as follows:
$1,250 ,. $70(PVIFA n 5 20
) 1 $1000 (PVIF n 5 20
) r 5 6% r 5 6%
$1,250 ,. $70(11.470) 1 $1,000(0.3118) $1,250 ,. $802.90 1 $311.80 $1,250 . $1,114.70
So, 6% was too high. If 5% were the second guess, the two sides would be equal, and the problem would be solved. However, because the problem was set up using semiannual payments, the 5% yield must be doubled to arrive at the annual yield to maturity of 10%. (continued)
n = 0
$1,000
$70 $70 $7n = 20
r = ?
n = 19n = 1
P 0 = $1,250
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CHAPTER 5Section 5.2 Bonds
Demonstration Problem 5.2: Bonds and YTM (continued)
Calculator Solution
Clear Registers
21250 [PV]
70 [PMT]
1000 [FV]
20 [N]
[I%]
The answer given is 4.9948, but this is based on semiannual coupon payment. The yield must be doubled to obtain an annual yield to maturity of 9.9896%.
Bond Prices and Interest Rates
Bonds are useful for illustrating the relationship between the time 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 initially 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 dramatically, 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 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 (Excel formula: 5 PV(6%,40,50,1000,0) 5 $849.54). If rates had dropped to 8%, for example, the bond would sell for $1,197.93 (Excel formula 5 PV(4%, 40,50,1000,0) 5 $1197.93). Thus the price of a bond moves in the opposite direction from interest rates.
Now let’s consider a bond identical to the one just described, except that 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. If we solve for the price of the bond using a 12% discount rate and an 8% discount rate, the prices are, respectively, $838.39 and $1,226.23.
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 sensitive 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 inter- est 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 (see Try It: Calculator Key Strokes and Excel Functions—Where Do These Bond Prices Come From for an explanation of the rating of bond risk).
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CHAPTER 5Section 5.2 Bonds
Try It: Calculator Key Strokes and Excel Functions—Where Do These Bond Prices Come From?
We can find these bond prices easily in Excel:
Excel formula: 5PV(6%,40,50,1000,0) 5 $849.54
Excel formula: 5PV(4%,40,50,1000,0) 5 $1197.93
The table shows how the prices of bonds vary with maturity and market interest rates. Notice that the longer the maturity, the greater the price change owing to shifts in market interest rates.
Prices of a 10% Coupon (semiannual payments) Bond
Market rate
Years to maturity 6.00% 8.00% 10.00% 12.00% 14.00%
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: (1) Bond prices move in the opposite direction as movements in interest rates; and (2) The longer the maturity of the bond, the greater the change in its price for a given change in rates. This relationship is illustrated in Figure 5.7, which shows a teeter-totter. When the interest rate side goes down, the price side goes up. The length of the right side of the teeter-totter may be thought of as the time until the bond matures. The longer the right side, the greater the movement in price for a given move- ment in interest rates.
Figure 5.7: The interest rate-bond price teeter-totter
Interest rates
Bond prices
i = 12%
i = 1 0%
P = $1,0
00 P =
$1,0 00
20 y ears
30 y ears
P = $849.54 P = $838.39
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CHAPTER 5Section 5.2 Bonds
Therefore, the longer the maturity, the more risk there is of a large adverse price change. This risk is termed interest rate risk.
Callable Bonds
Many bonds are callable. With callable bonds, a call feature is included that gives the issu- ing corporation the option to repurchase the bond at a price above the bond’s par value at some date prior to the bond’s maturity. These features are attractive to corporations because if the firm issues bonds at a time when interest rates are high, the call allows the company to repurchase the bonds early and avoid high interest payments in the future. Corporations 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 call- able bond carries the risk that the corporation may repurchase the bond prior to maturity and the bondholder, therefore, will not collect the high interest payment for the length of time initially anticipated. Thus, the yield to maturity for a callable bond may be mislead- ing, 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 that the time until call is substituted for the time until matu- rity (thus the number of coupon payments is reduced), and the call price is substituted for the call premium (thus the ending cash flow is greater than par).
Let’s demonstrate finding the yield to call by using a bond that sells for $800, pays cou- pons semiannually, matures in 10 years, carries a 9% coupon rate, and has a $1,000 par value. We assume that the call price is $1,100, and the call date is 5 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 dis- count 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 dis- count, meaning that market rates are currently above the coupon rate offered by the bond. Thus, the corporation would not choose to refund such an issue because it would have to issue bonds carrying a higher yield to replace these existing bonds. It is cheaper for the corporation to let the bonds mature rather than to call them. This is the case whenever the yield to maturity is below the yield to call.
Most callable bonds have a period during which they cannot be redeemed, usually the first 3 to 7 years after 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 pay- ment penalty. Usually the call premium is an extra year’s interest.
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CHAPTER 5Section 5.3 Solving for Expected Returns
5.3 Solving for Expected Returns
In the preceding section we solved for the value of preferred stock, common stock, and bonds. Many issues of these securities are actively traded in financial markets. It is often more useful for investors to solve for the returns they might expect to realize from an investment in such securities than to solve for their value. After all, prices are generally known in the marketplace, so investors would be more interested in expected returns on competing securities than in prices. Similarly, corporate managers can com- pare expected returns from prospective left-hand-side (asset-side) 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 substitute the known quantities into the perpetuity formula:
(5.1) P0 5 D r
Solving for r gives
r 5 D P0
5 $4.50
$53.50 5 0.0841
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 Equation (5.4), if we assume the stock’s dividends will grow at a constant rate:
(5.4) P0 5 D1
r 2 gN
We can rearrange this equation in several ways:
(5.10) (r 2 g N )P
0 5 D
1
(5.11) r 2 gN 5 D1 P0
(5.12) r 5 D1 P0
1 gN
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CHAPTER 5Section 5.3 Solving for Expected Returns
Equation (5.12) 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 can be calculated as follows:
r 5 D1 P0
1 gN
5 $3
$35 1 0.06
5 0.1457
Therefore r is 14.57%.
The second use of Equation (5.12) is to illustrate the sources of the expected return. The first term to the right of the equals sign in Equation (5.12) is the dividend yield, D
1 /P
0 . The
second term, g N , 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 appreciation.
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. From Equation (5.9) we have
$800 5 1$452 11 2 31/ 11 1 r2 20 4 2
r 1
$1,000 11 1 r2 20
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%. Then
1$452 11 2 31/ 11.0552 20 4 2 0.055
1 $1,000 11.0552 20
5 $880
Because $880 is above the actual price, we know we must raise the interest rate, lowering the value of the right side of the equation. This time let’s try 6%. Using 6%, we get a value of $827.95, which is still too high but closer. Not let’s try 6.25%. This time the answer is $803.29, which is 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.
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CHAPTER 5Section 5.3 Solving for Expected Returns
Try It: Calculator Key Strokes and Excel Functions—Corporate Bond YTM
What is the YTM (yield to maturity) of a bond with a 9% coupon rate and $1,000 face value, if its price today is $800, and it pays interest semiannually?
TI Business Analyst
20 [N]
45 [PMT]
800 [1/2] [PV]
1000 [FV]
[CPT] [I/Y]
5 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% 5 6.284 3 2.
Excel
5RATE(20,45,2800,1000,0,5%) 5 6.284%
This is the semiannual rate, so the annual return (yield to maturity) is 12.568%.
GUESS can be left out as in: 5RATE20,45,2800,1000,0,).
Investment Projects and Internal Rate of Return
We now assume the role of a corporate manager considering an investment in a project. Suppose we are considering buying a delivery truck and estimate the truck will produce cash flows of $3,000 a year for the next 5 years. At the end of the 5-year period, we expect to sell the truck for $5,000. The truck costs $13,500. What is the expected return on the truck investment? First, let’s illustrate the cash flows using the timeline in Figure 5.8.
Figure 5.8
The initial cash flow (CF 0 ) is $13,500, indicating the cost of the truck. Subsequent cash
flows are five annual receipts of $3,000 each, and one additional receipt of $5,000. Cash flow 5 (CF
5 ) is broken down to show both the $3,000 cash flow generated by operating the
truck during the fifth year and the $5,000 generated by the sale of the vehicle at the end of year 5. These cash flows represent a 5-year ordinary annuity of $3,000 per year and a single cash flow of $5,000.
n = 0
CF 0 =
– $1
3, 50
0
CF 1 =
$ 3,
00 0
CF 2 =
$ 3,
00 0
CF 3 =
– $3
,0 00
CF 4 =
$ 3,
00 0
CF 5 =
$ 3,
00 0
+ $5
,0 00
n = 2 n = 3n = 1 n = 4 n = 5
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CHAPTER 5Section 5.3 Solving for Expected Returns
Therefore, solving for the expected return from the truck investment is similar to finding the YTM on a bond. We must combine the annuity and single cash flow formula and use trial and error to find the rate that equates the present value of the future cash flows to the truck’s cost. Substituting into Equation (5.9) gives
$13,500 5 1$3,0002 11 2 31/ 11 1 r2 5 4 2
r 1
$5,000 11 1 r2 5
By arbitrarily guessing 10% as our initial estimate of r, we get
1$3,0002 11 2 31/ 11.102 5 4 2 0.10
1 $5,000 11.102 5
5 $11,372 1 $3,105 5 $14,477
It is apparent that 10% is too low a discount rate because $14,477 is greater than $13,500. The correct answer is r 5 12.4%. Thus, the expected return on the investment in the truck is 12.4%. The expected return from a project in which a firm is considering investing is also known as the project’s internal rate of return (IRR). (Note that computing IRRs is similar to computing YTMs, but convention gives the IRR of a bond the special name, “YTM.”) In this case the truck’s IRR is 12.4%.
If the firm could acquire capital from investors whose required return was less than the project’s IRR, then the firm should make the investment. For example, assume the firm could acquire financing at an 11% rate (investors’ required return on their investment). If so, the firm should proceed with the investment because its yield (12.4%) is more than the cost of funding the project (11%).
Try It: Calculator Key Strokes and Excel Functions—Truck Investment
TI Business Analyst
13500 [1/2][PV]
3000 [PMT]
5000 [FV]
5 [N]
[CPT] [I/Y]
Excel
5 RATE(5,3000,213500,5000,0,10%) 5 12.39%
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CHAPTER 5Key Terms
Summary
Chapter 5 has applied the time value skills from Chapter 3 to the valuation of cor-porate securities. Pricing preferred stock was shown to be an application 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 combination 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). Variations of the formulas were also used to solve for the expected returns of traded securities.
The ability to express equivalent values of cash flows at different points in time is a funda- mental skill in finance. As with any skill, practice increases proficiency and understanding in solving time value problems.
Key Terms
callable bonds Gives the issuing corpora- tion the option to repurchase the bond at a price above the bond’s par value at some date prior to the bond’s maturity.
call premium A bonus paid to investors over the par value of a callable bond when the security is called by the issuer.
capital gain A gain from the price appre- ciation of an asset.
constant-growth stock A stock whose valuation is found by applying the present value formula where dividends are mod- eled as a growing perpetuity.
coupon rate The fixed interest paid by a bond, stated as a percentage of par value.
dividend yield The return due to divi- dends received equals the annual dividend divided by share price.
expected returns The probability weighted value of an investment, com- puted by assigning a probability of occur- rence to the various possible future values.
growing perpetuity An infinite cash flow stream that makes payments at regular intervals (e.g., monthly, annually, etc.), with each payment equaling its predeces- sor times a fixed growth factor.
hybrid security An investment that combines features of both fixed claims and residual claims.
interest rate risk A measurement of the sensitivity of bond prices to changes in rates.
internal rate of return (IRR) The discount rate that equates the present value of an investment’s future cash flows with the investment’s cost.
par value (face amount) The amount a bond will repay to the bondholder when it matures. Corporate bonds often have a face amount of $1,000.
preferred stock The most common type of perpetuity; generally pays a fixed dividend.
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CHAPTER 5Key Formulas
refunding debt Corporations that call a bond issue usually finance the repurchase by issuing new bonds that carry lower yields.
selling at a discount Selling a bond below par value.
selling at a premium Selling a bond at a price above par.
selling at par Selling a bond at its face amount.
yield to call Calculation used to see what the return on bond investment for callable bonds will be.
yield to maturity (YTM) The yield of a debt security computed by considering its price and the timing of all cash flows, iden- tical to an IRR (internal rate of return).
zero coupon bond Bonds issued by corpo- rations and the government for which no payments are made to bondholders until maturity.
Key Formulas
Price of preferred stock:
(5.1) P0 5 D r
Price of common stock, constant-dividend growth formula:
(5.4) P0 5 D0 11 1 gN 2
r 2 gN 5
D1 r 2 gN
Price of common stock, abnormal growth period formula:
(5.7) P0 5 a t 5 1 A Dt
11 1 r2 t 1 PA
11 1 r2 A where
(5.8) PA 5 DA 1 1
r 2 gN
and A is the number of years until constant growth begins.
Price of bonds:
(5.9) PV0 5 1coupon2 11 2 31/ 11 1 r2 m 4 2
r 1
par value 11 1 r2 m
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CHAPTER 5Critical Thinking and Discussion Questions
Web Resources
Research PG&E nonredeemable preferred stock price and volume data for May 2012 at http://finance.yahoo.com/echarts?s=PCG-PA+Interactive#symbol=pcg-pa;range=5d;comp are=;indicator=volume;charttype=area;crosshair=on;ohlcvalues=0;logscale=off;source=u ndefined.
For a list of Caterpillar bonds, visit http://quicktake.morningstar.com/stocknet/bonds.aspx?symbol=cat.
Critical Thinking and Discussion Questions
1. Common stock is often modeled as a growing perpetuity in which the growth rate is constant. a. 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 price 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.)
b. Can you generalize the result of part a to a rule regarding the maximum value of g
N that should be used in the constant-growth model?
c. Negative values for g N are allowable. Under what circumstances might a
firm’s growth be seen as negative? d. If the constant-growth formula is applied to a stock whose growth rate is zero
( g N = 0), what will the formula resemble? Will D
1 differ from D
0 for a
zero-growth stock? 2. In efficient markets, the price of an asset equals its value. Efficient markets are
characterized by lots of competition among traders and lots of information being processed by traders. If the New York Stock Exchange is considered an efficient market and you calculate the expected return for an NYSE stock, how do you think that expected return will compare to investors’ required return for the stock?
3. Explain the relationship between investors’ return requirements and security values. If returns required by investors increase, then what is likely to happen to the prices of securities?
4. Explain why the perpetuity and growing perpetuity valuation formulas do, in fact, incorporate the present values of all these securities’ future cash flows.
5. Complete the following table by substituting for the question marks in each column. In columns 1 and 2 use ,, 5, or .; in column 3 use “par” or “discount.”
YTM . coupon rate Price ? par Bond sells at ?
YTM ? coupon rate Price 5 par Bond sells at ?
YTM ? coupon rate Price ? par Bond sells at a premium
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CHAPTER 5Practice Problems
Practice Problems
1. Zeta Enterprises’s common stock dividend is expected to grow at a long-run rate of 5% per year. The dividend recently paid was $1.40 per share. Investors require a 12% return from Zeta’s common stock. a. What is your estimate of its price? b. If Zeta’s dividends grow at 7% instead of 5%, what will be Zeta’s price? c. If investors’ required return on Zeta also increases by 2% (from 12% to 14%),
just as its growth rate increased from 5% to 7%, what will be the impact on Zeta’s share price?
2. Consider the three bonds described here.
Bond X Bond Y Bond Z
Maturity 10 years 10 years 20 years
Annual coupon rate (payable semiannually)
6% 8% 6%
Par value $1,000 $1,000 $1,000
a. If all three bonds have a required return of 8%, what will be each bond’s price? b. Which bonds are selling at a discount? At a premium? At par? c. If required returns on these bonds all rise to 10%, what are their new prices? d. If Bond X were selling for $1,163, what would its yield to maturity be?
3. To develop a newly discovered gold deposit would cost $10 million (payable immediately). It is estimated that, once developed, the mine would generate cash flows of $150,000 the first year (which would be low because most of the year would be spent in developing the mine), followed by cash flows of $12 million and $5 million. These cash flows are assumed to occur at the end of each year. After the third year, the mine would be worthless. What internal rate of return (IRR) would this mine generate?
4. A corporation can raise $2,112,000 in capital by issuing 5,000 zero coupon bonds, each having a $1,000 par value. The bonds will mature in 10 years (when the firm must repay $5,000,000). What rate of return are investors requiring on such bonds?
5. An 11-year bond pays interest of $42.50 semiannually and is selling for $862. Assume the face value is $1,000. a. What is its coupon rate and its yield to maturity? b. The bond’s current yield equals its annual coupon payment divided by the
current price of the bond. What is the current yield for the bond? 6. Acme paid a $2/share dividend yesterday. Its dividends are expected to grow
steadily at 8% per year. a. What are Acme’s dividends expected to be for each of the next 3 years? b. If the appropriate discount rate for Acme’s stock is 12%, what is your estimate
of its current price (P 0 )?
c. What is your estimate of the stock’s price 1 year from now (P 1 )?
d. If you buy the stock today (and pay P 0 for it) and hold the stock for 1 year, sell-
ing it for P 1 , what return would you realize?
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CHAPTER 5Practice Problems
7. Buggy Whip Industries is a company in a declining business. Dividends are expected to decrease at a 10% rate. Last year’s dividend was $3 per share. The required return for Buggy Whip’s stock is 18%. a. What would you pay for a share of the stock? b. If Buggy Whip’s stock sold for $13.50, what return must investors be requiring
of the security? 8. The Finance Club bought a soft drink vending machine to place in the lobby of
the business school building. The machine cost the club $2,500. The club esti- mates that the machine will last many decades and produce a cash flow of $375 next year, and that this cash flow will grow at a 6% rate for the foreseeable future (as soft drink prices increase with inflation and as student enrollments grow). What is the expected return for the club from the soft drink vending investment (i.e., what is its IRR)?
9. Beach Master Suntan Oil’s dividend is expected to grow at a 20% rate each of the next 2 years. After that, dividend growth is expected to normalize at about 6.5% annually. Beach Master just paid a $1.25 annual dividend per share. Investors require a 17% return on Beach Master’s stock. a. What is the forecasted dividend for each of the next 2 years (D
1 and D
2 )?
b. What is the forecasted dividend 3 years from now? c. At what price do you foresee Beach Master’s stock selling 2 years from now
(what’s your forecasted P 2 )?
d. What is the present value of D 1 , D
2 , and P
2 ?
e. What’s your estimate of today’s price? f. Suppose you revise your estimates for Beach Master and now expect the 20%
dividend growth to last 4 years. If all other assumptions remain the same (g
N 5 6.5% and r 5 17%), what’s your new estimate of Beach Master’s current
stock price? 10. Delta Computer Graphics Inc. currently pays no dividend. Profitability has been
phenomenal, and investors expect that once a dividend is initiated, it will be large. Investors believe that Delta’s first dividend will be $8 per share and will begin being paid 4 years from now. Furthermore, they expect these dividends to grow at an 8.5% annual rate once the firm begins the payments. If investors require a 14% annual return on Delta’s stock, what’s the current price per share?
11. Down the Drain Plumbing Equipment has raised capital by issuing bonds and stock. The bonds have 22 years left until they mature. They pay coupons semian- nually at an 8.5% coupon rate. These $1,000 par value bonds are currently selling for $675.52. Down the Drain’s stock paid a dividend of $2.75 a share last year. Its dividends are supposed to grow at a 14% annual rate for the next 2 years, followed by normal growth of 5% annually. The stock’s current market price is $49.27 per share. What is the expected return (YTM) for Down the Drain’s bonds, and what is the expected return for Down the Drain’s stock?
12. An analyst has made the following explicit estimates of Hee Haw Bridle Com- pany’s future dividends.
t 1 2 3 4 5
Dividend $1.25 $1.37 $1.40 $1.45 $1.50
After 5 years, Hee Haw is expected to stop growing and pay a constant dividend of $1.50 forever. If investors require a 10% return from a stock with Hee Haw’s risk, what should the stock’s price be?
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CHAPTER 5Practice Problems
13. A zero coupon bond matures in 15 years and has a $1,000 par value. The bond’s yield to maturity is 12%. a. What is the bond’s current price? b. What do you expect the bond’s price will be 1 year from now? c. If all happens just as you expect it will, what return would an investor who
buys the bond today and sells it in 1 year earn? d. What will the bond’s price be in 1 year if, at that time, investors require a 15%
yield to maturity for the bond? In this case, what would the investor in part c of the problem earn?
14. A stock is expected to pay a $2-per-share dividend next year. Dividends are expected to grow at a steady rate of 7% annually. Investors require an 18% return for investments with the stock’s risk. a. What should the stock’s current price be? b. At what price should the stock sell next year if conditions and forecasts do not
change? c. What would be your total return if you bought the stock today and sold it in
1 year? d. What would be your return from price appreciation and your dividend yield?
15. A bond is selling for $1,180. It carries an 8% coupon rate with coupon payments made semiannually. The bond matures in 13 years and has a $1,000 par value. The bond is callable at $1,125 in 9 years. a. Find the bond’s yield to maturity. b. Find the bond’s yield to call. c. Comment on whether or not you believe the bond will be called.
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