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CHAPTER 8 Bond Valuation Valuation and the Structure of Interest Rates Learning Objectives 1. Explain what an efficient capital market is and why “market efficiency” is important to financial managers.

2. Describe the market for corporate bonds and three types of corporate bonds. 3. Explain how to calculate the price of a bond and why bond prices vary negatively with interest rate movements.

4. Distinguish between a bond’s coupon rate, yield to maturity, and effective annual yield, and be able to calculate their values.

5. Explain why investors in bonds are subject to interest rate risk and why it is important to understand the bond theorems. 6. Discuss the the concept concept of default default risk and and know how how to compute compute a default default risk risk premium. premium. 7. Describe the factors factors that that determine determine the level level and shape of the yield curve. curve.

I.

Chapter Outline

8.1 Capital Market Efficiency  A.

Overview •

The supply and demand for securities are better reflected in organized

markets.

1

•

Any price that balances the overall supply and demand for a security is a

market equilibrium price. •

A security’s true value is the price that reflects investors’ estimates of the

value of the cash flows they expect to receive in the future. •

In an efficient capital market, security prices fully reflect the knowledge

and expectations of all investors at a particular point in time. If markets are efficient, investors and financial managers have no



reason to believe the securities are not priced at or near their true value. The more efficient a security market, the more likely securities are



to be priced at or near their true value. •

The overall efficiency of a capital market depends on its operational 

efficiency 

and its informational efficiency . Operational efficiency

focuses on bringing buyers and sellers

together at the lowest possible cost. 

Markets exhibit informational efficiency if market prices reflect

all relevant information about securities at a particular point in time. 

In an informationally efficient market, market prices adjust quickly

to new information about a security as it becomes available. 

Competition among investors is an important driver of 

informational efficiency.

2

 A.

Efficient Efficient Market Hypotheses Hypotheses •

Prices of securities adjust as the buying and selling from investors lead to the  price that truly reflects the market’s consensus. This reflects reflects the market’s efficiency.

•

Market efficiency can be explained at three levels—strong form, semistrong form, and weak form.

•

Strong form market efficiency states that the price of a security in the market

reflects all information—public as well as private or inside information. 

Strong form efficiency implies that it would not be possible to earn abnormally high returns (returns greater than those justified by the risks)  by trading on private on private information.

•

Semistrong

market efficiency implies that only public information that is

available to all investors is reflected in a security’s market price. 

Investors who have access to inside or private information will be able to earn abnormal returns.



Public stock markets in developed countries coun tries like the United States have a semistrong form of market efficiency.



•

 New information is immediately reflected reflected in a security’s market price.

In weak-form market efficiency, all information contained in past prices of a security is reflected in current prices. 

It would not be possible to earn abnormally high returns by looking for   patterns in security prices, but it would be possible to do so by trading on  public or private information.

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8.2 Corpor Corporate ate Bonds Bonds

 A.

Market for Corporate Corporate Bonds •

At the end of 2005, for example, the amount of corporate debt outstanding was $5.35 trillion, making it by far the largest U.S capital market.

•

The next largest market is the market for corporate stock with a value of $4.5 trillion, followed by the state and local government bond market valued at $1.86 trillion.

•

The largest investors in corporate bonds are life insurance companies and  pension funds, with trades in this market tending to be in very large blocks of  securities.

•

Less than 1 percent of all corporate bonds are traded on exchanges. Most secondary market transactions for corporate bonds take place through dealers in the over-the-counter (OTC) market.

•

Only a small number of the total bonds that exist actually trade on a single day. As a result, the market for corporate bonds is thin compared to the market for  money market securities or corporate stocks.

•

Corporate bonds are less marketable than the securities that have higher daily trading volumes.

•

Prices in the corporate bond market also tend to be more volatile than securities sold in markets with greater trading volumes.

•

The market for corporate bonds is not no t as efficient as that for stocks sold on the major stock exchanges or highly marketable money market instruments such as U.S. Treasury securities.

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 B.

Bond Price Price Information Information

•

The corporate bond market is not considered to be very transparent because it trades predominantly over the counter and investors do not find it easy to view  prices and trading volume.

•

In addition, many corporate bond transactions are negotiated between the  buyer and the seller, and there is little centralized reporting of these deals.

C.

Types of of Co Corporate Bo Bonds

•

Corporate bonds are long-term IOUs that represent claims against a firm’s assets.

•

Debt instruments, where the interest income paid to investors is fixed for the life of the contract, are called fixed-income securities .

•

Three types of corporate bonds—vanilla bonds, zero coupon bonds, and convertible bonds. 1. Vani anilla Bo Bonds •

These bonds have coupon payments that are fixed for the life of the

 bond, and at maturity, the principal is paid and the bonds are retired. •

Vanilla bonds have no special provisions, and the provisions they

do have are conventional and common to most bonds, such as a call  provision. •

Payments are usually made annually or semiannually.

•

The face value, or par value, for most corporate bonds is $1,000.

•

The bond’s coupon rate is calculated as the annual coupon

 payment (C) divided by the bond’s face value (F). 5

2. Ze Zero ro Coup Coupon on Bond Bondss

•

Corporations sometimes issue bonds that have no coupon cou pon

 payments over its life and only offer a single payment at maturity. •

Zero coupon bonds sell well below their face value (at a deep

discount) because they offer no coupons. •

The most frequent and regular issuer of zero coupon securities is

the U.S. Treasury Department. 3. Conv Conver erti tibl blee Bond Bondss •

These are bonds that can be converted into shares of common

stock at some predetermined ratio at the discretion of the bondholder. •

The convertible feature allows the bondholders to share in the good

fortunes of the firm if the firm’s stock rises above a certain level. •

The conversion ratio is set so that the firm’s stock price must

appreciate 15 to 20 percent before it is profitable to convert bonds into equity. •

To secure this advantage, bondholders would be willing to pay a

 premium. 8.3 Bond Valuation

•

The value, or price, of any an y asset is the present value of its future cash flows.

•

To calculate the price of the bond, we follow the same process as we would to value any financial asset.

6



Estimate the expected future cash flows—these are the coupons that the bond will  pay.



Determine the required rate of return, or discount rate. The required rate of return, or discount rate, for a bond is the market interest rate called the bond’s bon d’s yield to ). maturity (or more commonly, yield ).

This is the return one would earn from

 bonds that are similar in maturity and default risk. 

Compute the current value, or price, of a bond (PB) by calculating the present value of the bond’s expected cash flows: PB = PV (Coupon payments) + PV (Principal payments)

The general equation for the price of a bond can be written as follows in Equation



8.1: PB =

C1 (1 + i )

1

+

C2 (1 + i )

1  1 − (1 + i ) n = C× i  

 A.

2

+

C3 (1 + i )

3

+  +

Cn + F (1 + i ) n

  F + n  (1 + i ) 

Par, Premium, Premium, and Discount Discount Bonds Bonds •

If a bond’s coupon rate is equal e qual to the market rate, then the bond will sell at a  price equal to its face value. Such bonds are called par bonds.

•

If a bond’s coupon rate is less than the market rate, then the bond will sell at a  price that is less than its face value. Such bonds are called discount bonds.

7

•

If a bond’s coupon rate is greater g reater than the market rate, then the bond will sell at a price that is more than its face value. Such bonds are called premium  bonds.

 B.

Semiannual Semiannual Compounding  Compounding  •

While bonds in Europe pay annual coupons, bonds in the United States pay coupons semiannually.

•

In calculating the current price of a bond paying semiannual coupons, one needs to modify Equation 8.1. 

Each coupon payment is half of an annual coupon.



The number of payments is twice the number of years to maturity.



The discount rate used is then half of the annual rate.

 P   B = C.

C 1 m

+

C 2 m

(1 + i m) (1 + i m)

2

+

C 3 m

(1 + i m)

3

+ ....... +

( C mn m ) + F mn

(1 + i m) mn

(8.2)

Zero Coupon Bonds

•

Zero coupon bonds have no coupon payments but promise a single payment at maturity.

•

The price (or yield) of a zero coupon bond is simply a special case of  Equation 8.2, in that all the coupon payments are equal to zero.

•

Hence, the pricing equation is: PB = Fmn/(1 + i/m)mn

•

(8.3)

Zero coupon bonds, for which all the cash payments are made at maturity, must sell for less than similar bonds that make periodic coupon coupo n payments.

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8.4 8.4 Bond Bond Yield ieldss

Yield to Maturity •

The yield to maturity of a bond is the discount rate that makes the present value of the coupon and principal payments equal to the price of the bond.

•

It is the yield that the investor earns if the bond is held to maturity and all the coupon and principal payments are made as promised.

•

A bond’s yield to maturity changes daily as interest rates increase or decrease.

•

We can compute a bond’s yield to maturity using a trial-and-error approach.

 Effective  Effective Annual Yield  Yield 

•

As pointed out in Chapter 7, the correct way to annualize an interest rate (yields) is to compute the effective annual interest rate (EAR).

•

On Wall Street, the EAR is called the effective annual yield (EAY ) and EAR  = EAY.

•

The correct way to annualize the yield on a bond is as follows: EAY = (1 + Quoted rate/m)m – 1

•

The simple annual yield is the yield per period multiplied by the number of  compounding periods. For bonds with annual compounding, the simple annual yield = semiannual yield × 2.

 B.

Realized Yield  Yield  •

The realized yield is the return earned on a bond given the cash flows actually received by the investor.

9

•

The interest rate at which the present value of the actual cash flows generated  by the investment equals the bond’s price is the realized yield on an investment.

•

The realized yield is an important bond calculation because it allows investors to see the return they actually earned on their investment.

8.5 Inter Interest est Rate Rate Risk  Risk 

The prices of bonds fluctuate with changes change s in interest rates, giving rise to interest rate risk .  Bond Theorems Theorems 1.

Bond prices are negatively related to interest rate movements. •

As interest rates decline, the prices of bonds rise; and as interest rates rise,

the prices of bonds decline. 2.

For a given change in interest rates, the prices of long-term bonds will change more than the prices of short-term bonds. •

Long-term bonds have greater price volatility than short-term bonds.

•

All other things being equal, long-term bonds are more risky than

short-term bonds. •

Interest rate risk increases as maturity increases, but at a decreasing

rate. 3.

For a given change in interest rates, the prices of lower-coupon bonds change more than the prices of higher-coupon bonds. •

The lower a bond’s coupon coup on rate, the greater its price volatility, and

hence, lower coupon bonds have greater interest rate risk. 10

•

The lower the bond’s coupon rate, the greater the proportion of the

 bond’s cash flow investors will receive at maturity. maturity. •

All other things being equal, a given change in the interest rates will

have a greater impact on the price of a low-coupon bond than a higher-coupon  bond with the same maturity.  Bond Theorem Theorem Applications Applications •

If rates are expected to increase, a portfolio manager should avoid investing in long-term securities. The portfolio could see a significant decline in value.

•

If you are an investor and you expect interest rates to decline, you may well want to invest in long-term zero coupon bonds. As interest rates decline, the  price of long-term zero coupon bonds will increase more than that of any other  type of bond.

8.6

The St Struc ructure ure of of Int Inteerest est Ra Rates tes •

Market analysts have identified four risk characteristics of debt instruments that are responsible for most of the differences in corporate borrowing costs: the security’s marketability, call feature, default risk,

 A.

and term to maturity .

Marketability Marketability •

Marketability

refers to the ability of an investor to sell a security quickly, at a

low transaction cost, and at its fair market value. •

The lower these costs are, the greater a security’s marketability.

•

The interest rate, or yield, on a security varies inversely with its degree of  marketability.

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•

The difference in interest rates or yields between a marketable security (imarkt) and a less marketable security (iless) is known as the marketability risk   premium (MRP).

MRP = ilow mkt –  ihigh mkt > 0 •

U.S. Treasury bills have the largest and most active secondary market and are considered to be the most marketable of all securities.

 B.

Call Provision Provision •

A call provision gives the firm issuing the bonds the option to purchase the  bond from an investor at a predetermined price (the call price); the investor  must sell the bond at that price.

•

When bonds are called, investors suffer a financial loss because b ecause they are forced to surrender their high-yielding bonds and reinvest their funds at the lower   prevailing market rate of interest.

•

Bonds with a call provision sell at higher market yields than comparable noncallable bonds.

•

The difference in interest rates between a callable bond and a comparable noncallable bond is called the call interest premium (CIP) and can be defined as follows: CIP = icall - incall > 0

•

Bonds issued during periods when interest is high are likely to be called when interest rates decline, and as a result, these bonds have a high CIP.

 A.

Default Risk  Risk 

12

•

The risk that the lender may not receive payments as promised is called default d efault risk.

•

Investors must be paid a premium to purchase a security that exposes them to default risk.

•

The default risk premium (DRP) can thus be defined as follows: DRP = idr  - irf 

•

U.S. Treasury securities do not have any default risk and are the best proxy measure for the risk-free rate.

 B.

Bond Ratings Ratings •

Individuals and small business have to rely on outside agencies to provide them information on the default potential of bonds.

•

The two most prominent credit rating agencies are Moody’s Investors Service (Moody’s) and Standard and Poor’s (S&P). Both credit rating services rank  bonds in order of their expected probability of default and publish the ratings as letter grades.

•

The rating schemes used are shown in Exhibit 8.5.

•

The highest-grade bonds, those with the lowest default risk, are rated Aaa (or  AAA).

•

Bonds in the top four rating categories are called investment-grade bonds—  AAA to Baa.

•

State and federal laws typically require commercial banks, bank s, insurance companies, pension funds, other financial institutions, and government agencies to purchase securities rated only as investment grade.

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C.

The Term Stru Struccture ture of Int Inteerest rest Rates •

The relationship between yield and term to maturity is known as the term structure of interest rates .

•

Yield curves show graphically how market yields vary as term to maturity changes.

•

The shape of the yield curve is not constant over time.

•

As the general level of interest rises and falls over time, the yield curve shifts up and down and has different slopes.

•

There are three basic shapes (slopes) of yield curves in the marketplace. 

Ascending or normal yield curves are upward-sloping yield curves that occur when an economy is growing.



Descending or inverted yield curves are downward-sloping yield curves that occur when an economy is declining or heading into a recession.



Flat yield curves imply that interest rates are unlikely to change in the near  future.

 D.

The Shape of the Yield Yield Curve

•

Three economic factors determine the shape of the yield curve: (1) the real rate of interest, (2) the expected rate of inflation, and (3) interest rate risk. 1. Th Thee Real Real Rate Rate of Inter Interest est •

The real rate of interest varies with the business cycle, with the

highest rates seen at the end of a period of business expansion and the lowest at the bottom of a recession.

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•

Changes in the expected future real rate of interest can affect the

slope of the yield curve. 2. Th Thee Expe Expecte cted d Rate Rate of Infl Inflat ation ion

•

If investors believe that inflation will be increasing in the future,

the yield curve will be upward sloping because long-term interest rates will contain a larger inflation premium than short-term interest rates. •

If investors believe inflation will be subsiding in the future, the

 prevailing yield curve will be downward sloping. 3. Inte Intere rest st Rate Rate Risk  Risk 

•

The longer the maturity of a security, the greater its interest rate

risk, and the higher the interest rate. •

The interest risk premium always adds an upward bias to the slope

of the yield curve.  E.

The Cumulative Cumulative Effect  Effect  •

Exhibit 8.6 shows the cumulative effect of the three economic econ omic factors that influence the shape of the yield curve: the real rate of interest, the inflation  premium, and the interest rate risk premium.

•

In a period of economic expansion, both the real rate of interest and the inflation premium tend to increase monotonically over time.

•

In a period of contraction, both b oth the real rate of interest and the inflation  premium decrease monotonically over time.

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II. II.

Sugg Su gges ested ted and and Alte Altern rnat ativ ivee Appr Approa oach ches es to to the the Mat Mater eria iall

This chapter is all about bonds and how they are valued or priced in the marketplace. The bond valuation models presented in this chapter are derived from the present value materials discussed in Chapters 5 and 6. In the first part of the chapter, the authors a uthors discuss the concept of efficient  capital markets. Next, the discussion centers on the corporate bond market, bond price

information that is available, and the types of bonds found in the market. The development of the  basic equation used to calculate bond prices follows and leads to the steps required to compute a  bond’s: (1) yield to maturity, (2) realized yield, and (3) effective effective annual yield. This is followed  by a discussion on interest rate risk, and the three bond theorems that describe how bond prices respond to changes in interest rates are identified. The chapter concludes with a discussion on the term structure of interest rates and yield curves. Instructors have the flexibility of covering the chapter in its entirety or of focusing on  bonds alone. In a first course in finance, the discussion on term structure of interest interest rates may be ignored so as to place a heavier emphasis on bonds and their valuation.

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III. III. Su Summ mmar ary y of of Lear Learni ning ng Ob Obje jecti ctive vess 1.

Explain what an efficient capital market is and why “market efficiency” is important to financial managers.

An efficient capital market is a market in which w hich security prices reflect the knowledge and expectations of all investors. Public markets, for example, are more efficient than private markets because issuers of public securities are required to disclose a great deal of  information about these securities to investors and investors are constantly evaluating the  prospects for these securities and acting on the conclusions from their analyses by trading them. Market efficiency is important to investors because it assures them that the securities they buy for the firm’s portfolio are priced close to their true equilibrium price.

2.

Descr Describe ibe the the mar marke kett for for corp corpora orate te bonds bonds and and thre threee type typess of of corp corpor orate ate bonds bonds..

The market for corporate bonds is a very large market in which the most important investors are large institutions. Most trades in this market take place through dealers in the OTC market, and the corporate bond market is relatively thin. Prices of corporate  bonds tend to be more volatile than prices of securities that trade more frequently, such as stock and money markets, and the corporate bond market tends to be less efficient than markets for these other securities. A vanilla bond has fixed regular coupon payments over the life of the bond, and the entire principal is repaid at maturity. A zero coupon bond pays all interest and all  principal at maturity. Since there are no payments before maturity, zero coupon bonds are issued at a price well below their face value. Convertible bonds can be exchanged for  common stock at a predetermined ratio.

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3.

Explain how to calculate the price of a bond and why bond prices vary negatively with interest rate movements.

The price of a bond is equal to the present value of the future cash flows (coupons and  principal repayment) discounted at the market rate of interest for bonds with similar  similar  characteristics. Bond prices vary negatively with interest rates because the coupon rate on most bonds is fixed at the time the bond is issued. Therefore, as interest rates go up, investors seek other forms of investment that will allow them to take advantage of higher  returns. Because the bond’s coupon payments are fixed, the only way their yields can be adjusted to the current market rate of interest is to reduce the bond’s price. Similarly, when interest rates are declining, the yield on fixed income securities will be higher  relative to yield on similar securities price to market; the favorable yield will increase the demand for these securities, increasing their price and lowering their yield to the market yield. Read the first part of Section 8.3 and then work through Learn by Doing Application 8.1.

4.

Distinguish between a bond’s coupon rate, yield to maturity, realized yield, and effective annual yield, and be able to calculate their values.

A bond’s coupon rate is the stated interest rate on the bond when it is issued. U.S bonds typically pay interest semiannually, whereas European bonds pay once a year. The yield to maturity is the expected return on a bond if it is held to maturity date, whereas the realized yield is the return earned on a bond given the cash flows actually received by the investor. If the bond is held to maturity and the issuer does not default, these two rates are

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the same. Effective annual interest rate is the yield an investor actually earns in one year, adjusting for the effects of compounding. If the bond pays coupon payments more often than annually, the effective annual annua l yield will be higher than the simple annual yield  because of compounding. Work through Learn by Doing Applications 8.2, 8.3, and 8.4 to master these calculations.

5.

Expl Explain ain why inves investor torss in in bon bonds ds are are sub subjec jectt to inter interest est rate rate risk risk and and why it is important to understand the bond theorems.

Because interest rates are always changing in the market, all investors who hold bonds are subject to interest rate risk. Interest rate risk is the change in bond prices caused by by fluctuations in interest rates. Three of the most important bond theorems theore ms can be summarized as follows: 1. Bond prices prices are are inversel inverselyy related related to interest interest rate rate movement movementss 2.

For a given change in interest rates, the prices of long-term bonds will change more than the prices of short-term bonds.

3.

For a given change in interest rates, the prices of lower-coupon bonds will change more than the prices of higher-coupon bonds.

Understanding these relationships is important because it helps investors to better  understand why bond prices change and thus to make better decisions regarding the  purchase or sale of bonds and other fixed-income securities.

6.

Discuss the concept of default risk and know how to compute a default risk  premium.

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Default risk is the risk that the issuer will be unable to pay its debt obligation. Since investors are risk averse, they must be paid a premium to purchase a security that exposes them to default risk. The default risk premium has two components: (1) compensation for  the expected loss if a default occurs and (2) compensation for bearing the risk that a default could occur. All factors held constant, the degree of default risk a security  possesses can be measured as the difference between the interest rate on a risky security and the interest rate on a default-free security. The default risk is also reflected in a company’s bond rating. The highest-grade h ighest-grade bonds, those with the lowest default risk, are rated Aaa (or AAA). The default risk premium on corporate bonds increases as the bond rating becomes lower. Check Exhibit 8.5 8. 5 to see how risk premiums are calculated.

7.

Review the factors that determine the level and shape of the yield curve.

The shape of the yield curve is determined by three economic eco nomic factors: (1) the real rate of  interest, (2) the expected rate of inflation, and (3) interest rate risk. The real rate of  interest reflects people’s preference for consumption, which changes with the business cycle—in times of economic expansion, people tend to spend more, whereas during recession they save more. Even though this rate is not affected by the term to maturity, changes in the expected real rate of interest can have an effect on the slope of the yield curve. Similarly, the expected rate of inflation affects the slope of the yield curve in the same way. If investors believe inflation will be increasing in the future, the curve will be upward sloping as long-term rates will contain a larger inflation premium than short-term rates. Finally, interest rate risk, which increases with a security’s maturity, adds an upward bias to the yield curve.

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21

IV. IV. Su Summ mmar ary y of Ke Key y Eq Equa uati tion onss

Equation

8.1

Description

Price of a bond Price of a bond semiannual

8.2

8.3

compounding Price of zero coupon bond

Formula

PB=

PB =

C1 (1 + i )1

+

$C1

m

(1+ i

m)

1

C2 (1 + i )2 +

C2 (1 (1+ i

PB = Fmn /(1 + i/m)mn

22

+... + m m)

2

Cn + Fn (1 + i )n +

C3 (1+ i

m 3

m)

+...+

Cmn

m

(1+ i

+Fmn mn

m)

V.

Befo Be fore re You You Go On Qu Ques esti tion onss and and Ans Answers wers

Section 8.1

1.

How How does does inf infor orma matition on abou aboutt a fir firm’ m’ss pros prospe pect ctss get get refl reflec ecte tedd in its its shar sharee pric price? e?

Investors act upon the expectations of a firm’s prospects through trading of the securities. The buying and selling then causes the price of the security to reflect their assessment of  its value.

2.

What is strong-form market efficiency? semistrong-form market efficiency? weak-form market efficiency?

Strong-form market efficiency is a market in which all a ll information, private and public, is reflected in the price of the security. The semistrong-form of market efficiency suggests that only public information is reflected in a security’s price, while the weak-form market efficiency holds that both public and private information is reflected in the current price of a security, but also both public p ublic and private information has not been taken into account.

Section 8.2

1.

What Wh at are are the the mai mainn diff differ eren ence cess betw betwee eenn the the bon bondd mar market ketss and and sto stock ck mar marke kets ts??

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A corporate bond market is much larger than the stock market. The biggest investors in corporate bonds are mutual funds, life insurance companies, and pension funds, and given the size of these investors, the trades are conducted in much larger blocks than in the stock market. Also, while most stocks are traded in organized securities markets, most  bonds transactions take place through dealers in the OTC market.

2.

A bond has a 7 percent coupon rate, a face value of $1,000, and a maturity of four years. On a time line, lay out ou t the cash flows for the bond.

The annual payments for the bond will be $70 ($1,000 x 7%); thus the time line for cash inflows would be as follows: 0

3.

1

_2

$ 70

$ 70

_____ __ ___ __3

$ 70

_____ __ __4 4

$1,070 ($1,000 + $70)

Explain what a convertible bond is.

Convertible bonds are bonds that can be converted into shares of common stock at some  predetermined ratio at the discretion of the bondholder. The convertible feature allows the bondholder to take advantage of the firm’s prosperity if the share prices rises above a certain value.

Section 8.3

1.

Explain conceptually how bonds are priced.

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The current price of a bond is equal to the present value of all the cash flows that will be received from the investment. There are two sets of cash flows from a bond investment. First, there are the coupon payments to be received either annually or semiannually throughout the life of the bond. Second, there is the principal or face value of $1,000 that will be received when the bond matures. In order to find the price of the bond, we must find the present value of the coupons and the present value of the face value. We do this  by discounting the entire cash flow stream at the current market rate and adding them up. This gives us the current price of the bond. Recognize that the coupons represent an annuity and that we can use the equation for the present value of an annuity from Chapter  7 to calculate the present value of this cash flow stream.

2.

What is the compounding period for most bonds sold in the United States?

Most bonds sold in the United States pay interest semiannually, whereas European bonds typically only pay interest once a year.

3.

What hat ar are zer zeroo cou coupo ponn bon bonds ds,, and and how how ar are the theyy pri price ced? d?

Zero coupon bonds are debt instruments that do not pay coupon interest but promise a single payment (interest earned plus principal) paid at maturity. The price of a zero coupon bond can be calculated using the same equation as used for coupon bonds, but setting the coupon payments to zero. z ero. The resulting formula is as follows:

25

PB = Fmn/(1 + i/m)mn Because zero coupon bonds offer the entire payment at maturity, for a given change in interest rates, their price fluctuates more than coupon bonds with a similar maturity.

Section 8.4

1.

Explain how bond yields are calculated.

A bond’s yield can be defined as the interest rate that equates a bond’s price to the  present value of its interest payments and principal amount. The calculation of a bond’s yield, or its yield to maturity, takes into account acco unt the bond’s time to maturity, the coupon cou pon rate, and par 

Section 8.5 1.

What is interest rate risk?

Bond prices are negatively related to interest rate movements. As interest rates rise, bond  prices fall, and vice versa. Interest rate risk simply simply recognizes the fact that bond prices fluctuate as interest rates change, and, if you sell a bond before maturity, you may sell the  bond for a price other than what you paid for it. The greater the fluctuation in bond prices due to changes in interest rates, the greater the interest rate risk.

2.

Explain why long-term bonds with zero coupons are riskier than short-term bonds that  pay coupon interest.

26

According to bond theorems number two and three, for a given change in interest rates, longer-term bonds with low coupon rates have ha ve greater price changes than shorter-term  bonds with higher coupon rates. Thus, long-term zero coupon bonds have greater interest rate risk—greater price swings—than short-term bonds that pay coupon payments.

Section 8.6

1.

What are default risk premiums, and what do they measure?

Default risk premiums are the amount of return that investors must be paid to purchase a security that possesses default risk compared to a similar risk-free investment. Default risk premiums, at any point in time, represent compensation for the expected financial injury for owning a bond plus some additional premium for bearing risk.

2.

Describe the two most prominent bond rating systems. Default risk premiums tend to increase during periods of economic decline and to narrow n arrow during periods of economic expansion. This phenomenon is due to changes in investors’ willingness to own bonds with different credit ratings over the business cycle, the socalled flight to quality argument. Specifically, during periods of expansion when few defaults take place, investors are willing to invest in bonds with low credit quality to gain higher yields. In contrast, during tough economic eco nomic times when many businesses fail, investors are concerned with safety. Accordingly, they adjust their portfolios to include more high-quality credits and sell sell off bonds with low low credit ratings. The two most

27

 prominent credit rating agencies are Moody’s Investors Service (Moody’s) and Standard & Poor’s (S&P).

3.

What are the three factors that most affect the level and shape of the yield curve?

The three factors that most affect the shape of the yield curve are the real rate of interest, the expected rate of inflation, and an d interest rate risk. If the future real rate of interest is expected to rise, it will result in an upward slope of the real rate of interest and consequently in an upward bias to the market yield curve. Similarly, increasing the expected rate of inflation will result in an upward-sloping yield curve, because long-term interest rates will contain a larger inflation premium than short-term interest rates. If  these two variables are expected to decline in the future, the result will be a downward  bias to the yield curve. In contrast, the longer a bond’s maturity, the greater the bond’s interest rate risk. Thus, interest rate risk premium always adds an upward bias to the slope of the yield curve, since the longer the maturity of a security, the greater its interest rate risk.

28

VI. VI. Se Sellf Stu tud dy Pro Problem blemss

8.1

Calculate the price of a five-year bond that has a coupon of 6.5 percent and pays annual interest. The current market rate is 5.75 percent. p ercent.

Solution:

0 5.75%

1

2

3

4

5

├───────┼────────┼───────┼──── ├───────┼────── ──┼───────┼────────┼───────┤ ────┼───────┤ $ 65 PB =

=

$ 65

$6 5

$ 65

$ 1 , 0 65

C3 C1 C2 C C4 + F + + + + (1 + i)1 (1 + i) 2 (1 + i) 3 (1 + i ) 4 (1 + i) 5 $65 $65 $65 $65 ($65 + $1,000) + + + + 1 2 3 4 (1 + 0.0575) (1.0575) (1.0575) (1.0575) (1.0575) 5

= $61.47 + $58.12 + $54.96 + $51.95 + $805.28 = $1,031.81

8.2

Bigbie Corp issued a four-year bond a year ago with a coupon of 8 percent. The bond  pays interest semiannually. If the yield to maturity on this bond is 9 percent, what is the  price of the bond?

Solution:

0 9% 1

2

3

4

5

6

7

8

├───┼───┼───┼────┼───┼───┼───┼────┤

29

PB =

=

C1 / m (1 + i / m)

1

+

C2 / m (1 + i / m )

2

+

C3 / m (1 + i / m)

3

+ .......... + +

C8 + F (1 + i / m)8

$80 / 2 $40 $40 ($40 + $1,000) ........ + + + + (1 + 0.09 / 2)1 (1.045) 2 (1.045) 3 (1.045) 8

= $38.28 + $36.63 + $35.05 + $33.54 + $32.10 + $30.72 + $29.39 + $731.31 = $967.02

Alternatively, we can use the present value annuity factor from Chapter 6 and the present value equation from Chapter 5 to solve for the price of the bond: PB = C ( PVIFA

i ,n

) + F ( PVIF i , n )

1  − 1  (1 + 0.045 ) 8 = $ 40 *  0 .045   = $ 263 263 .84 + $ 703 .19 = $ 967 967 .03

8.3

  $1,000 000 + 8  (1 .045 ) 

Rockwell Industries has a three-year bond outstanding that pays a 7.25 percent coupon and is currently priced at $913.88. $913.8 8. What is the yield to maturity of this bond? b ond? Assume annual coupon payments.

Solution:

0

1

2

3

├───────┼────────┼───────┤ PB = $913.88

$ 7 2 .5 0

$ 7 2 .5 0

$ 1 , 0 7 2 .5 0

Use the trial-and error approach error approach to solve for YTM. Since the bond is selling at a discount, we know that the yield to maturity is higher than the coupon rate. Try YTM = 10%. 30

PB = C ( PVIFA

i,n

) + F( PVIF i ,n )

1   1 − (1 + 0 .10 ) 3  $1,000 + $913 .88 ≠ $72 .50 *  3 0.10   (1.10 )   ≠ $180 .30 + $ 793 793 .83 ≠ $931 .61

Try a higher rate, say YTM = 11%. PB = C( PVIFA

i,n

) + F( PVIF i ,n )

≠ $177 .17 + $731 731 .19 ≠ $908 .36

Since this is less than the price of the bond, we know that the YTM is between 10 and 11  percent and closer to 11 percent. Try YTM = 10.75%. PB = C ( PVIFA

i,n

) + F( PVIF i ,n )

≠ $177 177 .94 + $736 736 .15 ≠ $914 914 .09 ≅ 913 913 .88

Thus, the YTM is approximately 10.75 percent. Using a financial calculator provided an exact YTM of 10.7594 percent.

8.4

Hindenberg, Inc., has a 10-year bond that is priced at $1,100.00. It has a coupon of 8  percent paid semiannually. What is the yield to maturity on this bond?

Solution:

0

1

2

3

4

5

├───┼────┼───┼───┼───┼────┼── $ 40

$4 0

$ 40

$ 40

$4 0 31

$4 0

6

20 ─────┤ $4 0

$1,000 The easiest way to calculate the yield to maturity is with a financial calculator. The inputs are as follows: Enter

20 N

Answer

i 3.31

40

−1,100

1 ,0 0 0

PMT

PV

FV

The answer we get is 3.31 percent, which is the semiannual interest rate. To obtain an annualized yield to maturity, we multiply this by two: YTM = 3.31% × 2 YTM = 6.62%

8.5

Highland Corp., a U.S. company, has a five-year bond whose yield to maturity is 6.5  percent. The bond has no coupon payments. What is the price of this zero coupon bond?

Solution:

You are given the following information: YTM = 6.5%;

m=1

 No coupon payments Most U.S. bonds pay interest semiannually. Thus m x n = 5 x 2 = 10 and i/2 = 0.065/2 = 0.0326. Using Equation 8.3, we obtain the following calculation: PB = Fmn/(1 + i/m)mn PB = $1,000/(1 + 0.0325)10 PB = $726.27

32

VII. VII. Critic Critical al Think Thinking ing Ques Questio tions ns

8.1

You believe that you can make abnormally profitable trades by observing that the CFO of  a certain company always wears wea rs his green suit on days that the firm is about to release  positive information about his company. Describe which form of market efficiency is consistent with your belief.

You believe that the stock price of this company is affected by the choice of the CFO’s wardrobe and that this private information will grant you abnormally high returns. Therefore, your belief is consistent with the semistrong form of market efficiency, according to which it is possible to earn abnormally high returns by trading on private p rivate information.

8.2

Describe the informational differences that separate the three forms of market efficiency.

The strong-form of market efficiency states that all information is reflected in the security’s price. In other words, there is no private or inside information that, if released, would potentially change the price. The semistrong-form holds that all public information available to investors is reflected in the security’s price. Therefore, insiders with access to  private information could potentially profit from trading on this knowledge before it  becomes public. Finally, the weak form of market efficiency holds that there is both  pubic and private information that is not reflected in the security’s price and having access to it can lead to abnormal profits.

33

8.3

What economic conditions would prompt investors to take advantage of a bond’s convertibility feature?

A bond’s convertibility feature becomes attractive when the company’s stock price rises above the bond’s price. This usually happens in times of economic expansion when the stock market is booming and interest rates are decreasing, hence lowering the bond’s  price.

8.4

Define yield to maturity. Why is it important?

Yield to maturity (YTM) is the rate of return earned by

investors if they buy a bond today

at its market price and hold it to maturity. It is important because it represents the opportunity cost to the investor or the discount rate that makes the present value of the  bond’s cash flows (i.e., its coupons and its principal) equal to the market price. So, YTM is also referred to as the going market rate or the appropriate discount rate for a bond’s cash flows. It is important to understand that any investor who buys a bond and holds it to maturity will have a realized gain equal to the yield to maturity. If the investor sells  before the maturity date, then realized gain will not be equal to the YTM, but will only be  based on cash flows earned to that point. Similarly, for callable bonds, investors are guaranteed a gain to the point in time when the bond is first called, but they cannot be assured of the yield to maturity because the issuer could call the bond before maturity!

34

8.5

Define interest rate risk . How can the CFOs manage this risk?

The change in a bond's prices caused by changes in interest rates is called interest rate risk . In other words, we can measure the interest rate risk to a bond’s investor by measuring the  percentage  percentage change change in in the bond’s price caused caused by a 1 percent percent change change in the market market interest interest rates. The key to managing interest rate risk is to understand the relationships between interest rates, bond prices, the coupon rate, and the bond’s term to maturity. Portfolio managers need to understand that as interest rates rise bond prices decline, and it declines more for low-coupon bonds and longer-term bonds than for the others. In such a scenario,  bond portfo portfolio lio manager managerss can reduce reduce the the size size and maturity maturity of of their their portfo portfolio lio to to reduce reduce the the impact of interest rate increases. When interest rates decline, bond prices increase and rise more for longer-term bonds and higher coupon bonds. At such times, CFOs can increase the size and maturity of their portfolios p ortfolios to take advantage of the inverse relationship between interest rates and bond prices.

8.6

Explain why bond prices and interest rates are negatively related. What is the role of the coupon rate and term-to-maturity term-to-maturity in this relationship?

Bond prices and interest rates are negatively related because the market rate varies, while the coupon rate is constant over the life of the bond. Thus, as rates increase, demand and bond

35

 prices  prices of exist existing ing bonds bonds decline decline,, while while newer newer bonds bonds with with coupon coupon rates rates at the current current rate are in greater demand. o

For a given change in interest rates, longer-term bonds experience greater price changes ( price  price volatili volatility ty) than shorter-term bonds. Longer-term bonds have more of  their cash flows farther in the future, and their present value will be lower due to the compounding effect. In addition, the longer it takes for investors to receive the cash flows, the more uncertainty they have to deal with and hence the more price-volatile the bond will be.

o

Lower coupon bonds are more price volatile than higher coupon bonds. The same argument used above also explains this relationship. The lower the coupon on a  bond, the the greater greater the proport proportion ion of cash flows flows that investors investors receive receive at maturity maturity..

8.7

If rates are expected to increase, should investors look to long-term bonds or short-term securities? Explain.

As interest rates increase, bond prices decrease with longer-term bonds, bo nds, experiencing a  bigger decline than shorter-term securities. So, investors expecting an increase in interest rates should choose short-term securities over long-term securities and reduce their  interest rate risk.

8.8

Explain what you would assume the yield curve would look like during economic expansion and why.

36

At the beginning of an economic expansion, the yield curve tends to be rather steep as the rates begin to rise once the demand for capital is beginning to pick up due to growing economic activity. The yield curve will retain its positive slope during the economic expansion, which reflects the investors’ expectations that the economy will grow in the future and that the inflation rates will also rise in the future.

An investor holds a 10-year bond paying a coupon of 9 percent. The yield to maturity of 

8.9

the bond is 7.8 percent. Would you expect the investor to be holding a par-value,  premium, or discount bond? What if the yield to maturity was 10.2 percent? Explain.

Since the bond’s coupon of 9 percent is greater than the yield to maturity, the bond will  be a premium bond. As market market rates rates of intere interest st drop drop below below the coupon rate of the the 9 percent percent  bond, demand demand for the the bond increases increases,, driving driving up the price price of of the bond above above face face value. value. If the yield to maturity is at 10.2 percent, then the bond is paying a lower coupon than the going market rate and will be less attractive to investors. The demand for the 9  percent  percent bond will decline, decline, driving driving its price price below below the face value. value. This will be a discount  bond.

8.10 a.

Investor A holds a 10-year bond while investor B has an 8-year bond. If interest rate increases by 1 percent, which investor will have the higher interest rate risk? Explain.

Since A holds the longer-term bond, he or she will face the higher interest rate risk. Longer-term bonds are more price volatile than shorter-term bonds.

37

b.

Investor A holds a 10-year bond paying 8 percent a year, while investor B also has a 10-year bond that pays a 6 percent coupon. Which investor will have the higher interest rate risk? Explain.

Investor B will have the higher interest rate risk since lower coupon bonds have a higher interest rate risk than higher coupon bonds of the same maturity.

38

VIII.

Questions and Problems

BASIC 8.1

Bond price:

BA Corp is issuing a 10-year bond with a coupon rate of 8 percent. The

interest rate for similar bonds is currently 6 percent. Assuming annual payments, p ayments, what is the present value of the bond?

Solution:

Years to maturity = n = 10 Coupon rate = C = 8% Annual coupon = $1,000 x 0.08 = $80 Current market rate = i = 6% Present value of bond = PB 0 6% 1

2

3

4

5

6

10

├───┼────┼───┼───┼───┼────┼──

─────┤

$8 0

$8 0

$ 80

$80

$8 0

$ 80

$8 0

$1,000 C10 + F (1 + i )1 (1 + i ) 2 (1 + i ) 3 (1 + i )10 1  1    − − 1 1  (1 + i ) n   (1.06)10  $1,000 F = C× + + n = $80 ×  10 i   (1 + i )  0.06  (1.06)

PB =

C1



+

C2

+

C3

+





+



= $588.81 + $558.39 = $1,147.20

39



8.2

Bond price:

Pierre Dupont just received a gift from his grandfather. He plans to invest in

a five-year bond issued by Venice Corp. that pays annual coupons of 5.5 percent. If the current market rate is 7.25 percent, what is the maximum amount Pierre should be willing to pay for this bond?

Solution:

0

7.25%

1

2

3

4

5

├───────┼────────┼───────┼──── ├───────┼────── ──┼───────┼────────┼───────┤ ────┼───────┤ $5 5

$5 5

$ 55

$55

$ 1 ,0 5 5

Coupon rate = C = 5.5% Annual coupon = $1,000 x 0.055 = $55 Current market rate = i = 7.25% Present value of bond = PB

PB

8.3

1  1    − − 1 1  (1 + i ) n   (1.0725) 5  F $1,000 $ 55 = C× = × + +   n 5 i   (1 + i )  0.0725  (1.0725)     = $224.01 + $704.72 = $928.72

Bond price:

Knight, Inc., has issued a three-year bond that pays a coupon of 6.10

 percent. Coupon payments are made semiannually. Given the market rate of interest of  5.80 percent, what is the market value of the bond?

Solution:

Years to maturity = n = 3 40

Coupon rate = C = 6.1% Frequency of payment = m = 2 Semiannual coupon = $1,000 x (0.061/2) = $30.50 Current market rate = i = 5.8% Present value of bond = PB 0 5.8% 1

2

3

4

5

6

├───┼────┼───┼───┼───┼────┤ $30.5 $30 .500 $30. $30.50 50 $30.5 $30.500 $30 $30.5 .500 $30. $30.50 50 $30.5 $30.500 $1,000

PB

8.4

1   1 − 1   2n   1 − i  (1.029) 6  $1,000 F  (1 + 2 )  C = 2× + 2 n = $30.50 ×  6 + i 0 . 029 ( 1 . 029 ) i   ( ) 1 +   2 2       = $165.77 + $842.38 = $1,008.15

Bond price:

Regatta, Inc., has seven-year bonds outstanding that pay a 12 percent

coupon rate. Investors buying the bond today can expect to earn a yield to maturity of  8.875 percent. What is the current value of these bonds? Assume annual coupon  payments.

Solution:

Years to maturity = n = 7 Coupon rate = C = 12% Annual coupon = $1,000 x 0.12 = $120

41

Current market rate = i = 8.875% Present value of bond = PB 0

1

2

3

4

5

6

7

├───┼────┼───┼───┼───┼────┼───┤ $ 1 20 $ 1 20 $ 1 2 0 $ 12 0 $ 12 0 $ 12 0

$ 1 20 $1,000

PB

8.5

1  1    1 1 − − n 7  (1 + i )   (1.08875)  F $1,000 $ 120 = C× = × +  + n 7 i   (1 + i )  0.08875  (1.08875)     = $606.50 + $551.14 = $1,157.94

Bond price:

You are interested in investing in a five-year bond that pays 7.8 percent

coupon with interest to be received semiannually. Your required rate of return is 8.4  percent. What is the most you would be willing to pay for this bond?

Solution:

Years to maturity = n = 5 Coupon rate = C = 7.8% Frequency of payment = m = 2 Semi-annual coupon = $1,000 x (0.078/2) = $39.00 Current market rate = i = 8.4% Present value of bond = PB 0 8.4% 1

2

3

4

5

6

├───┼────┼───┼───┼───┼────┼── 42

10 ─────┤

$3 9

$ 39

$39

$3 9

$ 39

$3 9

$3 9 $1,000

1   − 1 1   2 n   − 1 i  (1.042)10  F $1,000  (1 + 2 )  PB = C ×  $ 39 + = × +   2n  2 i 0.042 (1.042)10 i )   ( + 1   2 2       = $313.20 + $662.71 = $975.91

8.6

Zero co coupon bonds:

Diane Carter is interested in buying a five-year zero coupon bond

whose face value is $1,000. She understands that the market interest rate for similar  investments is 9 percent. Assume annual compounding for payments. What is the current  price of this bond?

Solution:

Years to maturity = n = 5 Coupon rate = C = 0% Current market rate = i = 9% 0

1

2

3

4

5

├───┼────┼───┼───┼───┤ $0

$0

$0

$0

$0 $1,000

PB =

Fmn

(1 + i m) mn

=

$1,000 5 = $649.93 (1.09)

43

8.7

Zero co coupon bonds:

Ten-year zero coupon bonds issued by the U.S. Treasury have a

face value of $1,000 and interest is compounded semiannually. If similar bonds in the market yield 10.5 percent, what is the value of these bonds?

Solution:

Years to maturity = n = 10 Frequency of payment = m = 2 Coupon rate = C = 0% Current market rate = i = 10.5% 0

1

2

3

4

5

6

20

├───┼────┼───┼───┼───┼────┼──

─────┤

$0

$0

$0

$0

$0

$0

$0

$1,000 PB =

8.8

Fmn

(1 + i m) mn

=

$1,000 = $359.38 (1.0525) 20

Zero co coupon bonds:  Northrop Real Estate Company is planning to fund a development

 project by issuing 10-year zero coupon bonds with a face value of $1,000. Assuming semiannual compounding, what will be the price of these bonds if the appropriate discount rate is 14 percent?

Solution:

Years to maturity = n = 10

44

Coupon rate = C = 0% Current market rate = i = 14% Assume semiannual coupon payments. 0

1

2

3

4

5

6

20

├───┼────┼───┼───┼───┼────┼──

─────┤

$0

$0

$0

$0

$0

$0

$0

$1,000 PB =

8.9

Fmn

(1 + i m)

mn

=

$1,000 = $258.42 (1.07) 20

Yield to maturity:

Ruth Hornsby is looking to invest in a three-year bond that pays

semiannual coupons at a coupon rate of 5.875 percent. If these bonds have a market price of $981.13, what yield to maturity and effective annual yield can she expect to earn?

Solution:

Years to maturity = n = 3 Coupon rate = C = 5.875% Frequency of payment = m = 2 Semi-annual coupon = $1,000 x (0.05875/2) = $29.375 Yield to maturity = i Present value of bond = PB = $981.13 Use the trial-and-error approach trial-and-error approach to solve for YTM. Since the bond is selling at a discount, we know that the yield to maturity is higher than the coupon rate.

45

Try YTM = 6%. PB = C ( PVIFA

i,n

) + F( PVIF i ,n )

1  − 1  (1 + 0 .03) 6 $981 981 .13 ≠ $29 .375 *  0.03  

  $ 1,000 + 6  (1 .03 ) 

837 .48 ≠ $159 .13 + $837 ≠ $996 .61

Try a higher rate, say YTM = 6.6%. PB = C ( PVIFA

i,n

) + F( PVIF i ,n )

1  − 1  (1 + 0 .033 ) 6 $981 .13 ≠ $29 .375 *  0 .033 033  

  $1,000 + 6 033 )  (1.033 

≠ $157 .56 + $823 .00 ≠ $980 .56

The YTM is approximately 6.6 percent. Using a financial calculator provided an exact YTM of 6.58 percent Enter

Answer

6

$29.375 -$981.13 $1,000 N 6.58%

i%

PMT

PV

FV

The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1 2

= (1 + 0.06578 2 ) − 1 = (1.0335) 2 − 1 = 0.06686 = 6.69%

8.10 .10

Yie Yield to mat maturi urity: Rudy Sandberg wants to invest

in four-year bonds that are currently

 priced at $868.43. These bonds have a coupon rate of 6 percent and pay semiannual coupons. What is the current market yield on this bond? 46

Solution:

Years to maturity = n = 4 Coupon rate = C = 6% Frequency of payment = m = 2 Semiannual coupon = $1,000 x (0.06/2) = $30 Yield to maturity = i Present value of bond = PB = $868.43 Use the trial-and-error approach to solve for YTM. Since the bond is selling at a discount, we know that the yield to maturity is higher than the coupon rate. Try YTM = 10%. PB = C ( PVIFA

i ,n

) + F( PVIF i ,n )

1  1 − (1 + 0.05 ) 8 $868 .43 ≠ $30 *  0.05   ≠ $193 .90 + $676 .84 ≠ $870 .74

  $1,000 000 + 8  (1.05 ) 

Try a higher rate, say YTM = 10.1%. PB = C ( PVIFA

i ,n

) + F( PVIF i ,n )

1  1 − (1 + 0.0505 ) 8 $868 868 .43 ≅ $30 *  0 .0505   ≅ $193 .51 + $ 674 .27 ≅ $867 .77

  $1,000 + 8  (1.0505 ) 

The YTM is approximately 10.1 percent. Using a financial calculator provided an exact YTM of 10.08 percent. Enter

8

$30

-$868.43 $1,000 47

8.11

Answer

N i% 10.08%

PMT

PV

FV

Realized yi yield:

Josh Kavern bought 10-year, 12 percent coupon bonds issued by the U.S.

Treasury three years ago at $913.44. If he sells these bonds, which have a face value of  $1,000, at the current price of $804.59, what is the realized return on these bonds? Assume annual coupons on similar coupon-paying bonds.

Solution:

Purchase price of bond = $913.44 Years investment held = n = 3 Coupon rate = C = 12% Frequency of payment = m = 1 Annual coupon = $1,000 x (0.12) = $120 Realized yield = i Selling price of bond = PB = $804.59 To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has declined, market rates must have increased. So, the realized return is going to be less than the bond’s bond ’s coupon. Try rates lower than the coupon rate. Try i = 10%.

48

1   − 1 n   FV PB = C ×  (1 + i )  + n i   (1 + i )   1   − 1  (1.10) 3  $804.59 $913.44 = $120 ×  + 3 0 . 10   (1.10)   = $298.42 + $604.50 ≠ $902.92 Try a lower rate, i = 9.5%. 1   − 1  (1 + i ) n  FV PB = C ×  + n i   (1 + i )   1   − 1  (1.095) 3  $804.59 $913.44 = $120 ×  + 3 0 . 095   (1.095)   = $301.07 + $612.82 = $913.89 The realized rate of return is approximately 9.5 percent. Using a financial calculator   provided an exact yield of 9.52 percent. Enter

8.12

3

$120 i%

- $ 91 3 . 4 4 PMT

$ 8 0 4 .5 9

Answer

N 9.52%

PV

FV

Realized yi yield:

Four years ago, Lisa Stills bought six-year, 5.5 percent coupon bonds

issued by the Fairways Corp. for $947.68. $94 7.68. If she sells these bonds at the current price of  $894.52, what is the realized return on these bonds? Assume annual coupons on similar  coupon-paying bonds.

49

Solution:

Purchase price of bond = $947.68 Years investment held = n = 4 Coupon rate = C = 5.5% Frequency of payment = m = 1 Annual coupon = $1,000 x (0.055) = $55 Realized yield = i Selling price of bond = PB = $894.52 To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has declined, market rates must have increased. So, the realized return is going to be less than the bond’s bond ’s coupon. Try rates lower than the coupon rate. Try i = 5%. 1   − 1  (1 + i ) n  FV PB = C ×  + n i   (1 + i )   1   − 1  (1.05) 4  $894.52 $947.68 = $55 ×  + 4 0 . 05   (1.05)   = $195.03 + $735.92 ≠ $930.95 Try a lower rate, i = 4.5%.

50

1   − 1 n   FV PB = C ×  (1 + i )  + n i   (1 + i )   1   − 1  (1.045) 4  $894.52 $947.68 = $55 ×  + 4 0 . 045   (1.045)   = $197.31 + $750.11 ≅ $947.42 The realized rate of return is approximately 4.5 percent. Using a financial calculator   provided an exact yield of 4.49 percent. Enter

4

Answer

$ 55 N 4.49%

-$947.68 i%

$ 8 9 4 .52

PMT

PV

FV

INTERMEDIATE

8.13

Bond price:

The International Publishing Group is raising $10 million by issuing 15-year 

 bonds with a coupon rate of 8.5 percent. Coupon payments will be annual. Investors  buying the bond currently will earn a yield to maturity of 8.5 percent. At what price will the bonds sell in the marketplace? Explain.

Solution:

Years to maturity = n = 15 Coupon rate = C = 8.5% Annual coupon = $1,000 x 0.085 = $85 Current market rate = i = 8.5% 51

Present value of bond = PB 0

1

2

3

4

15

├───┼────┼───┼───┼─── $ 85

$8 5

$ 85

─────┤

$ 85

$ 85 $1,000

n = 7;

C = 8.5%;

i = YTM = 8.85%

1  1    1 1 − − n 15  (1 + i )   (1.085)  $1,000 F PB = C ×  $ 85 = × +  + n 15 i i + ( 1 ) 0 . 085     (1.085)     = $705.86 + $294.14 = $1,000.00

This answer should have been intuitive. Since the bond is paying a coupon equal to the going market rate of 8.5 percent, the bond should be selling at its par value of $1,000. Enter

15

8.85% $85 N

i%

Answer

8.14

Bond price:

$1,000 PMT -$1,000

PV

FV

Pullman Corp issued 10-year bonds four years ago with a coupon rate of 

9.375 percent, paid semiannually. At the time of issue, the bonds sold at par. Today,  bonds of similar risk and maturity will pay a coupon rate of 6.25 percent. What will be the current market price of the firm’s bonds?

Solution:

Years to maturity = n = 10 Coupon rate = C = 9.375%

52

Semiannual coupon = $1,000 x (0.09375/2) = $46.875 Current market rate = i = 6.25% Present value of bond = PB 0

1

2

3

4

20

├───┼────┼───┼───┼───

─────┤

$46.875 $46.875………

$ 46 .87 5 $1,000

2; n = 10; m = 2;

C = 9.3 9.375 75%; %;

i = YTM = 6.25%

1   − 1 1   2 n   − 1 i  (1.03125) 20  F $1,000  (1 + 2 )  PB = C ×  $ 46 . 875 + = × +   2n  2 i 0.03125 (1.03125) 20 i )   1 ( +   2 2       = $689.39 + $540.41 = $1,229.80 Enter

20

3.125% $46.875 N

Answer

8.15

Bond price:

i%

$1,000 PMT PV -$1,229.80

FV

Marshall Company is issuing eight-year bonds with a coupon rate of 6.5

 percent and semiannual coupon payments. If the current market rate for similar bonds is 8  percent, what will be the price of the bond? If the company wants to raise $1.25 million, how many bonds does the firm have to sell?

Solution:

Years to maturity = n = 8 Coupon rate = C = 6.5% 53

Semiannual coupon = $1,000 x (0.065/2) = $32.50 Current market rate = i = 8% Present value of bond = PB 0 8% 1

2

3

4

16

├───┼────┼───┼───┼───

$ 3 2 .50

─────┤

$32.50………..$32.50

$ 3 2 .5 0 $1,000

1   1   2n  1 − − 1 i 16  (1.04)  $1,000 F  (1 + 2 )  + = × PB = C ×  $ 32 . 50  + 2n  2 i 0 . 04 (1.04)16 i   ( ) + 1   2 2       = $378.70 + $533.91 = $912.61

To raise $1.25 bonds, the firm would have to sell:  Number of bonds = $1,250,000 / $912.61 = 1,370 bond contracts Enter

16

4% N

Answer

8.16

Bond price:

$32.50 i%

$1,000 PMT -$912.61

PV

FV

Rockne, Inc., has 15-year bonds that will mature in six years and pay an 8

 percent coupon, interest being paid semiannually. If you paid $1036.65 today, and your  required rate of return was 6.6 percent, did you pay the right price for the bond?

Solution:

Years to maturity = n = 6 54

Coupon rate = C = 8% Semiannual coupon = $1,000 x (0.08/2) = $40 Current market rate = i = 6.6% Present value of bond = PB 0

1

2

3

12

├───────┼────────┼────────┼── $ 40

$ 40

─────────┤

$4 0

$ 40 $1,000

1   − 1 1   2 n   − 1 i  (1.033)12  $1,000 F  (1 + 2 )  PB = C ×  $ 40 + = ×  + 2n  2 i 0 . 033 (1.033)12 i   1 ( ) +   2 2       = $391.12 + $677.32 = $1,068.45

You paid less than what the bond is worth. That was a good price! Enter

12

3.3% N

Answer

8.17

$40 i%

$1,000 PMT PV -$1,068.45

FV

Bond price:  Nanotech, Inc., has a bond issue maturing in seven years and paying a

coupon rate of 9.5 percent (semiannual payments). The company wants to retire a portion of the issue by buying the securities in the open market. If it can refinance at 8 percent, how much will Nanotech pay to buy back its current outstanding bonds?

Solution:

Years to maturity = n = 7 55

Coupon rate = C = 9.5% Semi-annual coupon = $1,000 x (0.095/2) = $47.50 Current market rate = i = 8% Present value of bond = PB 0

1

2

3

14

├───────┼────────┼────────┼── $4 7 .5 0

$ 4 7 .50

─────────┤

$ 4 7 .5 0

$ 4 7 .50 $1,000

1   − 1 1   2 n   − 1 i  (1.04)14  $1,000 F  (1 + 2 )  PB = C ×  $ 47 . 50 + = ×  + 2n  2 i 0 . 04 (1.04)14 i   1 ( ) +   2 2       = $501.75 + $577.48 = $1,079.22

The firm will be willing to pay no more than $1,079.22 for their bond. Enter

14

4% N

Answer

8.18 .18

Zero cou coup pon bon bonds:

$47.50 i%

$1,000 PMT PV -$1,079.22

FV

Kintel, Inc., wants to raise $1 million by issuing six-year zero

coupon bonds with a face value of $1,000. Their investment banker informs them that investors would use an 11.4 percent discount rate on such bonds. At what price would these bonds sell in the marketplace? How many bonds would the firm have to issue to raise $1 million? Assume annual compounding for payments.

Solution:

56

Years to maturity = n = 6 Coupon rate = C = 0% Current market rate = i = 11.4% Assume semi-annual coupon payments. 0

1

2

3

4

5

6

12

├───┼────┼───┼───┼───┼────┼──

─────┤

$0

$0

$0

$0

$0

$0

$0

$1,000 PB =

Fmn

(1 + i m) mn

=

$1,000 = $514.16 (1.057) 12

At the price of $514.16, the firm needs to raise $1 million. To do so, the firm will have to issue:  Number of contracts = $1,000,000 / $514.16 = 1,945 contracts

8.19 .19

Zero cou coup pon bon bonds:

Rockinghouse Corp. plans to issue seven-year zero coupon bonds.

They have learned that these bonds will sell today at a price of $439.76. Assuming annual compounding for payments, what is the yield to maturity on these bonds?

Solution:

Years to maturity = n = 7 Coupon rate = C = 0% Current market rate = i Assume annual coupon payments.

57

Present value of bond = PB = $439.76 0

1

2

3

4

5

6

7

├───┼────┼───┼───┼───┼────┼───┤ $0

$0

$0

$0

$0

$0

$0 $1,000

To solve for the YTM, a trail-and-error approach has to be used. Try YTM = 10%. PB =

Fmn

(1 + i m)

mn

=

$1,000 (1.10) 7

$439.76 ≠ $513.16

Try a higher rate, YTM = 12%. PB =

Fmn

(1 + i m) mn

=

$1,000 (1.12) 7

=

$1,000

$439.76 ≠ $452.35 Try YTM=12.5%. PB =

Fmn

(1 + i m)

mn

(1.125) 7

$439.76 ≅ $438.46

The YTM is approximately 12.5 percent. Enter

Answer

8.20 .20

7

$0 N i% 12.453%

-$439.76 $1,000 PMT

PV

FV

Yie Yield to mat maturi urity: Electrolex, Inc., has four-year bonds outstanding that pay a coupon

rate of 6.6 percent semiannually. If these bonds are currently selling at $914.89, what is

58

the yield to maturity that an investor can expect to earn on these bonds? What is the effective annual yield?

Solution:

Years to maturity = n = 4 Coupon rate = C = 6.6% Current market rate = i Semiannual coupon payments = $1,000 x (0.066/2) = $33 Present value of bond = PB = $914.89 0

1

2

3

├───────┼────────┼────────┼── $3 3

$ 33

$3 3

8 ─────────┤ $ 33 $1,000

To solve for the YTM, a trail-and-error approach has to be used. Since this is a discount  bond, the market rate should be higher than 6.6 percent. Try i = 8% or i/2 = 4%. 1   1 −  (1 + i ) n  FV PB = C ×  + n i   (1 + i )   1   1 −  (1.04) 8  $1,000 $914.89 = $33 ×  + 8  0.04  (1.04)   = $222.18 + $730.69 ≠ $952.87

Try a higher rate, i = 9%, i/2 = 4.5%.

59

1   − 1  (1 + i ) n  FV PB = C ×  + n i   (1 + i )   1   − 1 8  (1.045)  $1,000 $914.89 = $33 ×  + 8 0 . 045   (1.045)   = $217.66 + $703.19 ≠ $920.85

Try a higher rate, i = 9.2%, i/2 = 4.6%. 1   − 1  (1 + i ) n  FV PB = C ×  + n i   (1 + i )   1   1 −  (1.046)8  $1,000 $914.89 = $33 ×  + 8 0 . 046   (1.046)   = $216.78 + $697.82 ≅ $914.60

The yield to maturity is approximately 9.2 percent . The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1

= (1.046) − 1 = 0.0941 = 9.41% 2

8

Enter

Answer

$ 33

-$914.89 $1,000

N i% 4.5954%

PMT

The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1

= (1.045954 ) − 1 = 0.09399 = 9.4% 2

60

PV

FV

8.21 .21

Yie Yield to mat maturi urity: Serengeti Corp. has five-year bonds outstanding that pay a coupon of 

8.8 percent. If these bonds are priced at $1,064.86, what is the yield to maturity on these  bonds? Assume semiannual coupon payments. What is the effective annual yield?

Solution:

Years to maturity = n = 5 Coupon rate = C = 8.8% Current market rate = i Semiannual coupon payments = $1,000 x (0.088/2) = $44 Present value of bond = PB = $1,064.86 0

1

2

3

├───────┼────────┼────────┼── $4 4

44

$ 44

10 ─────────┤ $ 44 $1,000

To solve for the YTM, a trail-and-error approach has to be used. Since this is a premium  bond, the market rate should be lower than 8.8 percent. Try i = 7% or i/2 = 3.5%.

61

1   1 −  (1 + i ) n  FV PB = C ×  + n i   (1 + i )   1   − 1 10  (1.035)  $1,000 $1,064.86 = $44 ×  + 10 0 . 035   (1.035)   = $365.93 + $708.92 ≠ $1,074.85

Try a higher rate, i = 7.2%, i/2 = 3.6%. 1   − 1  (1 + i ) n  FV PB = C ×  + n i   (1 + i )   1   − 1  (1.036)10  $1,000 $1,068.86 = $44 ×  + 10  0.036  (1.036)   = $364.09 + $702.11 ≅ $1,068.04

The YTM is approximately 7.2 percent. The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1

= (1.036) − 1 = 0.0733 = 7.33% 2

10

Enter

Answer

$44

-$1,064.86 $1,000

N i% 3.6156%

PMT

The effective annual yield can be computed as: EAY = (1 + Quotedrate m) m − 1

= (1.036156 ) − 1 = 0.0736 = 7.36% 2

62

PV

FV

8.22 .22

Yie Yield to mat maturi urity: Adrienne Dawson is planning to buy 10-year zero coupon bonds

issued by the U.S. Treasury. If these bonds with a face value of $1,000 are currently selling at $404.59, what is the expected return on these bonds? Assume that interest compounds semiannually on similar coupon-paying bonds.

Solution:

Years to maturity = n = 10 Coupon rate = C = 0% Current market rate = i Assume annual coupon payments. Present value of bond = PB = $404.59 0

1

2

3

├───────┼────────┼────────┼── $0

$0

$0

20 ─────────┤ $0 $1,000

To solve for the YTM, a trail-and-error approach has to be used. Try YTM = 10%. PB =

Fmn

(1 + i m ) mn

=

$1,000 (1.05) 20

$404.59 ≠ $376.89

Try a lower rate, YTM = 9%. PB =

Fmn

(1 + i m )

mn

=

$1,000 (1.045) 20

$404.59 ≠ $414.64

63

Try YTM=9.25%. PB =

Fmn

(1 + i m)

mn

$1,000

=

(1.04625) 20

$404.59 ≅ $404.85

The YTM is approximately 9.25 percent . EAY = (1 + Quoted rate m ) m − 1

= (1.04625) 2 − 1 = 0.09464 = 9.46%

The expected return from this investment is 9.46 percent. 20

Enter

Answer

$0 N 4.63%

-$404.59 $1,000 i%

PMT

PV

FV

The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1

= (1.046283) − 1 = 0.0947 = 9.47% 2

8.23

Realized yield:

Brown & Co. issued seven-year bonds two years ago. You bought the

 bond at that time for its face value of $1,000. The bond makes semiannual coupon  payments at a coupon rate of 7.875 percent. The bond has a market value of o f $1,053.40, and the call price is $1,078.75. $1,0 78.75. If the bonds are called by the firm,, what is the investor’s realized yield?

Solution:

Purchase price of bond = $1,000

64

Years investment held = n = 2 Coupon rate = C = 7.875% Frequency of payment = m = 2 Annual coupon = $1,000 x (0.07875/2) = $39.375 Realized yield = i Selling price of bond = PB = $1,053.40 To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has increased, market rates must have decreased. So, the realized return is going to be greater than the bond’s coupon. Try rates higher than the coupon rate. Try i = 10%, or i/2 = 5%. 1   1 −  (1 + i ) m×n  C  FV + 2 PB = ×  (1 + i ) m×n i 2  2 2     1   − 1  (1.05) 4  $1,053.40 $1,000 = $39.375 ×  + 0 . 05 (1.05) 4     = $139.62 + $866.63 ≠ $1,006.26

Try a higher rate, i = 10.3% or i/2 = 5.15%.

65

1   − 1  (1 + i ) m×n  C  FV + 2 PB = ×  (1 + i ) m×n i 2  2 2     1   − 1 4  (1.0515)  $1,053.40 $1,000 = $39.375 ×  + 4 0 . 0515   (1.0515)   = $139.14 + $861.70 ≅ $1,000.84 EAY = (1 + Quoted rate m ) m − 1

= (1.0515) − 1 = 0.1057 = 10.57% 2

The realized rate of return is approximately 10.57 percent. Using a financial calculator   provided an exact yield of 10.35 percent. 4

Enter

Answer

$ 3 9 .3 7 5 - $ 1 , 0 0 0 N i% 5.173%

$ 1 ,0 5 3 . 4 0

PMT

PV

FV

The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1

= (1.05173) − 1 = 0.1061 = 10.61% 2

8.24

Realized yi yield:

Trevor Price Trevor Price bought 10-year bonds issued by Harvest Foods five years

ago for $936.05. The bonds make semiannual coupon payments at a rate of 8.4 percent. If  the current price of the bond is $1,048.77, what is the yield that Trevor would earn by selling the bonds today?

Solution:

66

Purchase price of bond = $936.05 Years investment held = n = 5 Coupon rate = C = 8.4% Frequency of payment = m = 2 Annual coupon = $1,000 x (0.084/2) = $42 Realized yield = i Selling price of bond = PB = $1,048.77 To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has increased, market rates must have decreased. So, the realized return is going to be greater than the bond’s coupon. Try rates higher than the coupon rate. Try i = 11%, or i/2 = 5.5%. 1   1 −  (1 + i ) m×n  C FV + 2 PB = ×   (1 + i ) m×n i 2  2 2     1   1 −  (1.055)10  $1,048.77 $936.05 = $42 ×  + 14  0.055  (1.055)   = $316.58 + $613.98 ≠ $930.56

Try a lower rate, i = 10.8% or i/2 = 5.4%.

67

1   − 1  (1 + i ) m×n  C  FV + 2 PB = ×  (1 + i ) m×n i 2  2 2     1   1 − 10  (1.054)  $1,048.77 $936.05 = $42 ×  + 14 0 . 054   (1.054)   = $318.10 + $619.83 ≅ $937.94 EAY = (1 + Quoted rate m ) m − 1

= (1.054) − 1 = 0.1109 = 11.09% 2

The realized rate of return is approximately 11.1 percent. Using a financial calculator   provided an exact yield of 11.14 percent. 10

Enter

Answer

$42

- $ 93 6 . 0 5

N i% 5.425%

$ 1 , 0 4 8 .7 7

PMT

PV

FV

The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1

= (1.05425) − 1 = 0.1114 = 11.14% 2

8.25

Realized yi yield:

You bought a six-year bond issued by Runaway Corp. four years ago. At

that time, you paid $974.33 for the bond. The bond pays a coupon rate of 7.375 percent, and interest is paid semiannually. Currently, the bond is priced at $1,023.56. What is the return that you can expect to earn on this bond if you sold it today?

Solution:

68

Purchase price of bond = $974.33 Years investment held = n = 4 Coupon rate = C = 7.375% Frequency of payment = m = 2 Annual coupon = $1,000 x (0.07375/2) = $36.875 Realized yield = i Selling price of bond = PB = $1,023.56 To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has increased, market rates must have decreased. So, the realized return is going to be greater than the bond’s coupon. Try rates higher than the coupon rate. Try i = 9%, or i/2 = 4.5%. 1  1 −  (1 + i ) m×n C  2 PB = ×  i 2 2  

  FV +  (1 + i ) m×n 2  

1   1 −  (1.045)8  $1,023.56 $974.33 = $36.875 ×  + 8  0.045  (1.045)   = $243.22 + $719.75 ≠ $962.98

Try a lower rate, i = 8.6% or i/2 = 4.3%.

69

1   − 1  (1 + i ) m×n  C  FV + 2 PB = ×  (1 + i ) m×n i 2  2 2     1   1 − 8  (1.043)  $1,023.56 $974.33 = $36.875 ×  + 8 0 . 043   (1.043)   = $245.22 + $730.87 ≅ $976.09 EAY = (1 + Quoted rate m ) m − 1

= (1.043) − 1 = 0.08785 = 8.79% 2

The realized rate of return is approximately 8.79 percent. Using a financial calculator   provided an exact yield of 8.84 percent.

70

8

Enter

Answer

$36.875 875 -$974.3 4.33 N i% 4.327%

$1,023 023.56

PMT

PV

FV

The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1

= (1.04327 ) − 1 = 0.0884 = 8.84% 2

ADVANCED 8.26

Lopez Information Systems is planning to issue 10-year bonds. b onds. The going market rate for  such bonds is 8.125 percent. Assume that coupon payments will be semiannual. The firm is trying to decide between issuing an 8 percent coupon bond or a zero coupon bond. The company needs to raise $1 million. a.

What will be the price of the 8 percent coupon bonds?

b.

How many coupon bonds would have to be issued?

c.

What will be the price of the zero coupon bonds?

d.

How many zero coupon bonds will have to be issued?

Solution: a.

Years to maturity = n = 10 Coupon rate = C = 8.125% Semiannual coupon = $1,000 x (0.08/2) = $40 Current market rate = i = 8.125% Present value of bond = PB 71

0

1

2

3

14

├───────┼────────┼────────┼── $ 40

$ 40

─────────┤

$4 0

$ 40 $1,000

1   − 1 1   2 n   − 1 i  (1.040625) 20  F $1,000  (1 + 2 )  PB = C ×  $ 40 + = × +   2n  2 i 0.040625 (1.040625) 20 i )   1 ( +   2 2       = $540.62 + $450.94 = $991.55

The firm can sell these bonds at $991.55 . 20

Enter

4.0625% $40 N

i%

PMT -$991.55

Answer

b.

$1,000 PV

FV

Amount needed to be raised = $1,000,000  Number of bonds sold = $1,000,000 / $991.55 = 1,009

c.

Years to maturity = n = 10 Coupon rate = C = 0% Current market rate = i = 8.125% Assume semiannual coupon payments. 0

1

2

3

4

5

6

20

├───┼────┼───┼───┼───┼────┼──

─────┤

$0

$0

$0

$0

$0

$0

$0

$1,000 72

PB =

Fmn

(1 + i m) 20

Enter

mn

=

$1,000 = $450.94 (1.040625) 20

4.0625% $0 N

i%

Answer

d.

$1,000 PMT -$450.94

PV

FV

At the price of $450.94, the firm needs to raise $1 million. To do so, the firm will have to issue:  Number of contracts = $1,000,000 / $450.94 = 2,218 contracts

8.27

Showbiz, Inc., has issued eight-year bonds with a coupon of 6.375 percent and semiannual coupon payments. The market’s required rate of return on such bonds is 7.65  percent. a. b.

What is the market price of these bonds?  Now, assume that the above bond, which was purchased for $1,000, is sold after  five years at $885. What is the realized return on this bond?

Solution: a.

Years to maturity = n = 8 Coupon rate = C = 6.375% Semiannual coupon = $1,000 x (0.06375/2) = $31.875 Current market rate = i = 7.65% Present value of bond = PB

73

0

1

2

3

├───────┼────────┼────────┼── $3 1 .8 75

$ 3 1 .8 7 5

$ 3 1 .8 7 5

16 ─────────┤ $ 3 1 .8 7 5 $1,000

1   − 1 1   2 n   − 1 i  (1.03825)16  F $1,000  (1 + 2 )  PB = C ×  $ 31 . 875 + = × +   2n  2 i 0.03825 (1.03825)16 i )   1 ( +   2 2       = $376.26 + $548.49 = $924.75

The firm can sell these bonds at $924.75 .

b.

Purchase price of bond = $1,000 Years investment held = n = 5 Coupon rate = C = 6.375% Semiannual coupon = $1,000 x (0.06375/2) = $31.875 Frequency of payment = m = 2 Realized yield = i Selling price of bond = PB = $885 To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has decreased, market rates must have increased. So, the realized return is going to be less than the bond’s bond ’s coupon. Try rates lower than the coupon rate. Try i = 4%, or i/2 = 2%.

74

1   − 1  (1 + i ) m×n  C  FV + 2 PB = ×  (1 + i ) m×n i 2  2 2     1   1 − 10  (1.02)  $885 $1,000 = $31.875 ×  + 10  0.02  (1.02)   = $286.32 + $726.01 ≠ $1,012.33

Try a higher rate, i = 4.2% or i/2 = 2.1%. 1  1 − (1 + i ) m×n C 2 PB = ×  i 2  2  

  FV +  (1 + i ) m×n 2  

1   1 − (1.021)10  $885 $1,000 = $31.875 ×  + 10  0.021  (1.021)   = $284.83 + $718.93 ≅ $1,003.76 EAY = (1 + Quoted rate m) m − 1 2

= (1.021) − 1 = 0.04244 = 4.24%

The realized rate of return is approximately 4.24 percent. Using a financial calculator   provided an exact yield of 4.33 percent. Enter

Answer

10

$ 3 1 . 8 75 - $ 1 ,0 0 0 N i% 2.144%

$ 88 5

PMT

The effective annual yield can be computed as:

75

PV

FV

EAY = (1 + Quotedrate m) m − 1 2

= (1.02144 ) − 1 = 0.0433 = 4.33%

8.28

Peabody Corp. has seven-year bonds outstanding. The bonds pay a coupon of 8.375  percent semiannually and are currently worth $1,063.49. The bonds were sold today after  holding them for three years. Assume the bonds were purchased at their par p ar value. a.

What is the yield to maturity on the bond?

b.

What is the effective annual yield?

c.

What is the realized yield on the bonds if they are called?

d.

If you plan to invest in this bond today and hold it to maturity, what is the expected yield on the investment? Explain.

Solution: a.

Years to maturity = n = 7 Coupon rate = C = 8.375% Current market rate = i Semiannual coupon payments = $1,000 x (0.08375/2) = $41.875 Present value of bond = PB = $1,063.49 0

1

2

3

├───────┼────────┼────────┼── $4 1 .8 7 5

$ 4 1 .8 7 5

$ 41 . 8 7 5

14 ─────────┤ $4 1 .8 75 $1,000

76

To solve for the YTM, a trail-and-error approach has to be used. Since this is a premium  bond, the market rate should be lower than 8.375 percent. Try i = 8% or i/2 = 4%. 1   1 −  (1 + i ) n  FV 2 + PB = C ×    (1 + i ) n i 2 2     1   − 1  (1.04)14  $1,000 $1,063.49 = $41.875 ×  + 14  0.04  (1.04)   = $442.33 + $577.48 ≠ $1,019.81

Try a lower rate, i = 7.2%, or i/2 = 3.6%. 1   − 1 n  (1 + i 2)  FV PB = C ×  + n i   (1 + i 2 ) 2   1   − 1 14  (1.036)  $1,000 $1,063.49 = $41.875 ×  + 14 0 . 036   (1.036)   = $454.24 + $609.49 ≅ $1,063.73

The yield-to maturity is approximately 7.2 percent.

b.

The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1

= (1.036) − 1 = 0.0941 = 7.3% 2

Enter

14

$41.875 N

i%

-$1,063.49 $1,000 PMT

77

PV

FV

Answer

3.5998%

The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1

= (1.035998) − 1 = 0.073292 = 7.3% 2

c.

Purchase price of bond = $1,000 Years investment held = n = 3 Coupon rate = C = 8.375% Semiannual coupon payments = $1,000 x (0.08375/2) = $41.875 Frequency of payment = m = 2 Realized yield = i Selling price of bond = PB = $1,063.49 To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has increased, market rates must have decreased. So, the realized return is going to be higher than the bond’s coupon. Try rates higher than the coupon rate. Try i = 10%, or i/2 = 5%. 1   1 −  (1 + i ) m×n  C  FV + 2 PB = ×  (1 + i ) m×n i 2  2 2     1   − 1  (1.05) 6  $1,063.49 $1,000 = $41.875 ×  + 0 . 05 (1.05) 6     = $212.55 + $793.59 ≠ $1,006.14

78

Try a higher rate, i = 10.2% or i/2 = 5.1%. 1  1 −  (1 + i ) m×n C  2 PB = ×  i 2 2  

  FV +  (1 + i ) m×n 2  

1   − 1  (1.051) 6  $1,063.49 $1,000 = $41.875 ×  + 6  0.051  (1.051)   = $211.87 + $789.07 ≅ $1,000.94 EAY = (1 + Quoted rate m ) m − 1 2

= (1.051) − 1 = 0.1046 = 10.5%

The realized rate of return is approximately 10.2 percent. Using a financial calculator   provided an exact yield of 10.5 percent. 6

Enter

Answer

$ 4 1 .8 7 5 - $ 1 ,0 0 0 N i% 5.118%

$ 1 ,0 6 3 . 4 9

PMT

PV

FV

The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1 2

= (1.05118) − 1 = 0.10498 = 10.5%

d.

Purchase price of bond = PB = $1,063.49 Years to maturity = n = 4 Coupon rate = C = 8.375% Semiannual coupon payments = $1,000 x (0.08375/2) = $41.875 Frequency of payment = m = 2 79

Maturity value = FV = $1,000 Use the trial-and-error approach to compute the yield to maturity. Since we have a  premium bond, market rates are lower than the bond’s coupon. Try i = 7%, or i/2 = 3.5%. 1   1 −  (1 + i ) m×n  C  FV + 2 PB = ×  (1 + i ) m×n i 2  2 2     1   − 1 8  (1.035)  $1,000 $1,063.49 = $41.875 ×  + 8 0 . 035   (1.035)   = $287.85 + $759.41 ≠ $1,047.26

Try a lower rate, i = 6.6% or i/2 = 3.3%. 1  − 1  (1 + i ) m×n C  2 PB = ×  i 2 2  

  FV +  (1 + i ) m×n 2  

1   1 −  (1.033)8  $1,000 $1,063.49 = $41.875 ×  + 8  0.033  (1.033)   = $290.26 + $771.25 ≠ $1,061.52

Try a slightly lower rate i = 6.5% or i/2 = 3.25%.

80

1   − 1  (1 + i ) m×n  C  FV + 2 PB = ×  (1 + i ) m×n i 2  2 2     1   1 − 8  (1.0325)  $1,000 $1,063.49 = $41.875 ×  + 8  0.0325  (1.0325)   = $290.87 + $774.25 ≅ $1,065.12 EAY = (1 + Quoted rate m ) m − 1 2

= (1.0325) − 1 = 0.0661 = 6.61%

The realized rate of return is approximately 6.61 percent. Using a financial calculator   provided an exact yield of 6.65 percent. 8

Enter

Answer

$41.875 -$1,063.49 $1,000 N i% 3.273%

PMT

PV

FV

The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1 2

= (1.03273) − 1 = 0.06653 = 6.65%

8.29

Maryland Department of Transportation has issued 25-year bonds bond s that pay semiannual coupons at a rate of 9.875 percent. The current market rate for similar securities is 11  percent. a.

What is the bond’s current market value?

b.

What will be the bond’s price if rates in the market (i) decrease to 9 percent; (ii) increase to 12 percent? 81

c.

Refer to your answers in part b. How do the interest rate changes affect premium p remium  bonds and discount bonds?

d.

Suppose the bond were to mature in 12 years. How do the interest rate changes in  part b affect the bond prices?

Solution: a.

Years to maturity = n = 25 Coupon rate = C = 9.875% Semiannual coupon = $1,000 x (0.09875/2) = $49.375 Current market rate = i = 11% Present value of bond = PB 0

1

2

3

50

├───────┼────────┼────────┼── $ 49 .3 75

$ 4 9 .3 7 5

─────────┤

$ 4 9 .3 7 5

$ 4 9 .3 7 5 $1,000

1   − 1 1   2n   1− i 50  F $1,000  (1 + 2 )  (1.055)  C PB = $49.375 ×  × + = +  2 n   2 i 0.055 (1.055) 50 i )   1 ( +   2 2       = $835.99 + $68.77 = $904.76

The Maryland bonds will sell at $904.76 . Enter

50

5.5% $49.375 N

Answer

i%

$1,000 PMT -$904.76

82

PV

FV

b.

(i)

Current market rate = i = 9% 1   − 1 1   2 n   − 1 i  (1.045) 50  F $1,000  (1 + 2 )  + = × + PB = C ×  $ 49 . 375   2n  2 i 0.045 (1.045) 50 i )   ( + 1   2 2       = $975.75 + $110.71 = $1,086.46

The Maryland bonds will increase in price to sell at $1,086.46 . 50

Enter

4.5% $49.375 N

i%

Answer

(ii)

$1,000 PMT PV -$1,086.46

FV

Current market rate = i = 12% 1   − 1 1   2n   − 1 i 50  ( ) 1+ F (1.06)  $1,000   2 C × = $49.375 ×  PB = + 2n + 2  i 0 . 06 (1.06) 50 i   ( ) 1 +   2 2       = $778.24 + $54.29 = $832.53

The Maryland bonds will drop in price to $832.53 . Enter

50

6% $49.375 N

i%

Answer

c.

$1,000 PMT -$832.53

PV

FV

Bonds, in general, decrease in price when interest rates go up. When interest rates decrease, bond prices increase.

83

d.

(i)

Current market rate = i = 9% Term to maturity = 12 years 1   1 1 −   2n   1 − i 24  ( ) 1 + F $1,000 (1.045)    2 C × + = × + PB = $ 49 . 375   2n  2 i 0.045 (1.045) 24 i )   ( + 1   2 2       = $715.71 + $347.70 = $1,063.42

The Maryland bonds will increase in price to sell at $1,063.42 . 24

Enter

4.5% $49.375 N

i%

Answer

(ii)

$1,000 PMT PV -$1,063.42

FV

Current market rate = i = 12% 1   1 1 −   2 n   1 − i 24  (1.06)  $1,000 F  (1 + 2 )  PB = C ×  $ 49 . 375 + = ×  + 2n  2 i 0 . 06 (1.06) 24 i   ( ) + 1   2 2       = $619.67 + $246.98 = $866.65

The Maryland bonds will drop in price to $866.65 . Enter

24

6% $49.375 N

Answer

i%

$1,000 PMT -$866.65

PV

FV

With shorter maturity, bond prices react the same way as in part b, but to a lesser extent. When interest rates increase, the bond’s price declines; but b ut the decline in price is less

84

than that for a longer term bond. When interest rates decrease, bond prices increase with longer-term bonds, increasing more than shorter-term bonds.

8.30

Rachette Corp. issued 20-year bonds five years ago. These bonds, which pay semiannual coupons, have a coupon rate of 9.735 percent and a yield to maturity of 7.95 percent.

a.

Compute the bond’s current price.

b.

If the bonds can be called after five more years at a premium of 13.5 percent over par  value, what is the investor’s realized yield?

c.

If you bought the bond today, what is your expected rate of return? Explain.

Solution: a.

Years to maturity = n = 15 Coupon rate = C = 9.735% Semiannual coupon = $1,000 x (0.09735/2) = $48.675 Current market rate = i = 7.95% Present value of bond = PB 0

1

2

3

├───────┼────────┼────────┼── $ 4 8 .67 5

$ 4 8 .67 5

$ 48 .6 7 5

30 ─────────┤ $ 4 8 .6 7 5 $1,000

85

1   − 1 1   2n   − 1 i 30   ( ) + 1 F $1,000 (1.03975)  2 + PB = C ×  $ 48 . 675 = × +   2n  2 i 0.03975 (1.03975) 30 i )   ( + 1   2 2       = $844.25 + $310.55 = $1,154.80

The bond’s current price is at $1,154.80. Enter

30

3.975% $48.675 N

i%

Answer

b.

$1,000 PMT PV -$1,154.80

FV

Purchase price of bond = $1,000 Years investment held = n = 5 Coupon rate = C = 9.735% Semiannual coupon = $1,000 x (0.09735/2) = $48.675 Frequency of payment = m = 2 Realized yield = i Selling price of bond = PB = $1,057.25 To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has increased, market rates must have decreased. So, the realized return is going to be higher than the bond’s coupon. Try rates higher than the coupon rate. Try i = 10%, or i/2 = 5%.

86

1   − 1  (1 + i ) m×n  C  FV + 2 PB = ×  (1 + i ) m×n i 2  2 2     1   1 − 10  (1.05)  $1,057.25 $1,000 = $48.675 ×  + 10 0 . 05   (1.05)   = $375.86 + $649.06 ≠ $1,024.92

Try a higher rate, i = 10.5% or i/2 = 5.25%. 1  1 − (1 + i ) m×n C 2 PB = ×  i 2  2  

  FV +  (1 + i ) m×n 2  

1   1 − (1.053)10  $1,057.25 $1,000 = $48.675 ×  + 10 0 . 053   (1.0535)   = $370.44 + $630.80 ≅ $1,001.24 EAY = (1 + Quoted rate m ) m − 1 2

= (1.053) − 1 = 0.1088 = 10.9%

The realized rate of return is approximately 10.9 percent. Using a financial calculator   provided an exact yield of 10.54 percent. Enter

Answer

10

$48.675 N i% 5.316%

-$1,000 PMT

The effective annual yield can be computed as:

87

$1 ,05 7 .2 5 PV

FV

EAY = (1 + Quoted rate m ) m − 1 2

= (1.05136) − 1 = 0.1054 = 10.54%

c.

Purchase price of bond = PB = $1,057.25 Years to maturity = n = 15 Coupon rate = C = 9.735% Semi-annual coupon = $1,000 x (0.09735/2) = $48.675 Frequency of payment = m = 2 Maturity value = FV = $1,000 Use the trial-and-error approach to compute the yield to maturity. Since we have a  premium bond, market rates are lower than the bond’s coupon. Try i = 9%, or i/2 = 4.5%. 1   − 1  (1 + i ) m×n  C  FV + 2 PB = ×  (1 + i ) m×n i 2  2 2     1   1 −  (1.045) 30  $1,000 $1,057.25 ≠ $48.675 ×  + 30  0.045  (1.045)   ≠ $792.86 + $267.00 = $1,059.86 EAY = (1 + Quoted rate m ) m − 1

= (1.045) − 1 = 0.0920 = 9.20% 2

The realized rate of return is approximately 9.2 percent. Using a financial calculator   provided an exact yield of 9.23 percent. 88

30

Enter

Answer

$48.675 -$1,057.25 $1,000 N i% 4.515%

PMT

The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1 2

= (1.04515) − 1 = 0.0923 = 9.23%

89

PV

FV

Sample Test Problems 8.1

Torino Foods issued 10-year bonds three years ago with a coupon of 6 percent. If the current market rate is 8.5 percent and the bonds pay annual coupons, what is the current market price of this bond?

Solution:

Years to maturity = n = 7 Coupon rate = C = 6% Annual coupon = $1,000 x 0.06 = $60 Current market rate = i = 8.5% Present value of bond = PB 0

1

2

3

4

7

├───┼────┼───┼───┼─── $ 60

$6 0

$ 60

─────┤

$ 60

$ 60 $1,000

n = 7;

C = 6%;

i = YTM = 8.5%

1  1    − − 1 1 7   (1 + i ) n   F $1,000 (1.085) PB = C ×  $ 60 = × + +   n 7 i   (1 + i )  0.085  (1.085)     = $307.11 + $564.93 = $872.04

8.2

Kim Sundaram recently bought a 20-year zero coupon bond which compounds interest semiannually. If the current market rate is 7.75 percent, what is the maximum price he should have paid for this bond? 90

Solution:

Years to maturity = n = 20 Coupon rate = C = 0% Current market rate = i = 7.75% 0

1

2

3

4

5

6

20

├───┼────┼───┼───┼───┼────┼──

─────┤

$0

$0

$0

$0

$0

$0

$0

$1,000 PB =

8.3

Fmn

(1 + i m)

mn

=

$1,000 = $224.73 (1.0775) 20

Five-year bonds of Infotech Corporation are currently priced at $1,065.23. They pay semiannual coupons of 8.5 percent. If you bought these bonds today, what would be the yield to maturity and effective annual yield that you would earn?

Solution:

Years to maturity = n = 5 Coupon rate = C = 8. 5% Current market rate = i Semiannual coupon payments = $1,000 x (0.085/2) = $42.50 Present value of bond = PB = $1,065.23 0

1

2

3

91

10

├───────┼────────┼────────┼── $4 2 .5 0

$ 4 2 .5 0 $4 2 . 5 0

─────────┤ $ 4 2 .5 0 $1,000

To solve for the YTM, a trail-and-error approach has to be used. Since this is a premium  bond, the market rate should be lower than 8. 5 percent. Try i = 8% or i/2 = 4%. 1  1 −  (1 + i ) n 2 PB = C ×   i 2  

   + FV  (1 + i ) n 2  

1   1 − 10  (1.04)  $1,000 $1,065.23 = $42.50 ×  + 10 0 . 04   (1.04)   = $344.71 + $675.56 ≠ $1,020.28

Try a lower rate, i = 7 %, or i/2 = 3.5%. 1   1 −  (1 + i ) n  FV 2 + PB = C ×    (1 + i ) n i 2 2     1   1 −  (1.035)10  $1,000 $1,065.23 = $42.50 ×  + 10 0 . 035   (1.035)   = $349.92 + $708.92 ≠ $1,062.37

The yield to maturity is approximately 7 percent. The effective annual yield can be computed as:

92

EAY = (1 + Quoted rate m ) m − 1 2

= (1.035) − 1 = 0.0712 = 7.12%

10

Enter

Answer

$42.50 -$1,065.23 -$1,065.23 $1,000 N 3.467%

i%

PMT

PV

FV

The effective annual yield can be computed as: EAY = (1 + Quoted rate m ) m − 1 2

= (1.03467 ) − 1 = 0.07054 = 7.05%

8.4

The Gold Company wants to borrow on a five-year term from its bank. The lender  determines that the firm should pay a default risk premium of 1.75 percent over the Treasury rate. The five-year Treasury rate is currently 5.65 5.6 5 percent. The firm also faces a marketability risk premium of 0.80 percent. What is the total borrowing cost to the firm?

Solution:

Risk-free real rate of interest =

5.65%

Market risk premium

=

0.80%

Default risk premium

=

1.75%

Using Equation 8.6: kcorp

=

irf  + risk premium adjustments

=

irf  + MRP + DRP

6.5% = =

5.65%+0.80%+1.75 8.2%

93

The company’s borrowing cost is 8.2 percent .

8.5

Trojan Corp. has issued seven-year bonds that are paying 7 percent semiannual coupons. If the opportunity cost for Brianna Lindner is 8.25 8 .25 percent, what is the maximum price that she would be willing to pay for this bond?

Solution:

Years to maturity = n = 7 Coupon rate = C = 7% Semi-annual coupon payments = $1,000 x (0.07/2) = $35 Current market rate = i = 8.25% Present value of bond = PB 0

1

2

3

4

14

├───┼────┼───┼───┼─── $ 35

$3 5

$ 35

─────┤

$ 35

$ 35 $1,000

n = 7; m = 2;

C = 7%; 7%;

i = YTM = 8.25%

1  1    1 1 − − n 14  (1 + i )   (1.04125)  F $1,000 PB = C ×  $ 35 = × +  + n 14 i   (1 + i )  0.04125  (1.04125)     = $366.68 + $567.84 = $934.52

Brianna Lindner should be willing to pay no more than $934.52.

94