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Copyright Information (bibliographic)
Document Type: Book Chapter
Title of Book: Financial Management Theory and Practice (16th Edition)
Author(s) of Book: Eugene F. Brigham, Michael C. Ehrhardt
Chapter Title: Chapter 5 Bonds, Bond Valuation, and Interest Rates
Author(s) of Chapter: Eugene F. Brigham, Michael C. Ehrhardt
Year: 2020
Publisher: Cengage Learning
Place of Publishing: the United States of America
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Bonds, Bond Valuation, and Interest Rates
A lot of bonds have been issued in the United States, and we mean a lot! According
to the Federal Reserve, in 2017 there were $15.8 trillion of outstanding U.S. Treasury
securities, $3.8 trillion of municipal securities, and $5.2 trillion of corporate bonds. Not
only is the dollar amount mind-boggling, but so is the variety.
Bonds aren't the only way to borrow. In addition to their bonds, corporations
owe $2.9 trillion in short-term debt. Noncorporate businesses, which include small
businesses, owe $5.2 trillion.
Let's not ignore households, which owe $3.8 trillion in consumer debt, such as car
loans, student loans, and credit cards. This works out to about $30,000 per household,
and this doesn't even include the $9.2 trillion in mortgages owed by homeowners.
Given the enormous amount of debt in the modern world, it is vital for everyone to
understand debt and interest rates.
Sources: Board of Governors of the Federal Reserve, "Financial Accounts of the United States-Z.l", www
.federalreserve.gov/releases/zl/default.htm; Federal Reserve Bank of New York at www.newyorkfed
.org/microeconomics/hhdc.html; and Federal Reserve Bank of St. Louis, "Total Households,"
https://fred.stlouisfed.org/series/TTLHH.
195
196 Part 2 Fixed Income Securities
Intrinsic Value and the Cost of Debt
This chapter explains bond pricing and bond risk, which
affect the return demanded by a firm's bondholders.
A bondholder's return is a cost from the company's point
of view. This cost of debt affects the firm's weighted
average cost of capital (WACC), which in turn affects the
company's intrinsic value. Therefore, it is important for all
managers to understand the cost of debt, which we explain
in this chapter.
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The textbook's Web site
contains an Excel file that will guide you through the chapter's calculations.
The file far this chapter is
ChOS Taal Kit.xlsx, and we
encourage you to open the file and fallow along as
you read the chapter.
Net operating profit after taxes
Required investments in operating capital
Free cash flow (FCF)
�� IT� �� Value=-----+-----+ ... +-----
(1 + WACC)l (1 + WACC)2 (1 + WACC)�
Fir.m's debt/equity mix
firm's business risk
Growing companies must acquire land, buildings, equipment, inventory, and other oper ating assets. The debt markets are a major source of funding for such purchases. Therefore, every manager should have a working knowledge of the types of bonds that companies and government agencies issue, the terms that are contained in bond contracts, the types of risks to which both bond investors and issuers are exposed, and procedures for deter mining the values of and rates of return on bonds.
5-1 Who Issues Bonds?
A bond is a long-term contract under which a borrower agrees to make payments of inter est and principal, on specific dates, to the holders of the bond. For example, on January 1, 2019, MicroDrive Inc. issued $170 million of bonds.1 For convenience, we assume that Mi croDrive sold 170,000 individual bonds for $1,000 each. (Actually, it could have sold one $170 million bond, 10 bonds with a $17 million face value, or any other combination that totals to $170 million.) In exchange for 170 million, MicroDrive promised to make annual interest payments and to repay the 170 million on a specified maturity date.
Investors have many choices when investing in bonds, but bonds are classified into four main types: Treasury, corporate, municipal, and foreign. Each type differs with re spect to expected return and degree of risk.
'The bonds would actually be issued on the first business day of the year. To reduce unnecessary complications,
we will assume that they were issued on January 1.
0
0
C
Chapter 5 Bonds, Bond Valuation, and Interest Rates 197
Treasury bonds (T-bonds) and Treasury bills (T-bills), sometimes referred to as government bonds, are issued by the U.S. federal government.2 It is reasonable to assume that the federal government will make good on its promised payments, so these bonds have almost no default risk. However, Treasury bond prices decline when interest rates rise, so they are not free of all risks.
Federal agencies and other government-sponsored enterprises (GSEs) include the Ten nessee Valley Authority, the Small Business Administration, Fannie Mae, Freddie Mac, and the Federal Home Loan Bank System, among others. Agency debt and GSE debt are not officially backed by the full faith and credit of the U.S. government, but investors as sume that the government implicitly guarantees this debt, so these bonds carry interest rates only slightly higher than Treasury bonds. In 2008, the implicit guarantee became much more explicit as the government placed several GSEs into conservatorship, includ ing Fannie Mae and Freddie Mac.
Corporate bonds, as the name implies, are issued by corporations. Unlike Treasury bonds, corporate bonds are exposed to default risk-if the issuing company gets into trou ble, it may be unable to make the promised interest and principal payments. Different corporate bonds have different levels of default risk, depending on the issuing company's characteristics and the terms of the specific bond. Default risk is often referred to as "credit risk," and the larger the credit risk, the higher the interest rate the issuer must pay.
Municipal bonds, or "mun is," are issued by state and local governments. Like corporate bonds, munis have default risk. However, munis offer one major advantage: The interest earned on most municipal bonds is exempt from federal taxes and also from state taxes if the holder is a resident of the issuing state. Consequently, municipal bonds carry interest rates that are considerably lower than those on corporate bonds with the same default risk.
Foreign bonds are issued by foreign governments or foreign corporations. Foreign corporate bonds are, of course, exposed to default risk, and so are some foreign govern ment bonds, as became apparent in the spring of2012 when Greece forced its bondholders into an exchange of securities that reduced the value of their holdings of Greek govern ment debt by more than 50%. An additional risk exists if the bonds are denominated in a currency other than that of the investor's home currency. For example, if a U.S. investor purchases a corporate bond denominated in Japanese yen and if the yen subsequently falls relative to the dollar, then the investor will lose money even if the company does not de fault on its bonds.
SELF -TEST
What is a bond?
What are the four main types of bonds?
Why are U.S. Treasury bonds not risk less?
To what types of risk are investors of foreign bonds exposed?
5-2 Key Characteristics of Bonds Although all bonds have some common characteristics, they do not always have identical contractual features, as described here.
2The U.S. Treasury actually issues three types of securities: bills, notes, and bonds. A bond makes an equal pay
ment every 6 months until it matures, at which time it makes an additional payment. If the maturity at the time of issue is less than 10 years, the security is called a note rather than a bond. AT-bill has a maturity of 52 weeks
or less at the time of issue, and it makes no payments at all until it matures. Thus, T-bills are sold initially at a
discount to their face, or maturity, value.
198 Part 2 Fixed Income Securities
Betting With or Against the U.S. Government: The Case of Treasury Bond Credit Default Swaps
It might be hard to believe, but there is actually a market for
U.S. Treasury bond insurance. In July 2011, investors worried
that Congress would not extend the debt ceiling, inducing de
faults in Treasury securities. At that time, a credit default swap
(CDS) on a 5-year T-bond was selling for 63.5 basis points.
(A basis point is 1/100 of a percentage point.) This means
that you could pay $6.35 a year to a counterparty who would
promise to insure $1,000 of T-bond principal against default.
Considering that the T-bond was yielding an amount equal
to about $15 a year, the insurance would eat up a lot of the
annual return for an investor who owned the bond. However,
most of the trading in this CDS is by speculators and hedgers
who don't even own the T-bond but are simply betting for or
against the financial soundness of the U.S. government.
But it does make you wonder: "If the United States fails,
who will be around to pay off the CDS?"
Note: For updates on the 5-year CDS, go to www.cnbc.com/ld/38451750,
and scroll down to •us CDSSYR."
WWW
An excellent site for information on mony types of bonds is the 'FINRA Web page, WWW .flnra.org. Select "VIEW ALL MARKET DATA" at the bottom of the page; then select "Bands" at the top left. The site has o great deal af infarmation about corporates, municipals, Treasuries, and bond funds. It includes free band searches, through which the user specifies the attributes desired in a band and then the search returns the publicly traded bonds meeting the criteria.
5-2a Par Value
The par value is the stated face value of the bond; for illustrative purposes, we generally assume a par value of $1,000. In practice, some bonds have par values that are multiples of $1,000 (for example, $5,000), and some have par values ofless than $1,000. (Treasury bonds can be purchased in multiples of $100.) The par value generally represents the amount of money the firm borrows and promises to repay on the maturity date.
5-2b Coupon Interest Rate
MicroDrive's bonds require the company to pay a fixed number of dollars of interest every year (or, more typically, every 6 months). When this coupon payment, as it is called, is divided by the par value, the result is the coupon interest rate. For example, MicroDrive's bonds have a $1,000 par value, and they pay $100 in interest each year. The bond's coupon interest is $100, so its coupon interest rate is $100/$1,000 = 10%. The coupon payment, which is fixed at the time the bond is issued, remains in force during the life of the bond. 3
Typically, at the time a bond is issued, its coupon payment is set at a level that will enable the bond to be issued at or near its par value.
In some cases, a bond's coupon payment will vary over time. For these floating-rate bonds, the coupon rate is set for, say, the initial 6-month period, after which it is adjusted every 6 months based on some market rate. Some corporate issues are tied to the Trea sury bond rate; other issues are tied to other rates, such as LIBOR (the London Interbank Offered Rate). Many additional provisions can be included in floating-rate issues. For ex ample, some are convertible to fixed-rate debt, whereas others have upper and lower limits (caps and floors) on how high or low the rate can go.
Floating-rate debt is popular with investors who are worried about the risk of rising in terest rates because the interest paid on such bonds increases whenever market rates rise. This stabilizes the market value of the debt, and it also provides institutional buyers, such as banks, with income that is better geared to their own obligations. Banks' deposit costs rise with interest rates, so the income on floating-rate loans they have made rises at the
'At one time, bonds literally had a number of small coupons attached to them, and on each interest payment date the owner would clip off the coupon for that date and either cash it at the bank or mail it to the company's paying agent, who would then mail back a check for the interest. For example, a 30-year, semiannual bond would start with 60 coupons. Today, most new bonds are registered-no physical coupons are involved, and interest checks automatically are mailed to the registered owners or directly deposited in their bank accounts.
wurce
For more on zero coupon
bonds, including U.S.
Treasury STRIP bonds, see
Web Extension SA on the textbook's Website.
Chapter 5 Bonds, Bond Valuation , and Interest Rates 199
same time as their deposit costs rise. The savings and loan industry was almost destroyed as a result of its former practice of making fixed-rate mortgage loans but borrowing on floating-rate terms. If you earn 6% fixed but pay 10% floating (which they were), you will soon go bankrupt (which they did). Moreover, floating-rate debt appeals to corporations that want to issue long-term debt without committing themselves to paying a historically high interest rate for the entire life of the loan.
Some bonds pay no coupons at all but are offered at a substantial discount below their par values and hence provide capital appreciation rather than interest income. These se curities are called zero coupon bonds ("zeros"). Most zero coupon bonds are Treasury bonds, although some are issued by banks. Some bonds are issued with a coupon rate too low for the bond to be issued at par, so the bond is issued at a price less than its par value. In general, any bond originally offered at a price significantly below its par value is called an original issue discount (OID) bond.
Some bonds don't pay cash coupons but pay coupons consisting of additional bonds (or a percentage of an additional bond). These are called payment-in-kind (PIK) bonds. PIK bonds are usually issued by companies with cash flow problems, which makes them risky.
Some bonds have a step-up provision: If the company's bond rating is downgraded, then it must increase the bond's coupon rate. Step-ups are more popular in Europe than in the United States, but that is beginning to change. Note that a step-up is quite dangerous from the company's standpoint. The downgrade means that it is having trouble servicing its debt, and the step-up will exacerbate the problem. This combination has led to a num ber of bankruptcies.
5-2c Maturity Date
Bonds generally have a specified maturity date on which the par value must be repaid. MicroDrive bonds issued on January 1, 2019, will mature on December 31, 2033; thus, they have a 15-year maturity at the time they are issued. Most bonds have an original maturity (the maturity at the time the bond is issued) ranging from 10 to 40 years, but any maturity is legally permissible.4 Of course, the effective maturity of a bond declines each year after it has been issued. Thus, MicroDrive's bonds have a 15-year original maturity, but in 2020, a year later, they will have a 14-year maturity, and so on.
Some bonds have no maturity date. For example, the British government issued some bonds in the mid-1700s with a constant payment each period but without a stated matu rity date. The government used the proceeds to pay off other British bonds. Because this action consolidated the government's debt, the new bonds were called "consols." The term stuck, and now any bond that promises to pay interest perpetually is called a consol. The terms of the bonds allowed the U.K. to redeem them for a certain amount. Although the consols were around for over 200 years, the U.K. redeemed the last one in 2015. Although the U.K. no longer has consols, some banks in India offer perpetual bonds, though most are callable, a feature described in the next section.
5-2d Provisions to Call or Redeem Bonds
Most corporate bonds contain a call provision, which gives the issuing corporation the right to call the bonds for redemption. 5 The call provision generally states that the company
' In July 1993, Walt Disney Co., attempting to lock in a low interest rate, issued the first 100-year bonds to be sold by any borrower in modern times. Soon after, Coca-Cola became the second company to stretch the mean ing of"long-term bond" by selling $150 million of 100-year bonds.
5 A majority of municipal bonds also contain call provisions. Although the U.S. Treasury no longer issues call able bonds, some past Treasury issues were callable.
200
You Can Take That to the Bank
To finance canals and dykes in the Netherlands, local com
munities would band together and create Water Board Au
thorities. One such board issued a bond in 1648 that had no
maturity and would pay 5% interest perpetually. Here's the
catch: To collect the interest payment, the bondholder would
have to take the bond, written on goatskin, to the board.
As you might expect, many of the boards no longer ex
ist, and many of the goatskin bonds have not survived the
years. An exception is the bond purchased by the Beinecke
Rare Book & Manuscript Library at Yale in 2003. The bond
Part 2 Fixed Income Securities
had 26 years of accrued interest, so a Yale finance professor
took the bond to the Netherlands and collected the interest.
Yale keeps the bond in the museum except for infrequent
trips to collect the interest. The last trip was in 2015 to col
lect 12 years of accrued interest, which was about 136 euros.
That certainly didn't pay for airfare or other expenses, but it
proved that the bond is a living artifact.
Source: See https://news.yale.edu/2015/09/22/llving-artifact
-dutch-goldan-aga-yale-s-367-year-old-water-bond-still
-pays-Interest.
must pay the bondholders an amount greater than the par value if they are called. The ad ditional sum, which is termed a call premium, is often set equal to 1 year's interest if the bonds are called during the first year, and the premium declines at a constant rate of INT/N each year thereafter (where INT = annual interest and N = original maturity in years). For example, the call premium on a $1,000 par value, 10-year, 10% bond would gen erally be $100 if it were called during the first year, $90 during the second year (calculated by reducing the $100, or 10%, premium by one-tenth), and so on. However, bonds are often not callable until several years (generally 5 to 10) after they are issued. This is known as a deferred call, and the bonds are said to have call protection.
Suppose a company sold bonds when interest rates were relatively high. Provided the issue is callable, the company could sell a new issue of low-yielding securities if and when interest rates drop. It could then use the proceeds of the new issue to retire the high-rate issue and thus reduce its interest expense. This process is called a refunding operation.
A call provision is valuable to the firm but potentially detrimental to investors. If in terest rates go up, the company will not call the bond, and the investor will be stuck with the original coupon rate on the bond, even though interest rates in the economy have risen sharply. However, if interest rates fall, the company will call the bond and pay off investors, who then must reinvest the proceeds at the current market interest rate, which is lower than the rate they were getting on the original bond. In other words, the investor loses when interest rates go up but doesn't reap the gains when rates fall. To induce an in vestor to take this type of risk, a new issue of callable bonds must provide a higher coupon rate than an otherwise similar issue of noncallable bonds.
Bonds that are redeemable at par at the holder's option protect investors against a rise in interest rates. If rates rise, the price of a fixed-rate bond declines. However, if holders have the option of turning their bonds in and having them redeemed at par, then they are protected against rising rates. If interest rates have risen, holders will turn in the bonds and reinvest the proceeds at a higher rate.
Event risk is the chance that some sudden event will occur and increase the credit risk of a company, hence lowering the firm's bond rating and the value of its outstanding bonds. Investors' concern over event risk means that those firms deemed most likely to face events that could harm bondholders must pay extremely high interest rates. To reduce this interest rate, some bonds have a covenant called a super poison put, which enables a bondholder to turn in, or "put," a bond back to the issuer at par in the event of a takeover, merger, or major recapitalization.
Chapter 5 Bonds, Bond Valuation, and Interest Rates 201
Some callable bonds have a make-whole provision. This allows a company to call the bond, but it must pay a call price that is essentially equal to the market value of a similar noncallable bond. This provides companies with an easy way to repurchase bonds as part of a financial restructuring, such as a merger.
5-2e Sinking Funds
Some bonds include a sinking fund provision that facilitates the orderly retirement of the bond issue. On rare occasions, the firm may be required to deposit money with a trustee, which invests the funds and then uses the accumulated sum to retire the bonds when they mature. Usually, though, the sinking fund is used to buy back a certain percentage of the issue each year. A failure to meet the sinking fund requirement puts the bond into default, which may force the company into bankruptcy.
In most cases, the firm is given the right to administer the sinking fund in either of two ways.
1. The company can call in for redemption (at par value) a certain percentage of the bonds each year; for example, it might be able to call 5% of the total original amount of the issue at a price of $1,000 per bond. The bonds are numbered serially, and those called for redemption are determined by a lottery administered by the trustee.
2. The company may buy the required number of bonds on the open market.
The firm will choose the less costly method. If interest rates have risen, causing bond prices to fall, then it will buy bonds in the open market at a discount; if interest rates have fallen, it will call the bonds. Note that a call for sinking fund purposes is quite different from a refunding call, as discussed previously. A sinking fund call typically requires no call premium, but only a small percentage of the issue is normally callable in any one year. 6
Although sinking funds are designed to protect bondholders by ensuring that an issue is retired in an orderly fashion, you should recognize that sinking funds also can work to the detriment of bondholders. For example, suppose that the bond carries a 10% interest rate but that yields on similar bonds have fallen to 7.5%. A sinking fund call at par would require an investor to give up a bond that pays $100 of interest and then to reinvest in a bond that pays only $75 per year. This obviously harms those bondholders whose bonds are called. On balance, however, bonds that have a sinking fund are regarded as being safer than those without such a provision, so at the time they are issued sinking fund bonds have lower coupon rates than otherwise similar bonds without sinking funds.
5-2f Other Provisions and Features
Owners of convertible bonds have the option to convert the bonds into a fixed number of shares of common stock. Convertibles offer investors the chance to share in the upside if a company does well, so investors are willing to accept a lower coupon rate on convertibles than on an otherwise identical but nonconvertible bond.
Warrants are options that permit the holder to buy stock at a fixed price, thereby pro viding a gain if the price of the stock rises. Some bonds are issued with warrants. As with convertibles, bonds with warrants have lower coupon rates than straight bonds.
An income bond is required to pay interest only if earnings are high enough to cover the interest expense. If earnings are not sufficient, then the company is not required to pay interest, and the bondholders do not have the right to force the company into bankruptcy. Therefore, from an investor's standpoint, income bonds are riskier than "regular" bonds.
•some sinking funds require the issuer to pay a call premium.
202 Part 2 Fixed Income Securities
Indexed bonds, also called purchasing power bonds, first became popular in Brazil, Israel, and a few other countries plagued by high inflation rates. The interest payments and maturity payment rise automatically when the inflation rate rises, thus protecting the bondholders against inflation. In January 1997, the U.S. Treasury began issuing indexed bonds called Treasury Inflation-Protected Securities (TIPS). Later in this chapter we show how TIPS can be used to estimate the real risk-free rate.
5-2g Bond Markets
Corporate bonds are traded primarily in electronic markets rather than in organized ex changes. Most bonds are owned by and traded among a relatively small number of very large financial institutions, including banks, investment banks, life insurance companies, mutual funds, and pension funds. Although these institutions buy and sell very large blocks of bonds, it is relatively easy for bond dealers to arrange transactions because there are relatively few players in this market as compared with stock markets.
Information on bond trades is not widely published, and many bonds trade infrequently, if at all, but a representative group of bonds is listed and traded on the bond division of the NYSE and is reported on the bond market page of The Wall Street Journal. The most use ful Web site (as of early 2018) is provided by the Financial Industry Regulatory Authority (FINRA) at www.finra.org; look for "Investors," "View All Market Data," and then "Bonds."
SELF-TEST
Define "floating-rate bonds" and "zero coupon bonds."
Why is a call provision advantageous to a bond issuer?
What are the two ways a sinking fund can be handled? Which method will be chosen by the firm if interest rates have risen? If interest rates have fallen?
Are securities that provide for a sinking fund regarded as being riskier than those without this
type of provision? Explain.
What are income bonds and indexed bonds?
Why do convertible bonds and bonds with warrants have lower coupons than similarly rated bonds that do not have these features?
5-3 Bond Valuation
The value of any financial asset-a stock, a bond, a lease, or even a physical asset such as an apartment building or a piece of machinery-is simply the present value of the cash flows the asset is expected to produce. This is called discounted cash flow (DCF) analysis, as described in Chapter 4. The following section applies DCF analysis to a bond.
5-3a Time Line, Cash Flows, and Valuation Formulas for a Bond
For a standard coupon-bearing bond, the cash flows consist of interest payments dur ing the life of the bond plus the amount borrowed when the bond matures (usually a $1,000 par value):
0 rd% 1 2 3 N 1------+-----+----+---... --I
Bond's Value INT INT
The notation in the time line is explained next.
INT INT M
Chapter 5 Bonds, Bond Valuation, and Interest Rates
r d
= The required rate of return on debt, which is the rate of return that fairlycompensates an investor for purchasing or holding debt, taking into consideration its risk, timing, and the returns available on other similar investments. This is the discount rate that is used to calculate the presentvalue of the bond's cash flows. It is also called the debt's going interest rate, market interest rate, quoted interest rate, nominal annual interest rate, or yield. Note that r
d is not the coupon interest rate. It is equal to thecoupon rate only if (as in this case) the bond is selling at par. Generally, most coupon bonds are issued at par, which implies that the coupon rate is set at the r
d prevailing when the bond is issued. Thereafter, interest rates, asmeasured by r
d , will fluctuate, but the coupon rate is fixed, so after issue r
d will equal the coupon rate only by chance. We use the term "i" or "I" to designate the interest rate for many calculations because those terms areused on financial calculators, but "r," with the subscript "d" to designatethe rate on a debt security, is normally used in finance. N = Number of years until the bond matures. Note that N declines each year afterthe bond was issued, so a bond that had a maturity of 15 years when it was issued (original maturity= 15) will have N = 14 after 1 year, N = 13 after 2 years, and so on. Note also that for the sake of simplicity we assume in thissection that the bond pays interest once a year, or annually, so N is measuredin years. We consider bonds with semiannual payments later in the chapter.
203
INT = Dollars of interest paid each year = (Coupon rate)(Par value). For a bond with a9% coupon and a $1,000 par value, the annual interest is 0.09($1,000) = $90. In calculator terminology, INT = PMT = 90. If the bond had been a semiannual payment bond, the payment would have been $ 45 every 6 months. M = Par, or maturity, value of the bond. This amount must be paid off atmaturity, and it is often equal to $1,000.
The following general equation, written in several forms, can be used to find the value ofany bond, V 8
:
INT INT VB=---+---+(1 + rd)1 (1 + r)2 =f INT + M
t=I (1 + r)1 (1 + r)N
= INT[_!_ - --1--] + _M __r d
ril + r)N (1 + r)N
Observe that the cash flows consist of an annuity of N years plus an additional payment at the end of Year N. Equation 5-1 can be solved by using (1) a formula, (2) a financialcalculator, or (3) a spreadsheet. 5-3b Solving for the Bond Price
Recall that MicroDrive issued a 15-year bond with an annual coupon rate of 10% and a parvalue of $1,000. To find the value of MicroDrive's bond by using a formula, insert values
204
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See Ch05 Tool Kit.xlsx on
the textbook's Web site.
;,.Q,U,rce
See Ch05 Tool Kit.xlsx on
the textbook's Web site.
Part 2 Fixed Income Securities
for MicroDrive's bond into Equation 5-1. You could use the first line of Equation 5-1 to discount each cash flow back to the present and then sum these PVs to find the bond's value of $1,000; see Figure 5-1 and Equation 5-la: 15 $100 $1,000 V = ""---+---=$1 000 B &i (1 + 0.10)1 (1 + 0.10)15 ' I
This procedure is not very efficient, especially if the bond has many years to maturity. Alternatively, you could use the formula in the second line of Equation 5-1 with a simple or scientific calculator:
[ 1 1 ] $1,000 VB= $100 -0.1-0 - -0-.1 -0(_1_+_0. -10-)-15 + _(1_+_0 -.10-)-15 = $760.61 + $239.39 = $1,000
As shown in Equation 5-lb, the total bond value of $1,000 is the sum of the coupons' present values ($760.61) and the par value's present value ($239.39). This is easier than the step-by-step approach, but it is still somewhat cumbersome. A financial calculator is ideally suited for finding bond values. Here is the setup for MicroDrive's bond: Inputs 15 10
N I/YR
Output
PV
-1000
100
PMT
1000
FV
Input N = 15, I/YR= r d
= 10, INT = PMT = 100, and M = FV = 1000; then press the PV key to find the value of the bond, $1,000. Because the PV is an outflow to the investor, it is shown with a negative sign. The calculator is programmed to solve Equation 5-1: It finds the PV of an annuity of $100 per year for 15 years, discounted at 10%, then it finds the PV of the $1,000 maturity payment, and then it adds these two PVs to find the value of the bond. Notice that even though the bond has a total cash flow of $1,100 at Year 15, you should not enter FV = 1100! When you entered N = 15 and PMT = 100, you told the calculator that there is a $100 payment at Year 15. Thus, setting FV = 1000 accounts for any extra payment at Year 15, above and beyond the $100 payment. With Excel, it is easiest to use the PV function: =PV(I,N,PMT,FV,Type).7 For MicroDrive's bond, the function is =PV(0.10,15,100,1000,0) with a result of -$1,000. Like the financial calculator solution, the bond value is negative because PMT and FV are positive . Excel also provides specialized functions for bond prices based on actual dates. For example, in Excel you could find the MicroDrive bond value as of the date it was issued by using the function wizard to enter this formula:
= PRICE(DATE(2019,l,l),DATE(2033,12,31),10%,10%,100,l,l)
7In Chapter 4 we note that Type is O (or omitted) for payments at the end of the period and 1 for payments at the beginning of the period.
C
FIGURES-!
Chapter 5 Bonds, Bond Valuation, and Interest Rates 205
Finding the Value of MicroDrive's Bond (V 8 )
A B I C D E F I G 20 INPUTS:
-
21 Years to maturity= N = 15 -
22 Coupon payment = INT = $100 II -
II 23 Par value = M = $1,000 -
24 Required return = rd = 10%
25
26 1. Step-by-Step: Divide each cash flow by (1 + r .J t
Coupon PVof Coupon PVof
27 -
l'.i:ilL(tl Payment Payment ParYaJue ParVaJue 28 1 $100 $90.91
-
29 2 $100 $82.64 -
.1Q_ 3 $100 $75.13 31 4 $100 $68.30
-
..11.. 5 $100 $62.09 33 6 $100 $56.45
-
34 7 $100 $51.32 -
35 8 $100 $46.65 -
36 9 $100 $42.41 -
37 10 $100 $38.55 -
38 11 $100 $35.05 -
39 12 $100 $31.86 -
40 13 $100 $28.97 -
41 14 $100 $26.33 -
42 15 $100 $23.94 $1,000 $239.39
.Jl. Total = $760.61 44 -
45 V 8 = PV of all coupon payments + PV of par value = $1,000.00
46
47 Inputs: 15 0 100 1000
48 2. Financial Calculator: -
I N I/YR PV PMT I FV 49 Output: -$1,000.00
so
51 3. Excel: PV function: PVN= =PV(Rate,Nper,Pmt,Fv,Type) -
I52 Fixed inputs: PVN = =PV(10%,15,100,1000) -$1,000.00 -
I53 Cell references: PVN= =PV(C24,C21,C22,C23) -$1,000.00
Source: See the file ChOS Tool Kit.xlsx. Numbers are reported as rounded values for clarity but are calculated using Exce/'s full precision. Thus, intermediate calculations using the figure's rounded values will be inexact.
The first two arguments in the function are Excel's DATE function. The DATE func tion takes the year, month, and day as inputs and converts them into a date. The first argument is the date on which you want to find the price, and the second argument is the maturity date. The third argument in the PRICE function is the bond's coupon rate, followed by the required return on the bond, r
d . The fifth argument, 100, is the redemp
tion value of the bond at maturity per $100 of face value; entering "100" means that the bond pays 100% of its face value when it matures. The sixth argument is the number of payments per year. The last argument, 1, tells the program to base the price on the actual number of days in each month and year. This function produces a result based upon a face value of $100. In other words, if the bond pays $100 of face value at maturity, then the PRICE function result is the price of the bond. Because MicroDrive's bond pays $1,000
206 Part 2 Fixed Income Securities
of face value at maturity, we must multiply the PRICE function's result by 10. In this ex ample, the PRICE function returns a result of $100. When we multiply it by 10, we get the actual price of $1,000.8 This function is essential if a bond is being evaluated between coupon payment dates. See Ch05 Tool Kit.xlsx on the textbook's Web site for an example.9
5-3c Interest Rate Changes and Bond Prices
In this example, MicroDrive's bond is selling at a price equal to its par value. Whenever the going market rate of interest, r
d' is equal to the coupon rate, a fixed-rate bond will sell
at its par value. Normally, the coupon rate is set at the market rate when a bond is issued, causing it to sell at par initially.
The coupon rate remains fixed after the bond is issued, but interest rates in the market move up and down. Looking at Equation 5-1, we see that an increase in the market interest rate (r) will cause the price of an outstanding bond to fall, whereas a decrease in rates will cause the bond's price to rise. For example, if the market inter est rate on MicroDrive's bond increased by 5 percentage points to 15% immediately after it was issued, we would recalculate the price with the new market interest rate as follows:
Inputs 15 15
N I/YR
Output
PV
-707.63
100
PMT
1000
FV
The price would fall to $707.63. Notice that the bond would then sell at a price below its par value. Whenever the going rate of interest rises above the coupon rate, a fixed-rate bond's price will fall below its par value, and it is called a discount bond.
On the other hand, bond prices rise when market interest rates fall. For example, if the market interest rate on MicroDrive's bond decreased by 5 percentage points to 5%, then we would once again recalculate its price:
Inputs 15 5
N I/YR
Output
PV
-1518.98
100
PMT
1000
FV
In this case, the price rises to $1,518.98. In general, whenever the going interest rate falls below the coupon rate, a fixed-rate bond's price will rise above its par value, and it is called a premium bond.
'The value based on the PRICE function with these inputs is actually $0.01 lower than the par value because the function finds the price at the end of the settlement day, which means the times to the future payments are shorter than a full year by 1 day.
'The bond prices quoted by brokers are calculated as described and are called "clean" prices. However, if you bought a bond between interest payment dates, the amount you would actually have to pay would be the basic price plus accrued interest, which is called the "dirty" price. Thus, if you purchased a MicroDrive bond 6 months after it was issued, your broker would send you an invoice stating that you must pay $1,000 as the basic price of the bond plus $45 interest, representing one-half the annual interest of $90, for a "dirty" price of $1,045. The seller of the bond would receive $1,045. If you bought the bond the day before its interest payment date, you would pay $1,000 + (364/365)($90) = $1,089.75. You would receive an interest payment of$90 at the end of the next day. Unless otherwise stated, all prices quoted in this text are "clean" prices.
Chapter 5 Bonds, Bond Valuation, and Interest Rates
SELF -TEST
Why do the prices of fixed-rate bonds fall if expectations for inflation rise?
What is a discount bond? A premium bond?
207
A bond that matures in 6 years has a par value of $1,000, an annual coupon payment of $80, and a market interest rate of9%. What is its price? ($955.14)
A bond that matures in 18 years has a par value of $1,000, an annual coupon of 10%, and a market interest rate of 7%. What is its price? ($1,301.77}
5-4 Changes in Bond Values over Time At the time a coupon bond is issued, the coupon is generally set at a level that will cause the market price of the bond to equal its par value. If a lower coupon were set, investors would not be willing to pay $1,000 for the bond, and if a higher coupon were set, investors would clamor for the bond and bid its price up over $1,000. Investment bankers can judge quite precisely the coupon rate that will cause a bond to sell at its $1,000 par value. A bond that has just been issued is known as a new issue bond. (Investment bankers classify a bond as a new issue for about a month after it has first been issued. New issues are usually actively traded and are called on-the-run bonds.) Once the bond has been on the market for a while, it is classified as an outstanding bond, also called a seasoned bond. Newly issued bonds generally sell very close to par, but the prices of seasoned bonds can vary widely from par. Except for floating-rate bonds, coupon payments are constant, so when economic conditions change, a 10% coupon bond with a $100 coupon that sold at par when it was issued will sell for more or less than $1,000 thereafter, and the annual coupon will remain at $100. MicroDrive's bonds with a 10% coupon rate were originally issued at par. If rd remained constant at 10%, what would the value of the bond be 1 year after it was issued? Now the term to maturity is only 14 years-that is, N = 14. With a financial calculator, just override N = 15 with N = 14, press the PV key, and you find a value of $1,000. If we continued, setting N = 13, N = 12, and so forth, we would see that the value of the bond will remain at $1,000 as long as the going interest rate remains equal to the coupon rate, 10%. Now suppose interest rates in the economy fell drastically after the MicroDrive bonds were issued and, as a result, r
d fell below the coupon rate, decreasing from 10% to 5%. Both the coupon interest payments and the maturity value remain constant, but now 5% would have to be used for rd in Equation 5-1. The value of the bond at the end of the first year would be $1,494.93:
14 $100 $1,000 VB= � (1 + 0.05)' + (1 + 0.05)14
[ 1 1 ] $1,000 = $100 0.05 - 0.05(1 + 0.05)14 + (1 + 0.05)14 = $1,494.93
With a financial calculator, just change r d
= I/YR from 10 to 5, and then press the PV key to get the answer, $1,494.93. Thus, if rd fell below the coupon rate, the bond would sell above par, or at a premium. The arithmetic of the bond value increase should be clear, but what is the logic behind it? Because rd has fallen to 5%, with $1,000 to invest, you could buy new bonds like MicroDrive's (every day some 10 to 12 companies sell new bonds), except that these new bonds
208 Part 2 Fixed Income Securities
would pay $50 of interest each year rather than $100. Naturally, you would prefer $100 to $50, so you would be willing to pay more than $1,000 for a MicroDrive bond to obtain its higher coupons. All investors would react similarly; as a result, the MicroDrive bonds would be bid up in price to $1,494.93, at which point they would provide the same 4% rate of return to a potential investor as the new bonds.
Assuming that interest rates remain constant at 5% for the next 14 years, what would happen to the value of a MicroDrive bond? It would fall gradually from $1,494.93 to $1,000 at maturity, when MicroDrive will redeem each bond for $1,000. This point can be illus trated by calculating the value of the bond 1 year later, when it has 13 years remaining to maturity. With a financial calculator, simply input the values for N, I/YR, PMT, and FV, now using N = 13, and press the PV key to find the value of the bond, $1,469.68. Thus, the value of the bond will have fallen from $1,494.93 to $1,469.68, or by $25.25. If you were to calculate the value of the bond at other future dates, the price would continue to fall as the maturity date approached.
Note that if you purchased the bond at a price of $1,494.93 and then sold it 1 year later with r
d still at 5%, you would have a capital loss of$25.25, or a total dollar return of$100.00 -
$25.25 = $74.75. Your percentage rate of return would consist of the rate of return due to the interest payment (called the current yield) and the rate of return due to the price change (called the capital gains yield). This total rate of return is often called the bond yield, and it is calculated as follows:
Interest, or current, yield = $100/$1,494.93 = 0.0669 = 6.69%
Capital gains yield = -$25.25/$1,494.93 = -0.0169 = -1.69%
Total rate of return, or yield = $74.75/$1,494.93 = 0.0500 = 5.00%
Had interest rates risen from 10% to 15% during the first year after issue (rather than falling from 10% to 5%), then you would enter N = 14, I/YR = 14, PMT = 100, and FV = 1000, and then press the PV key to find the value of the bond, $713.78. In this case, the bond would sell below its par value, or at a discount. The total expected future return on the bond would again consist of an expected return due to interest and an expected re turn due to capital gains or capital losses. In this situation, the capital gains yield would be positive. The total return would be 15%. To see this, calculate the price of the bond with 13 years left to maturity, assuming that interest rates remain at 15%. With a calculator, enter N = 13, I/YR= 15, PMT = 100, and FV = 1000; then press PV to obtain the bond's value, $720.84.
Note that the capital gain for the year is the difference between the bond's value at Year 2 (with 13 years remaining) and the bond's value at Year 1 (with 14 years remaining), or $720.84 - $713.78 = $7.06. The interest yield, capital gains yield, and total yield are calculated as follows:
Interest, or current, yield =
Capital gains yield
Total rate of return, or yield
$100/$713.78 = 0.1401 14.01%
$7.06/$713.78 0.0099 = 0.99%
$107.06/$713.78 0.1500 15.00%
Figure 5-2 graphs the value of the bond over time, assuming that interest rates in the economy (1) remain constant at 10%, (2) fall to 5% and then remain constant at that level, or (3) rise to 15% and remain constant at that level. Of course, if inter est rates do not remain constant, then the price of the bond will fluctuate. However, regardless of what future interest rates do, the bond's price will approach $1,000 as it nears the maturity date (barring bankruptcy, which might cause the bond's value to fall dramatically).
:e.M J .urce
See Ch05 Tool Klt.xlsx for
oil colculotions.
Chapter 5 Bonds, Bond Valuation, and Interest Rates
FIGURES-2
209
Time Path of the Value of a 10% Coupon, $1,000 Par Value Bond When Interest Rates Are 5%, 10%, and 15%
Bond Value ($)
1,600 rd Falls and Stays at 5% (Premium Bond)
1,400 c----...:.. __
1,200 rd = Coupon Rate= 10% (Par Bond)
1,000 �--------------------3►
SOOL---- ---------
---� .
rd Rises and Stays at 15% (Discount Bond)600
400
200
0 .._....__.,____....__ ...... ....__,.___.__,.___,__...,_.....__ ............. _ ...... �
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 Years Remaining Until Maturity
Note: The curves for 5% and 15% have a slight bow.
Figure 5-2 illustrates the following key points.
I. Whenever the going rate of interest, r d , is equal to the coupon rate, a fixed-rate bond
will sell at its par value. Normally, the coupon rate is set equal to the going rate when a bond is issued, causing it to sell at par initially.
2. Interest rates do change over time, but the coupon rate remains fixed after the bond has been issued. Whenever the going rate of interest rises above the coupon rate, a fixed-rate bond's price will fall below its par value. Such a bond is called a discount bond.
3. Whenever the going rate of interest falls below the coupon rate, a fixed-rate bond's price will rise above its par value. Such a bond is called a premium bond.
4. Thus, an increase in interest rates will cause the prices of outstanding bonds to fall, whereas a decrease in rates will cause bond prices to rise.
5. The market value of a bond will always approach its par value as its maturity date ap proaches, provided the firm does not go bankrupt.
These points are very important, for they show that bondholders may suffer capital losses or make capital gains depending on whether interest rates rise or fall after the bond is purchased.
SELF-TEST
What is meant by the terms "new issue" and "seasoned issue"?
Last year, a firm issued 30-year, 8% annual coupon bonds at a par value of $1,000. (1) Suppose that 1 year later the going rate drops to 6%. What is the new price of the bonds, assuming that they now have 29 years to maturity? ($1,271.81) (2) Suppose instead that 1 year after issue, the going interest rate increases to 10% (rather than dropping to 6%}. What is the price? ($812.61)
210
Chocolate Bonds
In 2010, the Hotel Chocolat UK became the first company to
sweeten a bond offering with chocolate-literally! Instead of
cash coupons, investors received 6 boxes of chocolate a year
as interest if they bought a £2,000 bond. For purchasing a
£4,000 bond, an investor would receive an extra sweetener,
receiving 13 boxes a year. At maturity in 2014, investors could
choose to be repaid in cash for their original investment or
to renew their bonds annually. Most investors preferred
chocolate to cash and chose the latter.
Part 2 Fixed Income Securities
The chocolate bond offering raised about £4.2 mil
lion and was so successful that Hotel Chocolat decided to
offer new bonds in 2014. In addition to chocolate, inves
tors can choose to be repaid with gift certificates to Ho
tel Chocolat stores and properties, including its hotel in
St. Lucia. Sweet!
Sources: Hotel Chocolat's Web site, including www.hotelchocolat.com/uk
/tasting-club/our-story/chocolate-bonds and www.hotelchocolat.com
/uk/about.
5-5 Bonds with Semiannual Coupons Although some bonds pay interest annually, the vast majority actually pay interest semi annually. To evaluate semiannual payment bonds, we must modify the valuation model as follows. I. Divide the annual coupon interest payment by 2 to determine the dollars of interest
paid every 6 months. 2. Multiply the years to maturity, N, by 2 to determine the number of semiannual
periods. 3. Divide the nominal (quoted) interest rate, r
d , by 2 to determine the periodic (semian
nual) interest rate. By making these changes, we obtain the following equation for finding the value of a
bond that pays interest semiannually:
lN INT/2 M Vs = � _(l_+_r_/_2_)t + -(l_+_r_/_2_)2-N I
To illustrate, assume now that MicroDrive's bonds pay $50 interest every 6 months rather than $100 at the end of each year. Each semiannual interest payment is only half as large, but there are twice as many of them. The nominal, or quoted, coupon rate is "10%, semiannual payments."10
'0In this situation, the coupon rate of"l0% paid semiannually" is the rate that bond dealers, corporate treasurers, and investors generally would discuss. Of course, if this bond were issued at par, then its effective annual rate would be higher than 10%:
( rNOM)M ( 0.10)' EAR= EFF% = I+ M - I= I+ -2- - I= (1.05}2 - I= 10.25% Because 10.25% with annual payments is quite different from 10% with semiannual payments, we have assumed a change in effective rates in this section from the situation described in the previous section, where we assumed 10% with annual payments.
Chapter 5 Bonds, Bond Valuation, and Interest Rates 211
When the going (nominal) rate of interest is 5% with semiannual compounding, the value of this 15-year bond is found as follows:
Inputs 30 2.5
N I/YR
Output
PV
-1523.26
so
PMT
1000
FV
Enter N = 30, rd = I/YR = 2.5, PMT = 50, FV = 1000, and then press the PV key to obtain the bond's value, $1,523.26. The value with semiannual interest payments is slightly larger than the value when interest is paid annually, $1,518.98. This higher value occurs be cause interest payments are received somewhat sooner under semiannual compounding.
SELF-TEST
Describe how the annual bond valuation formula is changed to evaluate semiannual coupon bonds. Write out the revised formula.
A bond has a 25-year maturity, an 8% annual coupon paid semiannually, and a face value of $1,000. The going nominal annual interest rate (r) is 6%. What is the bond's price? ($1,257.30)
5-6 Bond Yields
Unlike the coupon interest rate, which is fixed, the bond's yield varies from day to day depending on current market conditions. Moreover, the yield can be calculated in three different ways, and three "answers" can be obtained. These different yields are described in the following sections.
5-6a Yield to Maturity
Suppose you purchased MicroDrive's bond at a price of$1,494.93 exactly 1 year after it was issued. The bond you now own has a 10% annual coupon, $1,000 par value, and a maturity of 14 years (because you bought it 1 year after it was issued with an original maturity of 15 years). What rate of interest would you earn on your investment if you bought the bond and held it to maturity? This rate is called the bond's yield to maturity (YTM), and it is the interest rate generally discussed by investors when they talk about rates of return. The yield to maturity is usually the same as the market rate of interest, r
d " To find the YTM for
a bond with annual interest payments, you must solve Equation 5-1 for r/ 1
N INT M
Bond price = � ---- + ---- � (1 + YTM)' (1 + YTM)N
For MicroDrive's yield, you must solve this equation:
$100 $100 $1,000 $1,494.93 = (1 + rl + ...
+ (1 + rd)l4
+ (1 + rl4
I
You could substitute values for r d
until you found a value that "works" and forces the sum of the PVs on the right side of the equal sign to equal $1,494.93, but this would be
''If the bond has semiannual payments, you must solve Equation 5-2 for r d .
212
ource
See ChOS Tool Kit.x/sx on
the textbook's Web site.
Part 2 Fixed Income Securities
tedious and time-consuming.12 As you might guess, it is much easier with a financial calculator. Here is the setup:
Inputs
Output
14
N I/YR
5
-1494.93
PV
100
PMT
1000
FV
Simply enter N = 14, PV = -1,494.93, PMT = 100, and FV = 1000, and then press the I/YR key for the answer of 5%.
You could also find the YTM with a spreadsheet. In Excel, you would use the RATE function for this bond, inputting N = 14, PMT = 100, PV = -1494.93, FV = 1000, 0 for Type, and leave Guess blank: =RATE{l4,100,-1494.93,1000,0). The result is 5%. The RATE function works only if the current date is immediately after either the issue date or a coupon payment date. To find bond yields on other dates, use Excel's YIELD function. See the ChOS Tool Kit.xlsx file for an example.
The yield to maturity can be viewed as the bond's promised rate of return, which is the return that investors will receive if all the promised payments are made. However, the yield to maturity equals the expected rate of return only if: (1) The probability of default is zero. (2) The bond cannot be called. If there is some default risk or if the bond may be called, then there is some probability that the promised payments to maturity will not be received, in which case the calculated yield to maturity will differ from the expected return.
The YTM for a bond that sells at par consists entirely of an interest yield, but if the bond sells at a price other than its par value, then the YTM will consist of the interest yield plus a positive or negative capital gains yield. Note also that a bond's yield to maturity changes whenever interest rates in the economy change, and this is almost daily. If you purchase a bond and hold it until it matures, you will receive the YTM that existed on the purchase date, but the bond's calculated YTM will change frequently between the pur chase date and the maturity date.13
5-Gb Yield to Call
If you purchased a bond that was callable and the company called it, you would not be able to hold the bond until it matured. Therefore, the yield to maturity would not be earned. For example, if MicroDrive's 10% coupon bonds were callable and if interest rates fell from 10% to 5%, then the company could call in the 10% bonds, replace them with 4% bonds, and save $100 - $50 = $50 interest per bond per year. This would be good for the com pany but not for the bondholders.
"Alternatively, you can substitute values of r d
into the third form of Equation 5-1 until you find a value that works.
"We often are asked by students if the purchaser of a bond will receive the YTM if interest rates subsequently change. The answer is definitely "Yes" provided the question means, "Is the realized rate of return on the in vestment in the bond equal to the YTM?" This is because the realized rate of return on an investment is by definition the rate that sets the present value of the realized cash flows equal to the price. If instead the question means, "Is the realized rate of return on the investment in the bond and the subsequent reinvestment of the coupons equal to the YTM?" then the answer is definitely "No." Thus, the question really is one about strategy and timing. The bond, in combination with a reinvestment strategy, is really two investments, and clearly the realized rate on this combined strategy depends on the reinvestment rate. (See Web Extension SC for more on investing for a target future value.) For the rest of the book, we assume that an investment in a bond is only an
investment in the bond and not a combination of the bond and a reinvestment strategy; this means the investor earns the expected YTM if the bond is held to maturity.
Chapter 5 Bonds, Bond Valuation, and Interest Rates 213
If current interest rates are well below an outstanding bond's coupon rate, then a call able bond is likely to be called, and investors will estimate its expected rate of return as the yield to call (YTC) rather than as the yield to maturity. To calculate the YTC, solve this equation for r
d :
N INT Call price Price ofcallable bond = � ( + N
1-1 1 + rl (1 + rd) I
Here N is the number of years until the company can call the bond, r d
is the YTC, and "Call price" is the price the company must pay in order to call the bond (it is often set equal to the par value plus 1 year's interest).
To illustrate, suppose MicroDrive's bonds had a provision that permitted the com pany, if it desired, to call the bonds 10 years after the issue date at a price of $1,100. Suppose further that 1 year after issuance the going interest rate had declined, causing the price of the bonds to rise to $1,494.93. Here is the time line and the setup for finding the bond's YTC with a financial calculator:
0 YTC=?
-1,494.93
Inputs 9
N
Output
1
100
I/YR
4.21=YTC
2
100
-1494.93
PV
8
100
100 PMT
9
100 1100
1100 FV
The YTC is 4.21%, which is the return you would earn if you bought the bond at a price of $1,494.93 and it was called 9 years from today. (The bond could not be called until 10 years after issuance, and 1 year has gone by, so there are 9 years left until the first call date.)
Do you think MicroDrive will call the bonds when they become callable? Micro Drive's actions will depend on the going interest rate when the bonds become callable. If the going rate remains at r
d = 5%, then MicroDrive could save 10% - 5% = 5%, or $50
per bond per year, by calling them and replacing the 10% bonds with a new 5% issue. There would be costs to the company to refund the issue, but the interest savings would probably be worth the cost, so MicroDrive would probably refund the bonds. Therefore, you would probably earn YTC = 4.21% rather than YTM = 5% if you bought the bonds under the indicated conditions.
In the balance of this chapter, we assume that bonds are not callable unless otherwise noted. However, some of the end-of-chapter problems deal with yield to call.
5-6c Current Yield
If you examine brokerage house reports on bonds, you will often see reference to a bond's current yield. The current yield is the annual interest payment divided by the bond's cur rent price. For example, if MicroDrive's bonds with a 10% coupon were currently selling at $985, then the bond's current yield would be $100/$985 = 0.1015 = 10.15%.
214 Part 2 Fixed Income Securities
Unlike the yield to maturity, the current yield does not represent the rate of return that investors should expect on the bond. The current yield provides information regard ing the amount of cash income that a bond will generate in a given year, but it does not provide an accurate measure of the bond's total expected return, the yield to maturity. In fact, here is the relation between current yield, capital gains yield (which can be negative for a capital loss), and the yield to maturity:
Current yield + Capital gains yield = Yield to maturity ■ 5-6d The Cost of Debt and Intrinsic Value
The "Intrinsic Value and the Cost of Debt" box at the beginning of this chapter high lights the cost of debt, which affects the weighted average cost of capital (WACC), which in turn affects the company's intrinsic value. The pre-tax cost of debt from the com pany's perspective is the required return from the debtholder's perspective. Therefore, the pre-tax cost of debt is the yield to maturity (or the yield to call if a call is likely). But why do different bonds have different yields to maturity? The following sections answer this question.
SELF-TEST
Explain the difference between the yield to maturity and the yield to call.
How does a bond's current yield differ from its total return?
Could the current yield exceed the total return?
A bond currently sells for $850. It has an 8-year maturity, an annual coupon of $80, and a par value of $1,000. What is its yield to maturity? (10.90%) What is its current yield? (9.41%)
A bond currently sells for $1,250. It pays a $110 annual coupon and has a 20-year maturity, but it can be called in 5 years at $1,110. What are its YTM and its YTC? (8.38%, 6.85%) Is the bond likely to be called if interest rates don't change?
5-7 The Pre-Tax Cost of Debt: Determinants
of Market Interest Rates
Until now we have given you the quoted market interest rate, which is the required rate of return on debt, r
d . But as we showed in Chapter 1, different debt securities often have very
different market rates. What explains these differences? In a nutshell, different types of debt have expected future cash flows that differ with respect to timing and risk. We can use a conceptual framework that decomposes the quoted market interest rate into a truly risk-free rate plus several premiums that reflect exposure to inflation risk, price volatility caused by interest rate volatility, default risk, and liquidity risk (stemming from a lack of trading activity):
Quoted market interest rate = r d
= r" +IP+ MRP + DRP + LP ■
WWW
See www.bloomberg .com and select MARKETS. Then select RATES AND
BONDS for a partial listing
of indexed Treasury bonds
and their interest rotes.
See http://online.wsj .com for a complete set of Treasury quotes. See
www.treasurydirect .gov/Ind Iv/products /products.htm for a complete listing of all
Treasury securities.
Chapter 5 Bonds, Bond Valuation, and Interest Rates
Here are definitions of the variables in Equation 5-6:
r d
= The quoted market rate, which is the required rate of return on a debt security. There are many different securities and hence many different quoted interest rates.
r" = Real risk-free interest rate. Pronounced "r-star," r' is the rate paid each moment on a hypothetical riskless security if zero inflation were expected.
215
IP = Inflation premium, which is equal to the average expected inflation rate over the life of the security. The expected future inflation rate is not necessarily equal to the current inflation rate, so IP is not necessarily equal to current inflation.
MRP = Maturity risk premium. Changes in market interest rates can cause large changes in the prices of long-term bonds, even Treasury bonds. Lenders charge a maturity risk premium to reflect this risk.
DRP = Default risk premium. This premium reflects the possibility that the issuer will not pay interest or principal at the stated time and in the stated amount. The DRP is zero for U.S. Treasury securities, but it rises as the riskiness of issuers increases.
LP = Liquidity, or marketability, premium. This is a premium charged by lenders to reflect the fact that some securities cannot be converted to cash on short notice at a "reasonable" price. The LP is very low for Treasury securities and for securities issued by large, strong firms, but it is relatively high on securities issued by very small firms.
We discuss the components whose sum makes up the quoted, or nominal, rate on a given security in the following sections.
SELF-TEST
Write out an equation for the nominal interest rate on any debt security.
5-8 The Risk-Free Interest Rate: Nominal (r RF) and Real (r*)
The phrases risk-free interest rate (r RF
) and risk-free rate of return are used frequently in business and in the financial press, but what do they actually mean? When the term "risk free rate" is not preceded by "real," people generally mean the nominal risk-free interest rate (r
RF ), which is the market rate observed on a Treasury security. In particular, the T-bill
rate is used for the short-term r RF
' and a T-bond rate is used for the long-term r RF
. Although T-bills and T-bonds are default-free and trade in very active markets, they are not truly risk less. Both are exposed to inflation risk, and T-bonds also experience price volatility due to interest rate volatility.14
14We assume U.S. Treasury securities will not default, but evidence suggests that investors don't always think so. For example, see Srinivas Nippani, Pu Liu, and Craig T. Schulman, "Are Treasury Securities Free of Default?" Journal of Financial and Quantitative Analysis, June 2001, pp. 251-266. For more recent evidence, see Srinivas Nippani and Stanley D. Smith, "The Increasing Default Risk of U.S. Treasury Securities due to the Financial Crisis," Journal of Banking and Finance, Vol. 34, 2010, pp. 2472-2480. Additionally, on August 5, 2011. S&P downgraded the U.S. credit rating from AAA to AA+. See the box "U.S. Treasury Bonds Downgraded!" later in the chapter for discussion of the downgrade.
216 Part 2 Fixed Income Securities
In contrast, the real risk-free interest rate (r·) is the rate that a hypothetical risk less security pays each moment if zero inflation were expected. The real risk-free rate is not constant; r· changes over time depending on economic conditions, especially: (1) the rate of return corporations and other borrowers expect to earn on productive assets and (2) people's time preferences for current versus future consumption. Therefore, r• might change from moment to moment.
Even though no such hypothetical security actually exists, it provides a good con ceptual basis for understanding why different securities have different yields. We start with r• and add required returns based on a security's probability of default, degree of market liquidity, exposure to inflation, and payment dates. We discuss these sources of risk in the following sections, but we need to immediately address the concept of a risk free security.
Although there is no security that truly has a real risk-free interest rate, Treasury Inflation-Protected Securities (TIPS) have payments that are indexed to inflation. (For details on how TIPS are adjusted to protect against inflation, see Web Extension SB on the textbook's Web site.) Because the payments (including the principal) are tied to inflation, the yield on a TIPS with 1 year until maturity is a good estimate of the real I-year risk-free rate. 15 In theory, we would like an even shorter maturity to estimate the real risk-free rate, but short-term TIPS are thinly traded and the reported yields are not as reliable.
In early 2018, the YTM was 0.07% on I-year TIPS. Historically, the real interest rate has averaged around 1.5% to 2.5%. Although unusual, negative real rates are possible and have occurred in the last decade. However, negative nominal market rates are impossible (or at least extraordinarily rare) because investors would just hold cash instead of buying a bond that is guaranteed to return less than its cost.
SELF-TEST
What security provides a good estimate of the short-term nominal risk-free rate?
What security provides a good estimate of the long-term nominal risk-free rate?
What security provides a good estimate of the real risk-free rate?
5-9 The Inflation Premium (IP) Inflation has a major effect on interest rates because it erodes the purchasing power of the dollar and lowers the real rate of return on investments. To illustrate, suppose you invest $3,000 in a default-free zero coupon bond that matures in 1 year and pays a 5% interest rate. At the end of the year, you will receive $3,150-your original $3,000 plus $150 of interest. Now suppose that the inflation rate during the year is 10% and that it affects all items equally. If gas had cost $3 per gallon at the beginning of the year, it would cost $3.30 at the end of the year. Therefore, your $3,000 would have bought $3,000/$3 = 1,000 gal lons at the beginning of the year but only $3,150/$3.30 = 955 gallons at the end. In real
"The real rate of interest as discussed here is different from the current real rate as often discussed in the press. The current real rate is often estimated as the current interest rate minus the current (or most recent) inflation rate, whereas the real rate, as used here (and in the fields of finance and economics generally) with out the word "current," is the current interest rate minus the expected future inflation rate over the life of the security. For example, suppose the current quoted rate for a I-year Treasury bill is 5%, inflation during the previous year was 2%, and inflation expected for the coming year is 4%. Then the current real rate would be approximately 5% - 2% = 3%, but the expected real rate would be approximately 5% - 4% = 1%.
(
Chapter 5 Bonds, Bond Valuation, and Interest Rates 217
terms, you would be worse off: You would receive $150 of interest, but it would not be suf ficient to offset inflation. You would thus be better offbuying 1,000 gallons of gas (or some other storable asset) than buying the default-free bond.
Investors are well aware of inflation's effects on interest rates, so when they lend money, they build in an inflation premium (IP) that is approximately equal to the aver age expected inflation rate over the life of the security.
Consider a U.S. Treasury bill, which is default-free, is very liquid, and has a short maturity. Note that the interest rate on the T-bill (r
T -bill) includes the premium for
expected inflation:
i;..bm = r• + IP
Therefore, if the real short-term risk-free rate of interest is r• = 0.6% and if expected inflation is 1.0% (and hence IP = 1.0%) during the next year, then the quoted rate of inter est on I-year T-bills would be 0.6% + 1.0% = 1.6%.16
It is important to note that the inflation rate built into interest rates is the inflation rate expected in the future, not the rate experienced in the past. Thus, the latest reported fig ures might show an annual inflation rate of 2%, but that is for the past year. But if people on average expect a 6% inflation rate in the future, then 6% would be built into the current interest rate.
Note also that the inflation rate reflected in the quoted interest rate on any security is the average rate of inflation expected over the security's life. Thus, the inflation rate built into a I-year bond is the expected inflation rate for the next year, but the inflation rate built into a 30-year bond is the average rate of inflation expected over the next 30 years. If I, is the expected inflation during year t, then the inflation premium for an N-year bond's yield (IP N) can be approximated as:
I I
+ 12 + + IN IP = ------- N N I
For example, if investors expect inflation to average 3% during Year 1 and 5% during Year 2, then the inflation premium built into a 2-year bond's yield can be ap proximated by:17
I I
+ 12 3% + 5% IP=--= = 4% 2
2 2
In the previous section, we saw that the yield on an inflation-indexed Treasury bond (TIPS) is a good estimate of the real interest rate. Because a regular (nonindexed) T-bond is similar to a TIPS in all respects except inflation protection, the difference in their yields provides an estimate of the inflation premium. For example, in early 2018, the yield on a 5-year nonindexed T-bond was 2.64%, and the yield on a 5-year TIPS was 0.62%. Thus, the 5-year inflation premium was 2.64% - 0.62% = 2.02%, implying that investors expected
16This is not technically correct. The quoted rate on the T-bill would be found by solving this equation: (1 + r
T bill) = (I + r')(l + IP). For our example, the result would be rT bill = (1.006)(1.01) - I = 0.01606 "" 1.6%. This is almost exactly equal to the approximation. Because real rates and inflation rates usually are small, we will continue to use the approximation instead of the exact formula in this footnote.
"To be mathematically correct, we should take the geometric average: (I + IP 2)2 = (I + I,)(I + 12). In this ex ample, we have (I + IP
2 )2 = (I + 0.03)(1 + 0.05). Solving for IP
2 yields 3.9952, which is close to our approxima•
lion of 4% in the example.
218 Part 2 Fixed Income Securities
inflation to average 2.02% per year over the next 5 years.18 Using the same approach, the results for several different maturities follow:
Yields on: Maturity
I Year 5 Years 20Years
Nonindexed U.S. Treasury bond (nominal rate) 2.05% 2.64% 3.00%
TIPS (real rate) 0.07% 0.62% 0.92%
Inflation premium 1.98% 2.02% 2.08%
Expectations for future inflation are closely but not perfectly correlated with rates experienced in the recent past. Therefore, if the inflation rate reported for last month increases, people often raise their expectations for future inflation, and this change in expectations will cause an increase in interest rates.
Note that Germany, Japan, and Switzerland have, over the past several years, had lower inflation rates than the United States, so their interest rates have generally been lower than ours. South Africa, Brazil, and most South American countries have experienced higher inflation, which is reflected in their interest rates.
SELF-TEST
Explain how a TIPS and a nonindexed Treasury security can be used to estimate the inflation
premium.
The yield on a 15-year TIPS is 3%, and the yield on a 15-year Treasury bond is 5%. What is the
inflation premium for a 15-year bond? (2%)
5-10 The Maturity Risk Premium {MRP) All bonds, even Treasury bonds, are exposed to two additional sources of risk: interest rate risk and reinvestment risk. The net effect of these two sources of risk upon a bond's yield is called the maturity risk premium (MRP). The following sections explain how interest rate risk and reinvestment risk affect a bond's yield.
5-10a Interest Rate Risk
Interest rates go up and down over time, and an increase in interest rates leads to a decline in the value of outstanding bonds. This risk of a decline in bond values due to rising interest rates is called interest rate risk. To illustrate, suppose you bought some 15-year 10% MicroDrive bonds at a price of$1,000, and then interest rates rose to 15% in the following year. As we saw earlier, the price of the bonds would fall to $713.78, so you would have a loss of $286.22 per bond.19 Interest rates can and do rise, and rising rates
18 As we noted in the previous footnote, the mathematically correct approach is to use a geometric average and solve the following equation: (l + IP)(l + 0.0062)= l + 0.0264. Solving for IP gives IP= 2.01%, which is very close to our approximation for the 5-year inflation premium in early 2018.
Note, though, that the difference in yield between a T-bond and a TIPS of the same maturity reflects both the expected inflation and any risk premium for bearing inflation risk. So the difference in yields is really an upper limit on the expected inflation.
''You would have an accounting (and tax) loss only if you sold the bond; if you held it to maturity, you would not have such a loss. However, even if you did not sell, you would still have suffered a real economic loss in an opportunity cost sense because you would have lost the opportunity to invest at 14% and would be stuck with a 10% bond in a 15% market. In an economic sense, "paper losses" are just as bad as realized accounting losses.
source
See Ch05 Tool Kit.x/sx.
Chapter 5 Bonds, Bond Valuation, and Interest Rates 219
cause a loss of value for bondholders. Thus, bond investors are exposed to risk from changing interest rates.
This point can be demonstrated by showing how the value of a I-year bond with a 10% annual coupon fluctuates with changes in r
d and then comparing these changes with
those on a 25-year bond. The I-year bond's value for r d
= 5% is shown here:
Inputs 1 5 100 1000
N I/YR PV PMT FV
Output (Bond Value) -1047.62
Using either a calculator or a spreadsheet, you could calculate the bond values for a I-year and a 25-year bond at several current market interest rates; these results are plotted in Figure 5-3. Note how much more sensitive the price of the 25-year bond is to changes in interest rates. At a 10% interest rate, both the 25-year and the I-year bonds are valued at $I,000. When rates rise to I5%, the 25-year bond falls to $676.79 but the I-year bond falls only to $956.52.
For bonds with similar coupons, this differential sensitivity to changes in interest rates always holds true: The longer the maturity of the bond is, the more its price changes in re sponse to a given change in interest rates. Thus, even if the risk of default on two bonds is exactly the same, the one with the longer maturity is exposed to more risk from a rise in interest rates.
The explanation for this difference in interest rate risk is simple. Suppose you bought a 25-year bond that yielded 10%, or $100 a year. Now suppose interest rates on bonds of comparable risk rose to I5%. You would be stuck with only $100 of interest for the next 25 years. On the other hand, had you bought a I-year bond, you would have a low return for only I year. At the end of the year, you would get your $I,000 back, and you could then
FIGURE 5-3
Value of Long-and Short-Term 10% Annual Coupon Bonds at Different Market Interest Rates
Bond Value ($)
1,800
1,600
1,400
1,200
1,000
800
600
400
200
------�--�-�-,......_;i�:-- ,..
- ....l./�.._
1- - Ye
_
a
_
r
_
Bo
-
nd
o�--�--__, ___ ....._ __ __. __ ___,
0% 5% 10% 15% 20% 25% Interest Rate, rd
220
f!.S.O u rce
For more on bond risk,
including duration
analysis, see Web
Extension SC on the
textbook's Web site.
Part 2 Fixed Income Securities
reinvest it and receive a 15% return ($150) for the next year. Thus, interest rate risk reflects the length of time one is committed to a given investment.
In addition to maturity, interest rate sensitivity reflects the size of coupon payments. Intuitively, this is because more of a high-coupon bond's value is received sooner than that of a low-coupon bond of the same maturity. This intuitive concept is measured by duration, which finds the average number of years that the bond's PV of cash flows (coupons and principal payments) remains outstanding; see Web Extension SC and ChOS Tool Kit.xlsx for the exact calculation. A zero coupon bond, which has no payments un til maturity, has a duration equal to its maturity. Coupon bonds have durations that are shorter than maturity, and the higher the coupon rate, the shorter the duration is.
Duration measures a bond's sensitivity to interest rates in the following sense: Given a change in interest rates, the percentage change in a bond's price is proportional to its duration:20
% change in V 8
= (% change in 1 + r d ) (- Duration)
Excel's DURATION function provides an easy way to calculate a bond's duration. See Web Extension SC and ChOS Tool Kit.xlsx for more discussion of duration and its use in measuring and managing interest rate risk.
5-l0b Reinvestment Rate Risk
As we saw in the preceding section, an increase in interest rates will hurt bondholders because it will lead to a decline in the value of a bond portfolio. But can a decrease in in terest rates also hurt bondholders? The answer is "Yes" because if interest rates fall, then a bondholder may suffer a reduction in his or her income. For example, consider a retiree who has a portfolio of bonds and lives off the income they produce. The bonds, on average, have a coupon rate of 10%. Now suppose that interest rates decline to 5%. The short-term bonds will mature, and when they do, they will have to be replaced with lower-yielding bonds. In addition, many of the remaining long-term bonds may be called, and as calls oc cur, the bondholder will have to replace 10% bonds with 5% bonds. Thus, our retiree will suffer a reduction of income.
The risk of an income decline due to a drop in interest rates is called reinvestment rate risk. Reinvestment rate risk is obviously high on callable bonds. It is also high on short maturity bonds because the shorter the maturity of a bond, the fewer the years when the relatively high old interest rate will be earned and the sooner the funds will have to be reinvested at the new low rate. Thus, retirees whose primary holdings are short-term se curities, such as bank CDs and short-term bonds, are hurt badly by a decline in rates, but holders of long-term bonds continue to enjoy their old high rates.
5-l0c Comparing Interest Rate Risk and Reinvestment Rate Risk: The Maturity Risk Premium
Note that interest rate risk relates to the value of the bonds in a portfolio, while reinvest ment rate risk relates to the income the portfolio produces. If you hold long-term bonds, then you will face a lot of interest rate risk because the value of your bonds will decline if interest rates rise; however, you will not face much reinvestment rate risk, so your income will be stable. On the other hand, if you hold short-term bonds, you will not be exposed to much interest rate risk because the value of your portfolio will be stable, but you will
"'This is true for the case in which the term structure (which we discuss in Section 5-13) is flat and can only shift up and down. However, other duration measures can be developed for other term structure assumptions.
()
Chapter 5 Bonds, Bond Valuation, and Interest Rates 221
be exposed to considerable reinvestment rate risk because your income will fluctuate with changes in interest rates. We see, then, that no fixed-rate bond can be considered totally riskless-even most Treasury bonds are exposed to both interest rate risk and reinvest ment rate risk. 21
Bond prices reflect the trading activities of the marginal investors, defined as those who trade often enough and with large enough sums to determine bond prices. Although one particular investor might be more averse to reinvestment risk than to interest rate risk, the data suggest that the marginal investor is more averse to interest rate risk than to reinvestment risk. To induce the marginal investor to take on interest rate risk, long-term bonds must have a higher expected rate of return than short-term bonds. Holding all else equal, this additional return is the maturity risk premium (MRP).
SELF -TEST
Differentiate between interest rate risk and reinvestment rate risk.
To which type of risk are holders of long-term bonds more exposed? Short-term bondholders?
Assume that the real risk-free rate is,. = 3% and that the average expected inflation rate is 2.5% for the foreseeable future. The applicable MRP is 2% for a 20-year bond. What is the yield on a 20-year T-bond (which is default free and trades in a very active market)? (7.5%)
5-11 The Default Risk Premium (DRP) If the issuer defaults on a payment, investors receive less than the promised return on the bond. The quoted interest rate includes a default risk premium (DRP)-the greater the default risk, the higher the bond's yield to maturity.22 The default risk on Treasury securi ties is virtually zero, but default risk can be substantial for corporate and municipal bonds. In this section, we consider some issues related to default risk.
5-lla Bond Contract Provisions That
Influence Default Risk
Default risk is affected by both the financial strength of the issuer and the terms of the bond contract, especially whether collateral has been pledged to secure the bond. Several types of contract provisions are discussed next.
BOND INDENTURES
An indenture is a legal document that spells out the rights of both bondholders and the is suing corporation. A trustee is an official (usually a bank) who represents the bondholders and makes sure the terms of the indenture are carried out. The indenture may be several hundred pages in length, and it will include restrictive covenants that cover such points as the conditions under which the issuer can pay off the bonds prior to maturity, the levels at which certain ratios must be maintained if the company is to issue additional debt, and restrictions against the payment of dividends unless earnings meet certain specifications.
21 Although indexed Treasury bonds are almost riskless, they pay a relatively low real rate. Note also that risks have not disappeared; they have simply been transferred from bondholders to taxpayers.
"Suppose two bonds have the same promised cash flows, coupon rate, maturity, liquidity, and inflation expo sure, but one bond has more default risk than the other. Investors will naturally pay less for the bond with the greater chance of default. As a result, bonds with higher default risk will have higher yields.
222 Part 2 Fixed Income Securities
The Securities and Exchange Commission (1) approves indentures and (2) makes sure that all indenture provisions are met before allowing a company to sell new secu rities to the public. A firm will have different indentures for each of the major types of bonds it issues, but a single indenture covers all bonds of the same type. For example, one indenture will cover a firm's first mortgage bonds, another its debentures, and a third its convertible bonds.
SECURED DEBT AND MORTGAGE BONDS
Secured debt is any debt for which a corporation pledges a particular asset that may be claimed by the secured debtholder in the event of default. The pledged asset is said to have a lien against it because the corporation must satisfy the creditor before using proceeds from selling the asset for any other purpose.
A mortgage bond is a bond that is secured by property. If this bond is issued before other bonds, it is called a first-mortgage bond (also called a senior mortgage) and its claims are paid before those of other bonds. The company might also choose to issue second-mortgage bonds (also called junior mortgages) that are secured by the same assets as the first-mortgage bonds. However, in the event of liquidation, the holders of second-mortgage bonds would receive payments only after the first-mortgage bondhold ers are paid off in full. All mortgage bonds are subject to an indenture that usually limits the amount of new bonds that can be issued.
DEBENTURES AND SUBORDINATED DEBENTURES
A debenture is an unsecured bond, and as such it provides no lien against specific prop erty as security for the obligation. Debenture holders are, therefore, general creditors whose claims are protected by property not otherwise pledged.
There is a definite pecking order among debenture in terms of priority in a bank ruptcy. For example, some bonds are called senior bonds because they must be paid before any other general creditors. The term subordinate means "below," or "inferior to"; thus, in the event of bankruptcy, subordinated debt has claims on assets only after senior debt has been paid off. Subordinated debentures may be subordinated either to designated notes payable (usually bank loans), senior bonds, or to all other debt. In the event of liquidation or reorganization, holders of subordinated debentures cannot be paid until all senior debt, as named in the debentures' indentures, has been paid.
DEVELOPMENT BONDS
Some companies may be in a position to benefit from the sale of either development bonds or pollution control bonds. State and local governments may set up both industrial development agencies and pollution control agencies. These agencies are allowed, under certain circumstances, to sell tax-exempt bonds and then make the proceeds available to corporations for specific uses deemed (by Congress) to be in the public interest. For ex ample, a Detroit pollution control agency might sell bonds to provide Ford with funds for purchasing pollution control equipment. Because the income from the bonds would be tax exempt, the bonds would have relatively low interest rates. Note, however, that these bonds are guaranteed by the corporation that will use the funds, not by a governmental unit, so their rating reflects the credit strength of the corporation using the funds.
REVENUE BONDS AND PROJECT FINANCING
The payments for some bonds and their claims in the event of bankruptcy are limited to the income produced from a specific project. For example, a revenue bond is a type of mu nicipal bond that is secured by the revenues derived from a specific project such as roads
C
Chapter 5 Bonds, Bond Valuation, and Interest Rates
Insuring with Credit Default Swaps: Let the Buyer Beware!
223
The Global Economic Crisis
A credit default swap (CDS) is like an insurance policy. The investors. But how good was this type of insurance? As it
purchaser of the CDS agrees to make annual payments to a turned out, not very. For example, Lehman Brothers might
counterparty that agrees to pay if a particular bond defaults. have bought a CDS from AIG in order to sell a Lehman-created
During the 2000s, investment banks often would purchase MBS to an investor. But when the MBS began defaulting, nei-
CDS for the mortgage-backed securities (MBS) they were ther Lehman nor AIG was capable of making full restitution
creating in order to make the securities more attractive to to the investor.
and bridges, airports, water and sewage systems, and not-for-profit health care facilities. Project financing is a method in which a particular project's creditors do not have full recourse against the borrowers; the lenders and lessors must be paid from the project's cash flows and equity. Project financing often is used for large international projects such as oil refineries.
MUNICIPAL BOND INSURANCE
Municipalities can buy bond insurance, which means that an insurance company guar antees to pay the coupon and principal payments should the issuer default. This reduces risk to investors, who will thus accept a lower coupon rate for an insured bond than for a comparable but uninsured one. Even though the municipality must pay a fee to have its bonds insured, its savings due to the lower coupon rate often make insurance cost-effec tive. Keep in mind that the insurers are private companies and that the value added by the insurance depends on the creditworthiness of the insurer. The larger insurers are strong companies, and their own ratings are AAA.
5-llb Bond Ratings
A bond rating reflects the probability that a bond will go into default. The three major rating agencies are Moody's Investors Service (Moody's), Standard & Poor's Corporation (S&P), and Fitch Ratings. As shown in Columns (3) and (4) of Table 5-1, triple-A and dou ble-A bonds are extremely safe, rarely defaulting even within 5 years of being assigned a rating. Single-A and triple-B bonds are also strong enough to be called investment-grade bonds, and they are the lowest-rated bonds that many banks and other institutional inves tors are permitted by law to hold.Double-B and lower bonds are speculative bonds and are often called junk bonds. These bonds have a significant probability of defaulting.
5-llc Bond Rating Criteria, Upgrades, and Downgrades
Bond ratings are based on both quantitative and qualitative factors, as we describe next.
I. Financial Ratios. Many ratios potentially are important, but the return on invested capital, debt ratio, and interest coverage ratio are particularly valuable for predicting financial distress. For example, Columns (1), (5), and (6) in Table 5-1 show a strong relationship between ratings and the return on capital and the debt ratio.
2. Bond Contract Terms. Important provisions for determining the bond's rating in clude whether the bond is secured by a mortgage on specific assets, whether the bond
224
TABLE S·l
Part 2 Fixed Income Securities
is subordinated to other debt, any sinking fund provisions, guarantees by some other party with a high credit ranking, and restrictive covenants such as requirements that the firm keep its debt ratio below a given level or that it keep its times interest earned ratio above a given level.
3. Qualitative Factors. Included here would be such factors as sensitivity of the firm's earnings to the strength of the economy, how it is affected by inflation, whether it is having or is likely to have labor problems, the extent of its international operations (including the stability of the countries in which it operates), potential environmental problems, potential antitrust problems, and so on.
Rating agencies review outstanding bonds on a periodic basis and re-rate if necessary. Columns (7) and (8) in Table 5-1 show the percentages of companies in each rating cat egory that were downgraded or upgraded in 2016 by Fitch Ratings.
Over the long run, ratings agencies have done a reasonably good job of measuring the average credit risk of bonds and of changing ratings whenever there is a significant change in credit quality. However, it is important to understand that ratings do not adjust immediately to changes in credit quality, and in some cases there can be a considerable lag between a change in credit quality and a change in rating. Many abrupt downgrades occurred in 2007 and 2008, leading to calls by Congress and the SEC for changes in rat ing agencies and the way they rate bonds. Clearly, improvements can be made, but there will always be occasions when completely unexpected information about a company is released, leading to a sudden change in its rating.
Bond Ratings, Default Risk, and Yields
Percent Upgraded or Rating Agency• Percent Defaulting Within:b Median Ratios' Downgraded from 1990-2016b
S&P and Return on Total debt/ Fitch Moody's lyear Syears capital Total capital Up Down Vie[dd
(1) (2) (3) (4) (5) (6) (7) (8) (9)
Investment-grade bonds
AAA Aaa 0.13% 0.39% 27.6% 12.4% NA% 5.93% 2.99%
AA Aa 0.05 0.06 27.0 28.3 0.11 9.55 2.78
A A 0.06 0.30 17.5 37.5 1.73 6.00 3.06
BBB Baa 0.15 0.80 13.4 42.5 3.16 4.02 3.62
Junk bonds
BB Ba 0.73 3.28 11.3 53.7 7.78 8.04 4.28
B B 2.12 6.87 8.7 75.9 8.47 6.62 5.84
CCC Caa 21.24 31.61 3.2 113.5 19.75 21.24 10.81
Notes:
'The ratings agencies also use "modifiers" for bonds rated below triple-A. S&P and Fitch use a plus and minus system; thus, A+ designates the strongest A-rated bonds and A- the weakest. Moody's uses a 1, 2, or 3 designation, with 1 denoting the strongest and 3 the weakest; thus, within the double-A category, Aal is the best, Aa2 is average, and Aa3 is the weakest.
'Default data, downgrades, and upgrades are from Fitch Ratings Global Corporate Finance 2016 Transition and Default Study, August 2017: see www.fitchratings.com. Default data are for the period 1990-2016. A free registration is required to access the Fitch report.
'Median ratios are from Standard & Poor's 2006 Corporate Ratings Criteria, April 23, 2007: see www.standardandpoors.com/en_US/web/guest /article/-/view/type/HTML/id/785022. You must register (which is free) and log in to get access to this report.
'Data are from BofA Merrill Lynch, retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stloulsfed.org, December 1, 2017.
Chapter 5 Bonds, Bond Valuation, and Interest Rates 225
U.S. Treasury Bonds Downgraded!
The worsening recession that began at the end of 2007 led
Congress to pass a huge economic stimulus package in
early 2009. The combination of the stimulus package and
the government's bailouts of financial institutions caused
the U.S. government to increase its borrowing substan
tially. The current (October 2016) level of total debt is about
$19 trillion, about 101% of gross domestic product (GDP).
Any way you look at it, this is a lot of money, even by Wash
ington standards!
With so much debt outstanding and enormous an
nual deficits continuing, in mid-2011 Congress was faced
with the need to increase the amount of debt the federal
government is allowed to issue. Although Congress had in
creased the debt ceiling 74 times previously, and 10 times
since 2001, partisan and heated debate seriously delayed
approval of the measure and brought the federal govern
ment to the brink of default on its obligations by August. At
the last minute, Congress approved a debt ceiling increase,
narrowly avoiding a partial government shutdown. How
ever, the deficit reduction package that accompanied the
legislation was small, doing little to address the structural
revenue and spending imbalance the federal government
faces going forward.
On August S, 2011, the combination of a dysfunctional
political process apparently incapable of reliably performing
basic financial housekeeping chores and the lack of a clear
plan to address future deficits raised enough questions about
the U.S. government's financial stability to induce Standard
& Poor's (S&P), the credit rating agency, to downgrade U.S.
public debt from AAA to AA+, effectively removing it from
its list of risk-free investments. Financial markets quickly
responded to this dark assessment, with the Dow Jones In
dustrial Average plunging some 13% over the next week.
Moody's and Fitch, the other two major rating agencies, how
ever, kept their ratings of U.S. public debt at their highest
levels. With two out of three agencies rating U.S. debt at the
highest level, is the yield on U.S. debt still a proxy for the risk
less rate? Only time will tell, but since the initial downgrade in
2011, a bitterly divided Congress has brought the federal gov
ernment to the brink of default on its debt obligations several
more times. This behavior does not bode well for the pros
pect of maintaining an AAA rating.
5-lld Bond Ratings and the Default Risk Premium
Why are bond ratings so important? First, most bonds are purchased by institutional in vestors rather than by individuals, and many institutions are restricted to investment grade securities. Thus, if a firm's bonds fall below BBB, it will have a difficult time selling new bonds because many potential purchasers will not be allowed to buy them. Second, many bond covenants stipulate that the coupon rate on the bond automatically increases if the rating falls below a specified level. Third, because a bond's rating is an indicator of its default risk, the rating has a direct, measurable influence on the bond's yield. Column (9) of Table 5-1 shows that an AAA bond has a yield of2.99% and that yields increase as the rating falls. In fact, an investor would earn 10.81% on a CCC bond if it didn't default.
A bond spread is the difference between a bond's yield and the yield on some other se curity of the same maturity. Unless specified differently, the term "spread" generally means the difference between a bond's yield and the yield on a Treasury bond of similar maturity.
Figure 5-4 shows the spreads between an index of AAA bonds and a 10-year Treasury bond; it also shows spreads for an index of BAA bonds relative to the T-bond. Figure 5-4 illustrates three important points. First, the BAA spread always is greater than the AAA spread. This is because a BAA bond is riskier than an AAA bond, so BAA inves tors require extra compensation for their extra risk. The same is true for other ratings: Lower-rated bonds have higher yields.
Second, the spreads are not constant over time. For example, look at the AAA spread. It was exceptionally low during the boom years of 2005-2007 but rose dramatically as the economy declined in 2008 and 2009.
226
FIGURES-4
Bond Spreads
Spread {%)
7.00
6.00
Part 2 Fixed Income Securities
BAA-T-bond
5.00
4.00
3.00
2.00
1.00
0.00 .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... 0 0 0 0
9 0 0 0 9 9
0 0 0 0 0 0 0 I I 1
� 1 1 1 1 I I 1 1 1 I
"' "' U'l :8 ,- IX) 2l 0 .... "' "' ;:!j U'l "' ,-0 � �
0 0 0 .... .... .... .... .... .... .... 0 :=a :=a :=a :=a :=a :=a :=a :=a :=a :=a :=a :=a :=a "'
Note: All data are from the Federal Reserve Bank of St. Louis, Moody's Seasoned Aaa and Baa Corporate Bond Yield Relative to Yield on JO-Year Treasury Constant Maturity: https://fred.stlouisfed.org, December 1, 2017. The spreads are defined as the yield on the risky bond (AAA or BAA) minus the yield on a JO-year Treasury bond.
Third, the difference between the BAA spread and the AAA spread isn't constant over time. The two spreads were reasonably close to one another in 2005 but were very far apart in early 2009. In other words, BAA investors didn't require much extra return over that of an AAA bond to induce them to take on that extra risk most years, but in 2009 they required a very large risk premium.
The Few, the Proud, the ... AAA-Rated Companies!
AAA-rated companies are members of an elite group. Over
the last 20 years, this cream of the crop has included such
powerhouses as 3M, Abbott Labs, BellSouth, ExxonMobil,
GE, Kellogg, Microsoft, and UPS. Only large companies with
stable cash flows make it into this group, and for years they
guarded their AAA ratings vigilantly. In recent years, however,
the nonfinancialAAA-rated corporation has become a vanish
ing breed. In December 2017, Moody's and Standard & Poor's,
agreed on the highest rating for only two nonfinancial pub
licly traded companies: Johnson & Johnson and Microsoft.
Why do so few companies have AAA ratings? One reason
may be that the recent financial crisis and recession have hurt
the creditworthiness of even large, stable companies. Another
reason is that many of the top companies are choosing to be
rated by only one or two of the ratings agencies, rather than all
three. A third likely explanation is that in recent years, large, sta
ble companies have increased their debt levels to take greater
advantage of the tax savings that they afford. With higher debt
levels, these companies are no longer eligible for the highest rat
ing. In essence, they have sacrificed their AAA rating for lower
taxes. Does this sound like a good trade-off to you? We will dis
cuss how companies choose the level of debt in Chapter 15.
Source: Bond information from the Financial Industry Regulatory Authority,
www.flnra.org.
Chapter 5 Bonds, Bond Valuation, and Interest Rates 227
Fear and Rationality
The following graph shows two measures of fear. One is
the "Hi-Yield" spread between the yields on junk bonds
and Treasury bonds. The second is the TED spread, which
is the difference between the 3-month LIBOR rate and
the 3-month T-bill rate. Both are measures of risk aver
sion. The Hi-Yield spread measures the amount of extra
compensation investors need to induce them to take on
risky junk bonds. The TED spread measures the extra com
pensation that banks require to induce them to lend to one
another. Observe that the spreads were very low from mid-
2003 through the end of 2007. During these boom years,
investors and bankers had a voracious appetite for risk and
simply didn't require much extra return for additional risk.
But as the economy began to deteriorate in 2008, inves
tors and bankers reversed course and became extremely
risk averse, with spreads skyrocketing. Note that the pre
financial crisis appetite for risk seems to have returned,
with spreads again very low. It is hard to reconcile such
drastic changes in risk aversion with careful, deliberate,
and rational behavior!
Spread
(%)
20
18
16
14
12 Hi-Yield
10
8
6
4 Sources: The TED spread data are retrieved from FRED, Federal Reserve
Bank of St. Louis; https://fred.stloulsfed.org /series/TEDRATE, December 1,
2017. The Hi-Yield spread data are from BofA Merrill Lynch, BofA Merrill
Lynch US High Yield Option•Adjusted Spread JBAMLH0A0HYM2]. retrieved
from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org
/serles/BAMLH0A0HVM2, December 1, 2017.
Not only do spreads vary with the rating of the security, but they also usually increase as maturity increases. This should make sense. If a bond matures soon, investors are able to forecast the company's performance fairly well. But if a bond has a long time until it ma tures, investors have a difficult time forecasting the likelihood that the company will fall into financial distress. This extra uncertainty creates additional risk, so investors demand a higher required return.
SELF-TEST
Differentiate between mortgage bonds and debentures.
Name the major rating agencies, and list some factors that affect bond ratings.
What is a bond spread?
How do bond ratings affect the default risk premium?
A 10-year T-bond has a yield of 6%. A 10-year corporate bond with a rating of AA has a yield of
7.5%. If the corporate bond has excellent liquidity, what is an estimate of the corporate bond's
default risk premium? (1.5%)
5-12 The Liquidity Premium (LP} Financial assets generally have more liquidity than real assets, which means that securi ties can be converted to cash quickly at a "fair market value." However, not all securities have the same degree of liquidity, so investors include a liquidity premium (LP) when establishing a security's required rate of return. Although liquidity premiums are difficult to measure accurately, a differential of at least 2 percentage points (and perhaps up to 4 or
228 Part 2 Fixed Income Securities
5 percentage points) exists between the least liquid and the most liquid financial assets of similar default risk and maturity. Corporate bonds issued by small companies are traded less frequently than those issued by large companies, so small-company bonds tend to have a higher liquidity premium.
For example, liquidity in the market for mortgage-backed securities (MBS) evapo rated in 2008 and early 2009. The few transactions that occurred were priced so low that the yields on these MBS were extremely high, which was partially due to a much higher liquidity premium caused by the extremely low liquidity of MBS.
SELF -TEST
Which bond usually will have a higher liquidity premium: one issued by a large company or one
issued by a small company?
5-13 The Term Structure of Interest Rates
The term structure of interest rates describes the relationship between long-term and short-term rates. The term structure is important both to corporate treasurers deciding whether to borrow by issuing long-term or short-term debt and to investors who are de ciding whether to buy long-term or short-term bonds.
Interest rates for bonds with different maturities can be found in a variety of publi cations, including The Wall Street Journal and the Federal Reserve Bulletin, as well as on a number of Web sites, including Bloomberg, Yahoo!, CNN Financial, and the Federal Reserve Board. Using interest rate data from these sources, we can determine the term structure at any given point in time. For example, Figure 5-5 presents interest rates for different maturities on three different dates. The set of data for a given date, when plotted on a graph such as Figure 5-5, is called the yield curve for that date.
Maturity (yrs.) March 1980 February 2000 December 2017
0.5 15.0% 6.0% 0.48%
1 14.0% 6.2% 0.63%
5 13.5% 6.7% 1.18%
10 12.8% 6.7% 1.63%
30 12.3% 6.3% 2.34%
As the figure shows, the yield curve changes both in position and in slope over time. In March 1980, all rates were quite high because high inflation was expected. However, the rate of inflation was expected to decline, so the inflation premium (IP) was larger for short-term bonds than for long-term bonds. This caused short-term yields to be higher than long-term yields, resulting in a downward-sloping yield curve. By February 2000, inflation had indeed declined and thus all rates were lower. The yield curve had be come humped-medium-term rates were higher than either short- or long-term rates. In November 2017, all rates were below the 2000 levels. Because short-term rates had dropped below long-term rates, the yield curve was upward sloping.
Historically, long-term rates are generally higher than short-term rates owing to the maturity risk premium, so the yield curve usually slopes upward. For this reason, people often call an upward-sloping yield curve a normal yield curve and a yield curve that slopes downward an inverted yield curve or an abnormal yield curve. Thus, in Figure 5-5 the yield curve for March 1980 was inverted, whereas the yield curve in November 2017 was normal. As stated earlier, the February 2000 curve was humped.
For o discussion of the
expectations theory, see
Web Extension SD on the
textbook's Web site.
Chapter 5 Bonds, Bond Valuation, and Interest Rates
FIGURES-5
U.S. Treasury Bond Interest Rates on Different Dates
Interest Rate (%)
16
Yield Curve for March 1980
12
10
8 Yield Curve for February 2000
Yield Curve for December 2017
5 10 15 20 25
•
30
Years to Maturity
229
Sources: Board of Governors of the Federal Reserve System, "H.15 Selected Interest Rates," at www.federalreserve
,gov/releases/hlS/. For historical yield curve data, see www.treasury.gov/resource-center/data-chart-center
/interest-rates/Pages/TextView.aspx?data=yield.
A few academics and practitioners contend that large bond traders who buy and sell securities of different maturities each day dominate the market. According to this view, a bond trader is just as willing to buy a 30-year bond to pick up a short-term profit as to buy a 3-month security. Strict proponents of this view argue that the shape of the yield curve is therefore determined only by market expectations about future interest rates, a position that is called the pure expectations theory, or sometimes just the expectations theory. If this were true, then the maturity risk premium (MRP) would be zero and long-term interest rates would simply be a weighted average of current and expected future short-term inter est rates. See Web Extension SD for a more detailed discussion of the expectations theory.
SELF -TEST
What is a yield curve, and what information would you need to draw this curve?
Distinguish among the shapes of a "normal" yield curve, an "abnormal" curve, and a "humped" curve.
If the interest rates on 1-, 5-, 20-, and 30-year bonds are (respectively) 4%, 5%, 6%, and 7%, then how would you describe the yield curve? How would you describe it if the rates were reversed?
5-14 Financing with Junk Bonds Recall that bonds rated less than BBB are noninvestment-grade debt, also called junk bonds or high-yield debt. There are two ways that a bond can become a junk bond. First, the bond might have been investment-grade debt when it was issued, but its rating subse quently was cut because the issuing corporation had fallen on hard times. Such bonds are called "fallen angels".
230 Part 2 Fixed Income Securities
Some bonds are junk bonds at the time they are issued, but this was not always true. Prior to the 1980s, fixed-income investors such as insurance companies were generally unwilling to buy risky bonds. In the late 1970s, Michael Milken of the investment bank ing firm Drexel Burnham Lambert, relying on historical studies that showed risky bonds yielded more than enough to compensate for their risk, convinced institutional investors that junk-bond yields were worth their risk. Thus was born the junk-bond market.
In the 1980s, large investors like T. Boone Pickens and Henry Kravis thought that certain older established companies were run inefficiently and were financed too conser vatively. These corporate raiders were able to invest some of their own money, borrow the rest via junk bonds, and take over the target company, usually taking the company private. The fact that interest on the bonds was tax deductible, combined with the much higher debt ratios of the restructured firms, increased after-tax cash flows to stockholders and helped make the deals feasible. Because these deals used lots of debt, they were called lev eraged buyouts (LBOs). In recent years, private equity firms have conducted transactions similar to the LBOs of the 1980s, taking advantage of historically low junk-bond rates to help finance their purchases.
The 2017 Tax Cuts and Jobs Act reduced the federal corporate tax rate to 21% from 35%. The Act also placed limits on the amount of interest expense that can be immediately deducted. (Interest expenses over the limit can be carried forward indefinitely.) These provisions will certainly reduce the tax advantages of an LBO.
SELF -TEST
What are junk bonds?
5-15 Bankruptcy and Reorganization A business is insolvent if the book values of its liabilities are greater than the market value of its assets or if it does not have enough cash to meet its interest and principal pay ments. When this occurs, either the creditors or the company may file for bankruptcy in the United States Bankruptcy Court. After hearing from the creditors and the com pany's managers, a federal bankruptcy court judge decides whether to dissolve the firm through liquidation or to permit it to remain a viable business through a reorganization. Chapter 7 of the federal bankruptcy statutes addresses liquidation, and Chapter 11 ad dresses reorganization.
The decision to force a firm to liquidate versus permit it to reorganize depends on whether the value of the reorganized firm is likely to be greater than the value of the firm's assets if they are sold off piecemeal. In a reorganization, the firm's creditors negotiate with management on the terms of a potential reorganization. The reorganization plan may call for a restructuring of the firm's debt, in which case the interest rate may be reduced, the term to maturity may be lengthened, or some of the debt may be exchanged for equity. The point of the restructuring is to reduce the financial charges to a level that the firm's cash flows can support. Of course, the common stockholders also have to give up something: They often see their position diluted as a result of additional shares being given to debt holders in exchange for accepting a reduced amount of debt principal and interest. In fact, the original common stockholders often end up with nothing. The court may appoint a trustee to oversee the reorganization, but usually the existing management is allowed to retain control.
Liquidation occurs if the company is deemed to be too far gone to be saved-if it is worth more dead than alive. If the bankruptcy court orders liquidation, then assets are sold off, and the cash obtained is distributed as specified in Chapter 7 of the Bankruptcy
Chapter 5 Bonds, Bond Valuation, and Interest Rates 231
Act. Here is the priority of claims: (1) past-due property tax liens; (2) secured creditors who are entitled to the proceeds from the sale of collateral; (3) the trustee's costs of ad ministering and operating the bankrupt firm; (4) expenses incurred after bankruptcy was filed; (5) some wages due workers (capped at a maximum amount per worker and limited to wages earned within a specified period prior to the bankruptcy); (6) claims for unpaid contributions to employee benefit plans (capped at a maximum amount per worker and limited to wages earned within a specified period prior to the bankruptcy); (7) unsecured claims for customer deposits (capped at a maximum amount per cus tomer); (8) federal, state, and local taxes due; (9) unfunded pension plan liabilities (al though some limitations exist); (10) general unsecured creditors; (11) preferred stock holders (up to the par value of their stock); and (12) common stockholders (although usually nothing is left for them).
Here are the key points for you to know: (1) The federal bankruptcy statutes govern both reorganization and liquidation. (2) Bankruptcies occur frequently. (3) A priority of the specified claims must be followed when distributing the assets of a liquidated firm.
SELF-TEST
Differentiate between a Chapter 7 liquidation and a Chapter 11 reorganization.
List the priority of claims for the distribution of a liquidated firm's assets.
This chapter described the different types of bonds that governments and corporations issue, explained how bond prices are established, and discussed how investors estimate the rates of return they can expect to earn. The rate of return required by debtholders is the company's pre-tax cost of debt, and this rate depends on the risk that investors face when they buy bonds.
• A bond is a long-term promissory note issued by a business or governmental unit. The issuer receives money in exchange for promising to make interest payments and to repay the principal on a specified future date.
• Some special types of long-term financing include zero coupon bonds, which pay no annual interest but are issued at a discount; see Web Extension SA for more on zero coupon bonds. Other types are floating-rate debt, whose interest payments fluctuate with changes in the general level of interest rates, and junk bonds, which are high-risk, high-yield instruments issued by firms that use a great deal of financial leverage.
• A call provision gives the issuing corporation the right to redeem the bonds prior to maturity under specified terms, usually at a price greater than the maturity value (the difference is a call premium). A firm will typically call a bond if interest rates fall sub stantially below the coupon rate.
• A sinking fund is a provision that requires the corporation to retire a portion of the bond issue each year. The purpose of the sinking fund is to provide for the orderly re tirement of the issue. A sinking fund typically requires no call premium.
• The value of a bond is found as the present value of an annuity (the interest payments) plus the present value of a lump sum (the principal payment). The bond is evaluated at the appropriate periodic interest rate over the number of periods for which interest payments are made.
Part 2 Fixed Income Securities
• The equation used to find the value of an annual coupon bond is N INT M V =�--+--s � (1 + rl (1 + r
d )N
• An adjustment to the formula must be made if the bond pays interest semiannually: divide INT and r
d by 2, and multiply N by 2:
• 2N INT/2 M Value of semiannual bond = L ( / )1
+ ( / iN
t= 1 1 + rd 2 1 + rd 2) • The expected rate of return on a bond held to maturity is defined as the bond's yield
to maturity (YTM):
N INT M Bond price = � -----+ -----� (1 + YTM)1 (1 + YTM)N
• The expected rate of return on a callable bond held to its call date is defined as the yield to call (YTC).
• The required rate of return on debt (r d ) is the rate needed to fairly compensate inves
tors for purchasing or holding debt, taking into consideration its cash flows' risk and timing. It is the rate that is observed in the market, so it is also called the market interest rate, the quoted interest rate, or the nominal interest rate.
• The real risk-free interest rate (r*) is the rate that a hypothetical riskless security pays each moment if zero inflation were expected.
• The required rate of return on debt is composed of the real risk-free rate, r·, plus premiums that reflect inflation (IP), maturity risk (MRP), default risk (DRP), and liquidity (LP}:
r d
= r• +IP+ MRP + DRP + LP
• The risk-free interest rate (r RF
) is the quoted rate on a U.S. Treasury security, which is default-free and very liquid. The short-term risk-free rate is approximated by a T-bill's yield; the long-term risk-free rate is approximated by a T-bond's yield.
• Treasury Inflation-Protected Securities (TIPS) are U.S. Treasury bonds that have no inflation risk. See Web Extension SB for more discussion of TIPS.
• The longer the maturity of a bond, the more its price will change in response to a given change in interest rates; this is called interest rate risk. However, bonds with short maturities expose investors to high reinvestment rate risk, which is the risk that in come from a bond portfolio will decline because cash flows received from bonds will be rolled over at lower interest rates.
• Duration is a measure of interest rate risk. See Web Extension SC for a discussion of duration.
• Corporate and municipal bonds have default risk. If an issuer defaults, investors re ceive less than the promised return on the bond. Therefore, investors should evaluate a bond's default risk before making a purchase.
• A bond rating reflects the probability that a bond will go into default. The highest rating is AAA, and they go down to D. The higher a bond's rating, the lower its risk and therefore its interest rate.
• The relationship between the yields on securities and the securities' maturities is known as the term structure of interest rates, and the yield curve is a graph of this relationship.
Chapter 5 Bonds, Bond Valuation, and Interest Rates
• The shape of the yield curve depends on two key factors: (1) expectations about fu ture inflation and (2) perceptions about the relative risk of securities with different maturities.
233
• The yield curve is normally upward sloping; this is called a normal yield curve. How ever, the curve can slope downward (an inverted yield curve) if the inflation rate is expected to decline. The yield curve also can be humped, which means that interest rates on medium-term maturities are higher than rates on both short- and long-term maturities.
• The expectations theory states that yields on long-term bonds reflect expected future interest rates. Web Extension SD discusses this theory.
(5-1) Define each of the following terms:
a. Bond; Treasury bond; corporate bond; municipal bond; foreign bond b. Par value; maturity date; coupon payment; coupon interest rate c. Floating-rate bond; zero coupon bond; original issue discount bond (OID) d. Call provision; redeemable bond; sinking fund e. Convertible bond; warrant; income bond; indexed bond (also called a purchasing
power bond) f. Premium bond; discount bond g. Current yield (on a bond); yield to maturity (YTM); yield to call (YTC) h. Indentures; mortgage bond; debenture; subordinated debenture i. Development bond; municipal bond insurance; junk bond; investment
grade bond j. Real risk-free rate of interest, r·; nominal risk-free rate of interest, r
RF
k. Inflation premium (IP); default risk premium (DRP); liquidity; liquidity premium (LP)
l. Interest rate risk; maturity risk premium (MRP); reinvestment rate risk m. Term structure of interest rates; yield curve n. "Normal" yield curve; inverted ("abnormal") yield curve
( 5 -2) "Short-term interest rates are more volatile than long-term interest rates, so short-term bond prices are more sensitive to interest rate changes than are long-term bond prices." Is this statement true or false? Explain.
(5-3) The rate of return on a bond held to its maturity date is called the bond's yield to maturity. If interest rates in the economy rise after a bond has been issued, what will happen to the bond's price and to its YTM? Does the length of time to maturity affect the extent to which a given change in interest rates will affect the bond's price? Why or why not?
(5-4) If you buy a callable bond and interest rates decline, will the value of your bond rise by as much as it would have risen if the bond had not been callable? Explain.
(5-5) A sinking fund can be set up in one of two ways. Discuss the advantages and disadvantages of each procedure from the viewpoint of both the firm and its bondholders.
Part 2 Fixed Income Securities
S E L F - T E S T P R O B L E M S O L U T I O 1,1 S S H O W f\l I [\J J\ P P E [\J D , >'. i\
(ST-1) Bond Valuation
The Pennington Corporation issued a new series of bonds on January 1, 1997. The bonds were sold at par ($1,000), had a 12% coupon, and will mature in 30 years on December 31, 2026. Coupon payments are made semiannually (on June 30 and December 31).
a. What was the YTM on the date the bonds were issued? b. What was the price of the bonds on January 1, 2002 (5 years later), assuming that
interest rates had fallen to 10%? c. Find the current yield, capital gains yield, and total yield on January 1, 2002, given
the price as determined in Part b. d. Suppose you purchase a Pennington bond for $916.42 on July l, 2020 (6.5 years be
fore maturity). What are the YTM, the current yield, and the capital gains yield for that date?
e. Suppose instead that you purchase a Pennington bond on March 1, 2020, three months earlier than in Part d. If the going rate for a bond of this risk is 15.5%, how large a check must you write to complete the transaction? (Hint: Don't forget the accrued interest.)
P R O B L E M S A f\l S W t R S A R E I f\l f\ P P E r,1 D I X B
(5-1) Bond Valuation with
Annual Payments
(5-2) Yield to Maturity for
Annual Payments
(5-3) Current Yield for
Annual Payments
(5-4) Determinant of
Interest Rates
(5-5) Default Risk
Premium
(5-6) Maturity Risk
Premium
(5-7) Bond Valuation
with Semiannual Payments
EASY PROBLEMS 1-6
Jackson Corporation's bonds have 12 years remaining to maturity. Interest is paid annually, the bonds have a $1,000 par value, and the coupon interest rate is 8%. The bonds have a yield to maturity of 9%. What is the current market price of these bonds?
Wilson Corporation's bonds have 12 years remaining to maturity. Interest is paid annually, the bonds have a $1,000 par value, and the coupon interest rate is 10%. The bonds sell at a price of $850. What is their yield to maturity?
Heath Food Corporation's bonds have 7 years remaining to maturity. The bonds have a face value of $1,000 and a yield to maturity of 8%. They pay interest annually and have a 9% coupon rate. What is their current yield?
The real risk-free rate of interest is 4%. Inflation is expected to be 2% this year and 4% dur ing each of the next 2 years. Assume that the maturity risk premium is zero. What is the yield on 2-year Treasury securities? What is the yield on 3-year Treasury securities?
A Treasury bond that matures in 10 years has a yield of 6%. A 10-year corporate bond has a yield of 9%. Assume that the liquidity premium on the corporate bond is 0.5%. What is the default risk premium on the corporate bond?
The real risk-free rate is 3%, and inflation is expected to be 3% for the next 2 years. A 2-year Treasury security yields 6.3%. What is the maturity risk premium for the 2-year security?
INTERMEDIATE PROBLEMS 7-20
Renfro Rentals has issued bonds that have a 10% coupon rate, payable semiannually. The bonds mature in 8 years, have a face value of $1,000, and a yield to maturity of 8.5%. What is the price of the bonds?
(5-8) Yield to Maturity
and Call with Semiannual
Payments
(5-9) Bond Valuation and
Interest Rate Risk
(5-10) Yield to Maturity
and Required Returns
(5-11) Yield to Call and
Realized Rates of Return
(5-12) Bond Yields and Rates of Return
(5-13) Yield to Maturity
and Current Yield
(5-14) Current Yield
with Semiannual Payments
(5-15) Yield to Call, Yield
to Maturity, and Market Rates
(5-16) Interest Rate
Sensitivity
Chapter 5 Bonds, Bond Valuation, and Interest Rates 235
Thatcher Corporation's bonds will mature in 10 years. The bonds have a face value of $1,000 and an 8% coupon rate, paid semiannually. The price of the bonds is $1,100. The bonds are callable in 5 years at a call price of $1,050. What is their yield to maturity? What is their yield to call?
The Garraty Company has two bond issues outstanding. Both bonds pay $100 annual interest plus $1,000 at maturity. Bond L has a maturity of 15 years, and Bond S has a maturity of 1 year.
a. What will be the value of each of these bonds when the going rate of interest is (1) 5%, (2) 8%, and (3) 12%? Assume that there is only one more interest payment to be made on Bond S.
b. Why does the longer-term (15-year) bond fluctuate more when interest rates change than does the shorter-term bond (1 year)?
The Brownstone Corporation's bonds have 5 years remaining to maturity. Interest is paid annually, the bonds have a $1,000 par value, and the coupon interest rate is 9%.
a. What is the yield to maturity at a current market price of (1) $829 or (2) $1,104? b. Would you pay $829 for one of these bonds if you thought that the appropriate rate
of interest was 12%-that is, if r d
= 12%? Explain your answer.
Goodwynn & Wolflncorporated (G&W) issued a bond 7 years ago. The bond had a 20-year maturity, a 14% coupon paid annually, a 9% call premium and was issued at par, $1,000. Today, G&W called the bonds. If the original investors had expected G&W to call the bonds in 7 years, what was the yield to call at the time the bonds were issued?
A 10-year, 12% semiannual coupon bond with a par value of $1,000 may be called in 4 years at a call price of $1,060. The bond sells for $1,100. (Assume that the bond has just been issued.)
a. What is the bond's yield to maturity? b. What is the bond's current yield? c. What is the bond's capital gain or loss yield? d. What is the bond's yield to call?
You just purchased a bond that matures in 5 years. The bond has a face value of $1,000 and an 8% annual coupon. The bond has a current yield of 8.21 %. What is the bond's yield to maturity?
A bond that matures in 7 years sells for $1,020. The bond has a face value of $1,000 and a yield to maturity of 10.5883%. The bond pays coupons semiannually. What is the bond's current yield?
Absalom Energy's 14% coupon rate, semiannual payment, $1,000 par value bonds that mature in 30 years are callable 5 years from now at a price of $1,050. The bonds sell at a price of$1,353.54, and the yield curve is flat. Assuming that interest rates in the economy are expected to remain at their current level, what is the best estimate of the nominal interest rate on new bonds issued in 5 years?
A bond trader purchased each of the following bonds at a yield to maturity of 8%. Immediately after she purchased the bonds, interest rates fell to 7%. What is the
236
(5-17) Bond Value as Maturity Approaches
(5-18) Determinants of
Interest Rates
(5-19) Maturity Risk
Premiums
(5-20) Inflation Risk
Premiums
(5-21) Bond Valuation and Changes in
Maturity and Required Returns
Part 2 Fixed Income Securities
percentage change in the price of each bond after the decline in interest rates? Fill in the following table:
10-year, 10% annual coupon 10-year zero 5-year zero 30-year zero Perpetuity, $100 annual coupon
Price@SO/o Price @ 7% Percentage Change
An investor has two bonds in his portfolio. Each bond matures in 4 years, has a face value of $1,000, and has a yield to maturity equal to 9.6%. One bond, Bond C, pays an annual coupon of 10%; the other bond, Bond Z, is a zero coupon bond. Assuming that the yield to maturity of each bond remains at 9.6% over the next 4 years, what will be the price of each of the bonds at the following time periods? Fill in the following table:
T
0
1
2
3
4
Price of Bond C Price of Bond Z
The real risk-free rate is 2%. Inflation is expected to be 3% this year, 4% next year, and then 3.5% thereafter. The maturity risk premium is estimated to be 0.0005 X (t - 1), where t = number of years to maturity. What is the nominal interest rate on a 7-year Treasury security?
Assume that the real risk-free rate, r•, is 3% and that inflation is expected to be 8% in Year 1, 5% in Year 2, and 4% thereafter. Assume also that all Treasury securities are highly liquid and free of default risk. If 2-year and 5-year Treasury notes both yield 10%, what is the difference in the maturity risk premiums (MRPs) on the two notes; that is, what is MRP
5 minus MRP
2 ?
Because of a recession, the inflation rate expected for the coming year is only 3%. However, the inflation rate in Year 2 and thereafter is expected to be constant at some level above 3%. Assume that the real risk-free rate is r• = 2% for all maturities and that there are no maturity risk premiums. If 3-year Treasury notes yield 2 percentage points more than I-year notes, what inflation rate is expected after Year 1?
CHALLENGING PROBLEMS 21-23
Suppose Hillard Manufacturing sold an issue of bonds with a 10-year maturity, a $1,000 par value, a 10% coupon rate, and semiannual interest payments.
a. Two years after the bonds were issued, the going rate of interest on bonds such as these fell to 6%. At what price would the bonds sell?
b. Suppose that 2 years after the initial offering, the going interest rate had risen to 12%. At what price would the bonds sell?
c. Suppose that 2 years after the issue date (as in Part a) interest rates fell to 6%. Sup pose further that the interest rate remained at 6% for the next 8 years. What would happen to the price of the bonds over time?
(5-22) Yield to Maturity and Yield to Call
(5-23) Determinants of
Interest Rates
(5-24) Build a Model:
Bond Valuation
resource
Chapter 5 Bonds, Bond Valuation, and Interest Rates 237
Arnot International's bonds have a current market price of$1,200. The bonds have an 11% annual coupon payment, a $1,000 face value, and 10 years left until maturity. The bonds may be called in 5 years at 109% of face value (call price = $1,090).
a. What is the yield to maturity? b. What is the yield to call if they are called in 5 years? c. Which yield might investors expect to earn on these bonds, and why? d. The bond's indenture indicates that the call provision gives the firm the right to call
them at the end of each year beginning in Year 5. In Year 5, they may be called at 109% of face value, but in each of the next 4 years the call percentage will decline by 1 percentage point. Thus, in Year 6 they may be called at 108% of face value, in Year 7 they may be called at 107% of face value, and so on. If the yield curve is horizon- tal and interest rates remain at their current level, when is the latest that investors might expect the firm to call the bonds?
Suppose you and most other investors expect the inflation rate to be 7% next year, to fall to 5% during the following year, and then to remain at a rate of 3% thereafter. Assume that the real risk-free rate, r•, will remain at 2% and that maturity risk premiums on Trea sury securities rise from zero on very short-term securities (those that mature in a few days) to a level of 0.2 percentage points for I-year securities. Furthermore, maturity risk premiums increase 0.2 percentage points for each year to maturity, up to a limit of 1.0 percentage point on 5-year or longer-term T-notes and T-bonds.
a. Calculate the interest rate on 1-, 2-, 3-, 4-, 5-, 10-, and 20-year Treasury securities, and plot the yield curve.
b. Now suppose ExxonMobil's bonds, rated AAA, have the same maturities as the Treasury bonds. As an approximation, plot an ExxonMobil yield curve on the same graph with the Treasury bond yield curve. (Hint: Think about the default risk pre mium on ExxonMobil's long-term versus short-term bonds.)
c. Now plot the approximate yield curve ofLong Island Lighting Company, a risky nuclear utility.
Start with the partial model in the file Ch05 P24 Build a Model.xlsx on the textbook's Web site. A 20-year, 8% semiannual coupon bond with a par value of $1,000 may be called in 5 years at a call price of $1,040. The bond sells for $1,100. (Assume that the bond has just been issued.)
a. What is the bond's yield to maturity? b. What is the bond's current yield? c. What is the bond's capital gain or loss yield? d. What is the bond's yield to call? e. How would the price of the bond be affected by a change in the going market inter
est rate? (Hint: Conduct a sensitivity analysis of price to changes in the going market interest rate for the bond. Assume that the bond will be called if and only if the going rate of interest falls below the coupon rate. This is an oversimplification, but assume it for purposes of this problem.)
f. Now assume the date is October 25, 2020. Assume further that a 12%, 10-year bond was issued on July 1, 2020, pays interest semiannually (on January 1 and July 1), and sells for $1,100. Use your spreadsheet to find the bond's yield.
238 Part 2 Fixed Income Securities
Sam Strother and Shawna Tibbs are vice presidents of Mutual of Seattle Insurance Company and co-directors of the company's pension fund management division. An important new client, the North-Western Municipal Alliance, has requested that Mutual of Seattle present an investment seminar to the mayors of the represented cities, and Strother and Tibbs, who will make the actual presentation, have asked you to help them by answering the following questions.
a. What are the key features of a bond? b. What are call provisions and sinking fund provisions? Do these provisions make
bonds more or less risky? c. How does one determine the value of any asset whose value is based on expected
future cash flows? d. How is the value of a bond determined? What is the value of a 10-year, $1,000 par
value bond with a 10% annual coupon if its required rate of return is 10%? e. (1) What would be the value of the bond described in Part d if, just after it had been
issued, the expected inflation rate rose by 3 percentage points, causing investors to require a 13% return? Would we now have a discount or a premium bond?
(2) What would happen to the bond's value if inflation fell and r d
declined to 7%? Would we now have a premium or a discount bond?
(3) What would happen to the value of the 10-year bond over time if the required rate of return remained at 13%? If it remained at 7%? (Hint: With a financial cal culator, enter PMT, I/YR, FV, and N, and then change N to see what happens to the PV as the bond approaches maturity.)
f. (1) What is the yield to maturity on a 10-year, 9% annual coupon, $1,000 par value bond that sells for $887.00? That sells for $1,134.20? What does the fact that a bond sells at a discount or at a premium tell you about the relationship between r
d and the bond's coupon rate?
(2) What are the total return, the current yield, and the capital gains yield for the discount bond? (Assume the bond is held to maturity and the company does not default on the bond.)
g. How does the equation for valuing a bond change if semiannual payments are made? Find the value of a 10-year, semiannual payment, 10% coupon bond if the nominal r
d = 13%.
h. Suppose a 10-year, 10% semiannual coupon bond with a par value of $1,000 is cur rently selling for $1,135.90, producing a nominal yield to maturity of 8%. However, the bond can be called after 5 years for a price of $1,050. (1) What is the bond's nominal yield to call (YTC)? (2) If you bought this bond, do you think you would be more likely to earn the YTM
or the YTC? Why? i. Write a general expression for the yield on any debt security (r
d ) and define these
terms: real risk-free rate of interest (r•), inflation premium (IP), default risk premium (DRP), liquidity premium (LP), and maturity risk premium (MRP).
j. Define the real risk-free rate (r•). What security can be used as an estimate of r•? What is the nominal risk-free rate (r
RF )? What securities can be used as estimates
Of r R /
k. Describe a way to estimate the inflation premium (IP) for a t-year bond. l. What is a bond spread and how is it related to the default risk premium? How are
bond ratings related to default risk? What factors affect a company's bond rating?
0
0
Chapter 5 Bonds, Bond Valuation, and Interest Rates 239
m. What is interest rate (or price) risk? Which bond has more interest rate risk: an an nual payment I-year bond or a 10-year bond? Why?
n. What is reinvestment rate risk? Which has more reinvestment rate risk: a I-year bond or a 10-year bond?
o. How are interest rate risk and reinvestment rate risk related to the maturity risk premium?
p. What is the term structure of interest rates? What is a yield curve? q. Brief ly describe bankruptcy law. If a firm were to default on its bonds, would the
company be liquidated immediately? Would the bondholders be assured of receiving all of their promised payments?
The following cases from CengageCompose cover many of the concepts discussed in this chapter and are available at http://compose.cengage.com.
Klein-Brigham Series:
Case 3, "Peachtree Securities, Inc. (B)"; Case 72, "Swan Davis"; and Case 78, "Beatrice Peabody."
Brigham-Buzzard Series:
Case 3, "Powerline Network Corporation (Bonds and Preferred Stock)."
