Valuing Bonds Chapter 6 PDF

CH APTE R Valuing Bonds 6 AFTER A FOUR-YEAR HIATUS, THE U.S. GOVERNMENT BEGAN NOTATION issuing 30-year...

CH APTE R Valuing Bonds 6 AFTER A FOUR-YEAR HIATUS, THE U.S. GOVERNMENT BEGAN NOTATION issuing 30-year Treasury bonds again in August 2005. While the move was due in CPN coupon payment on a part to the government’s need to borrow to fund record budget deficits, the deci- bond sion to issue 30-year bonds was also a response to investor demand for long-term, n number of periods risk-free securities backed by the U.S. government. These 30-year Treasury bonds are part of a much larger market for publicly traded bonds. As of January 2022, y, YTM yield to maturity the value of traded U.S. Treasury debt was approximately $22.6 trillion, $12.6 P initial price of a bond trillion more than the value of all publicly traded U.S. corporate bonds. If we FV face value of a bond include bonds issued by municipalities, government agencies, and other issuers, YTM n y ield to maturity on investors had nearly $53 trillion invested in U.S. bond markets, compared with a zero-coupon bond $52 trillion in U.S. equity markets.1 with n periods to In this chapter, we look at the basic types of bonds and consider their valu- maturity ation. Understanding bonds and their pricing is useful for several reasons. First, rn interest rate or dis- the prices of risk-free government bonds can be used to determine the risk-free count rate for a cash interest rates that produce the yield curve discussed in Chapter 5. As we saw flow that arrives in there, the yield curve provides important information for valuing risk-free cash period n flows and assessing expectations of inflation and economic growth. Second, firms PV present value often issue bonds to fund their own investments, and the returns investors receive NPER a nnuity spreadsheet on those bonds is one factor determining a firm’s cost of capital. Finally, bonds notation for the num- provide an opportunity to begin our study of how securities are priced in a com- ber of periods or date of the last cash flow petitive market. The ideas we develop in this chapter will be helpful when we turn to the topic of valuing stocks in Chapter 9. RATE a nnuity ­spreadsheet notation for interest We begin the chapter by evaluating the promised cash flows for different rate types of bonds. Given a bond’s cash flows, we can use the Law of One Price to di- PMT a nnuity spreadsheet rectly relate the bond’s return, or yield, and its price. We also describe how bond notation for cash flow prices change dynamically over time and examine the relationship between the prices and yields of different bonds. Finally, we consider bonds for which there APR a nnual percentage rate is a risk of default, so that their cash flows are not known with certainty. As an important application, we look at the behavior of corporate and sovereign bonds during the recent economic crisis. 1 Securities Industry and Financial Markets Association, www.sifma.org. 209 M06_BERK6318_06_GE_C06.indd 209 26/04/23 6:07 PM  210 Chapter 6 Valuing Bonds 6.1 Bond Cash Flows, Prices, and Yields In this section, we look at how bonds are defined and then study the basic relationship between bond prices and their yield to maturity. Bond Terminology Recall from Chapter 3 that a bond is a security sold by governments and corporations to raise money from investors today in exchange for promised future payments. The terms of the bond are described as part of the bond certificate, which indicates the amounts and dates of all payments to be made. These payments are made until a final repayment date, called the maturity date of the bond. The time remaining until the repayment date is known as the term of the bond. Bonds typically make two types of payments to their holders. The promised interest pay- ments of a bond are called coupons. The bond certificate typically specifies that the coupons will be paid periodically (e.g., semiannually) until the maturity date of the bond. The principal or face value of a bond is the notional amount we use to compute the interest payments. Usually, the face value is repaid at maturity. It is generally denominated in standard increments such as $1000. A bond with a $1000 face value, for example, is often referred to as a “$1000 bond.” The amount of each coupon payment is determined by the coupon rate of the bond. This coupon rate is set by the issuer and stated on the bond certificate. By convention, the coupon rate is expressed as an APR, so the amount of each coupon payment, CPN, is Coupon Payment Coupon Rate × Face Value CPN = (6.1) Number of Coupon Payments per Year For example, a “$1000 bond with a 10% coupon rate and semiannual payments” will pay coupon payments of $1000 × 10% 2 = $50 every six months. Zero-Coupon Bonds The simplest type of bond is a zero-coupon bond, which does not make coupon pay- ments. The only cash payment the investor receives is the face value of the bond on the maturity date. Treasury bills, which are U.S. government bonds with a maturity of up to one year, are zero-coupon bonds. Recall from Chapter 4 that the present value of a future cash flow is less than the cash flow itself. As a result, prior to its maturity date, the price of a zero-coupon bond is less than its face value. That is, zero-coupon bonds trade at a discount (a price lower than the face value), so they are also called pure discount bonds. Suppose that a one-year, risk-free, zero-coupon bond with a $100,000 face value has an initial price of $96,618.36. If you purchased this bond and held it to maturity, you would have the following cash flows: 0 1 2$96,618.36 $100,000 Although the bond pays no “interest” directly, as an investor you are compensated for the time value of your money by purchasing the bond at a discount to its face value. Yield to Maturity. Recall that the IRR of an investment opportunity is the discount rate at which the NPV of the cash flows of the investment opportunity is equal to zero. So, the IRR of an investment in a zero-coupon bond is the rate of return that investors will earn on M06_BERK6318_06_GE_C06.indd 210 26/04/23 6:07 PM  6.1 Bond Cash Flows, Prices, and Yields 211 their money if they buy the bond at its current price and hold it to maturity. The IRR of an investment in a bond is given a special name, the yield to maturity (YTM) or just the yield: The yield to maturity of a bond is the discount rate that sets the present value of the promised bond payments equal to the current market price of the bond. Intuitively, the yield to maturity for a zero-coupon bond is the return you will earn as an investor from holding the bond to maturity and receiving the promised face value payment. Let’s determine the yield to maturity of the one-year zero-coupon bond discussed ear- lier. According to the definition, the yield to maturity of the one-year bond solves the ­following equation: 100,000 96,618.36 = 1 + YTM 1 In this case, 100,000 1 + YTM 1 = = 1.035 96,618.36 That is, the yield to maturity for this bond is 3.5%. Because the bond is risk free, investing in this bond and holding it to maturity is like earning 3.5% interest on your initial invest- ment. Thus, by the Law of One Price, the competitive market risk-free interest rate is 3.5%, meaning all one-year risk-free investments must earn 3.5%. Similarly, the yield to maturity for a zero-coupon bond with n periods to maturity, cur- rent price P, and face value FV solves2 FV P = n (6.2) ( 1 + YTM n ) Rearranging this expression, we get Yield to Maturity of an n -Year Zero -Coupon Bond FV  1/ n YTM n =  −1 (6.3)  P  The yield to maturity ( YTM n ) in Eq. 6.3 is the per-period rate of return for holding the bond from today until maturity on date n. Risk-Free Interest Rates. In earlier chapters, we discussed the competitive market interest rate rn available from today until date n for risk-free cash flows; we used this interest rate as the cost of capital for a risk-free cash flow that occurs on date n. Because a default-free zero-­coupon bond that matures on date n provides a risk-free return over the same period, the Law of One Price guarantees that the risk-free interest rate equals the yield to maturity on such a bond. Risk -Free Interest Rate with Maturity n rn = YTM n (6.4) Consequently, we will often refer to the yield to maturity of the appropriate maturity, zero- coupon risk-free bond as the risk-free interest rate. Some financial professionals also use the term spot interest rates to refer to these default-free, zero-coupon yields. 2 In Chapter 4, we used the notation FV n for the future value on date n of a cash flow. Conveniently, for a zero-coupon bond, the future value is also its face value, so the abbreviation FV continues to apply. M06_BERK6318_06_GE_C06.indd 211 26/04/23 6:07 PM  212 Chapter 6 Valuing Bonds In Chapter 5, we introduced the yield curve, which plots the risk-free interest rate for different maturities. These risk-free interest rates correspond to the yields of risk-free zero-coupon bonds. Thus, the yield curve we introduced in Chapter 5 is also referred to as the zero-coupon yield curve. EXAMPLE 6.1 Yields for Different Maturities Problem Suppose the following zero-coupon bonds are trading at the prices shown below per $100 face value. Determine the corresponding spot interest rates that determine the zero coupon yield curve. Maturity 1 Year 2 Years 3 Years 4 Years Price $96.62 $92.45 $87.63 $83.06 Solution Using Eq. 6.3, we have r1 = YTM 1 = ( 100 96.62 ) − 1 = 3.50% r2 = YTM 2 = ( 100 92.45 ) 1/ 2 − 1 = 4.00% r3 = YTM 3 = ( 100 87.63 ) 1/ 3 − 1 = 4.50% r4 = YTM 4 = ( 100 83.06 ) 1/ 4 − 1 = 4.75% FINANCE IN TIMES OF DISRUPTION Negative Bond Yields On December 9, 2008, in the midst of one of the worst finan- putting it “under the mattress” has a risk of theft!). Thus, we cial crises in history, the unthinkable happened: For the first can view the $25.56 as the price investors were willing to pay time since the Great Depression, U.S. Treasury Bills traded at to have the U.S. Treasury hold their money safely for them at a negative yield. That is, these risk-free pure discount bonds a time when no other investments seemed truly safe. traded at premium. As Bloomberg.com reported: “If you in- This phenomenon repeated itself in Europe starting in mid- vested $1 million in three-month bills at today’s negative dis- 2012. In this case, negative yields emerged due to a concern count rate of 0.01%, for a price of 100.002556, at maturity about both the safety of European banks as well as the stability you would receive the par value for a loss of $25.56.” of the euro as a currency. As investors in Greece or other coun- A negative yield on a Treasury bill implies that investors tries began to worry their economies might depart from the have an arbitrage opportunity: By selling the bill, and hold- euro, they were willing to hold German and Swiss government ing the proceeds in cash (paper currency or banknotes), they bonds even at negative yields as a way to protect themselves would have a risk-free profit of $25.56. Why did investors against the Eurozone unraveling. By mid-2015, some Swiss not rush to take advantage of the arbitrage opportunity and bonds had yields close to −1% and in 2016 Japanese govern- thereby eliminate it? ment bond yields also dropped below zero. Although yields Well, first, the negative yields did not last very long, sug- have increased since then, at the beginning of 2022 the amount gesting that, in fact, investors did rush to take advantage of invested in bonds with negative yields was still nearly $5 trillion. this opportunity. But second, after closer consideration, the At first blush, the persistence of negative yields are chal- opportunity might not have been a sure risk-free arbitrage. lenging to explain. But to take advantage of the arbitrage, in- When selling a Treasury security, the investor must choose vestors must hold physical currency. Obtaining, storing, and where to invest, or at least hold, the proceeds. In normal securing large quantities of cash is costly. For this reason, times investors would be happy to deposit the proceeds with banknotes account for only 6% of U.S. dollars in circulation. a bank, and consider this deposit to be risk free. But these (Indeed, Swiss banks have reportedly refused large cash with- were not normal times—many investors had great concerns drawals by hedge funds attempting to exploit the arbitrage about the financial stability of banks and other financial in- opportunity.) Bonds are also much easier to trade, and use as termediaries. Perhaps investors shied away from this “arbi- collateral, than giant vaults of cash. Presumably the combina- trage” opportunity because they were worried that the cash tion of safety and convenience must be worth the nearly 1% they would receive could not be held safely anywhere (even per year investors in these bonds were willing to sacrifice. M06_BERK6318_06_GE_C06.indd 212 26/04/23 6:07 PM  6.1 Bond Cash Flows, Prices, and Yields 213 Coupon Bonds Like zero-coupon bonds, coupon bonds pay investors their face value at maturity. In ad- dition, these bonds make regular coupon interest payments. Two types of U.S. Treasury coupon securities are currently traded in financial markets: Treasury notes, which have original maturities from one to 10 years, and Treasury bonds, which have original maturi- ties of more than 10 years. EXAMPLE 6.2 The Cash Flows of a Coupon Bond Problem The U.S. Treasury has just issued a five-year, $1000 bond with a 5% coupon rate and semiannual coupons. What cash flows will you receive if you hold this bond until maturity? Solution The face value of this bond is $1000. Because this bond pays coupons semiannually, from Eq. 6.1, you will receive a coupon payment every six months of CPN = $1000 × 5% 2 = $25. Here is the timeline, based on a six-month period: 0 1 2 3 10... $25 $25 $25 $25 1 $1000 Note that the last payment occurs five years (10 six-month periods) from now and is composed of both a coupon payment of $25 and the face value payment of $1000. We can also compute the yield to maturity of a coupon bond. Recall that the yield to maturity for a bond is the IRR of investing in the bond and holding it to maturity; it is the single discount rate that equates the present value of the bond’s remaining cash flows to its current price, shown in the following timeline: 0 1 2 3 N... 2P CPN CPN CPN CPN 1 FV Because the coupon payments represent an annuity, the yield to maturity is the interest rate y that solves the following equation:3 Yield to Maturity of a Coupon Bond 1 1  FV P = CPN × 1− N  + (6.5)  y ( 1 + y )  ( 1 + y )N Unfortunately, unlike in the case of zero-coupon bonds, there is no simple formula to solve for the yield to maturity directly. Instead, we need to use either trial-and-error or the annuity spreadsheet we introduced in Chapter 4 (or Excel’s IRR function). 3 In Eq. 6.5, we have assumed that the first cash coupon will be paid one period from now. If the first coupon is less than one period away, the cash price of the bond can be found by adjusting the price in Eq. 6.5 by multiplying by ( 1 + y ) , where f is the fraction of the coupon interval that has already elapsed. (Also, bond f prices are often quoted in terms of the clean price, which is calculated by deducting from the cash price P an amount, called accrued interest, equal to f × CPN. See the box on “Clean and Dirty” bond prices on page 219.) M06_BERK6318_06_GE_C06.indd 213 26/04/23 6:07 PM  214 Chapter 6 Valuing Bonds When we calculate a bond’s yield to maturity by solving Eq. 6.5, the yield we compute will be a rate per coupon interval. This yield is typically stated as an annual rate by multiplying it by the number of coupons per year, thereby converting it to an APR with the same com- pounding interval as the coupon rate. EXAMPLE 6.3 Computing the Yield to Maturity of a Coupon Bond Problem Consider the five-year, $1000 bond with a 5% coupon rate and semiannual coupons described in Example 6.2. If this bond is currently trading for a price of $957.35, what is the bond’s yield to maturity? Solution Because the bond has 10 remaining coupon payments, we compute its yield y by solving: 1 1  1000 957.35 = 25 × 1− 10  + y ( 1 + y )  ( 1 + y )10 We can solve it by trial-and-error or by using the annuity spreadsheet: NPER RATE PV PMT FV Excel Formula Given 10 ]957.35 25 1,000 Solve for Rate 3.00% 5RATE(10,25,]957.35,1000) Therefore, y = 3%. Because the bond pays coupons semiannually, this yield is for a six-month period. We convert it to an APR by multiplying by the number of coupon payments per year. Thus the bond has a yield to maturity equal to a 6% APR with semiannual compounding. We can also use Eq. 6.5 to compute a bond’s price based on its yield to maturity. We simply discount the cash flows using the yield, as shown in Example 6.4. EXAMPLE 6.4 Computing a Bond Price from Its Yield to Maturity Problem Consider again the five-year, $1000 bond with a 5% coupon rate and semiannual coupons pre- sented in Example 6.3. Suppose you are told that its yield to maturity has increased to 6.30% (expressed as an APR with semiannual compounding). What price is the bond trading for now? Solution Given the yield, we can compute the price using Eq. 6.5. First, note that a 6.30% APR is equiva- lent to a semiannual rate of 3.15%. Therefore, the bond price is 1  1  + 1000 = $944.98 P = 25 × 1−  0.0315  1.031510  1.031510 We can also use the annuity spreadsheet: NPER RATE PV PMT FV Excel Formula Given 10 3.15% 25 1,000 Solve for PV 2944.98 5PV(0.0315,10,25,1000) M06_BERK6318_06_GE_C06.indd 214 26/04/23 6:07 PM  6.2 Dynamic Behavior of Bond Prices 215 Because we can convert any price into a yield, and vice versa, prices and yields are often used interchangeably. For example, the bond in Example 6.4 could be quoted as having a yield of 6.30% or a price of $944.98 per $1000 face value. Indeed, bond traders generally quote bond yields rather than bond prices. One advantage of quoting the yield to maturity rather than the price is that the yield is independent of the face value of the bond. When prices are quoted in the bond market, they are conventionally quoted as a percentage of their face value. Thus, the bond in Example 6.4 would be quoted as having a price of 94.498, which would imply an actual price of $944.98 given the $1000 face value of the bond. CONCEPT CHECK 1. What is the relationship between a bond’s price and its yield to maturity? 2. The risk-free interest rate for a maturity of n-years can be determined from the yield of what type of bond? 6.2 Dynamic Behavior of Bond Prices As we mentioned earlier, zero-coupon bonds trade at a discount—that is, prior to m ­ aturity, their price is less than their face value. Coupon bonds may trade at a discount, at a p ­ remium (a price greater than their face value), or at par (a price equal to their face value). In this sec- tion, we identify when a bond will trade at a discount or premium as well as how the bond’s price will change due to the passage of time and fluctuations in interest rates. Discounts and Premiums If the bond trades at a discount, an investor who buys the bond will earn a return both from receiving the coupons and from receiving a face value that exceeds the price paid for the bond. As a result, if a bond trades at a discount, its yield to maturity will exceed its coupon rate. Given the relationship between bond prices and yields, the reverse is clearly also true: If a coupon bond’s yield to maturity exceeds its coupon rate, the present value of its cash flows at the yield to maturity will be less than its face value, and the bond will trade at a discount. A bond that pays a coupon can also trade at a premium to its face value. In this case, an investor’s return from the coupons is diminished by receiving a face value less than the price paid for the bond. Thus, a bond trades at a premium whenever its yield to maturity is less than its coupon rate. When a bond trades at a price equal to its face value, it is said to trade at par. A bond trades at par when its coupon rate is equal to its yield to maturity. A bond that trades at a discount is also said to trade below par, and a bond that trades at a premium is said to trade above par. Table 6.1 summarizes these properties of coupon bond prices. TABLE 6.1     Bond Prices Immediately After a Coupon Payment When the bond price is We say the bond trades This occurs when greater than the face value “above par” or “at a premium” Coupon Rate > Yield to Maturity equal to the face value “at par” Coupon Rate = Yield to Maturity less than the face value “below par” or “at a discount” Coupon Rate < Yield to Maturity M06_BERK6318_06_GE_C06.indd 215 26/04/23 6:07 PM  216 Chapter 6 Valuing Bonds EXAMPLE 6.5 Determining the Discount or Premium of a Coupon Bond Problem Consider three 30-year bonds with annual coupon payments. One bond has a 10% coupon rate, one has a 5% coupon rate, and one has a 3% coupon rate. If the yield to maturity of each bond is 5%, what is the price of each bond per $100 face value? Which bond trades at a premium, which trades at a discount, and which trades at par? Solution We can compute the price of each bond using Eq. 6.5. Therefore, the bond prices are 1  1  100 P (10% coupon ) = 10 × 1−  + = $176.86 (trades at a premium) 0.05  1.05 30  1.05 30 1  1  100 P (5% coupon ) = 5 × 1−  + = $100.00 (trades at par) 0.05  1.05 30  1.05 30 1  1  100 P (3% coupon ) = 3 ×  1 − 30  + = $69.26 (trades at a discount) 0.05 1.05 1.05 30 Most issuers of coupon bonds choose a coupon rate so that the bonds will initially trade at, or very close to, par (i.e., at face value). For example, the U.S. Treasury sets the coupon rates on its notes and bonds in this way. After the issue date, the market price of a bond generally changes over time for two reasons. First, as time passes, the bond gets closer to its maturity date. Holding fixed the bond’s yield to maturity, the present value of the bond’s remaining cash flows changes as the time to maturity decreases. Second, at any point in time, changes in market interest rates affect the bond’s yield to maturity and its price (the present value of the remaining cash flows). We explore these two effects in the remainder of this section. Time and Bond Prices Let’s consider the effect of time on the price of a bond. Suppose you purchase a 30-year, zero-coupon bond with a yield to maturity of 5%. For a face value of $100, the bond will initially trade for 100 P (30 years to maturity) = = $23.14 1.05 30 Now let’s consider the price of this bond five years later, when it has 25 years remaining until maturity. If the bond’s yield to maturity remains at 5%, the bond price in five years will be 100 P (25 years to maturity ) = = $29.53 1.05 25 Note that the bond price is higher, and hence the discount from its face value is smaller, when there is less time to maturity. The discount shrinks because the yield has not changed, but there is less time until the face value will be received. If you purchased the bond for $23.14 and then sold it after five years for $29.53, the IRR of your investment would be  29.53 1/ 5   − 1 = 5.0%  23.14  M06_BERK6318_06_GE_C06.indd 216 26/04/23 6:07 PM  6.2 Dynamic Behavior of Bond Prices 217 That is, your return is the same as the yield to maturity of the bond. This example illustrates a more general property for bonds: If a bond’s yield to maturity has not changed, then the IRR of an investment in the bond equals its yield to maturity even if you sell the bond early. These results also hold for coupon bonds. The pattern of price changes over time is a bit more complicated for coupon bonds, however, because as time passes, most of the cash flows get closer but some of the cash flows disappear as the coupons get paid. Example 6.6 illustrates these effects. EXAMPLE 6.6 The Effect of Time on the Price of a Coupon Bond Problem Consider a 30-year bond with a 10% coupon rate (annual payments) and a $100 face value. What is the initial price of this bond if it has a 5% yield to maturity? If the yield to maturity is unchanged, what will the price be immediately before and after the first coupon is paid? Solution We computed the price of this bond with 30 years to maturity in Example 6.5: 1  1  100 P = 10 × 1−  + = $176.86 0.05  1.05 30  1.05 30 Now consider the cash flows of this bond in one year, immediately before the first coupon is paid. The bond now has 29 years until it matures, and the timeline is as follows: 0 1 2 29... $10 $10 $10 $10 1 $100 Again, we compute the price by discounting the cash flows by the yield to maturity. Note that there is a cash flow of $10 at date zero, the coupon that is about to be paid. In this case, we can treat the first coupon separately and value the remaining cash flows as in Eq. 6.5: 1  1  100 P ( just before first coupon ) = 10 + 10 × 1 − + = $185.71 0.05  1.05 29  1.05 29 Note that the bond price is higher than it was initially. It will make the same total number of coupon payments, but an investor does not need to wait as long to receive the first one. We could also compute the price by noting that because the yield to maturity remains at 5% for the bond, investors in the bond should earn a return of 5% over the year: $176.86 × 1.05 = $185.71. What happens to the price of the bond just after the first coupon is paid? The timeline is the same as that given earlier, except the new owner of the bond will not receive the coupon at date zero. Thus, just after the coupon is paid, the price of the bond (given the same yield to maturity) will be 1  1  100 P ( just after first coupon ) = 10 × 1 − + = $175.71 0.05  1.05 29  1.05 29 The price of the bond will drop by the amount of the coupon ($10) immediately after the coupon is paid, reflecting the fact that the owner will no longer receive the coupon. In this case, the price is lower than the initial price of the bond. Because there are fewer coupon payments remaining, the premium investors will pay for the bond declines. Still, an investor who buys the bond initially, receives the first coupon, and then sells it earns a 5% return if the bond’s yield does not change: ( 10 + 175.71 ) 176.86 = 1.05. M06_BERK6318_06_GE_C06.indd 217 26/04/23 6:07 PM  218 Chapter 6 Valuing Bonds Figure 6.1 illustrates the effect of time on bond prices, assuming the yield to maturity re- mains constant. Between coupon payments, the prices of all bonds rise at a rate equal to the yield to maturity as the remaining cash flows of the bond become closer. But as each coupon is paid, the price of a bond drops by the amount of the coupon. When the bond is trading at a premium, the price drop when a coupon is paid will be larger than the price increase between coupons, so the bond’s premium will tend to decline as time passes. If the bond is trading at a discount, the price increase between coupons will exceed the drop when a coupon is paid, so the bond’s price will rise and its discount will decline as time passes. Ultimately, the prices of all bonds approach the bonds’ face value when the bonds mature and their last coupon is paid. For each of the bonds illustrated in Figure 6.1, if the yield to maturity remains at 5%, investors will earn a 5% return on their investment. For the zero-coupon bond, this return is earned solely due to the price appreciation of the bond. For the 10% coupon bond, this return comes from the combination of coupon payments and price depreciation over time. Interest Rate Changes and Bond Prices As interest rates in the economy fluctuate, the yields that investors demand to invest in bonds will also change. Let’s evaluate the effect of fluctuations in a bond’s yield to maturity on its price. Consider again a 30-year, zero-coupon bond with a yield to maturity of 5%. For a face value of $100, the bond will initially trade for 100 P (5% yield to maturity ) = = $23.14 1.05 30 But suppose interest rates suddenly rise so that investors now demand a 6% yield to maturity before they will invest in this bond. This change in yield implies that the bond price will fall to 100 P (6% yield to maturity ) = = $17.41 1.06 30 FIGURE 6.1 200 The Effect of Time on 180 Bond Prices The graph illustrates the effects 160 10% Coupon Rate Bond Price (% of face value) of the passage of time on bond 140 prices when the yield remains constant. The price of a zero- 120 5% Coupon Rate coupon bond rises smoothly. 100 The price of a coupon bond also rises between coupon 80 payments, but tumbles on the coupon date, reflecting the 60 3% Coupon Rate amount of the coupon pay- 40 ment. For each coupon bond, the gray line shows the trend 20 Zero Coupon of the bond price just after each coupon is paid. 0 0 5 10 15 20 25 30 Year M06_BERK6318_06_GE_C06.indd 218 26/04/23 6:07 PM  6.2 Dynamic Behavior of Bond Prices 219 Clean and Dirty Prices for Coupon Bonds As Figure 6.1 illustrates, coupon bond prices fluctuate Note that immediately before a coupon payment is made, around the time of each coupon payment in a sawtooth pat- the accrued interest will equal the full amount of the cou- tern: The value of the coupon bond rises as the next coupon pon, whereas immediately after the coupon payment is payment gets closer and then drops after it has been paid. made, the accrued interest will be zero. Thus, accrued inter- This fluctuation occurs even if there is no change in the est will rise and fall in a sawtooth pattern as each coupon bond’s yield to maturity. payment passes: Because bond traders are more concerned about changes in the bond’s price that arise due to changes in the bond’s CPN Accrued Interest yield, rather than these predictable patterns around coupon payments, they often do not quote the price of a bond in terms of its actual cash price, which is also called the dirty price or invoice price of the bond. Instead, bonds are often quoted in terms of a clean price, which is the bond’s cash 0 1 2 3 price less an adjustment for accrued interest, the amount of Time (coupon periods) the next coupon payment that has already accrued: As Figure 6.1 demonstrates, the bonds cash price also Clean price = Cash (dirty) price − Accrued interest has a sawtooth pattern. So if we subtract accrued interest from the bond’s cash price and compute the clean price, Accrued interest = Coupon amount × the sawtooth pattern of the cash price is eliminated. Thus,  Days since last coupon payment  absent changes in the bond’s yield to maturity, its clean price   converges smoothly over time to the bond’s face value, as  Days in current coupon period  shown in the gray lines in Figure 6.1. Relative to the initial price, the bond price changes by ( 17.41 − 23.14 ) 23.14 = −24.8%, a substantial price drop. This example illustrates a general phenomenon. A higher yield to maturity implies a higher discount rate for a bond’s remaining cash flows, reducing their present value and hence the bond’s price. Therefore, as interest rates and bond yields rise, bond prices will fall, and vice versa. The sensitivity of a bond’s price to changes in interest rates depends on the timing of its cash flows. Because it is discounted over a shorter period, the present value of a cash flow that will be received in the near future is less dramatically affected by interest rates than a cash flow in the distant future. Thus, shorter-maturity zero-coupon bonds are less sensitive to changes in interest rates than are longer-term zero-coupon bonds. Similarly, bonds with higher coupon rates—because they pay higher cash flows up front—are less sensitive to in- terest rate changes than otherwise identical bonds with lower coupon rates. The sensitivity of a bond’s price to changes in interest rates is measured by the bond’s duration.4 Bonds with high durations are highly sensitive to interest rate changes. EXAMPLE 6.7 The Interest Rate Sensitivity of Bonds Problem Consider a 15-year zero-coupon bond and a 30-year coupon bond with 10% annual coupons. By what percentage will the price of each bond change if its yield to maturity increases from 5% to 6%? 4 We define duration formally and discuss this concept more thoroughly in Chapter 30. M06_BERK6318_06_GE_C06.indd 219 26/04/23 6:08 PM  220 Chapter 6 Valuing Bonds Solution First, we compute the price of each bond for each yield to maturity: Yield to Maturity 15-Year, Zero-Coupon Bond 30-Year, 10% Annual Coupon Bond 100 1  1  100 5% = $48.10 10 × 1 − + = $176.86 1.0515 0.05  1.05 30  1.05 30 100 1  1  100 6% = $41.73 10 × 1 − + = $155.06 1.0615 0.06  1.06 30  1.06 30 The price of the 15-year zero-coupon bond changes by ( 41.73 − 48.10 ) 48.10 = −13.2% if its yield to maturity increases from 5% to 6%. For the 30-year bond with 10% annual coupons, the price change is ( 155.06 − 176.86 ) 176.86 = −12.3%. Even though the 30-year bond has a longer maturity, because of its high coupon rate, its sensitivity to a change in yield is actually less than that of the 15-year zero coupon bond. In actuality, bond prices are subject to the effects of both the passage of time and changes in interest rates. Bond prices converge to the bond’s face value due to the time effect, but simul- taneously move up and down due to unpredictable changes in bond yields. Figure 6.2 illustrates FIGURE 6.2 6.5 Yield to Maturity (%) 6.0 Yield to Maturity and Bond 5.5 Price Fluctuations over 5.0 Time 4.5 The graphs illustrate changes in price and yield for a 4.0 30-year zero-coupon bond 3.5 over its life. The top graph 3.0 illustrates the changes in 0 5 10 15 20 25 30 the bond’s yield to maturity Year over its life. In the bottom graph, the actual bond price 100 is shown in blue. Because Actual Bond Price 90 Price with 5% Yield the yield to maturity does Price with 4% Yield Bond Price (% of face value) not remain constant over the 80 Price with 6% Yield bond’s life, the bond’s price 70 fluctuates as it converges to 60 the face value over time. Also shown is the price if the yield 50 to maturity remained fixed at 40 4%, 5%, or 6%. 30 20 10 0 0 5 10 15 20 25 30 Year M06_BERK6318_06_GE_C06.indd 220 26/04/23 6:08 PM  6.3 The Yield Curve and Bond Arbitrage 221 this behavior by demonstrating how the price of the 30-year, zero-coupon bond might change over its life. Note that the bond price tends to converge to the face value as the bond approaches the maturity date, but also moves higher when its yield falls and lower when its yield rises. As Figure 6.2 demonstrates, prior to maturity the bond is exposed to interest rate risk. If an investor chooses to sell and the bond’s yield to maturity has decreased, then the investor will receive a high price and earn a high return. If the yield to maturity has increased, the bond price is low at the time of sale and the investor will earn a low return. In the appendix to this chapter, we discuss one way corporations manage this type of risk. CONCEPT CHECK 1. If a bond’s yield to maturity does not change, how does its cash price change between coupon payments? 2. What risk does an investor in a default-free bond face if he or she plans to sell the bond prior to maturity? 3. How does a bond’s coupon rate affect its duration—the bond price’s sensitivity to interest rate changes? 6.3 The Yield Curve and Bond Arbitrage Thus far, we have focused on the relationship between the price of an individual bond and its yield to maturity. In this section, we explore the relationship between the prices and yields of different bonds. Using the Law of One Price, we show that given the spot interest rates, which are the yields of default-free zero-coupon bonds, we can determine the price and yield of any other default-free bond. As a result, the yield curve provides sufficient information to evaluate all such bonds. Replicating a Coupon Bond Because it is possible to replicate the cash flows of a coupon bond using zero-coupon bonds, we can use the Law of One Price to compute the price of a coupon bond from the prices of zero-coupon bonds. For example, we can replicate a three-year, $1000 bond that pays 10% annual coupons using three zero-coupon bonds as follows: 0 1 2 3 Coupon bond: $100 $100 $1100 1-year zero: $100 2-year zero: $100 3-year zero: $1100 Zero-coupon bond portfolio: $100 $100 $1100 We match each coupon payment to a zero-coupon bond with a face value equal to the coupon payment and a term equal to the time remaining to the coupon date. Similarly, we match the final bond payment (final coupon plus return of face value) in three years to a three-year, zero-coupon bond with a corresponding face value of $1100. Because the cou- pon bond cash flows are identical to the cash flows of the portfolio of zero-coupon bonds, the Law of One Price states that the price of the portfolio of zero-coupon bonds must be the same as the price of the coupon bond. M06_BERK6318_06_GE_C06.indd 221 26/04/23 6:08 PM  222 Chapter 6 Valuing Bonds TABLE 6.2     ields and Prices (per $100 Face Value) Y for Zero-Coupon Bonds Maturity 1 year 2 years 3 years 4 years YTM 3.50% 4.00% 4.50% 4.75% Price $96.62 $92.45 $87.63 $83.06 To illustrate, assume that current zero-coupon bond yields and prices are as shown in Table 6.2 (they are the same as in Example 6.1). We can calculate the cost of the zero- coupon bond portfolio that replicates the three-year coupon bond as follows: Zero-Coupon Bond Face Value Required Cost 1 year 100 96.62 2 years 100 92.45 3 years 1100 11 × 87.63 = 963.93 Total Cost: $1153.00 By the Law of One Price, the three-year coupon bond must trade for a price of $1153. If the price of the coupon bond were higher, you could earn an arbitrage profit by selling the coupon bond and buying the zero-coupon bond portfolio. If the price of the coupon bond were lower, you could earn an arbitrage profit by buying the coupon bond and short selling the zero-coupon bonds. Valuing a Coupon Bond Using Zero-Coupon Yields To this point, we have used the zero-coupon bond prices to derive the price of the coupon bond. Alternatively, we can use the zero-coupon bond yields. Recall that the yield to matu- rity of a zero-coupon bond is the competitive market interest rate for a risk-free investment with a term equal to the term of the zero-coupon bond. Therefore, the price of a coupon bond must equal the present value of its coupon payments and face value discounted at the competitive market interest rates (see Eq. 5.7 in Chapter 5): Price of a Coupon Bond P = PV (Bond Cash Flows) CPN CPN CPN + FV (6.6) = + ++ n 1 + YTM 1 ( 1 + YTM 2 ) 2 ( 1 + YTM n ) where CPN is the bond coupon payment, YTM n is the yield to maturity of a zero-coupon bond that matures at the same time as the nth coupon payment, and FV is the face value of the bond. For the three-year, $1000 bond with 10% annual coupons considered earlier, we can use Eq. 6.6 to calculate its price using the zero-coupon yields in Table 6.2: 100 100 100 + 1000 P = + + = $1153 1.035 1.04 2 1.045 3 This price is identical to the price we computed earlier by replicating the bond. Thus, we can determine the no-arbitrage price of a coupon bond by discounting its cash flows using the zero-coupon yields. In other words, the information in the zero-coupon yield curve is sufficient to price all other risk-free bonds. M06_BERK6318_06_GE_C06.indd 222 26/04/23 6:08 PM  6.3 The Yield Curve and Bond Arbitrage 223 Coupon Bond Yields Given the yields for zero-coupon bonds, we can use Eq. 6.6 to price a coupon bond. In Section 6.1, we saw how to compute the yield to maturity of a coupon bond from its price. Combining these results, we can determine the relationship between the yields of zero- coupon bonds and coupon-paying bonds. Consider again the three-year, $1000 bond with 10% annual coupons. Given the zero- coupon yields in Table 6.2, we calculate a price for this bond of $1153. From Eq. 6.5, the yield to maturity of this bond is the rate y that satisfies 100 100 100 + 1000 P = 1153 = + + (1 + y ) (1 + y ) 2 ( 1 + y )3 We can solve for the yield by using the annuity spreadsheet: NPER RATE PV PMT FV Excel Formula Given 3 21,153 100 1,000 Solve for Rate 4.44% 5RATE(3,100,21153,1000) Therefore, the yield to maturity of the bond is 4.44%. We can check this result directly as follows: 100 100 100 + 1000 P = + 2 + = $1153 1.0444 1.0444 1.0444 3 Because the coupon bond provides cash flows at different points in time, the yield to maturity of a coupon bond is a weighted average of the yields of the zero-coupon bonds of equal and shorter maturities. The weights depend (in a complex way) on the magnitude of the cash flows each period. In this example, the zero-coupon bonds’ yields were 3.5%, 4.0%, and 4.5%. For this coupon bond, most of the value in the present value calculation comes from the present value of the third cash flow because it includes the principal, so the yield is closest to the three-year, zero-coupon yield of 4.5%. EXAMPLE 6.8 Yields on Bonds with the Same Maturity Problem Given the following zero-coupon yields, compare the yield to maturity for a three-year, zero- coupon bond; a three-year coupon bond with 4% annual coupons; and a three-year coupon bond with 10% annual coupons. All of these bonds are default free. Maturity 1 year 2 years 3 years 4 years Zero-coupon YTM 3.50% 4.00% 4.50% 4.75% Solution From the information provided, the yield to maturity of the three-year, zero-coupon bond is 4.50%. Also, because the yields match those in Table 6.2, we already calculated the yield to matu- rity for the 10% coupon bond as 4.44%. To compute the yield for the 4% coupon bond, we first need to calculate its price. Using Eq. 6.6, we have 40 40 40 + 1000 P = + 2 + = $986.98 1.035 1.04 1.045 3 M06_BERK6318_06_GE_C06.indd 223 26/04/23 6:08 PM  224 Chapter 6 Valuing Bonds The price of the bond with a 4% coupon is $986.98. From Eq. 6.5, its yield to maturity solves the following equation: 40 40 40 + 1000 $986.98 = + + ( 1 + y ) ( 1 + y )2 ( 1 + y )3 We can calculate the yield to maturity using the annuity spreadsheet: NPER RATE PV PMT FV Excel Formula Given 3 2986.98 40 1,000 Solve for Rate 4.47% 5RATE(3,40,2986.98,1000) To summarize, for the three-year bonds considered Coupon rate 0% 4% 10% YTM 4.50% 4.47% 4.44% Example 6.8 shows that coupon bonds with the same maturity can have different yields depending on their coupon rates. As the coupon increases, earlier cash flows become rela- tively more important than later cash flows in the calculation of the present value. If the yield curve is upward sloping (as it is for the yields in Example 6.8), the resulting yield to maturity decreases with the coupon rate of the bond. Alternatively, when the zero-coupon yield curve is downward sloping, the yield to maturity will increase with the coupon rate. When the yield curve is flat, all zero-coupon and coupon-paying bonds will have the same yield, independent of their maturities and coupon rates. Treasury Yield Curves As we have shown in this section, we can use the zero-coupon yield curve to determine the price and yield to maturity of other risk-free bonds. The plot of the yields of coupon bonds of different maturities is called the coupon-paying yield curve. When U.S. bond traders refer to “the yield curve,” they are often referring to the coupon-paying Treasury yield curve. As we showed in Example 6.8, two coupon-paying bonds with the same ma- turity may have different yields. By convention, practitioners always plot the yield of the most recently issued bonds, termed the on-the-run bonds. Using similar methods to those employed in this section, we can apply the Law of One Price to determine the zero-coupon bond yields using the coupon-paying yield curve (see Problem 25). Thus, either type of yield curve provides enough information to value all other risk-free bonds. CONCEPT CHECK 1. How do you calculate the price of a coupon bond from the prices of zero-coupon bonds? 2. How do you calculate the price of a coupon bond from the yields of zero-coupon bonds? 3. Explain why two coupon bonds with the same maturity may each have a different yield to maturity. 6.4 Corporate Bonds So far in this chapter, we have focused on default-free bonds such as U.S. Treasury securi- ties, for which the cash flows are known with certainty. For other bonds such as corporate bonds (bonds issued by corporations), the issuer may default—that is, it might not pay back M06_BERK6318_06_GE_C06.indd 224 26/04/23 6:08 PM  6.4 Corporate Bonds 225 the full amount promised in the bond prospectus. This risk of default, which is known as the credit risk of the bond, means that the bond’s cash flows are not known with certainty. Corporate Bond Yields How does credit risk affect bond prices and yields? Because the cash flows promised by the bond are the most that bondholders can hope to receive, the cash flows that a purchaser of a bond with credit risk expects to receive may be less than that amount. As a result, investors pay less for bonds with credit risk than they would for an otherwise identical default-free bond. Because the yield to maturity for a bond is calculated using the promised cash flows, the yield of bonds with credit risk will be higher than that of otherwise identical default-free bonds. Let’s il- lustrate the effect of credit risk on bond yields and investor returns by comparing different cases. No Default. Suppose that the one-year, zero-coupon Treasury bill has a yield to maturity of 4%. What are the price and yield of a one-year, $1000, zero-coupon bond issued by Avant Corporation? First, suppose that all investors agree that there is no possibility that Avant will default within the next year. In that case, investors will receive $1000 in one year for certain, as promised by the bond. Because this bond is risk free, the Law of One Price guarantees that it must have the same yield as the one-year, zero-coupon Treasury bill. The price of the bond will therefore be 1000 1000 P = = = $961.54 1 + YTM 1 1.04 Certain Default. Now suppose that investors believe that Avant will default with certainty at the end of one year and will be able to pay only 90% of its outstanding obligations. Then, even though the bond promises $1000 at year-end, bondholders know they will receive only $900. Investors can predict this shortfall perfectly, so the $900 payment is risk free, and the bond is still a one-year risk-free investment. Therefore, we compute the price of the bond by discounting this cash flow using the risk-free interest rate as the cost of capital: 900 900 P = = = $865.38 1 + YTM 1 1.04 The prospect of default lowers the cash flow investors expect to receive and hence the price they are willing to pay. Are Treasuries Really Default-Free Securities? Most investors treat U.S. Treasury securities as risk free, mean- President Franklin Roosevelt suspended bondholders’ right ing that they believe there is no chance of default (a conven- to be paid in gold rather than currency. tion we follow in this book). But are Treasuries really risk free? A new risk emerged in mid-2011 when a series of large The answer depends on what you mean by “risk free.” budget deficits brought the United States up against the No one can be certain that the U.S. government will debt ceiling, a constraint imposed by Congress limiting the never default on its bonds—but most people believe the overall amount of debt the government can incur. An act of probability of such an event is very small. More importantly, Congress was required by August 2011 for the Treasury to the default probability is smaller than for any other bond. meet its obligations and avoid a default. In response to the So saying that the yield on a U.S. Treasury security is risk political uncertainty about whether Congress would raise the free really means that the Treasury security is the lowest-risk ceiling in time, Standard & Poor’s downgraded its rating of investment denominated in U.S. dollars in the world. U.S. Government bonds. Congress ultimately raised the debt That said, there have been occasions in the past where ceiling and no default occurred. Given persistent budget Treasury holders did not receive exactly what they were deficits, similar debt ceiling debates recurred in 2013, 2015, promised: In 1790, Treasury Secretary Alexander Hamilton 2017, 2019, and 2021. These incidents serve as a reminder lowered the interest rate on outstanding debt and in 1933 that perhaps no investment is truly “risk free.” M06_BERK6318_06_GE_C06.indd 225 26/04/23 6:08 PM  226 Chapter 6 Valuing Bonds Given the bond’s price, we can compute the bond’s yield to maturity. When computing this yield, we use the promised rather than the actual cash flows. Thus, FV 1000 YTM = −1= − 1 = 15.56% P 865.38 The 15.56% yield to maturity of Avant’s bond is much higher than the yield to maturity of the default-free Treasury bill. But this result does not mean that investors who buy the bond will earn a 15.56% return. Because Avant will default, the expected return of the bond equals its 4% cost of capital: 900 = 1.04 865.38 Note that the yield to maturity of a defaultable bond exceeds the expected return of investing in the bond. Because we calculate the yield to maturity using the promised cash flows rather than the expected cash flows, the yield will always be higher than the expected return of invest- ing in the bond. Risk of Default. The two Avant examples were extreme cases, of course. In the first case, we assumed the probability of default was zero; in the second case, we assumed Avant would definitely default. In reality, the chance that Avant will default lies somewhere in between these two extremes (and for most firms, is probably much closer to zero). To illustrate, again consider the one-year, $1000, zero-coupon bond issued by Avant. This time, assume that the bond payoffs are uncertain. In particular, there is a 50% chance that the bond will repay its face value in full and a 50% chance that the bond will default and you will receive $900. Thus, on average, you will receive $950. To determine the price of this bond, we must discount this expected cash flow using a cost of capital equal to the expected return of other securities with equivalent risk. If, like most firms, Avant is more likely to default if the economy is weak than if the economy is strong, then—as discussed in Chapter 3—investors will demand a risk premium to invest in this bond. That is, Avant’s debt cost of capital, which is the expected return Avant’s debt holders will require to compensate them for the risk of the bond’s cash flows, will be higher than the 4% risk-free interest rate.5 Let’s suppose investors demand a risk premium of 1.1% for this bond, so that the ­appropriate cost of capital is 5.1%. Then the present value of the bond’s cash flow is 950 P = = $903.90 1.051 Consequently, in this case the bond’s yield to maturity is 10.63%: FV 1000 YTM = −1= − 1 = 10.63% P 903.90 Of course, the 10.63% promised yield is the most investors will receive. If Avant defaults, they will receive only $900, for a return of 900 903.90 − 1 = −0.43%. The average return is 0.50 ( 10.63% ) + 0.50 ( −0.43% ) = 5.1%, the bond’s cost of capital. Table 6.3 summarizes the prices, expected return, and yield to maturity of the Avant bond under the various default assumptions. Note that the bond’s price decreases, and its yield to maturity increases, with a greater likelihood of default. Conversely, the bond’s expected return, which is equal to the firm’s debt cost of capital, is less than the yield to maturity if there 5 We develop methods for estimating risk premia for risky bonds in the Chapter 3 Appendix, Chapter 12 (Section 4), and Chapter 20 (Section 6). M06_BERK6318_06_GE_C06.indd 226 26/04/23 6:08 PM  6.4 Corporate Bonds 227 TABLE 6.3    Bond Price, Yield, and Return with Different Likelihoods of Default Avant Bond (1-year, zero-coupon) Bond Price Yield to Maturity Expected Return Default Free $961.54 4.00% 4% 50% Chance of Default $903.90 10.63% 5.1% Certain Default $865.38 15.56% 4% is a risk of default. Moreover, a higher yield to maturity does not necessarily imply that a bond’s expected return is higher. Bond Ratings It would be both difficult and inefficient for every investor to privately investigate the default risk of every bond. Consequently, several companies rate the creditworthiness of bonds and make this information available to investors. The two best-known bond-rating companies are Standard & Poor’s and Moody’s. Table 6.4 summarizes the rating classes each company uses. Bonds with the highest rating are judged to be least likely to default. By consulting TABLE 6.4       Bond Ratings Rating* Description (Moody’s) Investment Grade Debt Aaa/AAA Judged to be of the best quality. They carry the smallest degree of investment risk and are generally referred to as “gilt edged.” Interest payments are protected by a large or an exceptionally stable margin and principal is secure. While the various protective elements are likely to change, such changes as can be visualized are most unlikely to impair the fundamentally strong position of such issues. Aa/AA Judged to be of high quality by all standards. Together with the Aaa group, they constitute what are generally known as high-grade bonds. They are rated lower than the best bonds because margins of protection may not be as large as in Aaa securities or fluctuation of protective elements may be of greater amplitude or there may be other elements present that make the long-term risk appear somewhat larger than the Aaa securities. A/A Possess many favorable investment attributes and are considered as upper-medium-grade obligations. Factors giving security to principal and interest are considered adequate, but elements may be present that suggest a susceptibility to impairment some time in the future. Baa/BBB Are considered as medium-grade obligations (i.e., they are neither highly protected nor poorly secured). Interest payments and principal security appear adequate for the present but certain protective elements may be lacking or may be characteristically unreliable over any great length of time. Such bonds lack outstanding investment characteristics and, in fact, have speculative characteristics as well. Speculative Bonds Ba/BB Judged to have speculative elements; their future cannot be considered as well assured. Often the protection of interest and principal payments may be very moderate, and thereby not well safeguarded during both good and bad times over the future. Uncertainty of position characterizes bonds in this class. B/B Generally lack characteristics of the desirable investment. Assurance of interest and principal payments of maintenance of other terms of the contr