Chapter 8: Bond Valuation and the Structure of Interest Rates
Instruction: After reading the chapter below, write a brief summary of the chapter and how the information in the chapter could be applied to meeting your life or career goals. If you cannot relate the chapter to your personal goals, write about the most important thing you learned from the chapter and how it could impact your life.
Chapter 8: Bond Valuation and the Structure of Interest Rates
8.1 CORPORATE BONDS
In this section we discuss the market for corporate bonds and some of the types of bonds that firms issue.
Market for Corporate Bonds
The market for corporate bonds is enormous. At the beginning of October 2013, for example, the value of corporate bonds outstanding in the United States was $11.1 trillion, almost 66 percent as large as the total U.S. gross domestic product in 2013 of $16.8 trillion. By comparison, the market for corporate equity was the largest part of the U.S. capital market with a value of $25.3 trillion and U.S. government long-term Treasury securities totaled $10.4 trillion. State and local government debt markets were much smaller at $3.6 trillion. The most important investors in corporate bonds are big institutional investors such as life insurance companies, pension funds, and mutual funds. Because the primary investors are so big, trades in this market tend to involve very large blocks of securities.
Most secondary market transactions for corporate bonds take place through dealers in the over-the-counter (OTC) market. An interesting characteristic of the corporate bond market is that there are a large number of different bond issues that trade in the market. The reason is that while a corporation typically has a single issue of common stock outstanding, it may have a dozen or more different notes and bonds outstanding. Therefore, despite the large overall trading volume of corporate bonds, the bonds from any particular issue will not necessarily trade on a given day. As a result, the market for corporate bonds is thin compared to the market for corporate stocks or money market securities. On Wall Street, the term thin means that secondary market trades of individual securities are relatively infrequent. Thus, corporate bonds are less marketable than the securities that have higher daily trading volumes.
Prices in the corporate bond market also tend to be more volatile than prices of securities sold in markets with greater trading volumes. This is because a few large trades can have a larger impact on a security’s price than numerous trades of various sizes. The result of this is that the market for corporate bonds is not as efficient (new information does not get incorporated into corporate bond prices as efficiently) as those for highly marketable stocks or money market instruments, such as U.S. Treasury securities.
Bond Price Information
The corporate bond market also has little transparency because it is almost entirely an OTC market. A financial market is transparent if it is easy to view prices and trading volume. An example of a transparent market is the New York Stock Exchange (NYSE), where price information on every trade and trade size are available for every transaction during the day. In contrast, corporate bond market transactions are widely dispersed, with dealers located all over the country, and there are an enormous number of different securities. Furthermore, many corporate bond transactions are negotiated directly between the buyer and the seller and there is limited centralized reporting of these sales. As a result, information on individual corporate bond transactions is not widely published as the transactions occur. This is another reason that the corporate bond market is not as efficient as the stock or money markets.
Types of Corporate Bonds
Corporate bonds are long-term IOUs that represent claims against a firm’s assets. Unlike stockholders’ returns, most bondholders’ returns are fixed; they receive only the interest payments that are promised plus the repayment of the loan amount when the bond matures. Debt instruments where the interest paid to investors is fixed for the life of the contract are called fixed-income securities. We examine three types of fixed-income securities in this section.
The most common bonds issued by corporations have coupon payments that are fixed for the life of the bond, and at maturity the entire original principal is paid and the bonds are retired. These bonds are known as vanilla bonds if they have no special characteristics, such as a conversion feature (discussed below).
The time line above shows the cash payments for a three-year vanilla bond with a $1,000 face value and an 8 percent coupon (interest) rate. is the price (value) of the bond, which will be discussed in the next section. The $80 cash payments made each year are called the coupon payments. Coupon payments are the interest payments made to bondholders. These payments are usually made annually or semiannually, and the payment amount (or rate) remains fixed for the life of the bond contract, which for our example is three years. The face value, or par value, for most corporate bonds is $1,000, and it is the principal amount owed to the bondholder at maturity. Finally, the bond’s coupon rate is the annual coupon payment (C) divided by the bond’s face value (F). Our vanilla bond pays $80 of coupon interest annually and has a face value of $1,000. The coupon rate is thus:
Zero Coupon Bonds
At times, corporations issue bonds that have no coupon payments but promise a single payment at maturity. The interest paid to a bondholder is the difference between the price paid for the bond and the face value received at maturity. These bonds are sold at a price below the amount that the investor receives at maturity because all of the interest is paid when the bonds are retired at maturity rather than in semiannual or yearly coupon payments. The face value of a zero coupon bond is different from that of a vanilla bond in that it includes both the interest and principal.
The most frequent and regular issuer of zero coupon securities is the U.S. Department of Treasury, and perhaps the best-known zero coupon bond is a United States Saving Bond. Corporations also issue zero coupon bonds from time to time. Firms that are expanding operations but have little cash available to make interest payments are especially likely to use zero coupon bonds for funding. In the 1990s, the bond market was flooded with zero coupon bonds issued by telecommunications firms. These firms were spending huge amounts to build fiber-optic networks, which generated few cash inflows until they were completed.
Corporate convertible bonds can be converted into shares of common stock at some predetermined ratio at the discretion of the bondholder. For example, a $1,000 face-value bond may be convertible into 100 shares of common stock. A conversion feature is valuable to bondholders because it allows them to share in the good fortunes of the firm if the firm’s stock price rises above a certain level. Specifically, the bondholders profit if they exchange their bonds for the company’s stock when the market value of the stock they receive exceeds the market value of the bonds.
Typically, the conversion ratio is set so that the firm’s stock price must appreciate at least 15 to 20 percent before it is profitable to convert the bonds into stock. As you would expect from our discussion, since a conversion feature is valuable to bondholders, firms that issue convertible bonds can do so at a lower interest rate. This reduces the amount of cash that the firms must use to make interest payments.
8.2 BOND VALUATION
We turn now to the topic of bond valuation—how bonds are priced. Throughout this book, we have stressed that the value, or price, of any asset is the present value of its future cash flows. The steps necessary to value an asset are as follows:
• 1.Estimate the expected future cash flows.
• 2.Determine the required rate of return, or discount rate. This rate depends on the riskiness of the future cash flows.
• 3.Compute the present value of the future cash flows. This present value is what the asset is worth at a particular point in time.
For bonds, the valuation procedure is relatively easy. The cash flows (coupon and principal payments) are contractual obligations of the firm and are known by market participants, since they are stated in the bond contract. Thus, market participants know the magnitude and timing of the expected cash flows as promised by the borrower (the bond issuer). The required rate of return, or discount rate, for a bond is the market interest rate, called the bond’s yield to maturity (or more commonly, simply its yield). This rate is determined from the market prices of bonds that have features similar to those of the bond being valued; by similar, we mean bonds that have the same term to maturity, have the same bond rating (default risk class), and are similar in other ways.
Notice that the required rate of return is the opportunity cost for the investors who purchase the bond. An opportunity cost is the highest alternative return that is given up when an investment is made. For example, if bonds identical to the bond being valued—having the same risk—yield 9 percent annually, the threshold yield or required return on the bond being valued is 9 percent. Why? An investor would not buy a bond with an 8 percent yield when an identical bond yielding 9 percent is available.
Given the above information, we can compute the current value, or price, of a bond by calculating the present value of the bond’s expected cash flows:
Next, we examine this calculation in detail.
The Bond Valuation Formula
To begin, refer to Exhibit 8.1, which shows the cash flows for a three-year corporate bond (a bond with three years to maturity) with an 8 percent coupon rate and a $1,000 face value. If the market rate of interest on similar bonds is 10 percent and interest payments are made annually, what is the market price of the bond? In other words, how much should you be willing to pay for the promised cash flow stream?
EXHIBIT 8.1 Cash Flows for a Three-Year BondThe exhibit shows a time line for a three-year bond that pays an 8 percent coupon rate and has a face value of $1,000. How much should we pay for such a bond if the market rate of interest is 10 percent? To solve this problem, we discount the promised cash flows to the present and then add them up.
There are a number of ways to solve this problem. Probably the simplest is to write the bond valuation formula in terms of the individual cash flows. Thus, the price of the bond is the sum of the present value calculations for the coupon payments (C) and the principal amount (F), discounted at the required rate . That calculation is:
Notice that you could have simplified the calculation by combining the final coupon payment and the principal payment , since both cash flows occur at time .
To develop the general bond pricing formula, we can write the equations for the price of a four-year, five-year, and six-year maturity bond, as follows:
If we continue the process for periods to maturity, we arrive at the general equation for the price of the bond:
In practice, the bond pricing equation is usually written with divided by rather than with multiplied by . Thus, the general equation for the price of a bond can be written as follows:
Note that there are five variables in the bond pricing equation. If we know any four of them, we can solve for the fifth.
Calculator Tip: Bond Valuation Problems
We can easily calculate bond prices using a financial calculator or a spreadsheet program. We solve for bond prices and bond yields in exactly the same way we solved for the present value (bond price) and discount rate (bond yield) in Chapter 6. We solve our example problem on a financial calculator as follows:
Several points are worth noting:
• 1.Always draw a time line for the cash flows. This simple step will significantly reduce mistakes.
• 2.The PMT key enters the dollar amount of an ordinary annuity for periods. In our example, keying in 3 with the N key and $80 with the PMT key enters an $80 annuity with the final payment made at the end of year 3.
• 3.Be sure that you enter the coupon and the principal payments separately. Do not enter the final coupon payment ($80) and principal amount ($1,000) as a single entry of $1,080 on the FV key. The reason is that the PMT key is the annuity key, and when you enter , the $80 is entered in the calculator as a three-year ordinary annuity with a final payment of $80 in period . If you then enter $1,080 on the FV key, you will have an extra $80 in the final period . For the example problem, we correctly entered the $80 coupon payments with the PMT key and the $1,000 principal payment with the FV key.
• 4.Finally, as we have mentioned in earlier chapters, you must be consistent throughout a problem in how you enter the signs (positive or negative) for cash inflows and cash outflows. For example, if you are a bond investor and decide to enter all cash inflows with a positive sign, then you must enter all coupon and principal payments with a positive sign. The price you paid for the bond, which is a cash outflow, must be entered as a negative number. This is the convention we will follow.
Par, Premium, and Discount Bonds
One of the mathematical properties of the bond pricing equation is that whenever a bond’s coupon rate is equal to the market rate of interest on similar bonds (the bond’s yield), the bond will sell at par value. We call such bonds par-value bonds. For example, suppose that you own a three-year bond with a face value of $1,000 and an annual coupon rate of 5 percent, when the yield or market rate of interest on similar bonds is 5 percent. The price of your bond, based on Equation 8.1, is:
As predicted, the bond’s price equals its par value.
Now assume that the market rate of interest rises overnight to 8 percent. What happens to the price of the bond? Will the bond’s price be below, above, or at par value?
When is equal to 8 percent, the price of the bond declines to $922.69. The bond sells at a price below par value; such bonds are called discount bonds.
Whenever a bond’s coupon rate is lower than the market rate of interest on similar bonds, the bond will sell at a discount. This is true because of the fixed nature of a bond’s coupon payments. Let’s return to our 5 percent coupon bond. If the market rate of interest is 8 percent and our bond pays only 5 percent, no economically rational person would buy the bond at its par value. This would be like choosing a bond with a 5 percent yield over one with an 8 percent yield. We cannot change the coupon rate to 8 percent because it is fixed for the life of the bond. That is why bonds are referred to as fixed-income securities. The only way to increase our bond’s yield to 8 percent is to reduce the price of the bond to $922.69. At this price, the bond’s yield will be precisely 8 percent, which is the current market rate for similar bonds. Through the price reduction of , the seller provides the new owner with additional “interest” in the form of a capital gain.
What would happen to the price of the bond if interest rates on similar bonds declined to 2 percent and the coupon rate remained at 5 percent? The price would rise to $1,086.52. At this price, the bond’s yield would be precisely 2 percent, which is the current market yield. The premium adjusts the bond’s yield to 2 percent, which is the current market yield for similar bonds. Bonds that sell at prices above par are called premium bonds. Whenever a bond’s coupon rate is higher than the market rate of interest, the bond will sell at a premium.
Our discussion of bond pricing can be summarized as follows, where is the market rate of interest:
This negative relation between changes in the level of interest rates and changes in the price of a bond (or any fixed-income security) is one of the most fundamental relations in corporate finance. The relation exists because the coupon payments on most bonds are fixed and the only way bonds can pay the current market rate of interest to investors is through an adjustment in the price of the bond. This is exactly what happened to the Greek government bonds discussed at the beginning of this chapter. As the risk of those bonds increased, their prices declined so that investors received a market rate of interest that reflected the bond’s higher risk.
In Europe, bonds generally pay coupon interest on an annual basis. In contrast, in the United States, most bonds pay coupon interest semiannually—that is, twice a year. Thus, if a bond has an 8 percent coupon rate (paid semiannually), the bondholder will in one year receive two coupon payments of $40 each, totaling . We can modify Equation 8.1 as follows to adjust for coupon payments made more than once a year:
where is the annual coupon payment, is the number of times coupon payments are made each year, is the number of years to maturity, and is the annual market interest rate. In the case of a bond with semiannual coupon payments, equals 2.
Whether we are computing bond prices annually, semiannually, quarterly, or for some other period, the computation is the same. We need only be sure that the bond’s yield, coupon payment, and maturity are adjusted to be consistent with the bond’s stated compounding period. Once that information is converted to the correct compounding period, it can simply be entered into Equation 8.1. Thus, there is really no need to memorize or use Equation 8.2 unless you find it helpful. Let’s work an example to demonstrate.
Earlier we determined that a three-year, 5 percent coupon bond will sell for $922.69 when the market rate of interest is 8 percent. Our computation assumed that coupon payments were made annually. What is the price of the bond if the coupon payments are made semiannually? The time line for the semiannual bond situation follows:
We convert the bond data to semiannual compounding as follows: (1) the market yield is 4 percent semiannually , (2) the coupon payment is $25 semiannually , and (3) the total number of coupon payments is . Plugging the data into Equation 8.1, we find that the bond price is:
Notice that the price of the bond is slightly less with semiannual compounding than with annual compounding . The slight difference in price reflects the change in the timing of the cash flows and the interest rate adjustment.1
Zero Coupon Bonds
As previously mentioned, zero coupon bonds have no coupon payments but promise a single payment at maturity. The price (or yield) of a zero coupon bond is simply a special case of Equation 8.2in which all the coupon payments are equal to zero.
Hence, the pricing equation for a zero coupon bond is:
Notice that if a zero coupon bond compounds annually, and Equation 8.3 becomes:
Now let’s work an example. What is the price of a zero coupon bond with a $1,000 face value, 10-year maturity, and semiannual compounding when the market interest rate is 12 percent? Since the bond compounds interest semiannually, the number of compounding periods is . The semiannual interest is . The time line for the cash flows is as follows:
Plugging the data into Equation 8.3, we find that the price of the bond is:
Notice that the zero coupon bond is selling at a very large (deep) discount. This should come as no surprise, since the bond has no coupon payment and all the dollars paid to investors are paid at maturity. Why are zero coupon bonds so heavily discounted compared with similar bonds that do have coupon payments? From Chapter 5, we know that because of the time value of money, dollars to be received in the future have less value than current dollars. Thus, zero coupon bonds, for which all the cash payments are made at maturity, must sell for less than similar bonds that make coupon payments before maturity.
8.3 BOND YIELDS
We frequently know a bond’s price from an offer to sell it, but not its yield to maturity. In this section, we discuss how to compute the yield to maturity and some other important bond yields.
Yield to Maturity
The yield to maturity of a bond is the discount rate that makes the present value of the coupon and principal payments equal to the price of the bond. The yield to maturity can be viewed as apromised yield because it is the annual yield that the investor earns if the bond is held to maturity and all the coupon and principal payments are made as promised. A bond’s yield to maturity changes daily as interest rates increase or decrease, but its calculation is always based on the issuer’s promise to make interest and principal payments as stipulated in the bond contract.
Let’s work through an example to see how a bond’s yield to maturity is calculated. Suppose you decide to buy a three-year bond with a 6 percent coupon rate for $960.99. For simplicity, we will assume that the coupon payments are made annually. The time line for the cash flows is as follows:
To compute the yield to maturity, we apply Equation 8.1 and solve for . We can set up the problem using Equation 8.1 as follows:
As we discussed in Chapter 6, we cannot solve for mathematically; we must find it by trial and error. We know that the bond is selling for a discount because its price is below par, so the yield must be higher than the 6 percent coupon rate. Let’s try 7 percent.
The computed price of $973.76 is still greater than our market price of $960.99; thus, we need to use a slightly larger discount rate. Let’s try 7.7 percent
Our computed value of $955.95 is now less than the market price of $960.99, so we need a lower discount rate. We’ll try 7.5 percent.
At a discount rate of 7.5 percent, the price of the bond is exactly equal to the market price, and thus, the bond’s yield to maturity is 7.5 percent.
We can, of course, also compute the bond’s yield to maturity using a financial calculator. Computing the yield in this way is no different from computing the price, except that the unknown is the bond’s yield. As with calculating the price of a bond, the major source of computational errors is failing to make sure that all the bond data are consistent with the bond’s compounding period. The three variables that may require adjustment are (1) the coupon payment, (2) the yield, and (3) the bond maturity.
For the three-year corporate bond discussed earlier, the bond data are already in a form that is consistent with the annual compounding period, so we enter it into the calculator and solve for , which is the yield to maturity:
The bond’s yield to maturity is 7.5 percent, which is identical to the answer from our hand calculation.
Effective Annual Yield
Up to now, when pricing a bond with a semiannual compounding period, we assumed the bond’s annual yield to be twice the semiannual yield. This is the convention used on Wall Street and by other practitioners who deal in bonds. However, notice that bond yields quoted in this manner are just like the APRs discussed in Chapter 6. For example, in Section 6.4 we showed that the APR for a bank credit card with a 1 percent monthly interest rate is simply the monthly interest rate multiplied by the number of months in a year, or 12 percent. As you recall, interest rates (or yields) annualized in this manner do not take compounding into account. Hence, the values calculated are not the true cost of funds, and their use can lead to decisions that are economically incorrect.
As a result, annualized yields calculated by multiplying a yield per period by the number of compounding periods is only acceptable for decision-making purposes when comparing bonds that have the same compounding frequencies. For example, an investor must be careful when comparing yields of European and U.S. bonds, since interest on a European bond is compounded annually while interest on a U.S. bond compounds twice a year.
The correct way to annualize an interest rate to make an economic decision is to compute the effective annual interest rate (EAR). On Wall Street, the EAR is called the effective annual yield (EAY); thus, EAR = EAY. Drawing on Equation 6.7 (see Chapter 6), we find that the correct way to annualize the yield on a bond is as follows:
Let’s work through an example to see how the EAY differs from the yield to maturity. Suppose an investor buys a 30-year bond with a $1,000 face value for $800. The bond’s coupon rate is 8 percent, and interest payments are made semiannually. What is the bond’s yield to maturity, and what is its EAY? To find out, we first need to convert the bond’s annual data into semiannual data: (1) the 30-year bond has 60 compounding periods and (2) the bond’s semiannual coupon payment is . The time line for this bond is:
We can set up the problem using Equation 8.1 (or 8.2) as:
However, solving an equation with so many terms can be time consuming. Therefore, we will solve for the yield to maturity using the yield function in a financial calculator as follows:
The answer is 5.07 percent. We then multiply the semiannual yield by 2 to convert it to an annual yield: . This is the bond’s yield to maturity.
If, instead of multiplying 5.07 percent by 2 we calculate the EAY for the semiannual yield of 5.07 percent, we will get:
The EAY of 10.40 percent is greater than the annual yield to maturity of 10.14 percent because the EAY takes into account the effects of compounding—earning interest on interest. As mentioned earlier, calculating the EAY is the economically correct way to annualize a bond’s yield because it takes compounding into account.
The yield to maturity (or promised yield) tells the investor the return on a bond if the bond is held to maturity and all the coupon and principal payments are made as promised. Quite often, however, the investor will sell the bond before maturity. The realized yield is the return earned on a bond given the cash flows actually received by the investor. More formally, it is the interest rate at which the present value of the actual cash flows from the investment equal the bond’s price. The realized yield allows investors to see the return they actually earned on their investment. It is the same as the holding period return discussed in Chapter 7.
Let’s return to the situation involving a three-year bond with a 6 percent coupon that was purchased for $960.99 and had a promised yield of 7.5 percent. Suppose that interest rates increased sharply and the price of the bond plummeted. Disgruntled, you sold the bond for $750.79 after having owned it for two years. The time line for the realized cash flows looks like this:
Substituting the cash flows into Equation 8.1 yields the following:
We can solve this equation for either by trial and error or with a financial calculator, as described earlier. Using a financial calculator, the solution is as follows:
The result is a realized yield of negative 4.97 percent. The difference between the promised yield of 7.50 percent and the realized yield of negative 4.97 percent is 12.47 percent , which can be accounted for by the capital loss of from the decline in the bond price.
8.4 INTEREST RATE RISK
As discussed previously, the prices of bonds fluctuate with changes in interest rates, giving rise to interest rate risk. Anyone who owns bonds is subject to interest rate risk because interest rates are always changing in financial markets. A number of relations exist between bond prices and changes in interest rates. These are often called the bond theorems, but they apply to all fixed-income securities. It is important that investors and financial managers understand these theorems.
• 1.Bond prices are negatively related to interest rate movements. As interest rates decline, the prices of bonds rise; and as interest rates rise, the prices of bonds decline. As mentioned earlier, this negative relation exists because the coupon rate on most bonds is fixed at the time the bonds is issued. Note that this negative relation is observed not only for bonds but also for all other financial claims that pay a fixed rate of interest to investors, such as bank loans and home mortgages.
• 2.For a given change in interest rates, the prices of long-term bonds will change more than the prices of short-term bonds. In other words, long-term bonds have greater price volatility (risk) than short-term bonds because, all other things being equal, long-term bonds have greater interest rate risk than short-term bonds. Exhibit 8.2 illustrates the fact that bond values are not equally affected by changes in market interest rates. The exhibit shows how the prices of a 1-year bond and a 30-year bond change with changing interest rates. As you can see, the long-term bond has much wider price swings than the short-term bond. Why? The answer is that long-term bonds receive most of their cash flows farther into the future, and because of the time value of money, these cash flows are heavily discounted. This makes the 30-year bond riskier than the 1-year bond.
• 3.For a given change in interest rates, the prices of lower-coupon bonds change more than the prices of higher-coupon bonds. Exhibit 8.3 illustrates the relation between bond price volatility and coupon rates. The exhibit shows the prices of three 10-year bonds: a zero coupon bond, a 5 percent coupon bond, and a 10 percent coupon bond. Initially, the bonds are priced to yield 5 percent (see column 2). The bonds are then priced at yields of 6 and 4 percent (see columns 3 and 6). The dollar price changes for each bond given the appropriate interest rate change are recorded in columns 4 and 7, and percentage price changes (price volatilities) are shown in columns 5 and 8.
EXHIBIT 8.2 Relation between Bond Price Volatility and MaturityThe prices of a 1-year and a 30-year bond respond differently to changes in market interest rates. The long-term bond has much wider price swings than the short-term bond, as predicted by the second bond theorem.
Price Change if Yield
Increases from 5% to 6% Price Change if Yield
Decreases from 5% to 4%
(1) (2) (3) (4) (5) (6) (7) (8)
Coupon Rate Bond Price at 5% Yield Bond Price at 6% Loss from Increase in Yield % Price Change Bond Price at 4% Gain from Decrease in Yield % Price Change
0% $613.91 $588.39 $25.52 -9.04% $675.56 $61.65 10.04%
5% $1,000.00 $926.40 $73.60 -7.36% $1,081.11 $81.11 8.11%
10% $1,386.09 $1,294.40 $91.69 -6.62% $1,486.65 $100.56 7.25%
Note: Calculations are based on a bond with a $1,000 face value and a 10-year maturity and assume annual compounding.
EXHIBIT 8.3 Relation between Bond Price Volatility and the Coupon RateThe exhibit shows the prices of three 10-year bonds: a zero coupon bond, a 5 percent coupon bond, and a 10 percent coupon bond. Initially, the bonds are priced at a 5 percent yield (column 2). The bonds are then priced at yields of 6 and 4 percent (columns 3 and 6). The price changes shown are consistent with the third bond theorem: the smaller the coupon rate, the greater the percentage price change for a given change in interest rates.
As shown in column 5, when interest rates increase from 5 to 6 percent, the zero coupon bond experiences the greatest percentage price decline, and the 10 percent bond experiences the smallest percentage price decline. Similar results are shown in column 8 for interest rate decreases. In sum, the lower a bond’s coupon rate, the greater its price volatility, and hence, lower-coupon bonds have greater interest rate risk.
The reason for the higher interest rate risk for low-coupon bonds is essentially the same as the reason for the higher interest rate risk for long-term bonds. The lower the bond’s coupon rate, the greater the proportion of the bond’s total cash flows investors will receive at maturity. This is clearly seen with a zero coupon bond, where all of the bond’s cash flows are received at maturity. The farther into the future the cash flows are to be received, the greater the impact of a change in the discount rate on their present value. Thus, all other things being equal, a given change in interest rates will have a greater impact on the price of a low-coupon bond than a higher-coupon bond with the same maturity.
Bond Theorem Applications
The bond theorems provide important information about bond price behavior for financial managers. For example, if you are the treasurer of a firm and are investing cash temporarily—say, for a few days—the last security you want to purchase is a long-term zero coupon bond. In contrast, if you are an investor and you expect interest rates to decline, you may well want to invest in a long-term zero coupon bond. This is because as interest rates decline, the prices of long-term zero coupon bonds will increase more than those of any other type of bond.
Make no mistake, forecasting interest rate movements and investing in long-term bonds is a very high-risk strategy. In 1990, for example, executives at Shearson Lehman Hutton (predecessor to Lehman Brothers, the firm that famously went bankrupt in 2008, the beginning of the great recession) made a huge bet on interest rate movements and lost. Specifically, over a number of months, the firm made investments in long-term bonds that totaled $480 million. The bet was that interest rates would decline. When interest rates failed to decline and losses mounted, the Shearson team sold the bonds at a loss totaling $115 million. The executives responsible were fired for “lack of judgment.”
The moral of the story is simple. Long-term bonds carry substantially more interest rate risk than short-term bonds, and investors in long-term bonds need to fully understand the magnitude of the risk involved. Furthermore, no one can predict interest rate movements consistently, including the Federal Reserve Bank (Fed)—and it controls the money supply.
8.5 THE STRUCTURE OF INTEREST RATES
In Chapter 2 we discussed the economic forces that determine the level of interest rates, and so far in this chapter, we have discussed how to price various types of debt securities. Armed with this knowledge, we now explore why, on the same day, different business firms have different borrowing costs. As you will see, market analysts have identified four risk characteristics of debt instruments that are responsible for most of the differences in corporate borrowing costs: the security’s marketability, call provision, default risk, and term to maturity.
The interest rate, or yield, on a security varies with its degree of marketability. Recall from Chapter 2 that marketability refers to the ease with which an investor can sell a security quickly at a low transaction cost. The transaction costs include all fees and the cost of searching for information. The lower the costs, the greater a security’s marketability. Because investors prefer marketable securities, they must be paid a premium to purchase otherwise similar securities that are less marketable. The difference in interest rates or yields between a highly marketable security and a less marketable security is known as the marketability risk premium (MRP).
U.S. Treasury bills have the largest and most active secondary market and are considered to be the most marketable of all securities. Investors can sell virtually any dollar amount of Treasury securities quickly without disrupting the market. Similarly, the securities of many well-known businesses enjoy a high degree of marketability, especially firms whose securities are traded on the major exchanges. For thousands of other firms whose securities are not traded actively, marketability can pose a problem and can raise borrowing costs substantially.
Most corporate bonds contain a call provision in their contract. A call provision gives the firm issuing the bonds the option to purchase the bond from an investor at a predetermined price (the call price). The investor must sell the bond at that price to the firm when the firm exercises this option. Bonds with a call provision pay higher yields than comparable noncallable bonds. Investors require the higher yields because call provisions work to the benefit of the borrower and the detriment of the investor. For example, if interest rates decline after the bond is issued, the issuer can call (retire) the bonds at the call price and refinance with a new bond issued at the lower prevailing market rate of interest. The issuing firm is delighted because the refinancing has lowered its interest expense. However, investors are less gleeful. When bonds are called following a decline in interest rates, investors suffer a financial loss because they are forced to surrender their high-yielding bonds and reinvest their funds at the lower prevailing market rate of interest.
The difference in interest rates between a callable bond and a comparable noncallable bond is called the call interest premium (CIP) and can be defined as follows:
where CIP is the call interest premium, is the yield on a callable bond, and is the yield on a noncallable bond of the same marketability, default risk, and term to maturity. Thus, the more likely a bond is to be called, the higher the CIP and the higher the bond’s market yield. Bonds issued during periods when interest rates are high are likely to be called when interest rates decline, and as a result, these bonds have a large CIP. Conversely, bonds sold when interest rates are relatively low are less likely to be called and have a smaller CIP.
Recall that any debt, such as a bond or a bank loan, is a formal promise by the borrower to make periodic interest payments and pay the principal as specified in the debt contract. Failure on the borrower’s part to meet any condition of the debt or loan contract constitutes default. As discussed in Chapter 4, default risk refers to the risk that the borrower will not be able to pay its debt obligations as they come due.
The Default Risk Premium
Because investors are risk averse, they must be paid a premium to purchase a security that exposes them to default risk. The size of the premium has two components: (1) compensation for the expected loss if a default occurs and (2) compensation for bearing the risk that a default could occur. The degree of default risk for a security can be measured as the difference between the interest rate on the risky security and the interest rate on a default-free security—all other factors, such as marketability, the existence of a call provision, or term to maturity are held constant. The default risk premium (DRP) is defined as follows:
where is the interest rate (yield) on the security that has default risk and is the interest rate (yield) on a risk-free security. U.S. Treasury securities are the best proxy measure for the risk-free rate. The larger the default risk premium, the higher the probability of default, and the higher the security’s market yield.
Many investors, especially individuals and smaller businesses, do not have the expertise to formulate the probabilities of default themselves, so they must rely on credit rating agencies to provide this information. The three most prominent credit rating agencies are Moody’s Investors Service (Moody’s), Standard & Poor’s (S&P), and Fitch. All three credit rating services rank bonds in order of their expected probability of default and publish the ratings as letter grades. The rating schemes used are shown in Exhibit 8.4. The highest-grade bonds, those with the lowest default risk, are rated Aaa (or AAA). The default risk premium on corporate bonds increases as the bond rating becomes lower.
EXHIBIT 8.4 Corporate Bond Rating SystemsMoody’s has a slightly different notation in their ratings of corporate bonds than do Standard & Poor’s and Fitch, but the interpretation is the same. Bonds with the highest credit standing are rated Aaa (or AAA) and have the lowest default risk. The credit rating declines as the default risk of the bonds increases.
Exhibit 8.4 also shows that bonds in the top four rating categories are called investment-grade bonds. Moody’s calls bonds rated below Baa (or BBB) noninvestment-grade bonds, but most Wall Street practitioners refer to them as speculative-grade bonds, high-yield bonds, or junk bonds. The distinction between investment-grade and noninvestment-grade bonds is important because state and federal laws typically require commercial banks, insurance companies, pension funds, other financial institutions, and government agencies to purchase securities rated only as investment grade.
Exhibit 8.5 shows default risk premiums associated with selected bonds with investment-grade bond ratings in December 2013. The premiums are the differences between yields on Treasury securities—which, as mentioned, are the proxy for the risk-free rate—and yields on riskier securities of similar maturity. The 0.63 percent default risk premium on Aaa-rated corporate bonds represents the market consensus of the amount investors must be compensated to induce them to purchase typical Aaa-rated bonds instead of a risk-free security. As credit quality declines from Aaa to Baa, the default risk premiums increase from 0.63 percent to 1.72 percent.
Credit Rating Security
(2) Default Risk:
(1) – (2)
Aaa 4.26 3.63 0.63
Aa 4.53 3.63 0.90
A 5.08 3.63 1.45
Baa 5.35 3.63 1.72
Sources: Federal Reserve Statistical Release H.15 (http://www.federalreserve.gov) and Bonds Online (http://www.bondsonline.com).
EXHIBIT 8.5 Default Risk Premiums for Selected Bond RatingsThe default risk premium (DRP) measures the yield difference between the yield on Treasury securities (the risk-free rate) and the yields on riskier securities of the same maturity.
The Term Structure of Interest Rates
The term to maturity of a loan is the length of time until the principal amount is payable. The relation between yield to maturity and term to maturity is known as the term structure of interest rates. We can view the term structure visually by plotting the yield curve, a graph with term to maturity on the horizontal axis and yield to maturity on the vertical axis. Yield curves show graphically how market yields vary as term to maturity changes.
For yield curves to be meaningful, the securities used to plot the curves should be similar in all features (for example, marketability, call provisions, and default risk) except for maturity. We do not want to confound the relation of yield and term to maturity with other factors that also affect interest rates. We can see the term structure relation by examining yields on U.S. Treasury securities with different maturities because their other features are similar.
Exhibit 8.6 shows data and yield curve plots for Treasury securities at various points in time in the 2000s. As you can see, the shape of the yield curve is not constant over time. As the general level of interest rises and falls, the yield curve shifts up and down and has different slopes. We can observe three basic shapes (slopes) of yield curves in the marketplace. First is the ascending, or upward-sloping, yield curve (May 2003 and February 2005), which is the yield curve most commonly observed. Second, descending, or downward-sloping, yield curves (September 2006) appear periodically and are characterized by short-term yields (for example, the six-month yield) exceeding long-term yields (for example, the 20-year yield). Downward-sloping yield curves often appear before the beginning of a recession. Finally, relatively flat yield curves are not common but do occur from time to time. Three factors affect the level and the shape (the slope) of the yield curve over time: the real rate of interest, the expected rate of inflation, and interest rate risk.
EXHIBIT 8.6 Yield Curves for Treasury Securities at Three Different Points in TimeThe shape, or slope, of the yield curve is not constant over time. The exhibit shows two shapes: (1) the curves for May 2003 and February 2005 are upward sloping, which is the shape most commonly observed, and (2) the curve for September 2006 is downward sloping for maturities out to 5 years.
The real rate of interest is the base interest rate in the economy and is determined by individuals’ time preference for consumption; that is, it tells us how much individuals must be paid to forgo spending their money today. The real rate of interest varies with the business cycle, with the highest rates seen at the end of a period of business expansion and the lowest at the bottom of a recession. The real rate is not affected by the term to maturity. Thus, the real rate of interest affects the level of interest rates but not the shape of the yield curve.
The expected rate of inflation can influence the shape of the yield curve. If investors believe that inflation will be increasing in the future, the yield curve will be upward sloping because long-term interest rates will contain a larger inflation premium than short-term interest rates. The inflation premium is the market’s best estimate of future inflation. Conversely, if investors believe inflation will be subsiding in the future, the prevailing yield will be downward sloping.
Finally, the presence of interest rate risk affects the shape of the yield curve. As discussed earlier, long-term bonds have greater price volatility than short-term bonds. Because investors are aware of this risk, they demand compensation in the form of an interest rate premium. It follows that the longer the maturity of a security, the greater its interest rate risk, and the higher the interest rate. It is important to note that the interest rate risk premium always adds an upward bias to the slope of the yield curve.
In sum, the cumulative effect of three economic factors determines the level and shape of the yield curve: (1) the cyclical movements of the real rate of interest affect the level of the yield curve, (2) the expected rate of inflation can bias the slope of the yield curve either positively or negatively, depending on market expectations of inflation, and (3) interest rate risk always provides an upward bias to the slope of the yield curve.
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