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Interest Rates Negotiation is a soft skill that might just be one of the most important skills you will ever learn. For this Discussion you will need to view the videos below to prepare for the rest of the Assignment. MonkeySee. (2009, August 5). How to Negotiate. Retrieved from http://www.5min.com/Video/How-to-Negotiate-155908733 MonkeySee. (2009, August 5). Powerful Negotiating Techniques. Retrieved from http://www.5min.com/Video/powerful-negotiating-techniques-155908736 MonkeySee. (2009, August 5). Practicing Negotiation Skills. Retrieved from http://www.5min.com/Video/Practicing-Negotiation-Skills-155908735 Now that you have a little information about how to negotiate, first discuss three concepts that interested you from the videos. Then describe a situation in which you recognized that you entered into a negotiation with another person. How did this go for you? If it went well, describe the tactics you used that contributed to the success of the negotiation. If it did not go well discuss the issues you now recognize that caused this outcome. Remember that negotiation should be a win-win situation. Now go practice this skill and return later in the week to discuss the skills you learned by telling us about the situation, other than purchasing an automobile,the actual negotiation that took place, and how what you learned contributed to your ability to effect an outcome.

PRINTED BY: irisgarcia3@student.kaplan.edu. Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Chapter 7 Interest Rates and Bonds LEARNING OBJECTIVES LO1 Zero Coupon Bond Features and Markets LO2 Zero Coupon BOnd Yields and Pricing LO3 Coupon Bond Features and Markets LO4 Coupon Bond Yields and Pricing LO5 Coupon Bond Price Properties Introduction Fixed income securities are a class of securities that pay a fixed income to the holder. The class includes bonds, which are the focus of this chapter. Bonds represent loans from the holder to the issuer. For example, when the U.S. government borrows money, it issues bonds, which are purchased by individual investors. The purchase of the bond represents a loan to the U.S. government. The bond stipulates how and when the money will be paid back as well as how much interest will be paid. The biggest difference between a loan and a bond is that the lender can get his money back early by selling the bond to another investor. With a loan, the lender has to wait until maturity to get his money back. This chapter focuses on the two main types of bonds: zero coupon bonds and coupon bonds. LO1 Zero Coupon Bond Features and Markets 1.1 Features of a Zero Coupon Bond An example of a zero coupon bond is shown in Figure 7.1. (Click on the image to learn the meaning of the various features of the bond.) Figure 7.1 Zero Coupon Bond This bond is called a zero coupon bond because it does not offer any coupons. (This will be clearer after we describe a coupon bond.) With a zero coupon bond, the holder (or lender) pays a price that is less than the face value, and then receives the face value at maturity. The difference between the price and the face value is the interest earned by the holder. 1.2 Zero Coupon Bond Markets Shortmaturity (less than 1 year) zero coupon bonds are traded on the money market. These bonds include bankers' acceptances, commercial paper, and government TBills. Table 7.1 shows the value of money market securities outstanding. At the end of 2010, there was almost $3T worth of money market securities outstanding, which is equal to 20% of the value of the U.S. gross domestic product. Buyers of money market securities include pension funds and mutual funds. Money market mutual funds are simply a portfolio of these securities. Table 7.1 U.S. Money MarketsFace Value Outstanding at Year End (USD Billions) Paper Bankers' Acceptance TBills Total 1996 779.4 23.6 761.2 1,564.2 2000 1,602.1 7.9 616.2 2,226.2 2010 1,057.5 0.0 1,788.5 2,846.0 Source: Data from Securities Industry and Financial Markets Association (SIFMA), www.sifma.org and the Department of the Treasury. Many corporations, states, and municipalities issue zero coupon bonds. The U.S. government does not issue zero coupon bonds with maturities greater than 1 year. However, many financial intermediaries strip TNote and TBonds and sell the stripped coupons and face value as zero coupon bonds. At the end of 2010, there was $892 billion of U.S. government TBonds outstanding and about 20% of that total had been stripped (over $181 billion).1 It's Time to Do a SelfTest Write your answers here. 1. A zero coupon bond: A. does not pay coupons and sells at a deep discount from face value B. typically sells at a premium to its face value C. increases in value when interest rates increase D. decreases in value when interest rates decrease E. pays coupons at regular intervals but does not have a face value Answer Answer 1U.S. Department of the Treasury. Bureau of the Public Debt. www.treasurydirect.gov Ready to do LO1 topic homework 1? LO2 Zero Coupon Bond Yields and Pricing 2.1 Zero Coupon Bond Yields A zero coupon bond promises the holder a fixed sum of money (the face value of $FV) at a fixed date in the future (date T). The timeline of cash flows from a zero coupon bond is shown below. Figure 7.2 Timeline of Cash Flows for Zero Coupon Bond If interest rates are positive (which they usually are), then the price of the bond is less than the face value. The dollar amount of interest earned on the bond is the difference between the face value and the price. The return on the bond is the rate that grows the price to equal the face value at maturity. If we think of the price as a present value and the face value as a future value, then we can use the future value formula to solve for the return. FV = Price (1 + k)n Eq. 7.1 We can solve for the return on the bond by simplifying Eq. 7.1. Eq. 7.2 The return is also called a yield to maturity (or yield) of the bond. The returns on U.S. government zero coupon bonds are also called Treasury spot rates. Treasury yields are the building block for all yields because they are essentially default free: there is almost no risk that the U.S. government will fail to pay its obligations. Example 7.1 The Yield on a Zero Coupon Bond Consider a zero coupon bond with a maturity in 2 years and a face value of $1,000. Assume that the price of the bond is $907.03. If you buy the bond at that price and hold it until maturity, then what is the yield on the bond? SOLUTION Write your answers here. It's Time to Do a SelfTest Write your answers here. 2. A zero coupon bond with a maturity in 5 years and a maturity value of $100 trades for a price of $78.353. If you buy the bond and hold it for 5 years, then what is the yield on the bond? Algebraic Answer Excel Answer Calculator Answer 2.2 The Term Structure of Rates and the Yield Curve Assume that you are an investor who is interested in buying U.S. government zero coupon bonds, but you are uncertain about what maturity you want. You call your broker for prices on five bonds with terms of 1, 2, 3, 4, and 5 years, and she replies with the information in Table 7.2. Table 7.2 Zero Coupon Bond Prices and Yields for Different Maturities Maturity Date Price ($100 face value) Yield to Maturity 1 $99.009 1% 2 $96.117 2% 3 $91.514 3% 4 $85.480 4% 5 $78.353 5% The yields across the different maturities are called the term structure of interest rates because they are the interest rates for different maturities (terms)! The Yield Curve If the term structure of rates is graphed against the respective maturities, then the resulting curve is called the yield curve. The yield curve of the Treasury spot rates is known as the Treasury Spot Rate Yield Curve. If we use the spot rates from Table 7.2, then the Treasury Spot Rate Yield Curve is shown in Figure 7.3. Figure 7.3 Treasury Spot Rate Yield Curve Each point on the yield curve represents the return earned on a U.S. government zero coupon bond with the indicated maturity. It's Time to Do a SelfTest Write your answers here. 3. Based on the table of bond prices below, what is the shape of the yield curve? (Each bond is a zero coupon bond with a face value of $100.) Zero Coupon Bond Prices Maturity (years) Price 1 $95.24 2 $90.70 3 $86.38 Algebraic Answer Excel Answer Calculator Answer 2.3 Determinants of the Shape of the Yield Curve The yield curve is not constant over time. In the early 1980s, shortterm rates were very high. However, investors expected them to fall in the future, so longterm rates were lower so the yield was downward sloping. Downward sloping yield curves are unusual. For this reason, an upwardsloping yield curve is called a normal yield curve and a downwardsloping yield curve is called an inverted yield curve. There are five factors that determine the position and shape of the spot rate yield curve: (1) Real interest rates (2) Inflation (3) Maturity Preference (4) Default risk and (5) Liquidity. We will discuss each in turn. Interest Rates and Inflation Interest rates include a component to compensate lenders for inflation. Infation drives a wedge between the real rate of interest, the rate that would prevail if there was no infation, and the nominal rate of interest which is the observed rate. If infation is zero, then the real rate equals the nominal rate. The relationship between the nominal rate, the real rate and infation was derived by an American economist called Irving Fisher. Eq. 7.3 is called the Fisher Equation. (1 + kn) = (1 + kr) (1 + ) Eq. 7.3 Where kn = the nominal interest rate kr = the real interest rate = the expected rate of infation We can simplify the Fisher Equation to obtain an expression for the nominal interest rate: kn = kr + (kr ) Eq. 7.4 Example 7.2 Nominal Rates, Real Rates, and Inflation If the real rate is 10% and inflation is 5%, then what is the nominal rate? SOLUTION Write your answers here. kr = 10% = 5% kn = 0.1 + 0.05 + (0.1 0.05) = 0.155 The nominal rate is 15.5%. Notice that is defned as the expected rate of infation. Lenders set interest rates before lending funds. Lenders set interest rates to compensate them for giving up the use of the money during the loan period. Of course, the actual infation rate for the next year is not known when the lender establishes the interest rate. The lender must guess. When infation rates are stable, as they have been since 1991, this is much easier to do. However, when they are volatile, as they were between 1970 and 1985, this is much more diffcult. Inflation rates can be estimated using the rate of change of the Consumer Price Index. Because we don't know what the market expects infation to be (how do you accurately measure expectations?), we cannot directly compute the real rate. Rather, we observe the real rate retrospectively once we observe the nominal rate and infation. It's Time to Do a SelfTest Write your answers here. 4. Historically, what factor has had the greatest effect on nominal interest rates? Answer 5. If the real rate is estimated to be 0.5% and inflation is expected to be 2%, then what is the nominal rate of interest? Algebraic Answer Excel Answer Calculator Answer 6. If the nominal interest rate is 3.5% and inflation is expected to be 2%, then estimate the real rate of interest. Algebraic Answer Excel Answer Calculator Answer Expectations and Monetary Policy The expectations theory holds that the yield on a longterm bond with a term greater than one year is the average of the individual yields expected during the long term.2 Consider the yield on a 2year bond. If the yield on a 1year zero coupon bond (which is issued today) is 10% and the expected yield on the 1year bond that will be issued next year is 12%, then the 2year rate would be 11%. 2The expectation theory predicts a geometric average and not an arithmetic average. The geometric average of two rates, k1 and k2, is: . As we showed in the previous section, nominal interest rates are a function of real interest rates and expected infation. Expectations of infation are strongly infuenced by monetary policy. As Milton Friedman, the Nobel Laureate in economics, famously put it: \"Infation is always and everywhere a monetary phenomenon...\"3 The Federal Reserve has a number of policy levers that it uses to affect the level of shortterm interest rates (e.g., overnight interest rates). Those policies affect the real rate of interest by changing supply and demand conditions in the market for overnight money, and they also affect the market's expectations of infation. In summary, according to the expectations theory, longterm bond yields are an average of the expected yields over the intervening years. The expected yields are determined by expectations of infation, and infation expectations are based on current (and anticipated) monetary policy. The Maturity Preference Theory The maturity preference theory holds that lenders prefer to make shortterm loans (bond investors prefer shorter maturities).4 To compensate for this preference, borrowers (bond issuers) must provide investors with higher yields to induce them to buy longerterm bonds. The premium builtin to longterm yields causes the yield curve to be a little steeper than predicted by the expectations hypothesis. We refer to the premium as the maturity risk premium (MRP). 3Friedman, Milton. The counterrevolution in monetary theory: first Wincott memorial lecture, delivered at the Senate House, University of London, 16 September, 1970 Institute of Economic Affairs. Occasional paper 33 London: Institute of Economic Affairs, 1970. 4The maturity preference theory is also known as the liquidity preference theory. We prefer to use the term maturity, as the term liquidity is better used to refer to the ease with which owners can sell their bonds. There are two strategies for achieving any given longterm maturity: invest in a series of shortterm bonds or invest for the long term directly. The longterm option is exposed to interest rate risk. That is the risk associated with fluctuations in the price of the longterm bond before it matures. An investor who needs to sell their bond before it matures is exposed to interest rate risk. The shortterm strategy faces a risk called reinvestment rate risk. With the shortterm strategy the investor doesn't know future rates with certainty. If future rates fall, then she could earn less than if she had invested in the longterm maturity directly. The maturity preference theory assumes that investors are more concerned with the interest rate risk than they are with the reinvestment rate risk. Default Risk Default means the failure to fulfill an obligation. In the context of loans and bonds, default can occur in a number of ways, such as failure of the borrower to make interest or principal payments or to follow the terms of the loan. Default is the major risk associated with corporate and foreign government bonds, and so a number of conventions have arisen in the bond markets to deal with it. These conventions are described in Figure 7.4 and the accompanying video. Figure 7.4 Default The only way that bond investors can be compensated for higher levels of default risk is by paying a lower price for the bonds and so earning a higher return. As a result, bonds from issuers with higher default probabilities (lower ratings) have higher yields than bonds from issuers with a lower likelihood of default (higher ratings). This difference in yields is called the default risk premium (DRP). Figure 7.5 shows the yield spread between corporate bonds with a rating of Baa and U.S. TBonds. A yield spread is just a difference in yields. This yield spread is the default risk premium associated with a Baa rating. The compensation for default risk varies with the probability of default. In turn, the probability of default varies with the business cycle. Figure 7.5 shows spikes in the default risk premium correspond with recessions (1982 and 2008) and stock market uncertainty (1987 and 2000). Figure 7.5 Yield Spread Between Baa and U.S. TBond Source: Data from Federal Reserve, www.federalreserve.gov. Liquidity Liquidity is defned as the ease with which an asset can be converted to cash. NYSE listed stocks are liquid assets because it is easy and cheap to sell a share on the NYSE. Real estate is illiquid because it is time consuming and expensive to sell a house or property. Bonds vary in their degree of liquidity. Recently issued TNotes and TBonds are very liquid. Older corporate bonds issued by small companies can be difficult and expensive to sell. Diffcult in the sense that you may have trouble finding a dealer who will buy them from you and expensive in the sense that you will receive less than fair value when you sell them. Liquidity is obviously not a problem for investors who plan to hold to maturity. Investors who buy illiquid bonds anticipate the costs of selling the illiquid assets and thus pay less for those bonds. As a result, illiquid bonds have a higher yield than equivalent liquid bonds. This difference in yields is called the liquidity risk premium (LRP). Reconciling the Theories In this section, we have identified fve factors that affect bond yields. Which of these factors influence the yield to maturity? They all do. The level of interest rates is always based on future expectations about the money supply and infation. The yield curve is normally upwardsloped because maturity premiums are added to the average expected interest rate. However, investor preferences for maturity can vary over time and so will the maturity risk premium. Buyers demand a liquidity risk premium to compensate them for the cost, time, and inconvenience associated with investments that may be hard to sell. Corporate bond investors are compensated for the risk of default through higher yieldsa default risk premium. In general, we can say that the yield on any bond is approximately equal to the following: YTM = kr + INF + MRP + LRP + DRP Eq. 7.5 where YTM = the yield to maturity of a bond kr = the real rate of return on an equivalent (default free) government bond INF = the inflation premium = + (kr ) (from the Fisher Equation, Eq. 7.4) = the expected rate of infation MRP = the maturity risk premium LRP = liquidity risk premium DRP = default risk premium It's Time to Do a SelfTest Write your answers here. 7. What is the nominal rate of interest for a 10year corporate bond if the real rate is 4%, expected inflation is 3% peryear, the liquidity risk premium is 0.5%, the default risk premium is 2%, and the maturity risk premium is 3%? Algebraic Answer Excel Answer Calculator Answer 8. What kind of risk increases when the number of buyers for an investment declines? Answer 9. According to the expectations theory, what is the relationship between future annual yields and future long term zero coupon bond yields? Answer 10. What prediction does the maturity preference theory make about the relationship of shortterm yields and longterm yields? Answer Ready to do LO2 topic homework 1? 2.4 Zero Coupon Bond Pricing If the yield is known, then the price of a zero coupon bond is the present value of the face discounted at the yield. We can solve for the price by rearranging Eq. 7.1. Eq. 7.6 Example 7.3 Price of Zero Coupon Bond If the 3year spot rate is 4%, then what is the price of a 3year U.S. government zero coupon bond with a face value of $100? $FV = 100 n = 3year k3 = 4% SOLUTION Write your answers here. The price of the bond is It's Time to Do a SelfTest Write your answers here. 11. The 2year spot rate is 5%. What is the price of a zero coupon bond with a maturity of 2 years and a face value of $1,000? Algebraic Answer Excel Answer Calculator Answer The Inverse Relationship Between Bond Prices and Yields Bond prices and interest rates move inversely: if yields rise, then bond prices fall. To see this relationship look at Eq. 7.6: the yield is in the denominator of the righthandside of the equationwhen the yield increases the ratio declines. You can also see this by increasing the yield in the spreadsheet that accompanies Figure 7.6. The graph in Figure 7.6 shows the prices of a 2year bond at different yields. Figure 7.6 Prices v. Yields for a 2Year Zero Coupon Bond Ready to do LO2 topic homework 2? LO3 Coupon Bond Features and Markets 3.1 Features of a Coupon Bond A coupon bond is one that makes periodic interest payments in addition to paying a lumpsum at maturity. A coupon bond is shown in Figure 7.7. The holder of the bond is promised to receive $20 on July 1, 1866. (History pop quiz: Why did the State of Arkansas need money in 1861? Did the holder receive the $20 at maturity?) The other features of the bond are shown when you click on the various parts of the bond. Figure 7.7 Coupon Bond Coupon bonds can also have several other features. 1. A convertible bond allows the holder of the bond to convert the face amount into a fixed number of common shares at any time before the maturity of the bond if they so choose. 2. A callable bond is one where the issuer has the right force early redemption of the bond (before the maturity date). When a bond is called, the holder must return the bond to the issuer. In exchange, the issuer pays the holder the face value of the bond. In some cases, the holder may receive a call premium, which is a small payment (over and above the face value) to compensate for the inconvenience of the call. 3. A foreign bond is a bond that is issued in a particular country and is denominated in that country's currency (but issued by a foreigner). It is generally kept distinct from domestically issued bonds for tax and disclosure purposes. 4. Finally, a bond may have a variable coupon. These bonds are called floating rate bonds. In this case, rather than having a fixed coupon rate, the coupon rate fluctuates with some benchmark rate, such as the LIBOR rate (London InterBank Offer). For example, if the coupon rate is stated as LIBOR plus 100 basis points, and LIBOR is 3%, then the coupon rate for the current period is 4%. 3.2 Coupon Bond Issuers Bonds may be issued by federal, state, and municipal governments, governmentsponsored enterprises (e.g., Fannie Mae and Freddie Mac), federal government owned corporations (e.g., Amtrak, Tennessee Valley Authority, PBS), and privatesector corporations. Table 7.3 shows the value of bonds issued by issuer type and by year. Table 7.3 Issuance in the U.S. Bond Markets (USD Billions) Municipal Treasury MortgageRelated Corporate Debt Federal Agency Securities AssetBacked Total 1996 185.2 612.4 479.7 343.7 277.9 168.4 2,067.2 2000 200.8 312.4 660.0 587.5 446.6 281.5 2,488.8 2005 408.2 746.2 2,182.4 752.8 669.0 753.5 5,512.1 2010 433.1 2,303.9 1,742.7 1,062.7 1,032.6 109.4 6,684.5 Source: Data from Securities Industry and Financial Markets Association (SIFMA), www.sifma.org. In 2010, new bond issues totaled $6.7 Trillion. To put this in perspective, the gross domestic product of the United States in 2010 was about $15T. Corporate issues accounted for only 16% of all new issues in the debt markets in 2010. The federal government was the largest issuer of bonds in that year. Federal government coupon bonds (Treasuries) with terms less than 10 years are called TNotes and those with terms greater than 10 years are called TBonds. 3.3 Coupon Bond Markets Coupon bonds trade in what is aptly called the bond market. The bond market is the market for both coupon and zerocoupon bonds with maturities ranging from one to thirty years and includes bonds issued by governments and corporations. The value of daily trading in the bond market is shown in Table 7.4. If there are 252 trading days in a year, then the annual value of bonds traded in 2010 was about $240 trillion. To put this in perspective, gross world product is estimated to be $63 trillion for 2010, so U.S. bond trading was approximately four times larger than gross world product! Table 7.4 U.S. Bond MarketsAverage Daily Trading Volume (USD Billions) Municipal Treasury Agency MBS Corporate Debt Federal Agency Securities Total 1996 1.1 203.7 38.1 - 31.1 274.0 2000 8.8 206.5 69.5 - 72.8 357.6 2010 13.3 528.2 320.6 16.3 71.5 949.8 Source: Data from Securities Industry and Financial Markets Association (SIFMA), www.sifma.org. Both the money market and the bond market are overthecounter or dealer markets. This means that there are no centralized physical or computerized exchanges for bonds. Transactions for bonds are privately negotiated through multiple dealers (or marketmakers), such as banks and large brokerages, which post prices and stand ready to buy and sell bonds at those prices. The prices are posted on proprietary computer systems, such as Reuters, and buyers and sellers communicate via telephone. The market maker's clients tend to be large institutions such as pensions and mutual funds. 3.4 Bond Price Reporting A selection of bond prices is printed in the financial daily press and shown on fnancial web sites. The prices are quotes from a particular marketmaker with which the media company has a contract. Table 7.5 shows price quotes from Bloomberg. Table 7.5 Bond Price Quote Government Bonds U.S. Treasuries 3Month 6Month 0.000 12Month 0.000 2Year Coupon 0.000 Maturity 10/20/2011 01/19/2012 06/28/2012 06/30/2013 Price/Yield 0.03/0.03 0.08/0.08 0.17/0.17 0.375 3Year 0.625 5Year 1.500 07/15/2014 06/30/2016 9931/0.39 9928+/0.66 9931/1.50 Price/Yield Change N.A./NA. 0.000/0.000 0.000/0.000 000/0.012 003/0.032 006/0.043 Time 07/22 07/22 07/22 07/22 07/22 07/22 Source: Data from Bloomberg.com, accessed July 22,2011. It's Time to Do a SelfTest Write your answers here. 12. The annual coupon of a bond divided by its face value is called the bond's: A. Coupon rate B. Capital gain rate C. Yield to maturity D. Coupon yield Answer 13. The 4year TNote has a Bloomberg price quote of 9928 1/2. What is the price as a percentage of face value? Algebraic Answer Excel Answer Calculator Answer Ready to do LO3 topic homework 1? LO4 Coupon Bond Yields and pricing This section starts by showing the cash flows of a coupon bond. A coupon bond can be thought of as a collection (or portfolio) of zero coupon bonds. This intuition leads to a basic relationship between zero coupon bond prices (which we just calculated at the end of the last section) and coupon bond prices. If a coupon bond's price is known, then we can calculate the yield to maturity, which is (approximately) the return an investor would earn if they bought the bond and held it to maturity. Conversely, if the yield to maturity for a bond is known, then we can calculate the bond's price. Finally, we show how to calculate the yield on a semiannual coupon bond. All of the yield and pricing formulas presented in this chapter are for the issue date or the point in time just after a coupon is paid. We leave pricing between coupon dates to a more advanced text. 4.1 Coupon Bond Cash Flows Coupon bonds pay the holder annual (or semiannual) coupons, (C), as well as the face value, (FV), at maturity. The coupons are usually expressed as a percentage of the face value and this percentage is known as the coupon rate. Consider a U.S. government bond with 2 years to maturity, a face value of $ FV and annual coupons of $C. The timeline for this bond's cash flows is shown in Figure 7.8. Figure 7.8 Timeline of 2Year Coupon Bond 4.2 The Relationship Between Coupon and Zero Coupon Bond Prices Pricing a coupon bond is straightforward if we think of the coupon bond as a portfolio of zero coupon bonds because we already know how to price zero coupon bonds. Example 7.4 The Prices of Two Zero Coupon Bonds The timeline shows the face values for zero coupon bonds. A 1year zero coupon bond has a face value of $C, where $ C = $10. A 2year zero coupon bond has a face value of ($ C+ $FV), where $ C = $10 and FV = $100. What are the prices of the bonds if the one year spot rate is 1% [k1 = 1%) and the 2year spot rate is 2% (k2 = 2%)? The cash flows of the two bonds are shown in the figure. SOLUTION Write your answers here. The 1year zero price,P1, is (from Eq. 7.6): The 2year zero price, P2, is: The cash flows from a portfolio comprising the two zero coupon bonds are equivalent to the cash fows from the 2 year coupon bond shown in Figure 7.8. Because the coupon bond and the package of zeros are equivalent (in cash fows, default risk, etc...), the price of the 2year coupon bond, Pbond, must equal the sum of the prices of the two zero coupon bonds. This is an application of the law of one price. The law of one price asserts that two identical securities which are sold in two different markets must sell at the same price. Pbond = P1 + P2 Eq. 7.7 If we substitute the equations for the two zeros from Example 7.4 into Eq. 7.7, then we get a pricing formula for a coupon bond. Eq. 7.8 Eq. 7.8 shows that a coupon bond price is equal to the present value of the bond's cash fows when each cash fow is discounted at the appropriate spot rate from the yield curve. Following Eq. 7.8 and Example 7.4, we now know that the price of a twoyear coupon bond with a face value of $100 and a 10% coupon rate is $115.63 1Pbond = $9.901 + $105.729 = $115.632. More generally, for any number of coupons, n, we can express the price of a coupon bond as follows: Eq. 7.9 It's Time to Do a SelfTest Write your answers here. 14. Consider a 3year U.S. government coupon bond with a 5% coupon rate. Assume that the coupons are paid annually and that the face value of the bond is $1,000. The term structure of interest rates is given in the table below. What is the price of the bond? Maturity Spot Rate 1 1% 2 2% 3 3% Algebraic Answer Excel Answer Calculator Answer 4.3 YieldtoMaturity Given a bond price, one obvious question is: \"What return will the bond holder earn if he buys the bond and holds it to maturity? \"The actual return is not known until the last coupon is received (and the interest from reinvested coupons is calculated), but a good estimate of the return is called the yieldtomaturity. The yield to maturity is the single rate which discounts the bond's cash fows so that the present value equals the bond's price. The yieldtomaturity is the solution to the following problem: Eq. 7.10 Where the bond price, Pbond, is known, n is the number of coupon periods to maturity, and kd is the yield to maturity which must be solved. Note that the discount rate in Eq. 7.10, kd, is different than the rates used in Eq. 7.9. In Eq. 7.9, each coupon is discounted at a different rate kt, which is the spot rate (from the spot rate yield curve) for that maturity. In Eq. 7.10, we are looking for a common rate for all maturities. Eq. 7.10 is tedious to use if there are many coupons. Since each of the coupon payments is the same, we can use the equation for an annuity. Using present value interest factor notation, Eq. 7.10 is rewritten as: Eq. 7.11 The yield is best solved using trial and error as the following example shows. Example 7.5 Calculate the YieldtoMaturity A 2year coupon bond has a coupon rate of 5% and a face value of $1,000. The price of the bond is $1,038.75. What is the yieldtomaturity of the bond? SOLUTION Write your answers here. The yieldtomaturity is the value for the variable,? kd, in the following equation: The solution value is kd = 2.976%. Some Insights about YieldtoMaturity Here are three insights about yield to maturity: 1. The yieldtomaturity is approximately equal to the return that the bond holder will earn if she pays the price (Pbond) and holds the bond to maturity. 2. The yieldtomaturity calculation implicitly assumes that intermediate coupons can be reinvested at the solution rate, or 2.976%. If the bondholder's reinvestment rate differs, then her actual return will not equal the yieldtomaturity. (And that is why we used the word \"approximately\" in the last point.) 3. The yield to maturity can only be calculated if the bond price is known. The yield is thus interchangeable with the price. In other words, if we are given the price, then we can solve for the yieldtomaturity. Or, if we are given the yieldtomaturity, then we can solve for the price. It's Time to Do a SelfTest Write your answers here. 15. Practice computing a coupon bond's yield to maturity. Answer Ready to do LO4 topic homework 1? 4.4 Pricing a Coupon Bond Given the YieldtoMaturity If the yieldtomaturity is given, then we can solve for the price of a coupon bond using Eq. 7.11 as in the following example. Example 7.6 Bond Valuation with Annual Coupons Suppose you found a bond in your great, great aunt's attic. It has 2years before it matures (it has been in that shoe box for 28 years now), a 10% coupon rate, and a $1,000 face value. If interest is paid annually and the yield to maturity of the bond is 9%, then what is the current price of this bond? SOLUTION Write your answers here. It's Time to Do a SelfTest Write your answers here. 16. Practice computing a coupon bond price. Answer Ready to do LO4 topic homework 2? 4.5 SemiAnnual Coupon Bonds Most bonds pay interest semiannually. To find the yield for a semiannual coupon bond, we can use a variation of Eq. 7.11, but we have to make three adjustments: 1. To fnd the semiannual coupon, divide the annual coupon payment by two since only half of the annual payment is paid each 6 months. If the annual coupon is C, then the semiannual coupon = C/2. 2. Discount the semiannual coupons with the semiannual yield. To fnd the yield for onehalf of the year, we divide the annual yield by two. The semiannual yield is kd/2. 3. Finally, double the number of periods since there will be two coupon payments per year. The number of periods = 2n. With semiannual compounding, Eq. 7.11 becomes the following: Eq. 7.12 The effective annual rate on a semiannual bond is bigger than kd, since we are compounding the periodic rate kd/2 twice (semiannually). Nonetheless, it is the market convention to solve for the yield as shown in Eq. 7.12 and then report the nominal annual rate (kd) rather than the effective annual rate. The nominal rate is known as the bond equivalent yield. If we are given the yield to maturity, then we can use Eq. 7.12 to solve for the bond's price, as is shown in the following example. Example 7.7 Bond Valuation with SemiAnnual Coupons Let us compute the price of a Chrysler bond. The bonds have a 10.95% coupon rate that is paid semi annually, a $1,000 face value, and mature in 20 years (n = 20). Assume semiannual compounding and that the yield to maturity is 12%. SOLUTION Write your answers here. Step 1: Calculate semiannual coupon amount: Annual coupon = C = 0.1095 $1,000 = $109.50 Semiannual coupon = Annual coupon/2 = C/2 = $109.50/2 = $54.75 Step 2: Calculate semiannual yield to maturity: Yield to maturity = kd = 12% Semiannual yield to maturity = kd/2 = 12%/2 = 6% Step 3: Note that with semiannual compounding the number of periods must be doubled. This means that there are 40 coupon payments and 40 periods (2n). It's Time to Do a SelfTest Write your answers here. 17. Practice computing a coupon bond price. Answer Ready to do LO4 topic homework 3? LO5 Coupon Bond Price Properties In this section, we present the properties of bond prices. We explain why some bonds are priced less than their face value and some bonds are priced higher. We show that bond prices and interest rates move inversely. We explain interest rate risk and show that longer maturity bonds and bonds with low coupon rates have more interest rate risk. Finally, we separate bond yields into their capital gain yield and coupon yield components and show how bond prices change as time passes to maturity. 5.1 Premiums and Discounts What determines whether a bond will sell for a premium or a discount? Suppose that you are asked to purchase an old bond that has a coupon rate of 10% and $1,000 face value. New bonds with similar risk yield 12%. You would not be willing to pay $1,000 for the old bond. Instead, you would want to pay less than $1,000 so that your yield is greater than 10%. In fact, the price would have to be low enough that your yield on the old bond exactly equals the yield on new bonds. Thus, the old bond would have to sell at a discount to its face value. Example 7.8 Discount and Premium Coupon Bonds Consider three annual coupon bonds each with a face value of $1,000 and 10years remaining to maturity. The first bond has a coupon rate of 8%, the second has a coupon rate of 10%, and the third has a coupon rate of 12%. Assume that the yield to maturity of the bonds is 10%. Which bond trades at a premium, which trades at a discount and which at par? SOLUTION Write your answers here. The 8% coupon bond trades at a discount, the 10% coupon bond trades at par, and the 12% coupon rate bond trades at a premium. It's Time to Do a SelfTest Write your answers here. 18. If the yield to maturity of a bond is greater than the coupon rate, then the price will be _____________ the face value. Answer 19. If the yield to maturity of a bond is less than the coupon rate, then the price will be _____________ the face value. Answer 20. If the yield to maturity of a bond is equal to the coupon rate, then the price will be _____________ the face value. Answer Ready to do LO5 topic homework 1? 5.2 Bond Prices and Interest Rates Move Inversely Bond prices and interest rates move inversely. We saw this for zero coupon bonds (Figure 7.6), and it is also true for coupon bonds. Figure 7.9 shows the prices of a 10year, 6% coupon bond at different yields. Figure 7.9 Prices v. Yields for a 10Year, 6% Coupon Bond The variation in bond prices associated with interest rate changes is called interest rate risk. Interest rate risk is the second biggest risk for corporate bond investors (behind default risk) and the largest source of risk for U.S. government bond investors. Interest rate risk is not a problem for investors who intend to hold the bond to maturity. Ready to do LO5 topic homework 2? 5.3 Longer Maturity Bonds Have More Interest Rate Risk When interest rates change, longterm bond prices change more than shortterm bond prices. In other words, long term bonds have more interest rate risk. This principle follows from the time value of money mathematics. The present value of a nearterm cash fow doesn't change much when rates change because it isn't discounted much. However, more distant cash fows are discounted many periods and the change in rates has a compound effect. Table 7.6 shows the market price of 10% coupon bonds with 1, 10, and 30 years to maturity at three different yields. Notice that the change in the value of the bonds is much greater for bonds with longer maturities. A 1% drop in yield (moving from the middle column to the lefthand column) results in a $9.17 increase in price if the bond has one year to maturity. That same 1% drop in yield results in a $102.74 increase in price if there are 30 years to maturity. We can conclude that interest rate risk increases with increasing maturity. Table 7.6 Price of a 10% Coupon Bond with Different Maturities and Yields Yield Years to Maturity 1 9% 10% 11% $1,009.17 $1,000.00 $990.99 10 1,064.18 1,000.00 941.11 30 1,102.74 1,000.00 913.06 Interest rate risk can be seen graphically in the following Explore It, which plots the market price of 1 and 30year maturity, 10% coupon, $1,000 face value bonds at varying market interest rates. We see that the 1year bond price changes relatively little when rates change, while the 30year bond price changes substantially. Bond portfolio managers take advantage of this bond pricing property by using a strategy called interest rate expectation. Interest rate expectation involves speculating on unanticipated interest rate changes. In effect, bond portfolio managers are using interest rate risk for profit. For example, consider a bond portfolio manager with a mix of short and longterm bonds. If the manager thought that interest rates were going to fall (more than anticipated), then she would sell some or all of her shortterm bonds and buy longer term bonds. If rates did fall, then the prices of all of her bonds would rise, but the longterm bonds prices would rise more than the original mix of short and longterm bonds. She would subsequently sell the longterm bonds and buy shortterm bonds thus returning her portfolio to its original maturity structure. By buying longterm bonds, she would have increased the value of the portfolio by more than if she had held the original mix of short and longterm bonds. It's Time to Do a SelfTest Write your answers here. 21. Consider the three bond quotes in the table for government bonds with maturities of 1 year, 10 years, and 30 years. All of the bonds pay annual coupons. If yields are forecast to rise (across all maturities) by 1%, then which bond's price will decline by the greatest percentage amount? Issuer Coupon Maturity Price (% of FV) Yield Government 3.00% 1 100.49 2.50% Government 3.00% 10 102.16 2.75% Government 3.00% 30 82.71 4.00% Algebraic Answer Excel Answer Calculator Answer Ready to do LO5 topic homework 3? 5.4 Bonds with Low Coupon Rates Have More Interest Rate Risk One last factor that affects the level of interest rate risk is the coupon rate. As we know, the cash flows from a coupon bond consist of a series of coupon payments followed by the final face value. Consider summing those cash fows and calculating the proportion attributable to coupons and the proportion attributable to the face value. For example, a 10 year 1% coupon bond with a face value of $100 will pay $10 of coupons ($1 10) and $100 of face value. Total cash fows from the bond are $110 and the face value accounts for 90.91% of the total cash fows. Alternatively, a 10year 10% coupon bond (with a $100 face value) pays the holder a total of $200 of which 50% is attributable to the face value. The low coupon rate bond pays proportionately more of its cash fow at maturity. The higher coupon rate bond spreads its cash fows out more smoothly over the life of the bond. We know from the last section that the present values of longer maturity cash fows change more (following a change in interest rates) than the present values of shorter maturity cash fows. Since low coupon rate bonds have more of their cash fows arriving at longer maturities the present value of their cash fows (the bond price) changes more for a given change in yield compared to an otherwise equivalent high couponrate bond. We can see this numerically in Table 7.7. Table 7.7 Prices of 30Year Bonds with Different Coupon Rates and a 1% Change in Yields Yield = Coupon Rate Yield > Coupon Rate Coupon Rate Yield Bond Price Yield Bond Price Change in Price 0% 0% $1,000 1% $741.92 25.8% 5% 5% $1,000 6% $862.35 13.8% 12% 12% $1,000 13% $925.04 7.5% In Table 7.7, the market price of 30year bonds is computed assuming yields increase by 1% above the bond's coupon rate. When rates go up, the price of the zero coupon bond decreases by 25.8%, compared to 13.8% for the 5% bond, and only 7.5% for the 12% bond. We can conclude from this that the lower the coupon rate, the greater the interest rate risk. It's Time to Do a SelfTest Write your answers here. 22. Consider the two bond quotes in the table. A 2year U.S. government bond and a 2year U.K. government bond. Which bond has greater interest rate risk? Issuer Coupon Maturity Price (% of FV) Yield U.S. Government 0.375% 2 years 99.91 U.K. Government 4.5% 107.58 0.67% 2 years Answer 5.5 Bond Price Changes Over Time First, we will calculate the capital gain yield, which is just the percentage change in a bond price. We will show you how the capital gain yield is related to the yield to maturity. Then, in the second subsection, we will explore how bond prices change as the maturity date approaches even when interest rates (and yields) do not change. Capital Gains, Coupon Yields, and the Yield to Maturity Consider an annual coupon bond issued by Chrysler Corp. The bond has a 10.95% coupon rate, a $1,000 face value, and matures in 20 years. The yield to maturity is 12%. The bond is priced at $921.57. Assume that you buy it today, hold it for 1 year and sell it after the next coupon. Assume that interest rates do not change during the year and so the yield is constant. What is the price change over the year, and what return do you earn on the bond? Figure 7.10 Timeline of Chrysler Bond Cash Flows To calculate the change in price over the year we first need to calculate the price after the frst coupon. Using Eq. 7.11, the data given above and n = 19, we find that the Chrysler bond price is $922.66 after the frst coupon is paid. The price of the bond rises over the year even though interest rates and the yield don't change. An investor who buys the bond at the beginning of the year and sells it after the frst coupon earns the following return: Eq. 7.13 The return on the bond is equal to the yield to maturity of the bond. The return can be divided into two components: Eq. 7.14 The frst part is the coupon yield. It is the coupon interest payment divided by the current market price of the bond. In this example, the coupon yield is $109.50/921.57 = 0.1188 or 11.88%. The second part is the capital gain yield which is just the percentage change in the price of the bond. In this example, the capital gain yield is $1.0885/921.57 = 0.00118 or 0.118%. Again, you can see why bonds with coupon rates lower than their yields trade at a discount. The yield to maturity in this example is 12%, but the coupon interest rate is only 10.95%. The coupons alone aren't enough to generate a sufficient return to satisfy investors. The low coupon rate has to be supplemented with a capital gain yield, so the bond trades at a discount to face value and slowly rises in price as the maturity date approaches (assuming no changes in the overall level of interest rates). When you buy a bond at a discount, you will receive the benefit of price appreciation as the maturity date approaches. This is in addition to the coupon payments. If you buy a premium bond, you will suffer a price decline as the maturity date approaches. Bond Prices and Time to Maturity The relationship between the time to maturity and the price of a discount bond, a premium bond and a par value bond is graphed in Figure 7.11. The lines in Figure 7.11 show bond prices over 30 years (on coupon dates) between issue and maturity. In each case, the bond has the same features: a $1,000 face value and a 10% coupon rate. The three lines correspond to three different yields. The top line shows the price path of the bond if the yield is 8%, the middle line shows the price path if the yield is 10%, and the bottom line shows the price path if the yield is 12%. Figure 7.11 Prices Over Time for a 10% Coupon Bond at 8%, 10%, and 12% Yields If the yield is 8%, then the 10% coupon bond initially sells for $1,225.16 when it is issued with 30 years remaining to maturity. The bond's price falls as it approaches maturity. If the yield is 12%, then the 10% coupon bond initially sells for $838.90 when it is issued. The bond's price rises as it approaches maturity. The 10% coupon bond has a constant value of $1,000 when the yield is also 10%. Ready to do LO5 topic homework 4? Chapter Summary Concepts You Solution Extra Practice Should Know Key Terms and Equations Tools MyFinanceLab LO1 Zero Coupon fixed income securities, bonds, face value, issuer, bond holder or owner, Bond maturity date, issue date, zero coupon bond, money market, bankers' Features acceptances, commercial paper, TBills, mutual funds, TNote, TBonds and Markets Study Plan 7.LO1 yield to maturity (or yield), Treasury spot rates, term structure of interest rates, yield curve, Treasury Spot Rate Yield Curve, inverted LO2 Zero yield curve, real rate of interest, nominal rate of interest, Consumer Coupon Price Index, expectations theory, overnight interest rates, maturity Bond preference theory, maturity risk premium (MRP), interest rate risk, Yields reinvestment rate risk, default, indenture, collateral, bond ratings, and covenants, trustee, secured bond, mortgage bond, debentures, ratings Pricing agencies, investment grade, junk, default risk premium (DRP), yield spread, basis points (bps), liquidity, liquidity risk premium (LRP) Study Plan 7.LO2 FV = Price (1 + k)n Eq. 7.1 The Yield on a Zero Coupon Bond Eq. 7.2 (1 + kn) = (1 + kr) (1 + ) Eq. 7.3 Nominal Rates, Real Rates, and Inflation kn = kr + (kr ) Eq. 7.4 YTM = kr + INF + MRP + LRP + DRP Eq. 7.5 Price of Zero Coupon Bond Eq. 7.6 LO3 Coupon coupon bond, convertible bond, callable bond, foreign bond, floating Bond rate bonds, LIBOR rate, Treasuries, bond market, overthecounter, Features dealer markets, dealers, marketmakers and Markets Study Plan 7.LO3 LO4 Coupon Bond Yields and Pricing coupon rate, law of one price, bond equivalent yield Study Plan 7.LO4 LO4 Coupon Bond Yields and Pricing coupon rate, law of one price, bond equivalent yield Study Plan 7.LO4 The Prices of Two Zero Coupon Bonds Pbond = P1 + P2 Eq. 7.7 Eq. 7.8 Eq. 7.9 Eq. 7.10 Calculate the Yield toMaturiy Eq. 7.11 Bond Valuation with Annual Coupons Bond Valuation with Semi Annual Coupons Eq. 7.12 Discount and Premium Coupon Bonds LO5 Coupon premium, discount, par, interest rate risk, interest rate expectation, Bond capital gain yield, coupon yield Price Properties Study Plan 7.LO5 Eq. 7.13 Eq. 7.14 It's time to do your chapter