This Document Contains Chapters 1 to 2 CHAPTER 1 INTRODUCTION CHAPTER SUMMARY This introductory chapter will focus on the fundamental features of bond, the type of issuers, and risk faced by investors in fixed-income securities. A bond is a debt instrument requiring the issuer to repay to the lender the amount borrowed plus interest over a specified period of time. A typical (“plain vanilla”) bond issued in the United States specifies (1) a fixed date when the amount borrowed (the principal) is due, and (2) the contractual amount of interest, which typically is paid every six months. The date on which the principal is required to be repaid is called the maturity date. Assuming that the issuer does not default or redeem the issue prior to the maturity date, an investor holding this bond until the maturity date is assured of a known cash flow pattern. since the early 1980s a wide range of bond structures has been introduced into the bond market. SECTORS OF THE U.S. BOND MARKET The U.S. bond market is divided into six sectors: U.S. Treasury sector, agency sector, municipal sector, corporate sector, asset-backed securities, and mortgage sector. The Treasury Sector The Treasury sector includes securities issued by the U.S. government. These securities include Treasury bills, notes, and bonds. This sector plays a key role in the valuation of securities and the determination of interest rates throughout the world. The Agency Sector The agency sector includes securities issued by federally related institutions and government- sponsored enterprises. The securities issued are not backed by any collateral and are referred to as agency debenture securities. The Municipal Sector The municipal sector is where state and local governments and their authorities raise funds. This sector is divided into two subsectors based on how the interest received by investors is taxed at the federal income tax level: the tax-exempt and taxable sectors. The municipal bond market includes two types of structures: tax-backed and revenue bonds. The Corporate Sector The corporate sector includes (i) securities issued by U.S. corporations and (ii) securities issued in the United States by foreign corporations. Issuers in the corporate sector issue bonds, medium- term notes, structured notes, and commercial paper. The corporate sector is divided into the investment grade and noninvestment grade sectors. The Asset-Backed Securities Sector This Document Contains Chapters 1 to 2 In the asset-backed securities sector, a corporate issuer pools loans or receivables and uses the pool of assets as collateral for the issuance of a security. The Mortgage Sector The mortgage sector is the sector where securities are backed by mortgage loans. These are loans obtained by borrowers in order to purchase residential property or an entity to purchase commercial property (i.e., income-producing property). The mortgage sector is then divided into the residential mortgage sector and the commercial mortgage sector. OVERVIEW OF BOND FEATURES A more detailed treatment of bond features is presented in later chapters. Type of Issuer There are three issuers of bonds: the federal government and its agencies, municipal governments, and corporations (domestic and foreign). Term to Maturity The maturity of a bond refers to the date that the debt will cease to exist, at which time the issuer will redeem the bond by paying the principal. There may be provisions in the indenture that allow either the issuer or bondholder to alter a bond’s term to maturity. Generally, bonds with a maturity of between one and five years are considered short-term. Bonds with a maturity between 5 and 12 years are viewed as intermediate -term, and those with a maturity of more than 12 years are called long-term. With all other factors constant, the longer the maturity of a bond, the greater the price volatility resulting from a change in market yields. Principal and Coupon Rate The principal of a bond is the amount that the issuer agrees to repay the bondholder at the maturity date. This amount is also referred to as the redemption value, maturity value, par value, or face value. The coupon rate, also called the nominal rate, is the interest rate that the issuer agrees to pay each year. The annual amount of the interest payment made to owners during the term of the bond is called the coupon. The holder of a zero-coupon bond realizes interest by buying the bond substantially below its principal value. Interest is then paid at the maturity date, with the exact amount being the difference between the principal value and the price paid for the bond. Floating-rate bonds are issues where the coupon rate resets periodically (the coupon reset date) based on a formula. The coupon reset formula has the following general form: reference rate + quoted margin. The quoted margin is the additional amount that the issuer agrees to pay above the reference rate. An important non-interest rate index that has been used with increasing frequency is the rate of inflation. Bonds whose interest rate is tied to the rate of inflation are referred to generically as linkers. The coupon on floating-rate bonds (which is dependent on an interest rate benchmark) typically rises as the benchmark rises and falls as the benchmark falls. Exceptions are inverse floaters whose coupon interest rate moves in the opposite direction from the change in interest rates. To reduce the burden of interest payments, firms involved in LBOs and recapitalizations, have issued deferred-coupon bonds that let the issuer avoid using cash to make interest payments for a specified number of years. Amortization Feature The principal repayment of a bond issue can call for either (1) the total principal to be repaid at maturity or (2) the principal repaid over the life of the bond. In the latter case, there is a schedule of principal repayments. This schedule is called an amortization schedule. For amortizing securities, a measure called the weighted average life or simply average life of a security is computed. Embedded Options It is common for a bond issue to include a provision in the indenture that gives either the bondholder and/or the issuer an option to take some action against the other party. The most common type of option embedded in a bond is a call provision. This provision grants the issuer the right to retire the debt, fully or partially, before the scheduled maturity date. An issue with a put provision included in the indenture grants the bondholder the right to sell the issue back to the issuer at par value on designated dates. A convertible bond is an issue giving the bondholder the right to exchange the bond for a specified number of shares of common stock. An exchangeable bond allows the bondholder to exchange the issue for a specified number of common stock shares of a corporation different from the issuer of the bond. Describing a Bond Issue There are hundreds of thousands of bonds issues. Most securities are identified by a nine character (letters and numbers) CUSIP number. CUSIP stands for Committee on Uniform Security Identification Procedures. The first six characters identify the issuer: the corporation, government agency, or municipality. The next two characters identify whether the issue is debt or equity and the issuer of the issue. The last character is simply a check character that allows for accuracy checking and is sometimes truncated or ignored. The CUSIP International Numbering System (CINS) is used to identify foreign securities and includes 12 characters. RISKS ASSOCIATED WITH INVESTING IN BONDS Bonds may expose an investor to one or more of the following risks: (1) interest-rate risk, (2) reinvestment risk, (3) call risk, (4) credit risk, (5) inflation risk, (6) exchange rate risk, (7) liquidity risk, (8) volatility risk, and (9) risk risk. In later chapters, other risks, such as yield curve risk, event risk, and tax risk, are also introduced. Interest-Rate Risk If an investor has to sell a bond prior to the maturity date, an increase in interest rates will mean the realization of a capital loss (i.e., selling the bond below the purchase price). This risk is referred to as interest-rate risk or market risk. Reinvestment Income or Reinvestment Risk Reinvestment risk is the risk that the interest rate at which interim cash flows can be reinvested will fall. Reinvestment risk is greater for longer holding periods, as well as for bonds with large, early, cash flows, such as high-coupon bonds. It should be noted that interest-rate risk and reinvestment risk have offsetting effects. That is, interest-rate risk is the risk that interest rates will rise, thereby reducing a bond’s price. In contrast, reinvestment risk is the risk that interest rates will fall. Call Risk Call risk is the risk investors have that a callable bond will be called when interest rates fall. Many bonds include a provision that allows the issuer to retire or “call” all or part of the issue before the maturity date. The issuer usually retains this right in order to have flexibility to refinance the bond in the future if the market interest rate drops below the coupon rate. For investors, there are three disadvantages to call provisions. First, the cash flow pattern cannot be known with certainty. Second, the investor is exposed to reinvestment risk. Third, the capital appreciation potential of a bond will be reduced. Even though the investor is usually compensated for taking call risk by means of a lower price or a higher yield, it is not easy to determine if this compensation is sufficient. Credit Risk Credit risk is the risk that the issuer of a bond will fail to satisfy the terms of the obligation with respect to the timely payment of interest and repayment of the amount borrowed. This form of credit risk is called default risk. Market participants gauge the default risk of an issue by looking at the default rating or credit rating assigned to a bond issue by rating companies. The yield on a bond issue is made up of two components: (1) the yield on a similar maturity Treasury issue and (2) a premium to compensate for the risks associated with the bond issue that do not exist in a Treasury issue—referred to as a spread. The part of the risk premium or spread attributable to default risk is called the credit spread. The risk that a bond price will decline due to an increase in the credit spread is called credit spread risk. An unanticipated downgrading of an issue or issuer increases the credit spread sought by the market, resulting in a decline in the price of the issue or the issuer’s debt obligation. This risk is referred to as downgrade risk. Consequently, credit risk consists of three types of risk: default risk, credit spread risk, and downgrade risk. Inflation Risk Inflation risk or purchasing-power risk arises because of the variation in the value of cash flows from a security due to inflation, as measured in terms of purchasing power. Exchange-Rate Risk A non-dollar-denominated bond (i.e., a bond whose payments occur in a foreign currency) has unknown U.S. dollar cash flows. The dollar cash flows are dependent on the exchange rate at the time the payments are received. The risk of the exchange rate causing smaller cash flows is referred to as exchange rate or currency risk. Liquidity Risk Liquidity or marketability risk depends on the ease with which an issue can be sold at or near its value. The primary measure of liquidity is the size of the spread between the bid price and the ask price quoted by a dealer. The wider the dealer spread, the more the liquidity risk. Marking a position to market, or simply marking to market, means that the portfolio manager must periodically determine the market value of each bond in the portfolio. Volatility Risk The value of an embedded option rises when expected interest-rate volatility increases. The risk that a change in volatility will affect the price of a bond adversely is called volatility risk. Risk Risk There have been new and innovative structures introduced into the bond market. Risk risk is defined as not knowing what the risk of a security is. There are two ways to mitigate or eliminate risk risk. The first way is to keep up with the literature on the state-of-the-art methodologies for analyzing securities. The second way is to avoid securities that are not clearly understood. ANSWERS TO QUESTIONS FOR CHAPTER 1 (Questions are in bold print followed by answers.) 1. What is the cash flow of a 10-year bond that pays coupon interest semiannually, has a coupon rate of 7%, and has a par value of $100,000? The principal or par value of a bond is the amount that the issuer agrees to repay the bondholder at the maturity date. The coupon rate multiplied by the principal of the bond provides the dollar amount of the coupon (or annual amount of the interest payment). A 10-year bond with a 7% annual coupon rate and a principal of $100,000 will pay semiannual interest of (0.07/2)($100,000) = $3,500 for 10(2) = 20 periods. Thus, the cash flow is an annuity of $3,500 made is 20 payments at the same two points in time for each of the ten years. In addition to this cash flow, the issuer of the bond is obligated to pay back the principal of $100,000 at the time the last $3,500 is paid. 2. What is the cash flow of a seven-year bond that pays no coupon interest and has a par value of $10,000? There is no periodic cash flow as found in the previous problem. Thus, the only cash flow will be the principal payment of $10,000 paid by the issuer (and received by the bond purchaser) at the end of seven years. This type of cash flow resembles a zero-coupon bond. The holder of such a bond realizes interest by buying the bond substantially below its principal value. Interest is then paid at the maturity date, with the exact amount being the difference between the principal value and the price paid for the bond. 3. Give three reasons why the maturity of a bond is important. There are three reasons why the maturity of a bond is important. First, the maturity gives the time period over which the holder of the bond can expect to receive the coupon payments and the number of years before the principal will be paid in full. Second, the maturity is important because the yield on a bond depends on it. The shape of the yield curve determines how the maturity affects the yield. Third, the price of a bond will fluctuate over its life as yields in the market change. The volatility of a bond’s price is dependent on its maturity. More specifically, with all other factors constant, the longer the maturity of a bond, the greater the price volatility resulting from a change in market yields. 4. Explain whether or not an investor can determine today what the cash flow of a floating- rate bond will be. Floating-rate bonds are issues where the coupon rate resets periodically based on a general formula equal to the reference rate plus the quoted margin. The reference rate is some index subject to change. The exact change is unknown and uncertain. Thus, an investor cannot determine today what the cash flow of a floating-rate bond will be in the future. 5. Suppose that coupon reset formula for a floating-rate bond is: 1-month LIBOR + 220 basis points. (a) What is the reference rate? The reference rate is the 1-month LIBOR. (b) What is the quoted margin? The quoted margin is the 220 basis points (or 2.20%). (c) Suppose that on a coupon reset date that 1-month LIBOR is 2.8%. What will the coupon rate be for the period? The coupon reset formula is: 1-month LIBOR + 220 basis points. Therefore, if 1-month LIBOR on the coupon reset date is 2.8%, the coupon rate is reset for that period at 2.80% + 2.20% = 5.00%. 6. What is a deferred coupon bond? Deferred-coupon bonds are coupon bonds that let the issuer avoid using cash to make interest payments for a specified number of years. There are three types of deferred-coupon structures: (1) deferred-interest bonds, (2) step-up bonds, and (3) payment-in-kind bonds. 7. What is meant by a linker? A linker is a bond whose interest rate is tied to the rate of inflation. The U.S. Treasury issues linkers, and they are referred to as Treasury Inflation Protection Securities (TIPS). 8. Answer the below questions. (a) What is meant by an amortizing security? The principal repayment of a bond issue can be for either the total principal to be repaid at maturity or for the principal to be repaid over the life of the bond. In the latter case, there is a schedule of principal repayments. This schedule is called an amortization schedule. Loans that have this amortizing feature are automobile loans and home mortgage loans. There are securities that are created from loans that have an amortization schedule. These securities will then have a schedule of periodic principal repayments. Such securities are referred to as amortizing securities. (b) Why is the maturity of an amortizing security not a useful measure? For amortizing securities, investors do not talk in terms of a bond’s maturity. This is because the stated maturity of such bonds or securities only identifies when the final principal payment will be made. For an amortized security, the repayment of the principal is made through multiple payments over its maturity and not just at the end of its term to maturity. Thus, the maturity is not a useful measure in terms of identifying when the principal is repaid. 9. What is a bond with an embedded option? A bond with an embedded option is a bond that contains a provision in the indenture that gives either the bondholder and/or the issuer an option to take some action against the other party. For example, the borrower may be given the right to alter the amortization schedule for amortizing securities. An issue may also include a provision that allows the bondholder to change the maturity of a bond. An issue with a put provision included in the indenture grants the bondholder the right to sell the issue back to the issuer at par value on designated dates. 10. What does the call provision for a bond entitle the issuer to do? A call provision grants the issuer the right to retire the debt, fully or partially, before the scheduled maturity date. 11. Answer the below questions. (a) What is the advantage of a call provision for an issuer? Inclusion of a call feature benefits bond issuers by allowing them to replace an old bond issue with a lower-interest cost issue if interest rates in the market decline. A call provision effectively allows the issuer to alter the maturity of a bond. The right to call an obligation is included in most loans and therefore in all securities created from such loans. This is because the borrower typically has the right to pay off a loan at any time, in whole or in part, prior to the stated maturity date of the loan. (b) What are the disadvantages of a call provision for the bondholder? From the bondholder’s perspective, there are three disadvantages to call provisions. First, the cash flow pattern of a callable bond is not known with certainty. Second, because the issuer will call the bonds when interest rates have dropped, the investor is exposed to reinvestment risk. Reinvestment risk is the risk that the investor will have to reinvest the proceeds when the bond is called at relatively lower interest rates. Finally, the capital appreciation potential of a bond will be reduced because the price of a callable bond may not increase much above the price at which the issuer will call the bond. 12. What does the put provision for a bond entitle the bondholder to do? An issue with a put provision included in the indenture grants the bondholder the right to sell the issue back to the issuer at par value on designated dates. The advantage to the bondholder is related to the possibility that if interest rates rise after the issue date (thereby reducing a bond’s price) the bondholder can force the issuer to redeem the bond at par value. 13. The Export Development Canada issued a bond on March 17, 2009. The terms were as follows: Currency of denomination: Japanese yen (JPY) Denomination: JPY100,000,000 Maturity date: March 18, 2019, or an optional redemption date Redemption/payment basis: Redemption at par value Interest payment dates: March 18 and September 18 in each year Optional redemption dates: The issuer has the right to call the instruments in whole (but not in part) at par starting on March 18, 2012 Interest rate: Fixed rate for the first three years up to but excluding March 18, 2012: 1.5% March 18, 2012-September 18, 2012 1.75% − 6 month JPY LIBORBBA September 18, 2012-March 18, 2013 1.75% − 6 month JPY LIBORBBA March 18, 2013-September 18, 2013 2.00% − 6 month JPY LIBORBBA September 18, 2013-March 18, 2014 2.00% − 6 month JPY LIBORBBA March 18, 2014-September 18, 2014 2.25% − 6 month JPY LIBORBBA September 18, 2014-March 18, 2015 2.25% − 6 month JPY LIBORBBA March 18, 2015-September 18, 2015 2.50% − 6 month JPY LIBORBBA September 18, 2015-March 18, 2016 2.50% − 6 month JPY LIBORBBA March 18, 2016-September 18, 2016 2.75% − 6 month JPY LIBORBBA September 18, 2016-March 18, 2017 2.75% − 6 month JPY LIBORBBA March 18, 2017-September 18, 2017 3.00% − 6 month JPY LIBORBBA September 18, 2017-March 18, 2018 3.00% − 6 month JPY LIBORBBA March 18, 2018-September 18, 2018 3.25% − 6 month JPY LIBORBBA September 18, 2018-March 18, 2019 3.25% − 6 month JPY LIBORBBA Answer the below questions. (a) What is meant by JPY LIBORBBA? The reference rate for most floating-rate securities is an interest rate or an interest rate index. The mostly widely used reference rate throughout the world is the London Interbank Offered Rate and referred to as LIBOR. In debt agreements LIBOR is often referred to as BBA LIBOR. The rate is reported for ten currencies including the Japanese yen (JPY). So, for example, the JPY BBA LIBOR is the rate for a LIBOR loan denominated in Japanese yens as computed by the British Bankers Association (BBA). (b) Describe the coupon interest characteristics of this bond. The characteristics are based on the floating-rate bonds, which are issues where the coupon rate resets periodically (the coupon reset date) based on a formula. The formula, referred to as the coupon reset formula, has the following general form: reference rate + quoted margin The quoted margin is the additional amount that the issuer agrees to pay above the reference rate. For example, suppose that the reference rate is 3.5% and the quoted margin is 150 basis points. Then the coupon reset formula is 1-month LIBOR + 150 basis points = 3.5% + 1.5% = 5.0% If the 1-month LIBOR on the coupon reset date is 3.5%, the coupon rate is reset for that period at 5.0% . (c) What are the risks associated with investing in this bond if the investor’s home currency is not in Japanese yen. From the perspective of a U.S. investor, a non-dollar-denominated bond (i.e., a bond whose payments occur in a foreign currency) has unknown U.S. dollar cash flows. The dollar cash flows are dependent on the exchange rate at the time the payments are received. For our situation, an investor purchases a bond whose payments are in Japanese yen. If the yen depreciates relative to the U.S. dollar, fewer dollars will be received. The risk of this occurring is referred to as exchange- rate or currency risk. Of course, should the yen appreciate relative to the U.S. dollar, the investor will benefit by receiving more dollars. 14. What are a convertible bond and an exchangeable bond? A convertible bond is an issue giving the bondholder the right to exchange the bond for a specified number of shares of common stock. Such a feature allows the bondholder to take advantage of favorable movements in the price of the issuer’s common stock. An exchangeable bond allows the bondholder to exchange the issue for a specified number of common stock shares of a corporation different from the issuer of the bond. 15. How do market participants gauge the default risk of a bond issue? It is common to define credit risk as the risk that the issuer of a bond will fail to satisfy the terms of the obligation with respect to the timely payment of interest and repayment of the amount borrowed. This form of credit risk is called default risk. Market participants gauge the default risk of an issue by looking at the default rating or credit rating assigned to a bond issue by one of the three rating companies—Standard & Poor’s, Moody’s, and Fitch. 16. Comment on the following statement: Credit risk is more than the risk that an issuer will default. There are risks other than default that are associated with investment bonds that are also components of credit risk. Even in the absence of default, an investor is concerned that the market value of a bond issue will decline in value and/or the relative price performance of a bond issue will be worse than that of other bond issues. The yield on a bond issue is made up of two components: (1) the yield on a similar maturity Treasury issue and (2) a premium to compensate for the risks associated with the bond issue that do not exist in a Treasury issue—referred to as a spread. The part of the risk premium or spread attributable to default risk is called the credit spread. The price performance of a non-Treasury debt obligation and its return over some investment horizon will depend on how the credit spread of a bond issue changes. If the credit spread increases—investors say that the spread has “widened”—the market price of the bond issue will decline. The risk that a bond issue will decline due to an increase in the credit spread is called credit spread risk. This risk exists for an individual bond issue, bond issues in a particular industry or economic sector, and for all bond issues in the economy not issued by the U.S. Treasury. 17. Explain whether you agree or disagree with the following statement: “Because my bond is guaranteed by an insurance company, I have eliminated credit risk.” Credit risk consists of three types of risk: default risk, credit spread risk, and downgrade risk. These risks are not necessarily eliminated if there is a financial guaranty by a nongovernment third-party entity such as a private insurance company. This is because insurance companies themselves can face financial difficulties. This fact was brought home to market participants at the end of 2007 when specialized insurance companies that provide financial guarantees faced financial difficulties and the downgrading of their own credit rating. Thus, one would disagree with the statement because one’s bond guarantee is only as good as the insurance company guaranteeing it. 18. Answer the below questions. (a) What is counterparty risk? Counterparty risk is a form of credit risk that involves transactions between two parties in a trade. The risk to each party of a contract is that the counterparty (or other party) will not be able to live up to its contractual obligations. In financial contracts, counterparty risk is known as “default risk.” (b) Give two examples of transactions where one faces counterparty risk. For a first example of counterparty risk, consider the strategy of a borrower using the borrowed funds from a lender to purchase another asset such as a bond. In this transaction, the lender is exposed to counterparty risk. Counterparty risk is the risk that the borrower will fail to repay the loan if his bond purchase defaults. A second example of counterparty risk involves a trade in a derivative (which is an investment that derives its value from the value of an underlying asset). A derivative, such as an option or a futures contract, is traded on an exchange that becomes the ultimate counterparty to the trade as it guarantees payments on money owed to the purchaser of the derivative instrument. For derivative instruments that are over-the-counter instruments, the counterparty is an entity other than an exchange. In such trades, there is considerable concern with counterparty risk. 19. Does an investor who purchases a zero-coupon bond face reinvestment risk? The calculation of the yield of a coupon paying bond assumes that the cash flows received are reinvested at the prevailing rate when the coupon payment is received. Because this rate is not known in advance it creates uncertainty and so it is called by the name of reinvestment risk to indicate there is risk or uncertainty in the reinvesting of coupon payments. For zero-coupon bonds, unlike bonds that pay a stream of coupon payments over time, the payment is reinvested at the same rate as the coupon rate. This eliminates any risk associated with the possibility that coupon payments will be reinvested at a lower rate. However, if rates go up, then the zero coupon bond will fall in value because its “locked-in” rate is below the higher market rate. 20. What is meant by marking a position to market? Marking a position to market means that periodically the market value of a portfolio must be determined. Thus, it can refer to the practice of reporting the value of assets on a market rather than book value basis. Marking to market can also refer to settling or reconciling changes in the value of futures contracts on a daily basis. 21. What is meant by a CUSIP number, and why is it important? By a CUSIP number, we mean a unique number assigned to a firm’s security to identify it. Thus, its importance lies in its ability to singularly identify each security. Most securities are identified by the characters (letters and numbers) found in its CUSIP number. CUSIP stands for Committee on Uniform Security Identification Procedures. For securities that have nine characters, the first six characters identify the issuer: the corporation, government agency, or municipality. The next two characters identify whether the issue is debt or equity and the issuer of the issue. The last character is simply a check character that allows for accuracy checking and is sometimes truncated or ignored. The CUSIP International Numbering System (CINS) is used to identify foreign securities and includes 12 characters. CHAPTER 2 PRICING OF BONDS CHAPTER SUMMARY This chapter will focus on the time value of money and how to calculate the price of a bond. When pricing a bond it is necessary to estimate the expected cash flows and determine the appropriate yield at which to discount the expected cash flows. Among other aspects of a bond, we will look at the reasons why the price of a bond changes REVIEW OF TIME VALUE OF MONEY Money has time value because of the opportunity to invest it at some interest rate. Future Value The future value of any sum of money invested today is: Pn = P0(1 + r)n where n = number of periods, Pn = future value n periods from now (in dollars), P0 = original principal (in dollars), r = interest rate per period (in decimal form), and the expression (1 + r)n represents the future value of $1 invested today for n periods at a compounding rate of r. When interest is paid more than one time per year, both the interest rate and the number of periods used to compute the future value must be adjusted as follows: r = annual interest rate / number of times interest paid per year, and n = number of times interest paid per year times number of years. The higher future value when interest is paid semiannually, as opposed to annually, reflects the greater opportunity for reinvesting the interest paid. Future Value of an Ordinary Annuity When the same amount of money is invested periodically, it is referred to as an annuity. When the first investment occurs one period from now, it is referred to as an ordinary annuity. The equation for the future value of an ordinary annuity is: Pn = (1+ ) −1 r n A r where A is the amount of the annuity (in dollars). Example of Future Value of an Ordinary Annuity Using Annual Interest: If A = $2,000,000, r = 0.08, and n = 15, then Pn = (1 r)n 1 A r + − ➔ P15 = 0.08 3.17217 1 $2,000,000 − = $2,000,000[27.152125] = $54,304.250. Because 15($2,000,000) = $30,000,000 of this future value represents the total dollar amount of annual interest payments made by the issuer and invested by the portfolio manager, the balance of $54,304,250 – $30,000,000 = $24,304,250 is the interest earned by reinvesting these annual interest payments. Example of Future Value of an Ordinary Annuity Using Semiannual Interest: Consider the same example, but now we assume semiannual interest payments. If A = $2,000,000 / 2 = $1,000,000, r = 0.08 / 2 = 0.04, n = 2(15) = 30, then Pn = (1 r)n 1 A r + − ➔ P30 = 0.04 (1 .04) 1 $1,000,000 30 − = 0.04 3.2434 1 $1,000,000 − = $1,000,000[56.085] = $56,085,000. The opportunity for more frequent reinvestment of interest payments received makes the interest earned of $26,085,000 from reinvesting the interest payments greater than the $24,304,250 interest earned when interest is paid only one time per year. Present Value The present value is the future value process in reverse. We have: ( ) 1 1 n PV r = + . For a given future value at a specified time in the future, the higher the interest rate (or discount rate), the lower the present value. For a given interest rate (discount rate), the further into the future that the future value will be received, then the lower its present value. Present Value of a Series of Future Values To determine the present value of a series of future values, the present value of each future value must first be computed. Then these present values are added together to obtain the present value of the entire series of future values. Present Value of an Ordinary Annuity When the same dollar amount of money is received each period or paid each year, the series is referred to as an annuity. When the first payment is received one period from now, the annuity is called an ordinary annuity. When the first payment is immediate, the annuity is called an annuity due. The present value of an ordinary annuity is: ( ) 1 1 PV = A 1 r n r − + where A is the amount of the annuity (in dollars). The term in brackets is the present value of an ordinary annuity of $1 for n periods. Example of Present Value of an Ordinary Annuity Using Annual Interest: If A = $100, r = 0.09, and n = 8, then: ( ) 1 1 PV = A 1 r n r − + = ( )8 1 1 $100 1.09 0.09 − 1 1 $100 1.99256 0.09 − = = $100 1 0.501867 0.09 − = $100[5.534811] = $553.48. Present Value When Payments Occur More Than Once Per Year If the future value to be received occurs more than once per year, then the present value formula is modified so that (i) the annual interest rate is divided by the frequency per year, and (ii) the number of periods when the future value will be received is adjusted by multiplying the number of years by the frequency per year. PRICING A BOND Determining the price of any financial instrument requires an estimate of (i) the expected cash flows, and (ii) the appropriate required yield. The required yield reflects the yield for financial instruments with comparable risk, or alternative investments. The cash flows for a bond that the issuer cannot retire prior to its stated maturity date consist of periodic coupon interest payments to the maturity date, and the par (or maturity) value at maturity. In general, the price of a bond can be computed using the following formula: 1 (1 ) (1 ) n t t n t= P = C + M + r + r . where P = price (in dollars), n = number of periods (number of years times 2), C = semiannual coupon payment (in dollars), r = periodic interest rate (required annual yield divided by 2), M = maturity value, and t = time period when the payment is to be received. Computing the Value of a Bond: An Example: Consider a 20-year 10% coupon bond with a par value of $1,000 and a required yield of 11%. Given C = 0.1($1,000) / 2 = $50, n = 2(20) = 40 and r = 0.11 / 2 = 0.055, the present value of the coupon payments is: ( ) 1 1 P = C 1 r n r − + = ( )40 1 1 $50 1.055 0.055 − = 1 1 $50 8.51332 0.055 − = 1 0.117463 $50 0.055 − = $50 16.046131 = $802.31. The present value of the par or maturity value of $1,000 is: (1+Mr)n = ( )40 $1,000 1 .055 = 8.51331 $1,000 = $117.46. The price of the bond (P) = present value coupon payments + present value maturity value = $802.31 + $117.46 = $919.77. Pricing Zero-Coupon Bonds For zero-coupon bonds, the investor realizes interest as the difference between the maturity value and the purchase price. The equation is: (1 )n M P r = + where M is the maturity value. Thus, the price of a zero-coupon bond is simply the present value of the maturity value. Zero-Coupon Bond Example Consider the price of a zero-coupon bond that matures 15 years from now, if the maturity value is $1,000 and the required yield is 9.4%. Given M = $1,000, r = 0.094 / 2 = 0.047, and n = 2(15) = 30, we have: (1 )n P = M + r = ($1, )30 000 1 .047 = 3.99644 $1,000 = $252.12. Price-Yield Relationship A fundamental property of a bond is that its price changes in the opposite direction from the change in the required yield. The reason is that the price of the bond is the present value of the cash flows. Relationship Between Coupon Rate, Required Yield, and Price When yields in the marketplace rise above the coupon rate at a given point in time, the price of the bond falls so that an investor buying the bond can realizes capital appreciation. The appreciation represents a form of interest to a new investor to compensate for a coupon rate that is lower than the required yield. When a bond sells below its par value, it is said to be selling at a discount. A bond whose price is above its par value is said to be selling at a premium. Relationship Between Bond Price and Time if Interest Rates Are Unchanged For a bond selling at par value, the coupon rate is equal to the required yield. As the bond moves closer to maturity, the bond will continue to sell at par value. Its price will remain constant as the bond moves toward the maturity date. The price of a bond will not remain constant for a bond selling at a premium or a discount. The discount bond increases in price as it approaches maturity, assuming that the required yield does not change. For a premium bond, the opposite occurs. For both bonds, the price will equal par value at the maturity date. Reasons for the Change in the Price of a Bond The price of a bond can change for three reasons: (i) there is a change in the required yield owing to changes in the credit quality of the issuer; (ii) there is a change in the price of the bond selling at a premium or a discount, without any change in the required yield, simply because the bond is moving toward maturity; or, (iii) there is a change in the required yield owing to a change in the yield on comparable bonds (i.e., a change in the yield required by the market). COMPLICATIONS The framework for pricing a bond assumes the following: (i) the next coupon payment is exactly six months away; (ii) the cash flows are known; (iii) the appropriate required yield can be determined; and, (iv) one rate is used to discount all cash flows. Next Coupon Payment Due in Less than Six Months When an investor purchases a bond whose next coupon payment is due in less than six months, the accepted method for computing the price of the bond is as follows: 1 1 =1 = (1 + r ) n v t v n t C M P (1 + r ) − + (1 + r ) (1 + r ) − where v = (days between settlement and next coupon) / (days in six-month period). Cash Flows May Not Be Known For most bonds, the cash flows are not known with certainty. This is because an issuer may call a bond before the stated maturity date. Determining the Appropriate Required Yield All required yields are benchmarked off yields offered by Treasury securities. From there, we must still decompose the required yield for a bond into its component parts. One Discount Rate Applicable to All Cash Flows A bond can be viewed as a package of zero-coupon bonds, in which case a unique discount rate should be used to determine the present value of each cash flow. PRICING FLOATING-RATE AND INVERSE-FLOATING-RATE SECURITIES The cash flow is not known for either a floating-rate or an inverse-floating-rate security; it will depend on the reference rate in the future. Price of a Floater The coupon rate of a floating-rate security (or floater) is equal to a reference rate plus some spread or margin. The price of a floater depends on (i) the spread over the reference rate and (ii) any restrictions that may be imposed on the resetting of the coupon rate. Price of an Inverse Floater In general, an inverse floater is created from a fixed-rate security. The security from which the inverse floater is created is called the collateral. From the collateral two bonds are created: a floater and an inverse floater. The price of a floater depends on (i) the spread over the reference rate and (ii) any restrictions that may be imposed on the resetting of the coupon rate. For example, a floater may have a maximum coupon rate called a cap or a minimum coupon rate called a floor. The price of a floater will trade close to its par value as long as the spread above the reference rate that the market requires is unchanged, and neither the cap nor the floor is reached. The price of an inverse floater equals the collateral’s price minus the floater’s price. PRICE QUOTES AND ACCRUED INTEREST Price Quotes A bond selling at par is quoted as 100, meaning 100% of its par value. A bond selling at a discount will be selling for less than 100; a bond selling at a premium will be selling for more than 100. Accrued Interest When an investor purchases a bond between coupon payments, the investor must compensate the seller of the bond for the coupon interest earned from the time of the last coupon payment to the settlement date of the bond. This amount is called accrued interest. For corporate and municipal bonds, accrued interest is based on a 360-day year, with each month having 30 days. The amount that the buyer pays the seller is the agreed-upon price plus accrued interest. This is often referred to as the full price or dirty price. The price of a bond without accrued interest is called the clean price. The exceptions are bonds that are in default. Such bonds are said to be quoted flat, that is, without accrued interest. KEY POINTS • The price of a bond is the present value of the bond’s expected cash flows, the discount rate being equal to the yield offered on comparable bonds. For an option-free bond, the cash flows are the coupon payments and the par value or maturity value. The higher (lower) the required yield, the lower (higher) the price of a bond. • For a zero-coupon bond, there are no coupon payments. The price of a zero-coupon bond is equal to the present value of the maturity value, where the number of periods used to compute the present value is double the number of years and the discount rate is a semiannual yield. • A bond’s price changes in the opposite direction from the change in the required yield. The reason is that as the required yield increases (decreases), the present value of the cash flow decreases (increases). • A bond will be priced below, at par, or above par depending the bond’s coupon rate and the required yield required by investors. When the coupon rate is equal to the required yield, the bond will sell at its par value. When the coupon rate is less (greater) than the required yield, the bond will sell for less (more) than its par value. • Over time, the price of a premium or discount bond will change even if the required yield does not change. Assuming that the credit quality of the issuer is unchanged, the price change on any bond can be decomposed into a portion attributable to a change in the required yield and a portion attributable to the time path of the bond. • The price of a floating-rate bond will trade close to par value if the spread required by the market does not change and there are no restrictions on the coupon rate. • The price of an inverse floater depends on the price of the collateral from which it is created and the price of the floater. • Accrued interest is the amount that a bond buyer who purchases a bond between coupon payments must pay the bond seller. The amount represents the coupon interest earned from the time of the last coupon payment to the settlement date of the bond. ANSWERS TO QUESTIONS FOR CHAPTER 2 (Questions are in bold print followed by answers.) 1. A pension fund manager invests $10 million in a debt obligation that promises to pay 7.3% per year for four years. What is the future value of the $10 million? To determine the future value of any sum of money invested today, we can use the future value equation, which is: Pn = P0 (1 + r)n where n = number of periods, Pn = future value n periods from now (in dollars), P0 = original principal (in dollars) and r = interest rate per period (in decimal form). Inserting in our values, we have: P4 = $10,000,000(1.073)4 = $10,000,000(1.325558466) = $13,255,584.66. 2. Suppose that a life insurance company has guaranteed a payment of $14 million to a pension fund 4.5 years from now. If the life insurance company receives a premium of $10.4 million from the pension fund and can invest the entire premium for 4.5 years at an annual interest rate of 6.25%, will it have sufficient funds from this investment to meet the $14 million obligation? To determine the future value of any sum of money invested today, we can use the future value equation, which is: Pn = P0 (1 + r)n where n = number of periods, Pn = future value n periods from now (in dollars), P0 = original principal (in dollars) and r = interest rate per period (in decimal form). Inserting in our values, we have: P4.5 = $10,400,000(1.0625)4.5 = $10,400,000(1.313651676) = $13,661,977.43. Thus, it will be short by: $13,661,977.43 – $14,000,000 = −$338,022.57. 3. Answer the below questions. (a) The portfolio manager of a tax-exempt fund is considering investing $500,000 in a debt instrument that pays an annual interest rate of 5.7% for four years. At the end of four years, the portfolio manager plans to reinvest the proceeds for three more years and expects that for the three-year period, an annual interest rate of 7.2% can be earned. What is the future value of this investment? At the end of year four, the portfolio manager’s amount is given by: Pn = P0 (1 + r)n. Inserting in our values, we have P4 = $500,000(1.057)4 = $500,000(1.248245382) = $624,122.66. In three more years at the end of year seven, the manager amount is given by: P7 = P4(1 + r)3. Inserting in our values, we have: P7 = $624,122.66(1.072)3 = $624,122.66(1.231925248) = $768,872.47. (b) Suppose that the portfolio manager in Question 3, part a, has the opportunity to invest the $500,000 for seven years in a debt obligation that promises to pay an annual interest rate of 6.1% compounded semiannually. Is this investment alternative more attractive than the one in Question 3, part a? At the end of year seven, the portfolio manager’s amount is given by the following equation, which adjusts for semiannual compounding. We have: Pn = P0(1 + r/2)2(n). Inserting in our values, we have P7 = $500,000(1 + 0.061/2)2(7) = $500,000(1.0305)14 = $500,000(1.522901960) = $761,450.98. Thus, this investment alternative is not more attractive. It is less by the amount of $761,450.98 – $768,872.47 = −$7,421.49. 4. Suppose that a portfolio manager purchases $10 million of par value of an eight-year bond that has a coupon rate of 7% and pays interest once per year. The first annual coupon payment will be made one year from now. How much will the portfolio manager have if she (1) holds the bond until it matures eight years from now, and (2) can reinvest all the annual interest payments at an annual interest rate of 6.2%? At the end of year eight, the portfolio manager’s amount is given by the following equation, which adjusts for annual compounding. We have: (1 + ) - 1 = Par Value n n r P A r + where A = coupon rate times par value. Inserting in our values, we have: 8 8 (1 + 0.062) 1 = 0.07($10,000,000) P 0.062 − + $10,000,000 = $700,000[9.9688005] + $10,000,000 = $6,978,160.38 + $10,000,000 = $16,978,160.38. 5. Answer the below questions. (a) If the discount rate that is used to calculate the present value of a debt obligation’s cash flow is increased, what happens to the price of that debt obligation? The price will fall. A fundamental property of a bond is that its price changes in the opposite direction from the change in the required yield. The reason is that the price of the bond is the present value of the cash flows. As the required yield increases, the present value of the cash flow decreases; thus the price decreases. The opposite is true when the required yield decreases: The present value of the cash flows increases, and therefore the price of the bond increases. (b) Suppose that the discount rate used to calculate the present value of a debt obligation’s cash flow is x%. Suppose also that the only cash flows for this debt obligation are $200,000 four years from now and $200,000 five years from now. For which of these cash flows will the present value be greater? Cash flows that come earlier will have a greater value. As long as x% is positive and the amount is the same, the present value will be greater for the $200,000 four years from now compared to five years from now. This can also be seen by noting that if x > 0 then ( )4 ( )5 1 1 1 x 1 x + + . The latter inequality implies ( )4 ( )5 1 1 $2,000 $2,000 1 x 1 x + + will hold. 6. The pension fund obligation of a corporation is calculated as the present value of the actuarially projected benefits that will have to be paid to beneficiaries. Why is the interest rate used to discount the projected benefits important? It is important because the present value increases as the discount rate (or interest rate) decreases and it decreases as the discount rate increases. Thus, in order to project the benefits accurately, we need an accurate estimate of the discount rate. If we underestimate the discount rate then we will be projecting more available pension funds than we will actually have. 7. A pension fund manager knows that the following liabilities must be satisfied: Years from Now Liability (in millions) 1 $2.0 2 $3.0 3 $5.4 4 $5.8 Suppose that the pension fund manager wants to invest a sum of money that will satisfy this liability stream. Assuming that any amount that can be invested today can earn an annual interest rate of 7.6%, how much must be invested today to satisfy this liability stream? To satisfy year one’s liability (n = 1), the pension fund manager must invest an amount today that is equal to the future value of $2.0 million at 7.6%. We have: ( ) 1 PV Pn 1 + r n = = (1.076) 1 $2,000,000 1 = $2,000,0000.929368030 = $1,858,736.06. To satisfy year two’s liability (n = 2), the pension fund manager must invest an amount today that is equal to the future value of $3.0 million at 7.6%. We have: ( ) 1 PV Pn 1 + r n = = $3,000,000 ( 1 2 1.076) = $3,000,000 0.863724935 = $2,591,174.80. To satisfy year three’s liability (n = 3), the pension fund manager must invest an amount today that is equal to the future value of $5.4 million at 7.6%. We have: ( ) 1 PV Pn 1 + r n = = $5,400,000 ( 1 3 1.076) = $5,400,000 0.802718341 = $4,334,679.04. To satisfy year four’s liability (n = 4), the pension fund manager must invest an amount today that is equal to the future value of $5.8 million at 7.6%. We have: ( ) 1 PV Pn 1 + r n = = (1.076) 1 $5,800,000 4 = $5,800,0000.746020763 = $4,326,920.42. If we add the four present values, we get $1,858,736.06 + $2,591,174.80 + $4,334,679.04 + $4,326,920.42 = $13,111,510.32, which is the amount the pension fund manager needs to invest today to cover the liability stream for the next four years. 8. Calculate for each of the following bonds the price per $1,000 of par value assuming semiannual coupon payments. Bond Coupon Rate (%) Years to Maturity Required Yield (%) A 8 9 7 B 9 20 9 C 6 15 10 D 0 14 8 Consider a 9-year 8% coupon bond with a par value of $1,000 and a required yield of 7%. Given C = 0.08($1,000) / 2 = $40, n = 2(9) = 18 and r = 0.07 / 2 = 0.035, the present value of the coupon payments is: ( ) 1 1 P = C 1 r n r − + = ( )18 1 1 $40 1.035 0.035 − = 1 1 $40 1.857489196 0.035 − 1 0.538361140 $40 0.035 − = = $40 13.189681727 = $527.587. The present value of the par or maturity value of $1,000 is: (1 )n M + r = $1,00018 (1.035) = $1,000 1.8574892 = $538.361. Thus, the price of the bond (P) = present value of coupon payments + present value of par value = $527.587 + $538.361 = $1,065.95. Consider a 20-year 9% coupon bond with a par value of $1,000 and a required yield of 9%. Given C = 0.09($1,000) / 2 = $45, n = 2(20) = 40 and r = 0.09 / 2 = 0.045, the present value of the coupon payments is: ( ) 1 1 P = C 1 r n r − + = ( )40 1 1 $45 1.045 0.045 − = 1 1 $45 5.81863645 0.045 − = 1 0.1719287 $45 0.045 − = $45[18.401584] = $828.071. The present value of the par or maturity value of $1,000 is: (1+Mr)n = ( )40 $1,000 1 .045 = $1,000 5.81863645 = $171.929. Thus, the price of the bond (P) = $828.071 + $171.929 = $1,000.00. [NOTE. We already knew the answer would be $1,000 because the coupon rate equals the yield to maturity.] Consider a 15-year 6% coupon bond with a par value of $1,000 and a required yield of 10%. Given C = 0.06($1,000) / 2 = $30, n = 2(15) = 30 and r = 0.10 / 2 = 0.05, the present value of the coupon payments is: ( ) 1 1 P = C 1 r n r − + = ( )30 1 1 $30 1.05 0.05 − = 1 1 $30 4.3219424 0.05 − = 1 0.2313774 $30 0.05 − = $30[15.372451] = $461.174. The present value of the par or maturity value of $1,000 is: (1+Mr)n = ( )30 $1,000 1 .05 = 4.3219424 $1,000 = $231.377. Thus, the price of the bond (P) = $461.174 + $231.377 = $692.55. Consider a 14-year 0% coupon bond with a par value of $1,000 and a required yield of 8%. Given C = 0($1,000) / 2 = $0, n = 2(14) = 28 and r = 0.08 / 2 = 0.04, the present value of the coupon payments is: ( ) 1 1 P = C 1 r n r − + = 28 1 1 (1.04) $0 0.04 − = 1 1 $0 2.998703319 0.055 − = 0.055 1 0.33477471 $0 − = $0[16.66306322] = $0. [NOTE. We already knew the answer because the coupon rate is zero.] The present value of the par or maturity value of $1,000 is: (1+Mr)n = ( )28 $1,000 1 .04 = $1,000 2.99870332 = $333.48. Thus, the price of the bond (P) = $0 + $333.48 = $333.48. 9. Consider a bond selling at par ($100) with a coupon rate of 6% and 10 years to maturity. (a) What is the price of this bond if the required yield is 15%? We have a 10-year 6% coupon bond with a par value of $1,000 and a required yield of 15%. Given C = 0.06($1,000) / 2 = $30, n = 2(10) = 20 and r = 0.15 / 2 = 0.075, the present value of the coupon payments is: ( ) 1 1 P = C 1 r n r − + = 20 1 1 (1.075) $30 0.075 − = 1 1 $30 4.2478511 0.075 − = 1 0.2354131 $30 0.075 − = $30[10.1944913] = $305.835. The present value of the par or maturity value of $1,000 is: (1+Mr)n = ( )20 $1,000 1 .075 = $1,000 4.2478511 = $235.413. Thus, the price of the bond (P) = $305.835 + $235.413 = $541.25. (b) What is the price of this bond if the required yield increases from 15% to 16%, and by what percentage did the price of this bond change? If the required yield increases from 15% to 16%, then we have: ( ) 1 1 P = C 1 r n r − + = 20 1 1 (1.08) $30 0.08 − = $30 9.8181474 = $294.544. The present value of the par or maturity value of $1,000 is: (1 )n M + r = ( )20 $1,000 1 .08 = $214.548. Thus, the price of the bond (P) = $294.544 + $214.548 = $509.09. The bond price falls with percentage fall is equivalent to $509.09 $541.25 $541.25 − = −0.059409 or about −5.94%. (c) What is the price of this bond if the required yield is 5%? If the required yield is 5%, then we have: ( ) 1 1 P = C 1 r n r − + = 20 1 1 (1.025) $30 0.025 − = $30 15.5891623 = $467.675. The present value of the par or maturity value of $1,000 is: (1+Mr)n = ( )20 $1,000 1 .025 = $610.271. Thus, the price of the bond (P) = $467.675 + $610.271 = $1,077.95. (d) What is the price of this bond if the required yield increases from 5% to 6%, and by what percentage did the price of this bond change? If the required yield increases from 5% to 6%, then we have: ( ) 1 1 P = C 1 r n r − + = 20 1 1 (1.03) $30 0.03 − = $30 14.87747486 = $446.324. The present value of the par or maturity value of $1,000 is: (1+Mr)n = 20 $1,000 (1.03) = $553.676. The price of the bond (P) = $446.324 + $553.676 = $1,000.00. [NOTE. We already knew the answer would be $1,000 because the coupon rate equals the yield to maturity.] The bond price falls with the percentage fall equal to ($1,000.00 – $1,077.95) / $1,077.95 = −0.072310 or about −7.23%. (e) From your answers to Question 9, parts b and d, what can you say about the relative price volatility of a bond in a high-interest-rate environment compared to a low-interest-rate environment? We can say that there is more volatility in a low-interest-rate environment because there was a greater fall (−7.23% versus −5.94%). 10. Suppose that you purchased a debt obligation three years ago at its par value of $100,000 and nine years remaining to maturity. The market price of this debt obligation today is $90,000. What are some reasons why the price of this debt obligation could have declined from time you purchased it three years ago? The price of a bond will change for one or more of the following three reasons: (i) There is a change in the required yield owing to changes in the credit quality of the issuer. (ii) There is a change in the price of the bond selling at a premium or a discount, without any change in the required yield, simply because the bond is moving toward maturity. (iii) There is a change in the required yield owing to a change in the yield on comparable bonds (i.e., a change in the yield required by the market). The first and third reasons are the likely reasons for the situation where the bond has plummeted from $100,000 to $90,000. The bond has plummeted in value because the credit quality of the issuer has fallen and/or the bond has plummeted because the yield on comparable bonds has increased. 11. Suppose that you are reviewing a price sheet for bonds and see the following prices (per $100 par value) reported. You observe what seem to be several errors. Without calculating the price of each bond, indicate which bonds seem to be reported incorrectly, and explain why. Bond Price Coupon Rate (%) Required Yield (%) U 90 6 9 V 96 9 8 W 110 8 6 X 105 0 5 Y 107 7 9 Z 100 6 6 If the required yield is the same as the coupon rate then the price of the bond should sell at its par value. This is the case of bond Z since par values are typical at or near a $100 quote. If the required yield decreases below the coupon rate then the price of a bond should increase. This is the case for bond W. This is not the case for bond V so this bond is not reported correctly. If the required yield increases above the coupon rate then the price of a bond should decrease. This is the case for bond U. This is not the case for bonds X and Y so these bonds are not reported correctly. Thus, bonds V, X, and Y are incorrectly reported because the change in the bond price is not consistent with the difference between the coupon rate and the required yield. 12. What is the maximum price of a bond? Consider an extreme case of a 100-year 20% coupon bond with a par value of $1,000 that after one year falls so that the required yield is 1%. Given C = 0.2($1,000) / 2 = $100, n = 2(99) = 198 and r = 0.01 / 2 = 0.005, the present value of the coupon payments is: ( ) 1 1 P = C 1 r n r − + = 198 1 1 (1.005) $100 0.005 − = 1 1 $100 2.684604 0.005 − 1 0.3724944 $100 0.005 − = = $1,000[1,125.51012] = $12,550.112. The present value of the par value of $1,000 is: (1+Mr)n 198 $1,000 = (1.005) $1,000 = 2.684604 = $372.494. Thus, the price of the bond (P) = $12,550.112 + $372.494 = $12,922.61. This is a percent increase of ($12,922.6 – $1,000) / $1,000 = 11.92606 or about 1,192.61%. If the required yield falls to 0.001%, then the bond price would increase to $20,778.33, which would be a percent increase of about 1,977.83%. If the required yield falls to 0.00001%, then the bond price would increase to $20,778.33, which would be a percent increase of about 1,977.83%. If the required yield falls to 0.0000000001%, then the bond price would increase to $20,801.76, which would be a percent increase of about 1,980.18%. Thus, we see that even for these extreme numbers (that are highly unlikely), we find there appears to be a limit on how high a bond price might rise assuming that rates do not reach negative numbers. If the required yield is a negative number then there would be no limit to how high a bond price might rise. For example, if the required yield becomes a negative 1%, then the bond price would increase to $70,468.18. If it becomes a negative 10%, then the bond price becomes $2,296,218,049,925.23 or about $2.3 trillion. 13. What is the “dirty” price of a bond? The “dirty” (or “full”) price is the amount that the buyer agrees to pay the seller, which is the agreed-upon price plus accrued interest. The price of a bond without accrued interest is called the clean price. The exceptions are bonds that are in default. Such bonds are said to be quoted flat, that is, without accrued interest. 14. Explain why you agree or disagree with the following statement: “The price of a floater will always trade at its par value.” One would disagree with the statement: “The price of a floater will always trade at its par value.” First, the coupon rate of a floating-rate security (or floater) is equal to a reference rate plus some spread or margin. For example, the coupon rate of a floater can reset at the rate on a three-month Treasury bill (the reference rate) plus 50 basis points (the spread). Next, the price of a floater depends on two factors: (1) the spread over the reference rate and (2) any restrictions that may be imposed on the resetting of the coupon rate. For example, a floater may have a maximum coupon rate called a cap or a minimum coupon rate called a floor. The price of a floater will trade close to its par value as long as (1) the spread above the reference rate that the market requires is unchanged and (2) neither the cap nor the floor is reached. However, if the market requires a larger (smaller) spread, the price of a floater will trade below (above) par. If the coupon rate is restricted from changing to the reference rate plus the spread because of the cap, then the price of a floater will trade below par. 15. Explain why you agree or disagree with the following statement: “The price of an inverse floater will increase when the reference rate decreases.” As explained below, one would disagree with the statement: “The price of an inverse floater will increase when the reference rate decreases.” The factors that affect the price of an inverse floater are affected by the reference rate only to the extent that it affects the restrictions on the floater’s rate. This is quite an important result. Some investors mistakenly believe that because the coupon rate rises, the price of an inverse floater should increase if the reference rate decreases. This is not true. The key in pricing an inverse floater is how changes in interest rates affect the price of the collateral. The reference rate is important only to the extent that it restricts the coupon rate of the floater. More details are given below. In general, an inverse floater is created from a fixed-rate security. The security from which the inverse floater is created is called the collateral. From the collateral two bonds are created: a floater and an inverse floater. The two bonds are created such that (1) the total coupon interest paid to the two bonds in each period is less than or equal to the collateral’s coupon interest in each period, and (2) the total par value of the two bonds is less than or equal to the collateral’s total par value. Suppose the total par value of the floater and inverse floater equals the par value of the collateral. Regardless of the level of the reference rate, the combined coupon rate for the two bonds is equal to the coupon rate of the collateral. However, if the reference rate exceeds a certain percentage, then the formula for the coupon rate for the inverse floater will be negative. To prevent this from happening, a floor is placed on the coupon rate for the inverse floater. Typically, the floor is set at zero. Because of the floor, the coupon rate on the floater must be restricted so that the coupon interest paid to the two bonds does not exceed the collateral’s coupon interest. Thus, when a floater and an inverse floater are created from the collateral, a floor is imposed on the inverse and a cap is imposed on the floater. The price of an inverse floater is found by determining the price of the collateral and the price of the floater. This can be seen as follows: collateral’s price = floater’s price + inverse’s price. Therefore, inverse’s price = collateral’s price – floater’s price. Solution Manual for Bond Markets, Analysis and Strategies Frank J. Fabozzi 9780132743549, 9780133796773
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