Interest Rates and Bond Valuation PDF

CHAPTER 7 I N T E R E S T R AT E S A N D B O N D V A L U AT I O N Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  LEARNING OBJECTIVES Define important bond features and types of bonds Expla...

CHAPTER 7 I N T E R E S T R AT E S A N D B O N D V A L U AT I O N Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  LEARNING OBJECTIVES Define important bond features and types of bonds Explain bond values and yields and why they fluctuate Describe bond ratings and what they mean Outline the impact of inflation on interest rates Illustrate the term structure of interest rates and the determinants of bond yields 1-2 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  CHAPTER OUTLINE Bonds and Bond Valuation More about Bond Features Bond Ratings Some Different Types of Bonds Bond Markets Inflation and Interest Rates Determinants of Bond Yields 1-3 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  BOND FEATURES AND PRICES When a corporation or government wishes to borrow money from the public on a long-term basis, it usually does so by issuing or selling debt securities that are generically called bonds Normally interest-only loans Suppose the Beck Corporation wants to borrow $1,000 for 30 years. The interest rate on similar debt issued by similar corporations is 12%. Beck will thus pay.12 × $1,000 = $120 in interest every year for 30 years. At the end of 30 years, Beck will repay the $1,000. Coupon is the stated interest payment made on a bond (i.e., $120) Face value, or par value, is the principal amount of a bond that is repair at the end of the term (i.e., $1,000) Coupon rate is the annual coupon divided by the face value of the bond (i.e., $120/$1,000 =.12, or 12%) 1-4 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  BOND VALUES AND YIELDS Over time, interest rates change in the marketplace, but the cash flows from a bond remain the same; as a result, the value of the bond will fluctuate When interest rates rise, the present value of the bond’s remaining cash flows declines, and the bond is worth less When interest rates fall, the bond is worth more To find the value of a bond at a particular point in time, we need the following pieces of information: Number of periods remaining until maturity Face value Coupon Yield to maturity (YTM), or yield, which is the rate 1-5 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  BOND VALUES AND YIELDS: AN EXAMPLE (PAR VALUE BOND) Suppose Xanth Co. were to issue a bond with 10 years to maturity. The Xanth bond has an annual coupon of $80. Similar bonds have a yield to maturity of 8 percent. What would this bond sell for? Bond will pay $80 per year for the next 10 years in coupon interest. In 10 years, Xanth will pay $1,000 to the bond owner. At the going rate of 8%, the PV of the $1,000 paid in 10 years is: Present value = $1,000/1.0810 = $1,000/2.1589 = $463.19 Bond offers $80 per year for 10 years; PV of this annuity stream is: Annuity present value = $80 × (1 − 1/1.0810) /.08 = $80 × (1 − 1/2.1589) /.08 = $80 × 6.7101 1-6 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  CASH FLOWS FOR XANTH CO. BOND Cash flows for the bond have an annuity component (the coupons) and a lump sum (the face value paid at maturity) Cash flows for the bond are shown below: 1-7 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  BOND VALUES AND YIELDS: AN EXAMPLE (DISCOUNT BOND) Suppose a year has gone by. The Xanth bond now has nine years to maturity. If the interest rate in the market has risen to 10%, what will the bond be worth? Present value of the $1,000 paid in nine years at 10 percent is: Present value = $1,000/1.109 = $1,000/2.3579 = $424.10 Bond now offers $80 per year for nine years; the present value of this annuity stream at 10% is: Annuity present value = $80 × (1 − 1/1.109) /.10 = $80 × (1 − 1/2.3579) /.10 = $80 × 5.7590 = $460.72 Add the values for the two parts together to get the 1-8 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  BOND VALUES AND YIELDS: AN EXAMPLE (PREMIUM BOND) What would the Xanth bond sell for if interest rates had dropped by 2% instead of rising by 2%? In other words, assume the bond has a coupon rate of 8% when the market rate is now only 6%. The present value of the $1,000 face amount is: Present value = $1,000/1.069 = $1,000/1.6895 = $591.90 The present value of the coupon stream is: Annuity present value = $80 × (1 − 1/1.069) /.06 = $80 × (1 − 1/1.6895) /.06 = $80 × 6.8017 = $544.14 Add the values for the two parts together to get the bond’s value: 1-9 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill. GENERAL EXPRESSION FOR VALUE OF A BOND If a bond has the following: A face value of F paid at maturity; A coupon of C paid per period; t periods to maturity; and A yield of r per period Its value is calculated as follows: 1-10 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  SEMIANNUA L COUPONS 1-11 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  INTEREST RATE RISK Interest rate risk arises from fluctuating interest rates, and the degree of interest rate risk a bond has depends on its sensitivity to interest rate changes, which directly depends on two things: 1. Time to maturity 2. Coupon rate All other things being equal: 1. The longer the time to maturity, the greater the interest rate risk 2. The lower the coupon rate, the greater the interest rate risk Interest rate risk increases at a decreasing rate We can illustrate the effect of interest rate risk using the 100-year BellSouth (AT&T) issue: 1-12 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill. INTEREST RATE RISK AND TIME TO MATURITY Notice the slope of the line connecting prices is much steeper for the 30- year bond than for the 1- year bond This tells us a relatively small change in interest rates will lead to a substantial change in the bond’s value The reason bonds with lower coupons have greatest interest rate risk is essentially the same The bond with the 1-13 higher Copyright coupon © 2022 McGraw Hill. All rightshas reserved.a No reproduction or distribution without the prior written consent of McGraw Hill. FINDING THE YIELD TO MATURITY: MORE TRIAL AND ERROR Suppose we are interested in a six-year, 8% coupon bond with annual payments. A broker quotes a price of $955.14. What is the yield on this bond? Price of a bond can be written as the sum of its annuity and lump sum components. Knowing there is an $80 coupon for six years and a $1,000 face value, we can say that the price is: $955.14 = $80 × [1 − 1/(1 + r)6]/r + 1,000/(1 + r)6 where r is the unknown discount rate, or yield to maturity; we cannot solve it for r explicitly so we must use trial and error Bond has an $80 coupon and is selling at a discount, so the yield is greater than 8 percent. If we compute the price at 10%, we will see 10% is too high because the value we calculate ($912.89) is lower than the actual price ($955.14): Bond value = $80 × (1 − 1/1.106) /.10 + $1,000 1-14 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill. 6 FINDING THE YIELD TO MATURITY: MORE TRIAL AND ERROR (CONTINUED) A bond’s yield to maturity should not be confused with its current yield, which is the bond’s annual coupon divided by its price In the example we just worked on the previous slide, the bond’s annual coupon was $80, and its price was $955.14 Current yield was $80/$955.14 =.0838, or 8.38 percent Yield to maturity was 9 percent (found by trial and error) 1-15 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  SUMMARY OF BOND VALUATION 1-16 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  CURRENT EVENTS 1-17 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  BOND YIELDS 1-18 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  HOW TO CALCULATE BOND PRICES AND YIELDS USING A Using FINANCIAL CALCULATOR Example 7.3 (on the previous slide), the first bond sells for $935.08 and has a 10% annual coupon rate. What’s the yield? For the second bond, we now know the relevant yield is 11%. It has a 12% annual coupon and 12 years to maturity, so what’s the price? Suppose we have a bond with a price of $902.29, 10 years to maturity, and a coupon rate of 6%. Assuming this bond pays semiannual interest, what’s the bond’s yield? 1-19 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill. MORE ABOUT BOND FEATURES Securities issued by corporations may be classified roughly as equity securities and debt securities Generally, equity represents an ownership interest, and it is a residual claim, meaning equity holders are paid after debt holders Differences between debt and equity are the following: Debt is not an ownership interest in the firm. Creditors generally do not have voting power. Corporation’s payment of interest on debt is considered a cost of doing business and is tax deductible (up to certain limits). Dividends paid to stockholders are not tax deductible. Unpaid debt is a liability of the firm. If it is not paid, the creditors can legally claim the assets of the firm. This action can result in liquidation or reorganization, two of the possible consequences of bankruptcy. 1-20 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill. LONG-TERM DEBT: THE BASICS All long-term debt securities are promises made by the issuing firm to pay principal when due and to make timely interest payments on the unpaid balance Maturity of a long-term debt instrument is the length of time the debt remains outstanding with some unpaid balance Short-term debt securities, or unfunded debt, have a maturity of one year or less Long-term debt securities have maturities of more than one year Debt securities are typically called notes, debentures, or bonds Difference between notes and bonds is original maturity, with those of 10 years or less being called notes and longer-term issues being called bonds 1-21 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill. FEATURES OF AN APPLE BOND 1-22 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  THE INDENTURE The indenture (i.e., deed of trust) is the written agreement between the corporation (the borrower) and the lender detailing the terms of the debt issue Indenture generally includes the following provisions: Basic terms of the bonds Total amount of bonds issued Description of property used as security Repayment arrangements Call provisions Details of the protective covenants Usually, a trustee (e.g., a bank) is appointed by the corporation to represent the bondholders and the trustee must: Make sure the terms of the indenture are obeyed Manage the sinking fund 1-23 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  TERMS OF A BOND Principal value is stated on bond certificate Par value of a bond is almost always the same as the face value, with the terms used interchangeably in practice Corporate bonds historically had a face value of $1,000 Municipal bonds often have par valued of $5,00 Treasury bonds with par values of $10,000 or $100,000 are common Corporate bonds are usually in registered form, meaning the registrar of the company records ownership of each bond with payment being made directly to the owner of record Bonds can also be in bearer form, in which case the bond is issued without record of the owner’s name with payment being made to whomever holds1-24 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  SECURITY Debt securities are classified according to the collateral and mortgages used to protect the bondholder Mortgage securities are secured by a mortgage on the real property of the borrower, with the property involved typically being real estate (e.g., land or buildings) Legal document that describes the mortgage is called a mortgage trust indenture or trust deed Sometimes mortgages are on specific property (e.g., railroad car) Blanket mortgage pledges all real property owned by the company Bonds usually represent unsecured obligations of the company A debenture is an unsecured debt, usually with a 1-25 maturity of 10 years or more Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  SENIORITY AND REPAYMENT Seniority indicates preference in position over other lenders, with debts sometimes labeled as senior or junior to indicate seniority Some debt is subordinated (e.g., subordinated debenture), and holders of such debt must give preference to other specified creditors Debt cannot be subordinated to equity Bonds can be repaid at maturity, at which time the bondholder will receive the stated, or face, value of the bond; or they may be repaid in part or in entirety before maturity Early repayment is common and is often handled through a sinking fund, an account managed by the bond trustee for early redemption There are various types of sinking fund arrangements; for example: 1-26 Some Copyright sinking © 2022 McGraw funds Hill. All rights reserved. Nostart about reproduction 10 or distribution years without after the prior written consentthe initial of McGraw Hill.  THE CALL PROVISION Call provision is an agreement giving the corporation the option to repurchase a bond at a specified price prior to maturity Corporate bonds are usually callable Call price is usually above the bond’s stated value (i.e., par value), with the difference between the call price and the stated value being the call premium Deferred call provisions prohibit the company from redeeming a bond prior toa certain date During period of prohibition, the bond is said to be call-protected “Make-whole” call is a new type of call provision where bondholders receive approximately what the bonds are worth if they are called To find the make-whole call price, calculate the PV of 1-27 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  PROTECTIVE COVENANTS A protective covenant limits certain actions that might be taken during the loan’s term, usually to protect the lender’s interest 1. Negative covenant is a “thou shalt not” type of covenant that limits or prohibits actions the company might take; examples include: Firm must limit amount of dividends paid, according to some formula Firm cannot pledge any assets to other lenders Firm cannot merge with another firm Firm cannot sell or lease any major assets without lender’s approval Firm cannot issue additional long-term debt 2. Positive covenant is a “thou shalt” type of covenant that specified an action the company agrees to take or a condition the company must abide by; examples include: 1-28 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  BOND RATINGS Firms often pay to have their debt rated, with ratings serving as an assessment of the creditworthiness of the corporate issuer Leading bond-rating firms are Moody’s and Standard & Poor’s (S&P) Definitions of creditworthiness used by Moody’s and S&P are based on how likely the firm is to default and the protection creditors have in the event of a default Ratings can change as the issuer’s financial strength changes Highest rating a firm’s debt can have is AAA or Aaa, and such debt is judged to be the best quality and to have the lowest degree of risk AA or Aa ratings indicate very good quality debt and are much more common Copyright Investment-grade bonds are rated at least BBB or Baa © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill. 1-29 BOND RATINGS (CONTINUED) 1-30 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  GOVERNMENT BONDS Biggest borrower in the world—by a wide margin— is everybody’s favorite family member, Uncle Sam When the government wishes to borrow money for more than one year, it sells Treasury notes and bonds to the public U.S. Treasury issues, unlike essentially all other bonds, have no default risk Treasury issues are exempt from state income taxes State and local governments also borrow money by selling notes and bonds (i.e., municipal notes and bonds, or “munis”) Have varying degrees of default risk and are rated much like corporate issues Almost always callable Coupons are exempt from federal income taxes; 1-31 because of the tax break they receive, yields on Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  TAXABLE VERSUS MUNICIPAL BONDS 1-32 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  ZERO COUPON BONDS Zero coupon bonds, or zeroes, are bonds that make no coupon payments and are thus initially priced at a deep discount For tax purposes, the issuer of a zero coupon bond deducts interest every year even though no interest is actually paid Similarly, the owner must pay taxes on interest accrued every year, even though no interest is actually received Attractive investment for tax-exempt investors with long-term dollar-denominated liabilities (e.g., pension funds) because the future dollar value is known with relative certainty 1-33 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  INTEREST EXPENSE FOR EIN’S ZEROES Suppose the Eight-Inch Nails (EIN) Company issues a $1,000 face value, five-year zero coupon bond. The initial price is set at $508.35 Even though no interest payments are made on the bond, zero coupon bond calculations use semiannual periods to be consistent with coupon bond calculations Using semiannual periods, it is straightforward to verify that, at this price, the bond yields about 14% to maturity The total interest paid over the life of the bond is $1,000 − 508.35 = $491.65. 1-34 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  FLOATING-RATE BONDS With floating-rate bonds (floaters), coupon payments are adjustable, with adjustments tied to an interest rate index (e.g., Treasury bill interest rate or 30-year Treasury bond rate) Value depends on how coupon payment adjustments are defined In most cases, the coupon adjusts with a lag to some base rate Most floaters have the following features: Holder has right to redeem note at par on the coupon payment date after some specified amount of time (i.e., put provision) Coupon rate has floor and ceiling, meaning the coupon is subject to a minimum and a maximum. In this case, the coupon rate is said to be “capped,” and the upper 1-35 and© 2022 Copyright lower rates McGraw Hill. mayNo reproduction All rights reserved. be called the or distribution collar without the prior written consent of McGraw Hill.  OTHER TYPES OF BONDS Catastrophe, or cat, bonds cover U.S.-named storms and earthquakes, with investors losing some or all of their money if one of these triggering events occurs In 2019, Conservation Capital issued Rhino impact bonds (RIB) designed to increase the population of black rhinos in Kenya and South Africa Bondholders receive payments based on the “outcome payments” model; in this case, payments will be based on the population of black rhinos in five years Coronavirus bonds came to market during the COVID-19 pandemic; proceeds were to go toward pandemic-related work A warrant is a bond feature that gives the buyer of the bond the right to purchase shares of stock in the company at a fixed price 1-36 Copyright Very valuable © 2022 feature McGraw Hill. All rights is theor distribution reserved. No reproduction stockwithout price climbs the prior written consent of McGraw Hill.  OTHER TYPES OF BONDS (CONTINUED) Income bonds are similar to conventional bonds, except that coupon payments depend on company income Coupons are paid to bondholders only if firm’s income is sufficient A convertible bond can be swapped for a fixed number of shares of stock anytime before maturity at the holder’s option A put bond allows the holder to force the issuer to buy back the bond at a stated price; this is the reverse of the call provision Reverse convertible is a relatively new item One type generally offers a high coupon rate, but the redemption at maturity can be paid in cash at par value or paid in shares of stock Death bonds involve companies purchasing life insurance policies from individuals who are expected 1-37 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  SUKUK Worldwide demand for assets that comply with sharia, or Islamic law and cultural tradition, has grown dramatically One of the major differences between Western financial practices and sharia is that Islamic law does not permit charging or paying riba, or interest To accommodate the restriction on interest payments, Islamic bonds, or sukuk, have been created There are many possible structures to sukuk, such as partial ownership in a debt (sukuk al-murabaha) or an asset (sukuk al-ijara) In the case of a sukuk al-ijara, there is a binding promise to repurchase a certain asset by the issuer at maturity Before the sukuk matures, rent is paid on the asset 1-38 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  HOW BONDS ARE BOUGHT AND SOLD Total volume of trading in bonds far exceeds that in stocks Most trading in bonds takes place over the counter, or OTC One reason the bond markets are so big is that the number of bond issues far exceeds the number of stock issues, and there are two reasons for this: 1. A corporation would typically have only one common stock issue outstanding, whereas a large corporation could easily have a dozen or more note and bond issues outstanding 2. Federal, state, and local borrowing is enormous A financial market is transparent if it is possible to easily observe its prices and trading volume On the New York Stock Exchange, for example, it is 1-39 possible Copyright toHill.see © 2022 McGraw All rights the reserved. price and No reproduction quantity or distribution for without the prior every written single consent of McGraw Hill.  BOND PRICE REPORTING In 2002, transparency in the corporate bond market began to improve, as new regulations required dealers to report trade information through the Trade Reporting and Compliance Engine (TRACE) TRACE bond quotes are available online Each day, representative prices for outstanding Treasury issues are reported Bid price is the price a dealer is willing to pay for a security Asked price is the price a dealer is willing to take for a security Bid-ask spread is difference between the bid price and asked price Treasury prices are quoted as a percentage of face value 1-40 If the bid price is 132.23 and the face value is $1,000, Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  SAMPLE WALL STREET JOURNAL U.S. TREASURY NOTE AND BOND PRICES 1-41 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  A NOTE ABOUT BOND PRICE QUOTES Clean price is the price of a bond net of accrued interest This is the price that is typically quoted Dirty price is the price of a bond including accrued interest, also known as the full or invoice price This is the price the buyer actually pays Suppose you buy a bond with a 12% annual coupon, payable semiannually. You actually pay $1,080 for this bond, so $1,080 is the dirty, or invoice, price. Further, on the day you buy it, the next coupon is due in four months, so you are between coupon dates. Notice that the next coupon will be $60. Accrued interest on a bond is calculated by taking the 1-42 fraction of the coupon period that has passed, in this Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill. REAL VERSUS NOMINAL RATES Real rates are interest rates or rates of return that have been adjusted for inflation Nominal rates are interest rates or rates of return that have not been adjusted for inflation Suppose prices are currently rising by 5% per year. In other words, the rate of inflation is 5%. An investment is available that will be worth $115.50 in one year. It costs $100 today. Notice that with a present value of $100 and a future value in one year of $115.50, this investment has a 15.5 percent rate of return In calculating this 15.5 percent return, we did not consider the effect of inflation, so this is the nominal return What is the impact of inflation here? Copyright Suppose pizzas © 2022 McGraw costNo$10 Hill. All rights reserved. each reproduction atwithout or distribution thethebeginning of the prior written consent of McGraw Hill. 1-43  THE FISHER EFFECT Fisher effect describes the relationship between nominal returns, real returns, and inflation Let R stand for the nominal rate, r stand for the real rate, and h stand for the inflation rate; Fisher effect can be written as: In the example on the previous slide, the nominal rate was 15.50% and the inflation rate was 5%. What was the real rate? 1 +.1550 = (1 + r) × (1 +.05) 1 + r = 1.1550/1.05 = 1.10 r =.10, or 10% We can rearrange the Fisher effect as follows: Nominal rate is approximately equal to real rate plus 1-44 inflation rate: Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  THE FISHER EFFECT: AN EXAMPLE 1-45 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  INFLATION AND PRESENT VALUES: DISCOUNTING NOMINAL CASH FLOWS What is the effect of inflation on PV calculations? You can either: Discount nominal cash flows at a nominal rate; or Discount real cash flows at a real rate Suppose you want to withdraw money each year for the next three years, and you want each withdrawal to have $25,000 worth of purchasing power as measured in current dollars. If the inflation rate is 4% per year, then the withdrawals will have to increase by 4% each year to compensate. The withdrawals each year will be: C1 = $25,000(1.04) = $26,000 C2 = $25,000(1.04)2 = $27,040 C3 = $25,000(1.04)3 = $28,121.60 What is the PV of these cash flows if the appropriate 1-46 nominal discount rate is 10%? Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  INFLATION AND PRESENT VALUES: DISCOUNTING REAL CASH FLOWS To calculate the PV using real cash flows, we need the real discount rate. Using the Fisher equation, the real discount rate is: 1 + R = (1 + r)(1 + h) 1 +.10 = (1 + r)(1 +.04) r =.0577, or 5.77% By design, the real cash flows are an annuity of $25,000 per year. So, the present value in real terms is: PV = $25,000[1 − (1/1.05773)]/.0577 = $67,111.65 You could also use the growing annuity equation; withdrawals are increasing at 4% per year, so the present value is: 1-47 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  TERM STRUCTURE OF INTEREST RATES At any point in time, short-term and long-term interest rates will generally be different The term structure of interest rates is the relationship between nominal interest rates on default-free, pure discount securities and time to maturity; that is, the pure time value of money When long-term rates are higher than short-term rates, we say that the term structure is upward sloping When short-term rates are higher than long-term rates, we say the term structure is downward sloping Term structure can also be “humped”, which is usually because rates increase at first, but then begin to decline as we look at longer- and longer-term rates 1-48 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  TERM STRUCTURE OF INTEREST RATES (CONTINUED) What determines the shape of the term structure? 1. Real rate of interest 2. Expected future inflation 3. Interest rate risk premium Inflation premium is the portion of a nominal interest rate that represents compensation for expected future inflation Upward-sloping term structure may reflect anticipated increases in inflation, while a downward-sloping term structure probably reflects the belief that inflation will be falling in the future Interest rate risk premium is the compensation investors demand for bearing interest rate risk Copyright Interest rate risk premium increases with maturity, © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill. 1-49 TERM STRUCTURE OF INTEREST RATES (CONCLUDED) 1-50 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill. BOND YIELDS AND THE YIELD CURVE: PUTTING IT ALL TOGETHER Treasury yield curve is a plot of the yields on Treasury notes and bonds relative to maturity Shape of yield curve reflects the term structure of interest rates Term structure is based on pure discount bonds, whereas the yield curve is based on coupon bond yields Treasuries depend on the three components that underlie the term structure: Real rate Expected future inflation Interest rate risk premium Recall Treasury notes and bonds are default-free,1-51 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  BOND YIELDS AND THE YIELD CURVE: PUTTING IT ALL TOGETHER (CONTINUED) A bond’s yield is calculated assuming all promised payments will be made If the issuer defaults, your actual yield will be lower Default risk premium is the portion of a nominal interest rate or bond yield that represents compensation for the possibility of default Lower-rated bonds have higher yields Taxability premium is the portion of a nominal interest rate or bond yield that represents compensation for unfavorable tax status Recall, municipal bonds are free from most taxes Liquidity premium is the portion of a nominal interest rate or bond yield that represents compensation for lack of liquidity 1-52 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  CONCLUSION Bond yields represent the combined effect of no fewer than six things: Real rate of interest Premiums representing compensation for the following: 1. Expected future inflation 2. Interest rate risk 3. Default risk 4. Taxability 5. Lack of liquidity Determining the appropriate yield on a bond requires careful analysis of each of these effects 1-53 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  SELECTED CONCEPT QUESTIONS What are the cash flows associated with a bond? What is the indenture? What are protective covenants? Give some examples. What is a junk bond? Why might an income bond be attractive to a corporation with volatile cash flows? Can you think of a reason why income bonds are not more popular? In general, what are bid and ask prices? What is the difference between a nominal and a real return? Which is more important to a typical investor? What is the term structure of interest rates? What determines its shape? 1-54 What is the Treasury yield curve? Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.  COMPREHENSIVE PROBLEM What is the price of a $1,000 par value bond with a 6% coupon rate paid semiannually, if the bond is priced to yield 5% and it has 9 years to maturity? What would be the price of the bond if the yield rose to 7%. What is the current yield on the bond if the YTM is 7%? 1-55  END OF CHAPTER CHAPTER 7 1-56 Copyright © 2022 McGraw Hill. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill.