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1 Bond Valuation Global Financial Management Campbell R. Harvey Fuqua School of Business Duke University charvey@mail.duke.edu http://www.duke.edu/~charvey

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2 Definition of a Bond n A bond is a security that obligates the issuer to make specified interest and principal payments to the holder on specified dates. Coupon rate Face value (or par) Maturity (or term) n Bonds are sometimes called fixed income securities.

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3 Types of Bonds n Pure Discount or Zero-Coupon Bonds Pay no coupons prior to maturity. Pay the bond ’ s face value at maturity. n Coupon Bonds Pay a stated coupon at periodic intervals prior to maturity. Pay the bond ’ s face value at maturity. n Perpetual Bonds (Consols) No maturity date. Pay a stated coupon at periodic intervals.

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4 Types of Bonds n Self-Amortizing Bonds Pay a regular fixed amount each payment period over the life of the bond. Principal repaid over time rather than at maturity.

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5 Bond Issuers n Federal Government and its Agencies n Local Municipalities n Corporations

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6 U.S. Government Bonds n Treasury Bills No coupons (zero coupon security) Face value paid at maturity Maturities up to one year n Treasury Notes Coupons paid semiannually Face value paid at maturity Maturities from 2-10 years

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7 U.S. Government Bonds n Treasury Bonds Coupons paid semiannually Face value paid at maturity Maturities over 10 years The 30-year bond is called the long bond. n Treasury Strips Zero-coupon bond Created by “ stripping ” the coupons and principal from Treasury bonds and notes.

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8 Agencies Bonds n Mortgage-Backed Bonds Bonds issued by U.S. Government agencies that are backed by a pool of home mortgages. Self-amortizing bonds. Maturities up to 20 years.

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9 U.S. Government Bonds n No default risk. Considered to be riskfree. n Exempt from state and local taxes. n Sold regularly through a network of primary dealers. n Traded regularly in the over-the-counter market.

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10 Municipal Bonds n Maturities from one month to 40 years. n Exempt from federal, state, and local taxes. n Generally two types: Revenue bonds General Obligation bonds n Riskier than U.S. Government bonds.

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11 Corporate Bonds n Secured Bonds (Asset-Backed) Secured by real property Ownership of the property reverts to the bondholders upon default. n Debentures General creditors Have priority over stockholders, but are subordinate to secured debt.

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12 Common Features of Corporate Bonds n Senior versus subordinated bonds n Convertible bonds n Callable bonds n Putable bonds n Sinking funds

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13 Bond Ratings

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14 Valuing Zero Coupon Bonds l What is the current market price of a U.S. Treasury strip that matures in exactly 5 years and has a face value of $1,000. The yield to maturity is r d =7.5%. l What is the yield to maturity on a U.S. Treasury strip that pays $1,000 in exactly 7 years and is currently selling for $591.11? 1000 1075 56 5. $696. 59111 1000 1 7. r d

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15 Bond Yields and Prices The case of zero coupon bonds l Consider three zero-coupon bonds, all with »face value of F=100 »yield to maturity of r=10%, compounded annually. We obtain the following table:

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16 l Suppose the yield would drop suddenly to 9%, or increase to 10%. How would prices respond? l Bond prices move up if the yield drops, decrease if yield rises l Prices respond more strongly for higher maturities The Impact of Price Responses

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17 l What is the market price of a U.S. Treasury bond that has a coupon rate of 9%, a face value of $1,000 and matures exactly 10 years from today if the required yield to maturity is 10% compounded semiannually? 0 6 12 18 24... 120Months 45 45 45 45 1045 Bond Valuation: An Example

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18 l What is the market price of a U.S. Treasury bond that has a coupon rate of 9%, a face value of $1,000 and matures exactly 10 years from today if the required yield to maturity is 10% compounded semiannually? 0 1 2 3 4... n C C C C C+F Valuing Coupon Bonds The General Formula

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19 Bond Yields and Prices The case of coupon bonds l Suppose you purchase the U.S. Treasury bond described earlier and immediately thereafter interest rates fall so that the new yield to maturity on the bond is 8% compounded semiannually. What is the bond ’ s new market price? l Suppose the interest rises, so that the new yield is 12% compounded semiannually. What is the market price now? l Suppose the interest equals the coupon rate of 9%. What do you observe? Note: »Coupon bonds can be regarded as portfolios of zero-coupon bonds (how?) »What implication does this have for price responses?

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20 n New Semiannual yield = 8%/2 = 4% n What is the price of the bond if the yield to maturity is 8% compounded semiannually? n Similarly: If r=12%: B=$ 827.95 If r= 9%: B=$1,000.00 Valuing Coupon Bonds (cont.)

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21 Relationship Between Bond Prices and Yields n Bond prices are inversely related to interest rates (or yields). n A bond sells at par only if its coupon rate equals the coupon rate n A bond sells at a premium if its coupon is above the coupon rate. n A bond sells a a discount if its coupon is below the coupon rate.

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22 Volatility of Coupon Bonds l Consider two bonds with 10% annual coupons with maturities of 5 years and 10 years. l The yield is 8% l What are the responses to a 1% price change? l The sensitivity of a coupon bond increases with the maturity?

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23 Bond Prices and Yields Bond Price F c Yield Longer term bonds are more sensitive to changes in interest rates than shorter term bonds.

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24 l Consider the following two bonds: »Both have a maturity of 5 years »Both have yield of 8% »First has 6% coupon, other has 10% coupon, compounded annually. l Then, what are the price sensitivities of these bonds to a 1% increase (decrease) in bond yields? l Why do we get different answers? Bond Yields and Prices The problem

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25 l Calculate the average maturity of a bond: »Coupon bond is like portfolio of zero coupon bonds »Compute average maturity of this portfolio »Give each zero coupon bond a weight equal to the proportion in the total value of the portfolio l Write value of the bond as: The factor: is the proportion of the t-th coupon payment in the total value of the bond. Duration Approximating the maturity of a bond

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26 l Duration is defined as a weighted average of the maturities of the individual payments: »This definition of duration is sometimes also referred to as Macaulay Duration. l The duration of a zero coupon bond is equal to its maturity. Duration: A Definition

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27 l Calculate the duration of the 6% 5-year bond: l Calculate the duration of the 10% 5-year bond: l The duration of the bond with the lower coupon is higher »Why? Calculating Duration

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28 Duration: An Exercise What is the interest rate sensitivity of the following two bonds. Assume coupons are paid annually. Bond A Bond B Coupon rate 10% 0% Face value $1,000 $1,000 Maturity 5 years 10 years YTM 10% 10% Price $1,000 $385.54

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29 Duration Exercise (cont.)

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30 Duration Exercise (cont.) n Percentage change in bond price for a small increase in the interest rate: Pct. Change = - [1/(1.10)][4.17] = - 3.79% Bond A Pct. Change = - [1/(1.10)][10.00] = - 9.09% Bond B

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31 l For a zero-coupon bond with maturity n we have derived: l For a coupon-bond with maturity n we can show: »The right hand side is sometimes also called modified duration. l Hence, in order to analyze bond volatility, duration, and not maturity is the appropriate measure. »Duration and maturity are the same only for zero-coupon bonds! Duration and Volatility

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32 Duration and Volatility The example reconsidered l Compute the right hand side for the two 5-year bonds in the previous example: » 6%-coupon bond: D/(1+r) = 4.44/1.08=4.11 »10%-coupon bond: D/(1+r) = 4.20/1.08=3.89 l But these are exactly the average price responses we found before! »Hence, differences in duration explain variation of price responses across bonds with the same maturity.

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33 Is Duration always Exact? l Consider the two 5-year bonds (6% and 10%) from the example before, but interest rates can change by moving 3% up or down: l This is different from the duration calculation which gives: » 6% coupon bond: 3*4.11%=12.33%<12.39% »10% coupon bond: 3*3.89%=11.67%<11.73% l Result is imprecise for larger interest rate movements »Relationship between bond price and yield is convex, but »Duration is a linear approximation

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34 The Term Structure of Interest Rates n The term structure of interest rates is the relationship between time to maturity and yield to maturity: Yield Maturity 123 5.00 5.75 6.00

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35 Spot and Forward Rates l A spot rate is a rate agreed upon today, for a loan that is to be made today. (e.g. r 1 =5% indicates that the current rate for a one- year loan is 5%). l A forward rate is a rate agreed upon today, for a loan that is to be made in the future. (e.g. 2 f 1 =7% indicates that we could contract today to borrow money at7% for one year, starting two years from today). »r 1 =5.00%, r 2 =5.75%, r 3 =6.00% »We can either: –Invest $100 for three years, or: –Invest $100 for two years, and contract (today) at the one year rate, two years forward

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36 Forward Rates A first look at arbitrage l Which investment strategy is optimal: »Invest $100 for three years: $100*(1.06) 3 = »Invest $100 for two years, and invest the proceeds at the two-year forward rate: $100*(1.0575) 2 (1+ 2 f 1 )= »Hence the first strategy is optimal if 2 f 1 6.50%. l Hence 2 f 1 =6.50% (Why?) »More generally: (1+r n+t ) n+t =(1+r n ) n (1+ n f t )

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37 When should you borrow? l Suppose you wish to borrow $20,000 in two years in order to borrow a car, and you know you can repay the loan in three years? You have two options: I.1. Borrow $17,884 now at 6%, repay $20,000*(1.06) 3 =$21,300.35 in three years. 2. Invest the proceeds from the loan for two years at 5.75% to have $17,884*(1.0575) 2 =$20,000 in two years. II. Wait for two years, borrow at the prevailing one year loan rate in 1 year? l When would you follow strategy I (lock in the current rate) rather than wait (strategy II)?

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38 When to borrow (cont.) l If you lock in the current rate, then you secure a borrowing rate of: $ 21,300.35/$20,000=1.065, i. e. 6.5% »This is exactly the forward rate we calculated above –Why? l Hence, you would borrow and lock in rates now, if you expect that the one-year interest rate is going to be higher than 6.5% i 1999. »When would you set the cut-off rate for waiting higher? (lower?) l If everybody invests this way, then the forward rate equals the expected future spot rate. »Why?

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39 Summary l Bonds can be valued by discounting future cash flows at the yield to maturity l Bond prices changes inverse with yield l Price response of bond to interest rates depends on term to maturity. »Works well for zero-coupon bond l Coupon bonds are like portfolios of zero-coupon bonds »Need duration as “ average maturity ” for coupon bonds »Only an approximation l The term structure implies terms for future borrowing: »Forward rates »Compare with expected future spot rates

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