Bond Valuation. FINANCE 100 Corporate Finance

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1 Bond Valuation FINANCE 100 Corporate Finance Prof. Michael R. Roberts 1 Bond Valuation An Overview Introduction to bonds and bond markets» What are they? Some examples Zero coupon bonds» Valuation» Interest rate sensitivity Coupon bonds» Valuation» Interest rate sensitivity Spot and Forward Rates Nominal and real interest rates 2 1 1

2 What is a Bond? Bonds (a.k.a fixed income securities) are simply loans. A bond is a security that obligates the issuer to make interest and principal payments to the holder on specified dates.» Maturity (or term)» Face value (or par)» Coupon rate Bonds differ in several respects:» Repayment type» Issuer» Maturity» Security» Priority in case of default 3 Repayment Schemes Bonds with a balloon payment» Pure discount or zero-coupon bonds Pay no coupons prior to maturity.» Coupon bonds Pay a stated coupon at periodic intervals prior to maturity.» Floating-rate bonds Pay a variable coupon, reset periodically to a reference rate. Bonds without a balloon payment» Perpetual bonds Pay a stated coupon at periodic intervals.» Annuity or self-amortizing bonds Pay a regular fixed amount each payment period. Principal repaid over time rather than at maturity

3 Issuers: U.S. Government Treasury bills» Short-term, zero coupon bonds Treasury notes» Medium-term, coupon bonds Treasury bonds» Long-term, coupon bonds Treasury strips» Wide-ranging maturity, zero-coupon bond 5 Issuers: Agencies and Municipalities Agency bonds: mortgage-backed bonds» Bonds issued by U.S. Government agencies that are backed by a pool of home mortgages.» Self-amortizing bonds. (mostly monthly payments)» Maturities up to 30 years.» Prepayment risk. Municipal bonds» Maturities from one month to 40 years.» Exempt from federal, state, and local taxes.» Generally two types: Revenue bonds General Obligation bonds» Riskier than U.S. Government bonds

4 Corporate Bonds Bonds issued by corporations» Bonds vs. Debentures» Indenture vs. Trust Deed» Fixed-rate versus floating-rate bonds.» Investment-grade vs. Below investment-grade bonds.» Additional features: call provisions convertible bonds puttable bonds 7 Bond Ratings Moody s S&P Quality of Issue Aaa AAA Highest quality. Very small risk of default. Aa AA High quality. Small risk of default. A A High-Medium quality. Strong attributes, but potentially vulnerable. Baa BBB Medium quality. Currently adequate, but potentially unreliable. Ba BB Some speculative element. Long-run prospects questionable. B B Able to pay currently, but at risk of default in the future. Caa CCC Poor quality. Clear danger of default. Ca CC High speculative quality. May be in default. C C Lowest rated. Poor prospects of repayment. D - In default

5 The US Bond Market Amount ($bil.). Source: U.S. Federal Reserve (June/05/2003) Debt Instrument U.S. Gov Municipal Corp./Foreign Bonds Consumer Credit Mortgages For comparison, the total market capitalization of NYSE firms (2,783 firms) in December/2002 was 9,603.3 billion dollars. 9 Bond Valuation: Zero Coupon Bonds V n = Market price of the Bond of bond in period n F = Face value R = Annual percentage rate m = compounding period (annual m = 1, semiannual m = 2, ) i = Effective periodic interest rate; i=r/m T = Maturity (in years) N = Number of compounding periods; N = T*m Two cash flows to purchaser of bond:» -V 0 at time 0» F at time T What is the price of a bond? Use present value formula: F V0 = 1+ ( i) N

6 Valuing Zero Coupon Bonds: An Example Value a 5 year, U.S. Treasury strip with face value of $1,000. The APR is R=7.5% with annual compounding? What about quarterly compounding? What is the APR on a U.S. Treasury strip that pays $1,000 in exactly 7 years and is currently selling for $ under annual compounding? Semi-annual compounding? 11 Interest Rate Sensitivity: Zero Coupon Bonds Consider the following 1, 2 and 10-year zero-coupon bonds, all with» face value of F=$1,000» APR of R=10%, compounded annually. We obtain the following table for increases and decreases of the interest rate by 1%: Interest Rate Bond 1 Bond 2 Bond 3 1-Year 2-Year 10-Year 9.0% $ $ $ % $ $ $ % $ $ $ Bond prices move up if interest rates drop, decrease if interest rates rise

7 Bond Prices and Interest Rates $1,200 $1,000 $800 $600 $400 $200 1-Year 2-Year 10-Year $0 0.0% 5.0% 10.0% 15.0% 20.0% 25.0% Bond prices are inversely related to IR Longer term bonds are more sensitive to IR changes than short term bonds The lower the IR, the more sensitive the price. 13 Measuring Interest Rate Sensitivity Zero Coupon Bonds We would like to measure the interest rate sensitivity of a bond or a portfolio of bonds.» How much do bond prices change if interest rates change by a small amount?» Why is this important? Use Dollar value of a one basis point decrease (DV01):» Basis point (bp): 1/100 of one percentage point =0.01%=0.0001» Calculate DV01: Method 1: Difference of moving one basis point down: DV01= B(R-0.01%)-B(R). Method 2: Use calculus: B DV 01 = R Method 3: Difference of moving 1/2bp down minus 1/2pb up: DV01=B(R-0.005%) -B(R+0.005%)

8 Computing DV01: An Example Reconsider the 1, 2 and 10- year bonds discussed before: Method 2: db dr 1 10,000 Interest Rate Bond 1 Bond 2 Bond 3 1-Year 2-Year 10-Year 9.990% $ $ $ % $ $ $ % $ $ $ % $ $ $ Method 1 $ $ $ Method 2 $ $ $ Method 3 $ $ $ $1,000 1 = T ,000 = T T *$0.10* 1.10 T DV01: A Graphical Approach 10-Year $1, $1, $ $ $ $ $0.00 Interest Rate DV01 estimates the change in the Price-Interest rate curve using a linear approximation. higher slope implies greater sensitivity

9 Valuing Coupon Bonds Example 1: Amortization Bonds Consider Amortization Bond» T=2» m=2» C=$2,000 c = C/m = $2,000/2 = $1,000» R=10% i = R/m = 10%/2 = 5% How can we value this security?» Brute force discounting» Similar to another security we already know how to value?» Replication 17 A First Look at Arbitrage Reconsider amortization bond; suppose bond trades at $3,500 (as opposed to computed price of $3,545.95)» Can we make a profit without any risk? What is the strategy? What is the profit?

10 Valuation of Coupon Bonds: Example 2: Straight Bonds 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 interest rate is 10% compounded semiannually? Months Valuing Coupon Bonds The General Formula What is the market price of a bond that has an annual coupon C, face value F and matures exactly T years from today if the required rate of return is R, with m-periodic compounding?» Coupon payment is: c = C/m» Effective periodic interest rate is: i = R/m» number of periods N = Tm N c c c c c+f V 0 = [ Annuity] + [ Zero] 1 (1 + i) = c i N + F i N ( 1+ )

11 The Concept of a Yield to Maturity So far we have valued bonds by using a given interest rate, then discounted all payments to the bond. Prices are usually given from trade prices» need to infer interest rate that has been used Definition: The yield to maturity is that interest rate that equates the present discounted value of all future payments to bondholders to the market price: Algebraic: B = c yield / m + ( 1+ yield/ m) N ( 1 yield/ m) N Can only be computed using numerical procedures when number of compounding periods is greater than 5. F 21 Interest Rate Sensitivity: Coupon Bonds Coupon bonds can be represented as portfolios of zero-coupon bonds» Implication for price sensitivity Consider purchasing the US Treasury bond discussed earlier (10 year, 9% coupon, $1,000 face)» Suppose immediately thereafter interest rates fall to 8%, compounded semiannually.» Suppose immediately thereafter interest rate rises to 12% compounded semiannually.» Suppose the interest rate equals 9%, compounded semiannually. What are the pricing implications of these scenarios?

12 Implication of Interest Rate Changes on Coupon Bond Prices Recall the general formula: 1 (1 + i) V0 = c i + N ( 1+ ) What is the price of the bond if the APR is 8% compounded semiannually? N F i Similarly: If R=12%: B=$ If R= 9%: B=$1, Relationship Between Coupon Bond Prices and Interest Rates Bond prices are inversely related to interest rates (or yields). A bond sells at par only if its interest rate equals the coupon rate A bond sells at a premium if its coupon rate is above the interest rate. A bond sells at a discount if its coupon rate is below the interest rate

13 DV01 and Coupon Bonds Consider two bonds with 10% annual coupons with maturities of 5 years and 10 years. The APR is 8% What are the responses to a.01% (1bp) interest rate change? Yield 5-Year Bond $ Change % Change 10-Year Bond $ Change % Change 7.995% $1, $ % $1, $ % 8.000% $1, $1, % $1, $ % $1, $ % DV01 $ $ Does the sensitivity of a coupon bond always increase with the term to maturity? 25 Bond Prices and Interest Rates $2, $2, Year Bond 10-Year Bond Price (P) $1, $1, $ $0.00 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% Interest Rate (R) 24% Longer term bonds are more sensitive to changes in interest rates than shorter term bonds, in general

14 Bond Yields and Prices 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. Then, what are the price sensitivities of these bonds, measured by DV01 as for zero coupon bonds? Yield 6%-Bond $ Change % change 10%-Bond $ Change % change 7.995% $ $ $1, $ % $ $1, % $ ($0.1891) $1, ($0.2101) % % DV01 $ $ Why do we get different answers for two bonds with the same yield and same maturity? 27 Maturity and Price Risk Zero coupon bonds have well-defined relationship between maturity and interest rate sensitivity: Coupon bonds can have different sensitivities for the same maturity» DV01 now depends on maturity and coupon Need concept of average maturity of coupon bond:» Duration

15 Duration The logical way to measure sensitivity of the bond price to changes in interest rates is to take the derivative of the price B with respect to effective rate i: N B = n c i n= 1 n 1 N 1 ( 1+ i) + N F (1 + i) This measure gives us the change in price for a one unit change in the periodic interest rate (or periodic yield). 29 Duration If we adjust the slope as follows: we get Duration, D, (Macaulay (1938)). Can also write duration as: Duration B D = i PV ( c1) PV ( c2 ) PV ( c N ) = T1 + T TN + TN B B B Duration is a weighted average term to maturity where the weights are relative size of the contemporaneous cash flow. Duration is quoted in years. ( 1+ i) mb 1 = B N 0 n= 1 T CF n n ( 1+ i) n PV ( F ) B

16 Duration (A picture) Probability Coupon Paying Bond Think of time as a random variable taking on values 1 T*m. The distribution of the random variable is given by the weights: PV(CF at t) / Price of Bond To get Duration, just find the expectation of the random variable Time (I.e. take a weighted average) Time 31 Duration (cont.) The duration of a bond is less than its time to maturity (except for zero coupon bonds). The duration of the bond decreases the greater the coupon rate. This is because more weight (present value weight) is being given to the coupon payments. As market interest rate increases, the duration of the bond decreases. This is a direct result of discounting. Discounting at a higher rate means lower weight on payments in the far future. Hence, the weighting of the cash flows will be more heavily placed on the early cash flows -- decreasing the duration. Modified Duration = Duration / (1+i) = percent change in price from a 1% change in yield

17 Spot Rates A spot rate is a rate agreed upon today, for a loan that is to be made today» r 1 =5% indicates that the current rate for a one-year loan is 5%.» r 2 =6% indicates that the current rate for a two-year loan is 6%.» Etc. The term structure of interest rates is the series of spot rates r 1,r 2,r 3,» We can build using STRIPS or coupon bond yields» Explanations of the term structure Yield T 33 Yield Curve A sample yield curve Yield Curve for US Treasuries Source: Bloomberg.com (October 22, 2003) 6.00% 5.00% 4.00% Yield 3.00% 2.00% 1.00% 0.00% Bond Maturity

18 Forward Rates A forward rate is a rate agreed upon today, for a loan that is to be made in the future. (Not necessarily equal to the future spot rate!)» f 2,1 =7% indicates that we could contract today to borrow money at 7% for one year, starting two years from today. (1+r 1 ) (1+f 1,2 ) 2 (1+r 1 ) (1+f 1,1 ) (1+f 2,1 ) (1+r 3 ) 3 (1+r 2 ) 2 (1+f 2,1 ) Time Forward Rates Example r 1 =5.00%, r 2 =5.75%, r 3 =6.00% Which investment strategy is optimal:» Invest $100 for three years how much do we have?» Invest $100 for two years, and invest the proceeds at the one-year forward rate, two periods hence how much do we have?» When are these two payoffs equal? General Forward Rate Relation: (1+r n+t ) n+t =(1+r n ) n (1+f n,t ) t» Think of the previous picture for intuition

19 Arbitraging Forward Rates What happens if the forward rate in the market is not equal to the implied forward rate?» Consider the following term structure of interest rates: r 1 =5.00%, r 2 =5.75%, r 3 =6.00% Above we calculated an implied forward rate, f 2,1 =6.5%. Assume that the prevailing forward rate in the market was 7%? Is there an arbitrage opportunity and if so, how would you take advantage of it? 37 Real Rates and Nominal Rates Money is of value because of its ability to purchase goods. Measure price changes by inflation defined as the percent change in the Consumer Price Index (CPI). CPI New CPIOld π = CPI Adjust the return for our investment by the rate of inflation. The adjusted rate is called the real rate.» Often see the real interest rate written as rr = R π Old 1+ R 1+ rr = = π ( R) CPI CPI Old New

20 Summary Bonds can be valued by discounting their future cash flows Bond prices change inversely with yield Price response of bond to interest rates depends on term to maturity.» Works well for zero-coupon bond, but not for coupon bonds Measure interest rate sensitivity using DV01 and duration. The term structure implies terms for future borrowing:» Forward rates» Compare with expected future spot rates Consumers care about real rates of interest

Bond Valuation. Capital Budgeting and Corporate Objectives

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