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  • 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 The term structure of interest rates 1
  • 2. What is a Bond? 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) Bonds are also called fixed income securities. Bonds differ in several respects: » Repayment type » Issuer » Maturity » Security » Priority in case of default 2
  • 3. Repayment Schemes Pure Discount or Zero-Coupon Bonds » Pay no coupons prior to maturity. » Pay the bond‟s face value at maturity. Coupon Bonds » Pay a stated coupon at periodic intervals prior to maturity. » Pay the bond‟s face value at maturity. Floating-Rate Bonds » Pay a variable coupon, reset periodically to a reference rate. » Pay the bond‟s face value at maturity. Perpetual Bonds (Consols) » No maturity date. » 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
  • 4. Types of Bonds: IssuersBonds IssuerGovernment Bonds US Treasury, Government AgenciesMortgage-Backed Securities Government agencies (GNMA etc)Municipal Bonds State and local governmentCorporate Bonds CorporationsAsset-Back Securities Corporations 4
  • 5. U.S. Government Bonds Treasury Bills » No coupons (zero coupon security) » Face value paid at maturity » Maturities up to one year Treasury Notes » Coupons paid semiannually » Face value paid at maturity » Maturities from 2-10 years 5
  • 6. U.S. Government Bonds (Cont.) Treasury Bonds » Coupons paid semiannually » Face value paid at maturity » Maturities over 10 years » The 30-year bond is called the long bond. Treasury Strips » Zero-coupon bond » Created by “stripping” the coupons and principal from Treasury bonds and notes. No default risk. Considered to be risk free. Exempt from state and local taxes. Sold regularly through a network of primary dealers. Traded regularly in the over-the-counter market. 6
  • 7. Agency and Municipal Bonds 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. » Usually exempt from federal, state, and local taxes. » Generally two types: – Revenue bonds – General Obligation bonds » Riskier than U.S. Government bonds. 7
  • 8. Corporate Bonds Bonds issued by corporations » Bonds vs. Debentures » Fixed-rate versus floating-rate bonds. » Investment-grade vs. Below investment-grade bonds. » Additional features: – call provisions – convertible bonds – puttable bonds 8
  • 9. Seniority of Corporate Bonds In case of default, different classes of bonds have different claim priority on the assets of a corporation. Secured Bonds (Asset-Backed) » Secured by real property. » Ownership of the property reverts to the bondholders upon default. Debentures » Same priority as general creditors. » Have priority over stockholders, but subordinate to secured debt. 9
  • 10. Bond RatingsMoody’s S&P Quality of IssueAaa 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. 10
  • 11. The US Bond Market Debt Instrument 2006 Q2 Treasury securities 4759.6 Municipal securities 2305.7 Corporate and foreign bonds 8705.3 Consumer Credit 2327.4 Mortgages 12757.7 Corporate equities 18684.5Amount ($bil.). Source: U.S. Federal Reserve (Table L.4, September/2006) 11
  • 12. Bond Valuation: Zero Coupon BondsB = Market price of the Bond of bondF = Face valueR = Annual percentage ratem = compounding period (annual  m = 1, semiannual  m = 2,…)i = Effective periodic interest rate; i=R/mT = Maturity (in years)N = Number of compounding periods; N = T*m Two cash flows to purchaser of bond: » -B at time 0 » F at time T What is the price of a bond? Use present value formula: F B N 1 i 12
  • 13. 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 $591.11 under annual compounding? Semi-annual compounding? 13
  • 14. 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% $917.43 $841.68 $422.41 10.0% $909.09 $826.45 $385.54 11.0% $900.90 $811.62 $352.18 Bond prices move up if interest rates drop, decrease if interest rates rise 14
  • 15. Bond Prices and Interest Rates$1,200  Bond prices are$1,000 inversely related to IR $800  Longer term $600 bonds are more sensitive to IR $400 1-Year changes than 2-Year short term $200 bonds 10-Year $0  The lower the 0.0% 5.0% 10.0% 15.0% 20.0% 25.0% IR, the more sensitive the price. 15
  • 16. 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: Difference of moving 1/2bp down minus 1/2pb up: DV01=B(R-0.005%) -B(R+0.005%). – Method 3: Use calculus: B DV 01 0.0001 R 16
  • 17. Computing DV01: An Example Reconsider the 1, 2 and 10- year bonds discussed before: Interest Rate Bond 1 Bond 2 Bond 3 1-Year 2-Year 10-Year 9.990% $909.1736 $826.5966 $385.8940 9.995% $909.1322 $826.5214 $385.7186 10.000% $909.0909 $826.4463 $385.5433 10.005% $909.0496 $826.3712 $385.3681 Method 1 $0.082652 $0.150283 $0.350669 Method 2 $0.082645 $0.150263 $0.350494 Method 3 $0.082645 $0.150263 $0.350494 Method 3: B $1,000 1 0.0001 T 0.0001 T * $0.10 * R 1.10T 1 1.10T 1 17
  • 18. DV01: A Graphical Approach 10-Year $1,200.00 $1,000.00 $800.00 $600.00 $400.00 $200.00 $0.00 Interest Rate DV01 estimates the change in the Price-Interest rate curve using a linear approximation. higher slope implies greater sensitivity 18
  • 19. 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 19
  • 20. Valuing Coupon Bonds Example 1: Amortization Bonds  Compare with a portfolio of zero coupon bonds: 0 1 2 3 4Buy Coupon Bond -$3,545.95 $1,000.00 $1,000.00 $1,000.00 $1,000.00Buy 6-Month Zero -$952.38Buy 1-Year Zero -$907.03Buy 1.5-Year Zero -$863.84Buy 2-Year Zero -$822.70Portfolio -$3,545.95 20
  • 21. 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? 21
  • 22. A First Look at Arbitrage Reconsider amortization bond; suppose bond trades at $3,500 (as opposed to computed price of $3,545.95) » Can make risk less profit – Buy low: buy amortization bond – Sell high: Sell portfolio of zero coupon bonds Time Period 0 1 2 3 4 Buy Coupon Bond -$3,500.00 $1,000.00 $1,000.00 $1,000.00 $1,000.00 Sell 6-Month Zero $952.38 -$1,000.00 $0.00 $0.00 $0.00 Sell 1-Year Zero $907.03 $0.00 -$1,000.00 $0.00 $0.00 Sell 1.5-Year Zero $863.84 $0.00 $0.00 -$1,000.00 $0.00 Sell 2-Year Zero $822.70 $0.00 $0.00 $0.00 -$1,000.00 Portfolio $3,545.95 -$1,000.00 -$1,000.00 -$1,000.00 -$1,000.00 Net Cash Flow $45.95 $0.00 $0.00 $0.00 $0.00 – riskless profit of $45.95 – no riskless profit if price is correct 22
  • 23. 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?0 6 12 18 24 ... 120 Months 45 45 45 45 1045 23
  • 24. 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 = Tm0 1 2 3 4 ... … N c c c c… … c+F B Annuity Zero c 1 F 1 N N i 1 i 1 i 24
  • 25. 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: c 1 F B 1 N N yield / m 1 yield / m 1 yield / m 25
  • 26. Yield to Maturity A Graphical Interpretation $2,500.00 $2,000.00 $1,500.00 $1,000.00 $500.00 $0.00 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% Consider a U.S. Treasury bond that has a coupon rate of 10%, a face value of $1,000 and matures exactly 10 years from now. » Market price of $1,500, implies a yield of 3.91% (semi-annual compounding); for B=$1,000 we obviously find R=10%. 26
  • 27. 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, $1,000 face) » Suppose immediately thereafter interest rates fall to 8%, compounded semiannually. » Suppose immediately thereafter interest rate rises to 12% compounded semiannually. » Suppose the interest rate equals 9%, compounded semiannually. What are the pricing implications of these scenarios? 27
  • 28. Implication of Interest Rate Changes on Coupon Bond Prices Recall the general formula: c 1 F B 1 N N i 1 i 1 i What is the price of the bond if the APR is 8% compounded semiannually? Similarly: If R=12%: B=$ 827.95 If R= 9%: B=$1,000.00 28
  • 29. 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. 29
  • 30. 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% $1,080.06 $0.21019 0.0195% $1,134.57 $0.36585 0.0323% 8.000% $1,079.85 $1,134.20 8.005% $1,079.64 -$0.21013 -0.0195% $1,133.84 -$0.36569 -0.0322% DV01 $0.42032 $0.73154 Does the sensitivity of a coupon bond always increase with the term to maturity? 30
  • 31. Bond Prices and Interest Rates $2,500.00 5-Year Bond $2,000.00 10-Year Bond Price (P) $1,500.00 $1,000.00 $500.00 $0.00 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% Interest Rate (R) Longer term bonds are more sensitive tochanges in interest rates than shorter term bonds, in general. 31
  • 32. 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% $920.33 $0.1891 $1,080.06 $0.2102 8.000% $920.15 $1,079.85 8.005% $919.96 ($0.1891) $1,079.64 ($0.2101) 0.0411% 0.0389% DV01 $0.3782 $0.4203 Why do we get different answers for two bonds with the same yield and same maturity? 32
  • 33. 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 33
  • 34. Duration Duration is a weighted average term to maturity where the weights are relative size of the contemporaneous cash flow. PV (c ) PV (c ) PV (c ) Duration T 1 T 2  T N T PV (F) 1 B 2 B N B N B Duration is a unitless number that quantifies the percentage change in a bond‟s price for a 1 percentage change in the interest rate. B B 1 R B Duration R B R 1 R 34
  • 35. 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+yield) 35
  • 36. A Few Bond Markets Statistics U.S. Treasuries, May 20th 2007.Bills MATURITY DISCOUNT/YIELD DISCOUNT/YIELD TIME DATE CHANGE3-Month 08/16/2007 4.72 / 4.84 0.01 / .010 13:416-Month 11/15/2007 4.78 / 4.98 0.01 / .015 13:41Notes/Bonds COUPON MATURITY CURRENT PRICE/YIELD TIME DATE PRICE/YIELD CHANGE2-Year 4.500 04/30/2009 99-121⁄4 / 4.84 -0-02 / .03514:083-Year 4.500 05/15/2010 99-081⁄2 / 4.77 -0-031⁄2 / .040 14:065-Year 4.500 04/30/2012 98-281⁄2 / 4.75 -0-06 / .04314:0710-Year 4.500 05/15/2017 97-15 / 4.82 -0-091⁄2 / .038 14:0730-Year 4.750 02/15/2037 96-17+ / 4.97 -0-17 / .03514:07 36
  • 37. Spot Rates A spot rate is a rate agreed upon today, for a loan that is to be made today » r1=5% indicates that the current rate for a one-year loan is 5%. » r2=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 r1, r2, r3,… » We can build using STRIPS or coupon bond yields. » Explanations of the term structure. 37
  • 38. The Term Structure of Interest Rates An Example Yield 6.00 5.75 5.00 1 2 3 Maturity 38
  • 39. Term Structure, July 1st 2005. 39
  • 40. Term Structure, September 12th, 2006 40
  • 41. Term Structure, May 20th, 2007 41
  • 42. Term Structure of Interest Rates 42
  • 43. 43
  • 44. 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 44