Bonds Finance 100 Prof Michael R Roberts Copyright © Michael R Roberts Copyright © Michael R Roberts Topic Overview Introduction to bonds and bond markets Zero coupon bonds » » » » Valuation Yield-to-Maturity & Yield Curve Spot Rates Interest rate sensitivity – DVO1 Coupon bonds » » » » » Valuation Arbitrage Bond Prices Over Time Yield Curve Revisited Interest rate sensitivity – Duration & Immunization Forward Rates 1 What is a Bond and What are its Features? 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): Notional amount used to compute interest payments » Coupon rate: Determines the amount of each coupon payment, expressed as an APR Coupon = Coupon Rate × Face Value Number of Coupon Payments per Year Bonds differ in several respects: » » » » » Repayment type Issuer Maturity Security Priority in case of default Copyright © Michael R Roberts Repayment Schemes Bonds with a balloon (or bullet) 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 Copyright © Michael R Roberts 2 Who Issues Bonds? US Government (Treasuries) » » » » » T-bills: 4,13,16-week maturity, zero coupon bonds T-notes: 2,3,5,10 year, semi-annual coupon bonds T-bonds: 20 & 30-year, semi-annual coupon bonds TIPS: 5,10,20-year, semi-annual coupon bond, principal π-adjusted Strips: Wide-ranging maturity, zero-coupon bond, IB-structured Foreign Governments Municipalities » » » » Maturities from one month to 40 years, semiannual coupons Exempt from federal taxes (sometimes state and local as well) Generally two types: Revenue bonds vs General Obligation bonds Riskier than government bonds (e.g., Orange County) Copyright © Michael R Roberts Who Issues Bonds? (Cont.) Agencies: » E.g Government National Mortgage Association (Ginnie Mae), Student Loan Marketing Association (Sallie Mae) » Most issues are mortgage-backed, pass-through securities » Typically 30-year, monthly paying annuities mirroring underlying securities » Prepayment risk Corporations » types: notes, debentures, mortgage, asset-backed » ~30 year maturity, semi-annual coupon set to price at par » Additional features/provisions: – Callable: right to retire all bonds on (or after) call date, for call price – convertible bonds – putable bonds Copyright © Michael R Roberts 3 Bond Ratings Moody’s S&P Aaa AAA Aa AA A A Baa BBB Ba BB Quality of Issue Highest quality Very small risk of default High quality Small risk of default High-Medium quality Strong attributes, but potentially vulnerable Medium quality Currently adequate, but potentially unreliable Some speculative element Long-run prospects questionable Able to pay currently, but at risk of default in the future B B Caa CCC Ca CC High speculative quality May be in default C C Lowest rated Poor prospects of repayment D - In default Poor quality Clear danger of default Copyright © Michael R Roberts The US Bond Market – Flows Amount ($bil.) Source: Flow of Funds Data 2005-2007 Debt Instrument 2005 2006 2007 U.S Gov 307.3 183.7 237.5 Municipal 195 177.3 214.6 Corporate 53.6 213.4 314.1 Consumer Credit 94.5 104.4 132.3 Mortgages 1417.5 1397.1 1053.2 Dollar volume of bonds traded daily is 10 times that of equity markets! Outstanding investment-grade dollar denominated debt is about $8.3 trillion (e.g., treasuries, agencies, corporate and MBSs Copyright © Michael R Roberts 4 Zero Coupon Bonds (a.k.a Pure Discount Bonds) Notation Reminder: » » » » » » » » Vn= Bn = Market price of the bond in period n F = Face value R= Annual percentage rate m = compounding periods (annual m = 1, semiannual i = Effective periodic interest rate; i=R/m T = Maturity (in years) N = Number of compounding periods; N = T*m r = discount rate m = 2,…) Two cash flows to buyer of a zero coupon bond (a.k.a “zero”): » -V0 at time » F at time T What is the price of a bond? V0 = B = F (1 + r ) T ⎛ F ⎜ or V0 = B0 = N ⎜ + i) ( ⎝ ⎞ ⎟ ⎟ ⎠ Copyright © Michael R Roberts Zero Coupon Bond Examples Value a year, U.S Treasury strip with face value of $1,000 The APR is 7.5% with quarterly compounding? » Approach 1: Using R (APR) and i (effective periodic rate) ? » Approach 2: Using r (EAR) ? » Approach 3: Using r (periodic discount rate) ? Copyright © Michael R Roberts 10 5 Yield to Maturity The Yield to Maturity (YTM) is the one discount rate that sets the present value of the promised bond payments equal to the current market price of the bond » Doesn’t this sound vaguely familiar… ? Example: Zero-Coupon Bond V0 = 1/ T ⎛F⎞ ⇒r=⎜ ⎟ ⎝ V0 ⎠ F (1 + r ) T − = YTM = y » But this is just the IRR since = V0 − 1/ T F (1 + IRR ) T ⎛F⎞ ⇒ IRR = ⎜ ⎟ ⎝ V0 ⎠ − = YTM = y Copyright © Michael R Roberts 11 Yields for Different Maturities Note: bonds of different maturities have different YTMs Copyright © Michael R Roberts 12 6 Spot Rates, Term Structure, Yield Curve A spot rate is the interest rate on a T-year loan that is to be made today » » » » r1=5% indicates that the current rate for a one-year loan today is 5% r2=6% indicates that the current rate for a two-year loan today is 6% Etc Spot rate = YTM on default-free zero bonds The term structure of interest rates is the series of spot rates r1, r2, r3,… relating interest rates to investment term The yield curve is just a plot of the term structure: interest rates against investment term (or maturity) » Zero-Coupon Yield Curve: built from zero-coupon bond yields (STRIPS) » Coupon Yield Curve: built from coupon bond yields (Treasuries) » Corporate Yield Curve: built from corporate bond yields of similar risk (i.e., credit rating) Copyright © Michael R Roberts 13 Term Structure of Risk-Free U.S Interest Rates, January 2004, 2005, and 2006 Copyright © Michael R Roberts 14 7 Using the Yield Curve We should discount each cash flow by its appropriate discount rate, governed by the timing of the cash flow Example: What is the present value of $100, 10 years from today ? (Use the term structure from January 2004) Generally speaking, we must use the appropriate discount rate for each cash flow: PV = C1 C2 + + + r1 (1 + r2 ) + CN = (1 + rN ) N N ∑ n =1 CN (1 + rn ) n Copyright © Michael R Roberts 15 A Cautionary Note All of our valuation formulas (e.g., perpetuity, annuity) assume a flat term structure » I.e., there is only one discount rate for cash flows received at any point in time Recall: » Growing Annuity: PV = C × (r − g ) » Growing Perpetuity: PV = N ⎛ ⎛ + g ⎞ ⎞ ⎜1 − ⎜ ⎟ ⎟ ⎜ ⎝ (1 + r ) ⎠ ⎟ ⎝ ⎠ C (r − g ) – “r” is implicitly assumed to be the same every period… Copyright © Michael R Roberts 16 8 Interest Rate Sensitivity Zero Coupon Bonds Why zero-coupon bond prices change? Interest rates change! F V0 = (1 + i ) N The price of a zero-coupon bond maturing in one year from today with face value $100 and an APR of 10% is: V0 = 100 (1 + 0.10 ) = $90.91 Example: Now imagine that immediately after you buy the bond, the interest rate increase to 15% What is the price of the bond now ? Copyright © Michael R Roberts 17 Characterizing the Price Rate Sensitivity of Zero Coupon Bonds Consider the following 1, and 10-year zero-coupon bonds, all with » F=$1,000 » APR of R=10%, compounded annually $1,200 $1,000 $800 $600 $400 1-Year $200 2-Year 10-Year $0 0.0% 5.0% 10.0% 15.0% 20.0% 25.0% Note things: Bond prices are inversely related to IR Fix the interest rate: Longer term bonds are less expensive Longer term bonds are more sensitive to IR changes than short term bonds The lower the IR, the more sensitive the price Copyright © Michael R Roberts 18 9 Quantifying the Interest Rate Sensitivity of Zero Coupon Bonds – DV01 What’s the natural thing to do? Compute the derivative V0 = F (1 + i ) N ∂V0 −( N +1) = − FN (1 + i ) < (Negative slope in i ) ∂i ∂ 2V −( N + ) ⇒ 20 = FN ( N + 1)(1 + i ) > (Convex function of i ) ∂i ⇒ » If we change the interest rate by a little (e.g., 0.0001 or basis point) than multiplying this number by the derivative should tell me how much the price will change, all else equal (i.e., DV01 = Dollar Value of Basis Point) Alternatively, we can just compute the prices at two different interest rates and look at the difference: B0(i) – B0 (i+0.0001) Copyright © Michael R Roberts 19 Valuing Coupon Bonds Amortization Bonds Consider an amortization bond maturing in two years with semiannual payments of $1,000 Assume that the APR is 10% with semiannual compounding How can we value this security? Brute force discounting 1000 1000 1000 1000 (1 + 0.1025) (1 + 0.1025) (1 + 0.1025 ) (1 + 0.1025 ) (i): V0 = + + + = $3545.95 (1 + 0.10 / ) (1 + 0.10 / )2 (1 + 0.10 / )3 (1 + 0.10 / )4 or 1000 1000 1000 1000 EAR (r): V0 = + + + = $3545.95 0.5 1.5 2 Recognize the stream of cash flows as an annuity V0 = 1000 (1 − (1 + 0.10 / 2)−4 ) = $3,545.95 0.10 / Copyright © Michael R Roberts 20 10 10 Replication Can we construct the same cash flows as our amortization bond using other securities? Copyright © Michael R Roberts 21 A First Look at Arbitrage What if the bond is selling for $3,500 in the market? Copyright © Michael R Roberts 22 11 11 Valuation of Straight Coupon Bond Example 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? 12 108 Timeline: Months Cash Flows 45 45 45 120 1045 Present Value = Current Price = ? Copyright © Michael R Roberts 23 Valuation of Straight Coupon Bond 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 V = [ Annuity ] + [Zero ] ⎤ ⎡ − (1 + i ) − N ⎤ ⎡ F = ⎢c ⋅ N ⎥ ⎥ + ⎢( i + i) ⎦ ⎣ ⎦ ⎣ » Note the assumption of a flat term structure… Copyright © Michael R Roberts 24 12 12 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 » Most bonds set the coupon rate at origination to sell at par 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 Copyright © Michael R Roberts 25 The Effect of Time on Bond Prices Copyright © Michael R Roberts 26 13 13 YTM and Bond Price Fluctuations Over Time Copyright © Michael R Roberts 27 Yield to Maturity Coupon Bonds Recall: The Yield to Maturity is the one discount rate that sets the present value of the promised bond payments equal to the current market price of the bond Prices are usually given from trade prices » need to infer interest rate that has been used B= » ⎛ c ⎜1 − ⎜ yield / m ⎝ (1 + yield / m )N ⎞ F ⎟+ ⎟ (1 + yield / m )N ⎠ This is not the annualized yield, which equals yield* = ( + yield / m)m-1 Typically must solve using a computer » E.g., IRR function in excel or your calculator since: B= ⎛ c ⎜1 − yield / m ⎜ (1 + yield / m )N ⎝ ⎞ F ⎟+ ⎟ (1 + yield / m )N ⎠ Copyright © Michael R Roberts 28 14 14 The Yield Curve Revisited Treasury Coupon-Paying Yield Curve » Often referred to as “the yield curve” » Same idea as the zero-coupon yield curve except we use the yields from coupon paying bonds, as opposed to zerocoupon bonds – Treasury notes and bonds are semi-annual coupon paying bonds » We often use On-the-Run Bonds to estimate the yields – On-the-Run Bonds are the most recently issued bonds Copyright © Michael R Roberts 29 Interest Rate Sensitivity Duration The Duration of a security is the percent sensitivity of the price to a small parallel shift in the level of interest rates Duration = DB = − dB B dy » A small uniform change dy across maturities might by basis point » Duration gives the proportionate decline in value associated with a rise in yield » Negative sign is to cancel negative first derivative Alternatively, given a duration DB of a security with price B, a uniform change in the level of interest rates brings about a change in value of dB = − DB × dy × B Copyright © Michael R Roberts 30 15 15 Duration of a Coupon Bond The mathematical expression for Duration is: − dB 1 ⎡ N ⎤ − n −1 = + N ⋅ F ⋅ (1 + y / m) − N −1 ⎥ ⎢ ∑ n ⋅ cn ⋅ (1 + y / m ) B dy B m ⎣ n =1 ⎦ which we can rearrange ⎡ N n c ⋅ (1 + y / m )− n N F ⋅ (1 + y / m) − N −1 D = (1 + y / m ) ⎢ ∑ ⋅ n + ⋅ B m B ⎢ n =1 m ⎣ ⎡ ⎤ ⎢N ⎥ PV ( cn ) N PV ( F ) ⎥ n −1 = (1 + y / m ) ⎢ ∑ ⋅ + ⋅ ⎢ n =1 m B m B ⎥ ⎢ ⎥ Time in Years "Weight" on ⎢ until nth payment nth payment ⎥ ⎣ ⎦ ⎤ ⎥ ⎥ ⎦ Copyright © Michael R Roberts 31 Duration of a Coupon Bond Example Compute the duration of a two-year, semi-annual, 10% coupon, par bond, with face value of $100 Copyright © Michael R Roberts 32 16 16 More on Duration Duration is a linear operator: D(B1 + B2) = D(B1) + D(B2) » The duration of a portfolio of securities is the value-weighted sum of the individual security durations » DVO1 is also a linear operator Duration is a local measure » Based on slope of price-yield relation at a specific point » Based on a bond of fixed maturity but maturity declines over time Duration of a zero is D = (1 + y / m ) −1 N m Copyright © Michael R Roberts 33 Duration Matching Example Bank of Philadelphia balance sheet (Figures in $billions, D=duration assuming flat spot rate curve) Assets Liabilities & Shareholders Equity Commercial Paper (D = 0.48) $10 2-Year Notes (D = 1.77) $10 Shareholder Equity Total Assets (D = 1) $5 25 Total Liabilities (D = ?) $25 Duration of liabilities = ? The problem: » Increases in interest rates will decrease value of liabilities by more than assets because of duration mismatch Copyright © Michael R Roberts 34 17 17 Duration Matching Example (Cont.) What is the change in assets value when interest rates change uniformly? What is the change in liability value when interest rates change uniformly ? We want our assets and liabilities to experience similar value changes when interest rates change, so set these two expressions to be equal and solve for DL (DA=1.0): ? Copyright © Michael R Roberts 35 Duration Matching Example (Cont.) What fraction of the bank’s liabilities should be in CP and Notes in order to get a liability duration of 1.25 ? How much money should the bank hold in CP and Notes in order to get a liability duration of 1.25? How should the bank alter their liabilities to achieve this structure ? Copyright © Michael R Roberts 36 18 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!) » f2,1=7% indicates that we could contract today to borrow money at 7% for one year, starting two years from today Example: Consider the following term structure r1=5.00%, r2=5.75%, r3=6.00% » Consider two investment strategies: Invest $100 for three years how much we have? Invest $100 for two years, and invest the proceeds at the one-year forward rate, two periods hence how much we have? » When are these two payoffs equal? (i.e what is the implied forward rate?) Copyright © Michael R Roberts 37 Forward Rates Strategy #1: Invest $100 for three years have ? how much we Strategy #2: Invest $100 for two years and then reinvest the proceeds for another year at the one year forward rate, two periods hence how much we have ? When are these two payoffs equal? (i.e what is the implied forward rate?) ? Copyright © Michael R Roberts 38 19 19 Arbitraging Forward Rates Example What if the prevailing forward rate in the market is 7%, as opposed to what calculated in the previous slide? Step 1: Is there a mispricing and, if so, what is mispriced ? Step 2: Is the forward loan cheap or expensive ? Step 3: Given your answer to Step 2, what is the first step in taking advantage of the mispricing ? Copyright © Michael R Roberts 39 Arbitraging Forward Rates Example Copyright © Michael R Roberts 40 20 20 General Forward Rate Relation Forward rates are entirely determined by spot rates (and vice versa) by no arbitrage considerations General Forward Rate Relation: (1+rn+t)n+t=(1+rn)n(1+fn,t)t Think of this picture for intuition: (1+f1,2)2 (1+r1) (1+r1) (1+f1,1) (1+f2,1) (1+r3)3 (1+r2)2 Time (1+f2,1) Copyright © Michael R Roberts 41 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 duration The term structure implies terms for future borrowing: » Forward rates » Compare with expected future spot rates Copyright © Michael R Roberts 42 21 21 ... discounting 1000 1000 1000 1000 (1 + 0.1025) (1 + 0.1025) (1 + 0.1025 ) (1 + 0.1025 ) (i): V0 = + + + = $3545.95 (1 + 0.10 / ) (1 + 0.10 / )2 (1 + 0.10 / )3 (1 + 0.10 / )4 or 1000 1000 1000 1000 EAR... rate Copyright © Michael R Roberts 25 The Effect of Time on Bond Prices Copyright © Michael R Roberts 26 13 13 YTM and Bond Price Fluctuations Over Time Copyright © Michael R Roberts 27 Yield... Copyright © Michael R Roberts 31 Duration of a Coupon Bond Example Compute the duration of a two-year, semi-annual, 10% coupon, par bond, with face value of $100 Copyright © Michael R Roberts 32