Econ4013 Bond Valuation, Yield and the Yield Curve PDF

Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Outline of this unit Econ4013:...

Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Outline of this unit Econ4013: Financial Markets and Corporate Finance Unit 4: Bond valuation, yield, and the yield curve Bond prices and yields Carsten Sprenger The yield curve ASBS, University of Glasgow Spring Term 2024/25 1 / 33 2 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Bond characteristics (1) Bonds are debt obligations of issuers (borrowers) to Bond prices and yields bondholders (creditors). In contrast to most loans, they are securities, i.e. they can be traded. The indenture is the contract between the issuer and the bondholder that specifies the main features of a bond such as face value, coupon rate, coupon frequency, and maturity date but also possible bond covenants, option features Reading: Bodie, Kane, Markus ch. 14 (see also (callable, puttable, convertible) and other characteristics. Berk/deMarzo ch. 6) The bond certificate contains the main information and the "coupons". Here is an example (omitted in handout): 3 / 33 4 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Bond characteristics (2) U.S. Treasury Bonds ("Treasuries") Face or par value F is the principal repaid at maturity T , typically $100 or $1,000. Bonds, notes and bills may be purchased directly from the The coupon rate c determines the interest payment Treasury or through banks, dealers or brokers. (“coupon payments”) per year. Treasury bills (short T-bills) have maturities of up to one If coupons are paid more than once per year, say, with a year, frequency of m times per year, then each coupon payment Treasure notes mature in 1-10 years, is Treasury bonds have maturities of 10-30 years. cF Coupon payment = Denomination can be as small as $100, but $1,000 is more m common. Q: For example, for an 8% coupon bond with face value $1,000 and semiannual coupon payments, how many dollar is each coupon? 5 / 33 6 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Price quotes for Treasuries WSJ Price quotes for Treasuries Bid price: Price at which you can sell to the dealer Ask price: Price at which you can buy from the dealer Quoted in % of par (=face value). For example a price quote of 106.21 for a bond with F = $1,000 means an actual price of 1062.1 7 / 33 8 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Clean (flat) and invoice price Clean and invoice price: Example Quoted prices are clean (flat) prices net of accrued interest Consider a bond with coupon rate of 8%. A buyer must pay the invoice price = clean price plus Annual coupon is $80 and semiannual coupon payment is accrued interest $40. Our annuity formula for the bond price gives the invoice If 30 days have passed since the last coupon payment, price. 30 accrued interest on the bond is: $40 × 182 = 6.59. Annual coupon pmt Days since last coupon pmt Accrued interest = × If the quoted price of the bond is $990, then the invoice 2 Days between coupon pmts price will be $990 + $6.59 = $996.59. (pmt = payment) 9 / 33 10 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Corporate Bonds Bond Pricing Some are traded at exchanges such as NYSE but most over the counter. Thin markets. Some bonds have options embedded: The (annualized) rate of return from zero-coupon bond with Callable bonds: Can be repurchased before the maturity maturity t is the market interest rate for that maturity, rt. date We are in a "chicken-and-egg" situation here: We could Puttable Bonds: Give the holder an option to retire or price bonds from market interest rates and then derive extend the bond market interest rates from bond prices. Convertible bonds: Can be exchanged for shares of the firm’s common stock Q: Which comes first, the market interest rates or the bond prices? Most corporate bonds have fixed coupon rates. But there are also floating-rate bonds with where the coupon rate is pegged to a reference interest rate (e.g., LIBOR). 11 / 33 12 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Present value formula for bonds Price and yield T X Ct F We can also write this formula with a single discount rate P= t + (1 + rt ) (1 + rT )T that, when applied to discount every cash payment, gives t=1 the market price of the bond. P is the price of the bond, We know already that this discount rate is called the yield F is the face value, to maturity y , or more general, the internal rate of return. Ct are the interest or coupon payments = cF where c is the Since typically all coupon payments are positive, the yield coupon rate, to maturity is a unique number. T T is the number of periods to maturity, X Ct F P= t + rt is the market interest rate for maturity t. Interest rates for (1 + y ) (1 + y )T t=1 different t’s will typically be not the same (more on the term structure of interest rates below). 13 / 33 14 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Annuity formula Example Bond Pricing We can use the annuity formula for the sum of discounted coupon payments since all coupons are the same, Ct = C = cF.   Price of a 30 year, 8% coupon bond (with face value 1,000)? C 1 F The market rate of interest is 10% for all maturities. Coupons P = 1− T + y (1 + y ) (1 + y )T are paid semiannually. = C × Annuity factor + F × Present value factor   $80 1 1, 000 P= 1− + = $810.71 If we measure T in years (not coupon periods), we need to 10% (1 + 5%)60 (1 + 5%)60 adjust the formula if the coupon frequency m is different from annual: " # C 1 F P= 1− y mT + y mT y (1 + m ) (1 + m ) 15 / 33 16 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Example Yield to maturity Inverse relationship between bond price and yield Now let’s proceed the opposite way. We compute the yield to maturity from an observed market price of a bond. Suppose an 8% coupon, 30 year bond is selling for $1276.76. What is its yield to maturity? " # $80 1 1, 000 $1276.76 = 1− y 60 + y (1 + 2 ) (1 + y2 )60 The solution is y = 6%. Q: What would be the effective annual rate? 17 / 33 18 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Long and short term bonds Pure discount bonds = Zero-coupon bonds As before, consider an 8% coupon bond with coupons paid semiannually, face value $1,000. T-bills with maturity less than one year are issued as zero coupon (= pure discount) bonds For longer maturities, pure discount bonds can be created by breaking down a coupon bond into many securities Each security is a claim to one of the payments of the original bond. For US Treasuries these are called Treasury strips STRIPS = Separate Trading of Registered Interest and Principal of Securities Long-term bonds are more sensitive the to changes in market interest rates. Q: Why? 19 / 33 20 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz The Price of a 30-Year Zero-Coupon Bond over Time The yield curve (term structure) Reading: Bodie, Kane, Markus ch. 15 (see also Berk/deMarzo ch. 6) 21 / 33 22 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz The yield curve Treasury yield curve, 15 Feb, 2019 1m 2m 3m 6m 1y 2y 3y 5y 7y 10y 20y 30y 2.43 2.43 2.43 2.50 2.55 2.52 2.50 2.49 2.57 2.66 2.84 3.00 The yield curve is a graph that displays the relationship between yield to maturity and time to maturity. This relationship is also referred to as the term structure of interest rates. We can extract information about expected future short-term rates from the yield curve. Let’s look at the yields of Treasury bills, notes and bonds for maturities from one month up to 30 years, as of Feb 15, 2019. 23 / 33 24 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Different yield curve shapes Two types of yield curves What we have just seen is a so-called on-the-run yield curve. It typically uses recently-issued coupon bonds selling at or near par. If maturities are not exactly observed, yields are obtained by interpolation between neighboring maturities. These yield curves are typically published by the financial press. A different type is the pure yield curve or zero yield curve. It uses yields of zero coupon bonds and, if they are not available, of stripped coupon bonds. These yields are not "contaminated" by intermediate cash flows; they refer to a single cash flows at the corresponding maturity. The zero Treasury yield curve gives the risk-free interest rates for each maturity. 25 / 33 26 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Zero yield curve: Example (1) Zero yield curve: Example (2) Prices and Yields to Maturities on zero-coupon bonds ($1,000 Value a 3 year, 10% coupon bond using discount rates from the Face Value) previous table. 100 100 1, 100 P= + + = 1082.17 1.05 1.062 1.073 The yield to maturity solves 100 100 1, 100 1082.17 = + 2 + 1+y (1 + y ) (1 + y )3 Task: Value a 3 year, 10% coupon bond using the discount Using the IRR function in Python, Excel or another software rates from the table. What is the yield to maturity? gives y = 6.88%. 27 / 33 28 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Further reading Read more on the yield curve, forward rates and future interest rates: BKM, ch. 15 interest rate risk, duration and convexity: BKM, ch. 16. Mini-quiz bond pricing formula between coupon dates, clean (flat) price and accrued interest and a lot more about bonds and fixed income markets: Smith, Donald J. (2014), Bond math: the theory behind the formulas, Wiley. 29 / 33 30 / 33 Bond prices and yields The yield curve Mini-quiz Bond prices and yields The yield curve Mini-quiz Question 1 Question 2 The yield to maturity of a bond is Accrued interest is a) the discount rate that makes the NPV of the payments of a) the present value of all interest payments of a bond, the bond equal to zero, b) the sum of all interest payments of a bond, b) the sum of all interest payments, c) the coupon payment multiplied by the fraction of the c) its rate of return, number of days since the last coupon payment of the total number of days in a coupon period. d) the geometric average of the yields for the maturity of each coupon payment. 31 / 33 32 / 33 Bond prices and yields The yield curve Mini-quiz Question 3 The price of a two-year bond with 10% coupon rate, semiannual coupon payments, face value 100, with a current market yield of similar bonds of zero, is a) 140 b) 120 c) 105 d) 100. 33 / 33