Bond Prices and Yields

Bond Characteristics 14-2 Bond Prices and Yields Bonds are debt. Issuers are borrowers and holders are creditors. The indenture is the contract between the issuer and the bondholder. The indenture gives the coupon rate, maturity date, and par value. Bond Characteristics 14-3 U.S. Treasury Bonds 14-4 Face or par value is typically $1000; this is the principal repaid at maturity. The coupon rate determines the interest payment. Interest is usually paid semiannually. The coupon rate can be zero. Interest payments are called coupon payments. Note maturity is 1-10 years Bond maturity is 10-30 years Bonds and notes may be purchased directly from the Treasury. Denomination can be as small as $100, but $1,000 is more common. Bid price of 100:08 means 100 8/32 or $1002.50

Corporate Bonds 14-5 Preferred Stock 14-6 Callable bonds can be repurchased before the maturity date. Convertible bonds can be exchanged for shares of the firm s common stock. Puttable bonds give the bondholder the option to retire or extend the bond. Floating rate bonds have an adjustable coupon rate Equity Fixed income Dividends are paid in perpetuity. Nonpayment of dividends does not mean bankruptcy. Preferred dividends are paid before common. No tax break. 14-7 14-8 Innovation in the Bond Market Inverse Floaters Asset-Backed Bonds Catastrophe Bonds Indexed Bonds Treasury Inflation Protected Securities (TIPS). P B C ParValue t T 1 (1 r) (1 r) T t Bond Pricing P B = Price of the bond C t = interest or coupon payments T = number of periods to maturity r = semi-annual discount rate or the semi-annual yield to maturity

Example 14.2: Bond Pricing 14-9 Bond Prices and Yields 14-10 Price of a 30 year, 8% coupon bond. Market rate of interest is 10%. 60 $40 $1000 Price t1 05 t 1.05 1. 60 Prices and yields (required rates of return) have an inverse relationship The bond price curve (Figure 14.3) is convex. The longer the maturity, the more sensitive the bond s price to changes in market interest rates. Figure 14.3 The Inverse Relationship Between Bond Prices and Yields 14-11 Table 14.2 Bond Prices at Different Interest Rates 14-12

Yield to Maturity Interest rate that makes the present value of the bond s payments equal to its price is the YTM. Solve the bond formula for r P B C ParValue t T 1 (1 r) (1 r) T t 14-13 Yield to Maturity Example Suppose an 8% coupon, 30 year bond is selling for $1276.76. What is its average rate of return? 60 $40 $ 1276.76 1000 t 60 t 1 (1 r) (1 r) 14-14 YTM vs. Current Yield 14-15 Yield to Call 14-16 YTM The YTM is the bond s internal rate of return. YTM is the interest rate that makes the present value of a bond s payments equal to its price. YTM assumes that all bond coupons can be reinvested at the YTM rate. Current Yield The current yield is the bond s annual coupon payment divided by the bond price. For bonds selling at a premium, coupon rate > current yield>ytm. For discount bonds, relationships are reversed. If interest rates fall, price of straight bond can rise considerably. The price of the callable bond is flat over a range of low interest rates because the risk of repurchase or call is high. When interest rates are high, the risk of call is negligible and the values of the straight and the callable bond converge.

Figure 14.4 Bond Prices: Callable and Straight Debt 14-17 Figure 14.6 Prices over Time of 30-Year Maturity, 6.5% Coupon Bonds 14-18 YTM vs. HPR 14-19 Figure 14.7 The Price of a 30-Year Zero- Coupon Bond over Time 14-20 YTM YTM is the average return if the bond is held to maturity. YTM depends on coupon rate, maturity, and par value. All of these are readily observable. HPR HPR is the rate of return over a particular investment period. HPR depends on the bond s price at the end of the holding period, an unknown future value. HPR can only be forecasted.

14-21 14-22 Default Risk and Bond Pricing Factors Used by Rating Companies Rating companies: Moody s Investor Service, Standard & Poor s, Fitch Rating Categories Highest rating is AAA or Aaa Investment grade bonds are rated BBB or Baa and above Speculative grade/junk bonds have ratings below BBB or Baa. Coverage ratios Leverage ratios Liquidity ratios Profitability ratios Cash flow to debt Table 14.3 Financial Ratios and Default Risk by Rating Class, Long-Term Debt 14-23 Figure 14.9 Discriminant Analysis 14-24

Protection Against Default Sinking funds a way to call bonds early Subordination of future debt restrict additional borrowing Dividend restrictions force firm to retain assets rather than paying them out to shareholders Collateral a particular asset bondholders receive if the firm defaults 14-25 Default Risk and Yield The risk structure of interest rates refers to the pattern of default premiums. There is a difference between the yield based on expected cash flows and yield based on promised cash flows. The difference between the expected YTM and the promised YTM is the default risk premium. 14-26 Figure 14.11 Yield Spreads 14-27 The Term Structure of Interest Rates

Overview of Term Structure 15-29 Figure 15.1 Treasury Yield Curves 15-30 The yield curve is a graph that displays the relationship between yield and maturity. Information on expected future short term rates can be implied from the yield curve. Bond Pricing Yields on different maturity bonds are not all equal. We need to consider each bond cash flow as a stand-alone zero-coupon bond. The value of the bond should be the sum of the values of its parts. 15-31 Table 15.1 Prices and Yields to Maturities on Zero-Coupon Bonds ($1,000 Face Value) 15-32

15-33 15-34 Example 15.1 Valuing Coupon Bonds Two Types of Yield Curves Value a 3 year, 10% coupon bond using discount rates from Table 15.1: $100 $100 $1100 Price 2 3 1.05 1.06 1.07 Price = $1082.17 and YTM = 6.88% 6.88% is less than the 3-year rate of 7%. Pure Yield Curve The pure yield curve uses stripped or zero coupon Treasuries. The pure yield curve may differ significantly from the on-the-run yield curve. On-the-run Yield Curve The on-the-run yield curve uses recently issued coupon bonds selling at or near par. The financial press typically publishes onthe-run yield curves. Yield Curve Under Certainty 15-35 Figure 15.2 Two 2-Year Investment Programs 15-36 Suppose you want to invest for 2 years. Buy and hold a 2-year zero -or- Rollover a series of 1-year bonds Equilibrium requires that both strategies provide the same return.

15-37 15-38 Yield Curve Under Certainty Spot Rates vs. Short Rates Buy and hold vs. rollover: (1 y ) (1 r ) x(1 r ) 2 2 1 2 1 y (1 r ) x(1 r ) 2 1 2 Next year s 1-year rate (r 2 ) is just enough to make rolling over a series of 1-year bonds equal to investing in the 2-year bond. 1 2 Spot rate the rate that prevails today for a given maturity Short rate the rate for a given maturity (e.g. one year) at different points in time. A spot rate is the geometric average of its component short rates. Short Rates and Yield Curve Slope 15-39 Figure 15.3 Short Rates versus Spot Rates 15-40 When next year s short rate, r 2, is greater than this year s short rate, r 1, the yield curve slopes up. May indicate rates are expected to rise. When next year s short rate, r 2, is less than this year s short rate, r 1, the yield curve slopes down. May indicate rates are expected to fall.

Forward Rates from Observed Rates 15-41 Example 15.4 Forward Rates 15-42 (1 y ) 1) n n ( 1 fn) n1 (1 yn f n = one-year forward rate for period n y n = yield for a security with a maturity of n n (1 y ) (1 y n ) (1 f ) n1 n1 n The forward interest rate is a forecast of a future short rate. Rate for 4-year maturity = 8%, rate for 3- year maturity = 7%. 4 4 1 y4 1.08 1 f4 1.1106 3 3 1 y 1.07 3 f4 11. 06% 15-43 15-44 Interest Rate Uncertainty Interest Rate Uncertainty Suppose that today s rate is 5% and the expected short rate for the following year is E(r 2 ) = 6%. The value of a 2-year zero is: The value of a 1-year zero is: The investor wants to invest for 1 year. Buy the 2-year bond today and plan to sell it at the end of the first year for $1000/1.06 =$943.40. 0r- Buy the 1-year bond today and hold to maturity.

Interest Rate Uncertainty What if next year s interest rate is more (or less) than 6%? 15-45 Interest Rate Uncertainty Investors require a risk premium to hold a longer-term bond. 15-46 The actual return on the 2-year bond is uncertain! This liquidity premium compensates short-term investors for the uncertainty about future prices. Theories of Term Structure 15-47 Expectations Theory 15-48 Expectations Liquidity Preference Upward bias over expectations Observed long-term rate is a function of today s short-term rate and expected future short-term rates. f n = E(r n ) and liquidity premiums are zero.

Liquidity Premium Theory 15-49 Figure 15.4 Yield Curves 15-50 Long-term bonds are more risky; therefore, f n generally exceeds E(r n ) The excess of f n over E(r n ) is the liquidity premium. The yield curve has an upward bias built into the long-term rates because of the liquidity premium. Figure 15.4 Yield Curves 15-51 Interpreting the Term Structure 15-52 The yield curve reflects expectations of future interest rates. The forecasts of future rates are clouded by other factors, such as liquidity premiums. An upward sloping curve could indicate: Rates are expected to rise And/or Investors require large liquidity premiums to hold long term bonds.

Interpreting the Term Structure The yield curve is a good predictor of the business cycle. Long term rates tend to rise in anticipation of economic expansion. Inverted yield curve may indicate that interest rates are expected to fall and signal a recession. 15-53 Figure 15.6 Term Spread: Yields on 10-year vs. 90-day Treasury Securities 15-54 Bond Pricing Relationships 16-56 Managing Bond Portfolios 1. Bond prices and yields are inversely related. 2. An increase in a bond s yield to maturity results in a smaller price change than a decrease of equal magnitude. 3. Long-term bonds tend to be more price sensitive than short-term bonds.

Bond Pricing Relationships 16-57 Figure 16.1 Change in Bond Price as a Function of Change in Yield to Maturity 16-58 4. As maturity increases, price sensitivity increases at a decreasing rate. 5. Interest rate risk is inversely related to the bond s coupon rate. 6. Price sensitivity is inversely related to the yield to maturity at which the bond is selling. Table 16.1 Prices of 8% Coupon Bond (Coupons Paid Semiannually) 16-59 Table 16.2 Prices of Zero-Coupon Bond (Semiannually Compounding) 16-60

Duration 16-61 Duration: Calculation 16-62 A measure of the effective maturity of a bond The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment Duration is shorter than maturity for all bonds except zero coupon bonds. Duration is equal to maturity for zero coupon bonds. w CF t t D T t1 t ( 1 y) Price t wt CF t =cash flow at time t Duration/Price Relationship 16-63 Example 16.1 Duration 16-64 Price change is proportional to duration and not to maturity P (1 y) Dx P 1 y D * = modified duration P D* y P Two bonds have duration of 1.8852 years. One is a 2-year, 8% coupon bond with YTM=10%. The other bond is a zero coupon bond with maturity of 1.8852 years. Duration of both bonds is 1.8852 x 2 = 3.7704 semiannual periods. Modified D = 3.7704/1+0.05 = 3.591 periods

Example 16.1 Duration 16-65 Example 16.1 Duration 16-66 Suppose the semiannual interest rate increases by 0.01%. Bond prices fall by: * P D y P =-3.591 x 0.01% = -0.03591% Bonds with equal D have the same interest rate sensitivity. Coupon Bond The coupon bond, which initially sells at $964.540, falls to $964.1942 when its yield increases to 5.01% percentage decline of 0.0359%. Zero The zero-coupon bond initially sells for $1,000/1.05 3.7704 = $831.9704. At the higher yield, it sells for $1,000/1.05 3.7704 = $831.6717. This price also falls by 0.0359%. Rules for Duration 16-67 Rules for Duration 16-68 Rule 1 The duration of a zero-coupon bond equals its time to maturity Rule 2 Holding maturity constant, a bond s duration is higher when the coupon rate is lower Rule 3 Holding the coupon rate constant, a bond s duration generally increases with its time to maturity Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond s yield to maturity is lower Rules 5 The duration of a level perpetuity is equal to: (1+y) / y

Figure 16.2 Bond Duration versus Bond Maturity 16-69 Table 16.3 Bond Durations (Yield to Maturity = 8% APR; Semiannual Coupons) 16-70 Convexity The relationship between bond prices and yields is not linear. 16-71 Figure 16.3 Bond Price Convexity: 30-Year Maturity, 8% Coupon; Initial YTM = 8% 16-72 Duration rule is a good approximation for only small changes in bond yields. Bonds with greater convexity have more curvature in the price-yield relationship.

16-73 16-74 Convexity Figure 16.4 Convexity of Two Bonds Convexity P 1 (1 y) n t 2 ( t t) 2 t t1 (1 y) CF Correction for Convexity: P D y Convexity y P 2 1 2 [ ( ) ] Why do Investors Like Convexity? 16-75 Callable Bonds 16-76 Bonds with greater curvature gain more in price when yields fall than they lose when yields rise. The more volatile interest rates, the more attractive this asymmetry. Bonds with greater convexity tend to have higher prices and/or lower yields, all else equal. As rates fall, there is a ceiling on the bond s market price, which cannot rise above the call price. Negative convexity Use effective duration: P/ P Effective Duration = r

Figure 16.5 Price Yield Curve for a Callable Bond 16-77 Passive Management Two passive bond portfolio strategies: 16-78 1.Indexing 2.Immunization Both strategies see market prices as being correct, but the strategies have very different risks. Bond Index Funds Bond indexes contain thousands of issues, many of which are infrequently traded. Bond indexes turn over more than stock indexes as the bonds mature. Therefore, bond index funds hold only a representative sample of the bonds in the actual index. 16-79 Figure 16.8 Stratification of Bonds into Cells 16-80

16-81 16-82 Immunization Immunization Immunization is a way to control interest rate risk. Widely used by pension funds, insurance companies, and banks. Immunize a portfolio by matching the interest rate exposure of assets and liabilities. This means: Match the duration of the assets and liabilities. Price risk and reinvestment rate risk exactly cancel out. Result: Value of assets will track the value of liabilities whether rates rise or fall. Table 16.4 Terminal value of a Bond Portfolio After 5 Years 16-83 Table 16.5 Market Value Balance Sheet 16-84

Figure 16.9 Growth of Invested Funds 16-85 Figure 16.10 Immunization 16-86 Cash Flow Matching and Dedication Cash flow matching = automatic immunization. Cash flow matching is a dedication strategy. Not widely used because of constraints associated with bond choices. 16-87