Bond Pricing Relationships 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. (price sensitivity inversely related to YTM) 3.Long term bonds tend to be more price sensitive than short term bonds. Bond Pricing Relationships 4. As maturity increases, price sensitivity increases at a decreasing rate. Figure 16.1 Change in Bond Price as a Function of Change in Yield to Maturity (1Y vs. 2Y has higher price sensitivity than 19 vs. 20Y because PV of CF far in the future are discounted more heavily) 5. Interest rate risk is inversely related to the bond s coupon rate. (higher coupon > higher portion of PV of CFs shift closer to present time > shorter duration & interest rate sensitivity) 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) Table 16.2 Prices of Zero Coupon Bond (Semiannually Compounding) 1
Macaulay Duration Macaulay Duration (1938) 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. First derivative of the price yield curve Duration: Calculation Why duration matter? It is a simple summary measure of the effective average maturity of the portfolio Is an essential tool in immunizing portfolios (more on this later this class) Where w t = weight of the CF D = duration = PV weighted average of each CFs Is a measure of interest rate sensitivity of a bond portfolio Duration/Price Relationship Where y = yield to maturity D* = Modified Duration = Macaulay Duration / (1+y) Example 11.1 Duration A bond with maturity of 30Y has annual coupon rate of 8% and YTM 9%. Its prices is $897.26, and its duration is 11.37 years. What will happen if the bond price if the bond s YTM increase to 9.1%? Sol. Delta y = +0.1%, P=897.26, Delta P =? Delta P / P = D* Delta y Delta P = (11.37/1.09) * 0.001 * 897.26 = $9.36 2
Rules for Duration 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 h when the coupon rate is lower (high coupon = more money coming back to you faster = shorter duration and vice versa) Rule 3 Holding the coupon rate constant, a bond s duration generally increases with its time to maturity Rules for Duration Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond s yield to maturity is lower (discount rate inversely related to duration) Rules 5 The duration of a level perpetuity is equal to: (1+y) / y (property from geometric sum The effect of discount of CFs far in the future is more severe) For example, at a 10% yield, the duration of perpetuity that pays $100 once a year forever will equal 1.10/.10 = 11 years, but at an 8% yield it will equal 1.08/.08 = 13.5 years. The maturity of the perpetuity is infinite, whereas the duration of the instrument at a 10% yield is only 11 years. Figure 16.2 Bond Duration versus Bond Maturity Table 16.3 Bond Durations (Yield to Maturity = 8% APR; Semiannual Coupons Convexity The relationship between bond prices and yields is not linear. Figure 16.3 Bond Price Convexity: 30 Year Maturity, 8% Coupon; Initial YTM = 8% 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. Second derivative of the price yield curve 3
Convexity Figure 16.4 Convexity of Two Bonds A bond with greater convexity is less affected by interest rates than a bond with less convexity Bond A has a higher convexity than Bond B, which means that all else being equal, Bond A will always have a higher price than Bond B as interest rates rise or fall (better risk characteristic) Why do Investors Like Convexity? Bonds with greater curvature gain more in price when yields fall than they lose when yields rise. The more volatile interest rates, themore attractive this asymmetry. Bonds with greater convexity tend to have higher prices and/or lower yields, all else equal. Callable Bonds 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: Figure 16.5 Price Yield Curve for a Callable Bond Passive Management Two passive bond portfolio strategies: 1. Immunization (Duration matching) 2. Dedication (CF matching) Both strategies see market prices as being correct 4
Immunization Many banks have a natural mismatch between the maturities of assets and liabilities Interest rate change affect the value of long term assets (such as mortgage loan) more severely than deposit (short duration) Immunization is a way to control interest rate risk. Widely used by pension funds, insurance companies, and banks. Immunization 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 Table 16.5 Market Value Balance Sheet Figure 11.3 Growth of Invested Funds Figure 11.4 Immunization When interest fall, the coupon grow less than in the base case, but the higher value of the bond offset this. When interest rise (in picture), the value of the bond falls but the coupon more than make up fir this loss because they are reinvested at the higher rates. The initial impact is capital loss but this loss eventually is offset by the now faster growth rate of reinvested funds. 5
Cash Flow Matching and Dedication Cash flow matching = automatic immunization. Cash flow matching is a dedication strategy. Not widely used becauseof constraints associated with bond choices. Ex. It is not possible to dedicate zero coupon bond for perpetuity payments of a pension fund. In addition, some bonds are believed to be undervalued. Hence firms prefer to immunize using these bonds. Homework Rank the following bonds in order of ascending (low to high) duration Bond Coupon Time to Maturity (Years) Yield to Maturity A 14% 15 10% B 14% 10 10% C 0% 15 10% D 10% 15 10% E 14% 10 15% Duration : C > D > A > B > E 6