IV. Fixed-Income Securities A. Bond Prices and Yields B. The Term Structure of Interest Rates C. Managing Bond Portfolios 1. Duration 2. Convexity 3. Applications

C. Managing Bond Portfolios 6 well known bond pricing relationships 1. Bond prices and yields are inversely related. 2. Yield increases cause proportionately smaller price changes than yield decreases. 3. Long-term bond prices tend to be more sensitive to interest rate changes than short-term bond prices. 4. As maturity increases, price sensitivity to yield changes increases at a decreasing rate. 5. Interest rate risk is inversely related to the bond’s coupon rate. 6. Bond prices are more sensitive to changes in yields when the bond is selling at a lower initial yield to maturity.

C. Managing Bond Portfolios 1. Duration Duration: a measure of the sensitivity of a bond price to interest rate changes. Macaulay duration: D = t t wt where wt = CFt/(1+y)t P Macaulay duration is a measure of the time flow of cash from a bond. It is the weighted average of the times to each coupon or principal payment made by the bond. The weights are related to the “importance” of the payment to the value of the bond.

C. Managing Bond Portfolios 1. Duration Duration is important for 3 primary reasons: (1) It is a simple summary statistic of the effective average maturity of a bond or a portfolio of bonds. Example: What is the duration of a 2 year coupon bond. Assume coupon rate is 8%, coupons are paid semiannually, yield to maturity is 10%. Compare this to the duration of a two-year zero. D = .5(40)/(1.05) + 1(40)/(1.05)2 + 1.5(40)/(1.05)3 + 2(1040)/(1.05)4 = 1.8853 P P P P D = 2(1000)/(1.05)4 = 2 P

C. Managing Bond Portfolios 1. Duration (2) Duration is a measure of the interest rate sensitivity of a bond. It can be shown that, P/P = -D[ (1+y)/(1+y) ] In words, the percent change in the price of a bond is equal to the percent change in the gross yield times duration. Practitioners commonly use a modified duration: D* = D/(1+y) Thus, P/P = -D* (1+y) = -D*  y

C. Managing Bond Portfolios 1. Duration (2) Duration is a measure of the interest rate sensitivity of a bond. Example: Consider a 2 year coupon bond with D = 1.8853, paying semi-annual coupons at an initial semi-annual interest rate or 5%, and a zero with D = 1.8853. Suppose the yield rises to 5.01%. Pc = 964.54  964.19 0.0359% change Pz = 1000/(1.05)3.7704 = 831.97  1000/(1.051) 3.7704 = 831.67 0.0359% change

C. Managing Bond Portfolios 1. Duration (3) Duration is an essential tool in immunizing portfolios from interest rate risk. In order to immunize an overall position from interest rate movements, one should construct portfolios that minimize the “duration gap” of assets and liabilities.

C. Managing Bond Portfolios 1. Duration Rules for duration: (1) Duration of a zero equals its time to maturity. (2) Holding maturity constant, a bond’s duration is higher when the coupon rate is lower. (3) Holding the coupon rate constant, duration generally increases with time to maturity. It always increases with maturity for bonds selling at par or at a premium to par. (4) Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower. (5) Duration of a level perpetuity is (1+y)/y.

C. Managing Bond Portfolios 2. Convexity The duration rule for the impact of interest rates on bond prices is only an approximation. P/P = -D[ (1+y)/(1+y) ]  a linear relationship, but the actual price-yield relationship is convex.

C. Managing Bond Portfolios 2. Convexity Remark: Convexity is generally considered a desirable trait. Duration: 1st derivative of price-yield curve expressed as a fraction of bond price. Convexity: 2nd derivative of price-yield curve divided by bond price. Convexity = d2P 1 dy2 P = 1/[ P(1+y)2] t[ CFt/(1+y)t (t2 + t) ] Thus, we can improve the duration approximation as follows: P/P = -D* y + 1/2 convexity (y )2

C. Managing Bond Portfolios 3. Applications (a) Passive Bond Management - takes no view on future movements in interest rates, and seeks to control risk of portfolio. 2 general strategies (1) indexing (2) immunization - insulate portfolio from interest rate risk. “Gap management.”

C. Managing Bond Portfolios 3. Applications Example: Immunize overall position by DA = DL or DAA = DLL Suppose DL = 6. Available assets are cash and a 10 year zero. Form a portfolio of assets with DA = DL. w 0 + (1-w) 10 = 6 (1-w) = .6 For every $100 of liabilities, invest $60 in bonds and $40 in cash.

C. Managing Bond Portfolios 3. Applications (b) Active Bond Management - attempts to beat the market by exploiting forecasts of interest rate changes. (1) Rate Anticipation Strategy: investment manager takes a view on the future course of the level of interest rates, and constructs a portfolio that benefits most from interest rate moves in the anticipated direction.

C. Managing Bond Portfolios 3. Applications (b) Active Bond Management - attempts to beat the market by exploiting forecasts of interest rate changes. (2) Spread Anticipation Strategy: investment manager takes a view on how the yield curve spread will vary over time and constructs a portfolio accordingly. No specific view is taken on how the level of rates will vary over time. Basic idea: if the spread is larger than historical levels, perhaps because short-term rates are artificially low due to Fed intervention, one may anticipate that it will decrease over time.

C. Managing Bond Portfolios 3. Applications (b) Active Bond Management - attempts to beat the market by exploiting forecasts of interest rate changes. (3) Interest Rate Volatility Anticipation Strategy: investment manager takes a view on the future volatility of interest rates and constructs a portfolio accordingly. No specific view is taken on how the level of rates will vary over time. Basic idea: suppose that the manager must construct a bond portfolio with a certain duration, and also expects that the volatility of interest rates will be high in the future, perhaps because of economic uncertainty. What sort of portfolio should he choose?

C. Managing Bond Portfolios Homework, ch. 16: 1-2, 5, 8a-f and h, 9-10 and For this question, you should assume that the current risk-free interest rate is 8%, and the term structure is flat for bonds of all maturities. (a) The bank you work for has entered into a Guaranteed Income Contract (GIC) that promises to pay $500,000 per year for the next 15 years. What is the duration and convexity of this liability? Assuming that the yield to maturity on risk-free bonds falls by 100 basis points, estimate your expected capital loss on the GIC using only its duration and using both its duration and convexity. What is the actual capital loss on the GIC? (For simplicity, assume that the first payment is due exactly one year from today.) (b) How well can you hedge the interest rate risk associated with the GIC in question using only cash and a 10-year zero-coupon bond? How well can you hedge this interest rate risk using cash, a 5-year zero-coupon bond, and a 10-year zero-coupon bond?