Foundations of Finance: Bond Portfolio Management Prof. Alex Shapiro 1 Lecture Notes 13 Bond Portfolio Management I. Readings and Suggested Practice Problems II. Risks Associated with Default-Free Bonds III. Duration: Details and Examples IV. Immunization Buzz Words: Interest Rate Risk, Reinvestment Risk, Liquidation Risk, Macaulay Duration, Modified Duration, Convexity, Target-Date Immunization, Net-Worth Immunization, Duration GapFoundations of Finance: Bond Portfolio Management 2 I. Readings and Suggested Practice Problems BKM, Chapter 16, Sections 16.1-16.3 Suggested Problems, Chapter 16: 16-18. E-mail: Open the Bond Immunization program to generate the examples in Section IV, and to construct your own examples of target-date immunization. II. Risks Associated with Default-Free Bonds A. Reinvestment Risk If an individual has a particular time horizon T and holds an instrument with a fixed cash flow received prior to T, then the investor faces uncertainty about what yields will prevail at the time of the cash flow. This uncertainty is known as reinvestment risk. Example Suppose an investor has to meet an obligation of $5M in two years time. If she buys a two year coupon bond to meet this obligation, there is uncertainty about the rate at which the coupons on the bond can be invested. This uncertainty is an example of reinvestment risk.Foundations of Finance: Bond Portfolio Management 3 B. Liquidation (Price) Risk If an individual has a particular time horizon T and holds an instrument which generates cash flows that are received after T, then the investor faces uncertainty about the price of the instrument at time T. This uncertainty is known as liquidation risk. Example Suppose an investor has to meet an obligation of $5M in two years time. If she buys a five year discount bond to meet this obligation, there is uncertainty about the price at which this bond will sell in two years time. This uncertainty is an example of liquidation risk. C. Yield Changes and Price Changes 1. If a bond sells at a premium or discount, its price will converge to par, even if the YTM y stays constant. (This price change is expected, and is not normally considered risk.) 2. Unexpected bond price changes will occur if market interest rates changes unexpectedly. This is interest rate risk, which causes the reinvestment risk and liquidation risk; It affects the rate at which coupon payments can be reinvested, and affects the price at which a bond can be sold (prior to maturity).Foundations of Finance: Bond Portfolio Management 4 Example Price path of a $100-par zero that matures in year 10 3. Can examine the plots of price P of any bond as a function of the YTM, y, using the following functional relation: P = ∑=+TtttyCF1)1( If you know how ∆y affects ∆P, you can tell how ∆y affects portfolios of bonds, and can decide how to manage your bond portfolio given your objectives. $0.00$20.00$40.00$60.00$80.00$100.00$120.000246810TimeBond PricePrice path if y=5% for 10 years Price path if y unexpected shifts to 10% in year 3. Unexpected price dropFoundations of Finance: Bond Portfolio Management 5 ∆P = dydP ∆y + 2221dyPd (∆y)2 + … Divide by P: PP∆ = P1dydP ∆y + 22121dyPdP (∆y)2 + … = -D* ∆y + 21 (Convexity) (∆y)2 + … = -yD+1 ∆y + 21 (Convexity) (∆y)2 + … A second order approximation for the impact of yield changes is therefore (using the definitions for D* and Convexity given below): PP∆ ≈ -D* ∆y + 21 (Convexity) (∆y)2 A first order approximation (a linear, and less precise one) is: PP∆ ≈ -D* ∆y = -yD+1 ∆y = -Dyy++1)1( with the Duration (D), the Modified Duration (D*), and Convexity defined as: D = D* (1 + y) = - P1dydP (1 + y) = ∑=+×TtttPyCFt1)1(/ D* = D / (1 + y) Convexity = 221dyPdP = P12)1(1y+∑=++TtttyttCF1)1()2(Foundations of Finance: Bond Portfolio Management 6 III. Duration: Details and Examples To understand the important features of managing fixed income portfolios, we will focus on the simpler, first-order approximation to the impact of yield changes on prices. A. Duration Duration (the D defined above is Macaulay’s first measure of duration) is used to measure the price risk of a bond (i.e., interest rate sensitivity). Duration relates the change in a bond price (∆P) to the associated change in the bond’s YTM (∆y). Duration is computed as the effective (weighted average) maturity of the bond. (This is distinct from the nominal maturity of the bond.) B. The price change caused by a given change in YTM: ∆p vs. ∆y The link between maturity and price risk is easiest to demonstrate for a zero: Example 15-year zero, $100 par, y = 8% ⇒ p = PV(100, 8%, 15) = 31.5242Foundations of Finance: Bond Portfolio Management 7 Suppose we have a small change in y: ∆y = -.01% = -0.0001 (“y drops by one basis point”). p = PV(100, 7.99%, 15) = 31.5680 ∆p = 31.5680 - 31.5242 = 0.0438 One way to relate ∆p and ∆y is to compute The example illustrates that for a zero coupon bond, we can estimate its relative price change (for a given change in y) by: For any bond : where D is the duration of the bond, and y is the YTM. ()yp+−1in change alProportionin change alProportion()()maturity) (the 150000926.0001389.008.10001.05242.310438.011Myypp===−−=++∆∆−()()yyMpp++∆×−≈∆11()()yyDpp++∆×−≈∆11Foundations of Finance: Bond Portfolio Management 8 C. Computing the Duration for Zeros For a zero-coupon bond: D = M (the stated maturity) For a portfolio of two zeros: D = the weighted average maturity of the two zeros: DP = w1 D1 + w2 D2 where the weights, w’s, are market-value weights. Example y = 10% Bond 1 is a 5-yr, 100 par zero, ⇒ P1=62.09 Bond 2 is a 10-yr, 100 par zero, ⇒ P2=38.55 ⇒ The total value of the portfolio is 100.64 Exercise: verify that if 1+y goes to 1.101, the value of the portfolio changes by -6.9∆(1+y)/(1+y) years 96103805620 106410055385641000962.......DP=×+×=+=Foundations of Finance: Bond Portfolio Management 9 D. The Duration of a Coupon Bond A coupon bond is simply a portfolio of zeros. Let CFt =