1. Fixed-Income Portfolio Management b Strategies Risk ManagementRisk Management Trade on interest rate predictionsTrade on interest rate predictions Trade on market inefficienciesTrade on market inefficiencies

2. Duration b A measure of the effective maturity of a bond b The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment b Duration = the 1st-order derivative of bond price with respect to yield. b Duration = bond’s sensitivity to interest-rate risk

3. Duration Formula b

4. Properties of Duration b The longer a bond’s maturity, the longer its duration and hence the more risky. b Duration is shorter than maturity for all bonds ( except zero coupon bonds ) b Duration = maturity, for zero -coupon bonds

5. Duration: Example b 8% Bond Time years PaymentPV of CF (10%) Weight t*wtt*wt.54038.095.0395.0198 14036.281.0376 1.5 2.0 40 1040 sum 34.553 855.611 964.540.0358.8871 1.000.0537 1.7742 1.8853

6. Duration/Price Relationship Price change is proportional to duration and not to maturity  P/P = -D {  (1+y) / (1+y) } or,  P/P = - D *  y where D * = D / (1+y), modified duration

7. 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 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

8. Passive (Risk) Management b Bond-Index Funds b Immunization of interest rate risk Net worth immunizationNet worth immunization Duration of assets = Duration of liabilities Target date immunizationTarget date immunization Holding Period matches Duration b Cash flow matching and dedication

9. Duration and Convexity b Price Duration Pricing Error from convexity Yield

10. Adjusting for Convexity b Thus, adjusting for Convexity, we have

11. Better Risk Management b For the previous types of funds/portfolios Duration Matching duration of assets = duration of liabilitiesDuration Matching duration of assets = duration of liabilities Convexity Matching: convexity of assets = convexity of liabilitiesConvexity Matching: convexity of assets = convexity of liabilities