Chapter 13

1. Chapter 13 Managing Bond Portfolios

2. Chapter Summary • Objective: To examine various strategies available to fixed-income portfolio managers. • Duration and Convexity • Passive Management • Active Management

3. Managing Fixed Income Securities: Basic Strategies • Active strategy • Trade on interest rate predictions • Trade on market inefficiencies • Passive strategy • Control risk • Balance risk and return

4. Managing interest rate risk • Bond price risk • Coupon reinvestment rate risk • Matching maturities to needs • The concept of duration • Duration-based strategies • Controlling interest rate risk with derivatives

5. Bond Pricing Relationships • Inverse relationship between price and yield • An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield • Long-term bonds tend to be more price sensitive than short-term bonds

6. Bond Pricing Relationships (cont’d) • As maturity increases, price sensitivity increases at a decreasing rate • Price sensitivity is inversely related to a bond’s coupon rate • Price sensitivity is inversely related to the yield to maturity at which the bond is selling

7. Percentage change in bond price A B C D Change in yield to maturity (%) Interest Rate Sensitivity 0

8. Summary Reminder • Objective: To examine various strategies available to fixed-income portfolio managers. • Duration and Convexity • Passive Management • Active Management

9. Duration • 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

10. Bond Duration = 5.97 years Example: 8-year, 9% annual coupon bond

11. Duration: Calculation

12. Duration Calculation: Example using Table 13.3 8% Time Payment PV of CF Weight C1 X Bond years (10%) C4 .5 40 38.095 .0395 .0198 1 40 36.281 .0376 .0376 1.5 40 34.553 .0358 .0537 2.0 1040 855.611 . 8871 1.7742 sum 964.540 1.000 1.8853

13. Why is duration a key concept? • It’s a simple summary statistic of the effective average maturity of the portfolio; • It is an essential tool in immunizing portfolios from interest rate risk; • It is a measure of interest rate risk of a portfolio

14. Duration/Price Relationship • Price change is proportional to duration and not to maturity • Or, if we denote D* = modified duration

15. Notes on duration • The duration formula is derived from the bond price-yield relationship • Calculate dP/dy and divide with P • Approximating the bond price change using duration is equivalent to moving along the slope of the bond price-yield curve

16. 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

17. Rules for Duration (cont’d) Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower Rule 5 The duration of a level perpetuity is equal to:

18. Rules for Duration (cont’d) Rule 6 The duration of a level annuity is equal to: Rule 7 The duration for a coupon bond is equal to:

19. Duration and Convexity Price Pricing error from convexity Yield Duration

20. Percentage change in bond price Bond A 0 Change in yield to maturity (%) Convexity of Two Bonds Bond B

21. Correction for Convexity Correction for Convexity:

22. 8% Time Payment PV of CF Weight t(t+1)x Bond Sem. (10%) C4 1 40 38.095 .0395 .0790 2 40 36.281 .0376 .2257 3 40 34.553 .0358 .4299 4 1040 855.611 . 8871 17.7413 sum 964.540 1.000 18.4759 Convexity calculation (from Table 13.3)

23. Convexity calculation (cont.) • Convexity is computed like duration, as a weighted average of the terms (t2+t) (rather than t) divided by (1+y)2 • Thus, in the above example, it is equal to 18.4759/1.052 = 16.7582 in semester terms.

24. Region of negative convexity Price-yield curve is below tangent Call Price Region of positive convexity Interest Rate 0 10% 5% Duration and Convexity of Callable Bonds

25. Summary Reminder • Objective: To examine various strategies available to fixed-income portfolio managers. • Duration and Convexity • Passive Management • Active Management

26. Passive Management • Bond-Index Funds • Immunization of interest rate risk • Net worth immunization Duration of assets = Duration of liabilities • Target date immunization Holding Period matches Duration • Cash flow matching and dedication

27. Notes on duration of bond portfolios • The duration of a bond portfolio is equal to the weighted average of the durations of the bonds in the portfolio • The portfolio duration, however, does not change linearly with time (see problem 16). The portfolio needs, therefore, to be rebalanced periodically to maintain target date immunization

28. Summary Reminder • Objective: To examine various strategies available to fixed-income portfolio managers. • Duration and Convexity • Passive Management • Active Management

29. Active Bond Management: Swapping Strategies • Substitution swap • Inter-market swap • Rate anticipation swap • Pure yield pickup • Tax swap

30. Yield to Maturity % 1.50 1.25 0.75 Maturity 3M 6M 9M Yield Curve Ride

31. Contingent Immunization • Combination of active and passive management • Strategy involves active management with a floor rate of return • As long as the rate earned exceeds the floor, the portfolio is actively managed • Once the floor rate or trigger rate is reached, the portfolio is immunized