Managing Bond Portfolios

1. Managing Bond Portfolios • Market / interest rate risk • Duration and Convexity • Passive Management • Active Management

2. Market/Interest Rate Risk • Biggest risk faced by bondholders • Two components: • Bond price risk • If selling before maturity, uncertainty about the price: capital gain/loss • Reinvestment risk • Reinvestment of coupons Note: “interest rate” and “yield” used interchangeably in the text

3. Drivers of interest rates

4. Coupons: Origin of the name

5. Bond Price Risk • Are long-term bonds more price sensitive to a change in yield, or short-term bonds? • Long-term bonds: more periods of discounting are affected (PV formula) • What other factors affect the price sensitivity of a bond? • Examples: coupon rate, initial yield (original yield before the change)

6. Bond features and bond price sensitivity

7. B. Duration and Convexity • Bond price sensitivity – need a formal measure • Duration is an elasticity concept – percentage change in bond price as a result of a one percentage change in yield • Macaulay Duration (1938) – Canadian from Montreal • Note: minus sign in front • Note: percentage change rather than absolute change

8. Duration • Recall the bond valuation/pricing model: • y is the yield to maturity of the bond (YTM)

9. Derive D using the bond valuation model Solution: where CFt = Cash flow to the bondholder in period t, y is the YTM Macaulay Duration

10. Numerical Example2-year, 8% bond (semi-annual), 10% YTM Time CF PV of CF Weight C1  C4 (in years) 0.5 40 38.095 .0395 .0198 40 36.281 .0376 .0376 1.0 1.5 40 34.553 .0358 .0537 2.0 1040 855.611 . 8871 1.7742 sum 964.540 1.000 1.8853 Duration of this bond is 1.8853 year

11. Numerical Example2-year, 8% bond (semiannual), 10% YTM Time CF PV of CF Weight C1  C4 (in half years) 1 40 38.095 .0395 .0395 2 40 36.281 .0376 .0752 3 40 34.553 .0358 .1074 4 1040 855.611 . 8871 3.5484 sum 964.540 1.000 3.7706 Duration of this bond is 1.8853 year

12. Another Interpretation of Duration • D is sometimes interpreted as the effective maturity of a bond • Weighted average maturity of the cash flow (coupons and par value), with the weights being proportional to the PV of the cash flow • For zero-coupon bonds: duration = term to maturity • For coupon bonds: duration < term to maturity

13. Taking frequency of coupon payments, k, explicitly into account in the definition of wt: If k = 2, y/k is the semiannual discount rate k in the denominator converts D into years More General Definition (e.g., Fabozzi)

14. Comparing Price Sensitivity Across Bonds • Duration allows comparison across bonds that differ in coupon rate, yield and maturity • D takes all of these parameters into account

15. Estimating Bond Price Changes Using Duration • Price change is proportional to duration (not to maturity): • Let D* (modified duration), taking into account the frequency of coupon payment, k, be:

16. Percentage Bond Price Change

17. Macaulay D = 10.98 years Initial YTM (before change in yield) = 6% Semiannual coupon Yield increases to 8% (i.e., +0.02) Percentage change in bond price Numerical example

18. Properties of 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 increases with its time to maturity

19. Properties of 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:

20. 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:

21. Bond Price Convexity(30-Year Term; 8% Coupon; Initial YTM 8%)

22. Duration vs. Convexity • Duration assumes a symmetric price-yield relationship. Therefore, it: • Underestimates price change if yield reduction • Overestimates price change if yield increase • To measure the curvature of the price-yield relationship – 2nd derivative of the PV formula. (Note that duration is based on the 1st derivative, i.e., the slope)

23. Correction for Convexity Estimating price change: correction for convexity (2nd order Taylor approximation):

24. Convexity • Duration: OK for small changes in the yield • Convexity-adjusted approximation provides a more accurate measure of the impact on bond prices for larger changes in yield

25. Estimating bond price changes with convexity • Using previous numerical example, but adding information on convexity • Yield change of +0.02 • Impact on bond price using modified duration = -21.32% • Suppose convexity of the bond = 164.106

26. Estimating bond price changes with convexity Taking into account convexity: Total impact on bond price: = -21.32% + 3.28% = -18.04% Instead of a yield increase, what if there had been a yield reduction of the same magnitude (-200 basis points)? Price change = +21.32% + 3.28% = +24.60%

27. Note on duration • Macaulay duration and modified duration only suitable for option-free bonds, or bonds with embedded options that are deep out-of-the-money • Recall the price-yield diagram for callable bonds. Bond price is capped at the call price • Use “effective duration” for bonds with embedded options (you are not responsible for this section in the appendix of this chapter)