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Managing Bond Portfolios

Managing Bond Portfolios. Fixed income portfolio management strategies Interest rate risk Duration and Convexity Passive Management Active Management. Interest rate risk. Interest rate (market) risk - biggest risk faced by bondholders. Two components: Bond price risk

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Managing Bond Portfolios

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  1. Managing Bond Portfolios Fixed income portfolio management strategies • Interest rate risk • Duration and Convexity • Passive Management • Active Management

  2. Interest rate risk • Interest rate (market) risk - biggest risk faced by bondholders. • Two components: • Bond price risk • If selling before maturity: capital gain/loss • Reinvestment risk • Reinvestment of coupons

  3. 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 in the PV formula • What other factors affect the price sensitivity of a bond? • Examples: coupon rate, initial yield (yield before the change)

  4. Change in bond price (%) A B C D Change in yield to maturity (%) Examples 0

  5. 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 • Note the minus sign in front • Why look at percentage change rather than absolute change?

  6. Macaulay duration • Solve for D using the bond pricing (PV) formula: where CFt = Cash flow to the bondholder in period t, y is the YTM

  7. Example2-year, 8% bond (semiannual), 10% YTM Time Payment PV of CF Weight C1 X C4 (in years) .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 Duration for this bond is 1.8853 year

  8. Example2-year, 8% bond (semiannual), 10% YTM Time Payment PV of CF Weight C1 X 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 for this bond is 1.8853 year

  9. 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

  10. More general definition (Fabozzi) • Taking frequency of coupon payments, k, explicitly into account • If k = 2, y/k is the semiannual discount rate • k in the denominator converts D into years

  11. 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

  12. Modified duration • Definition: D* = modified duration • To calculate % change in bond price

  13. Numerical example • Suppose the Macaulay D = 10.98 years • Initial yield (before change) = 6% • Yield increases to 8% (i.e., +0.02) • Percentage change in bond price

  14. 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 generally increases with its time to maturity

  15. 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:

  16. 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:

  17. Duration vs. convexity Price Pricing error from convexity Duration Yield

  18. 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)

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

  20. 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

  21. 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

  22. 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%

  23. 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 chapter)

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