1. comm 324 --- W. Suo Slide 1

2. comm 324 --- W. Suo Slide 2 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

3. comm 324 --- W. Suo Slide 3 Interest Rate Sensitivity BondCouponMaturityInitial YTM A12%5 years10% B12%30 years10% C3%30 years10% D3%30 years6% ABCDABCD Change in yield to maturity (%) Percentage change in bond price 0

4. comm 324 --- W. Suo Slide 4  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  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 Bond Pricing Relationships

5. comm 324 --- W. Suo Slide 5 Derivation of Formula For Macauley’s Duration  The slope of a bond’s price-yield relationship measures the bond’s sensitivity to YTM Modified duration (Mod) Macauley duration (MAC) Macaulay (1938) suggested studying a bond’s time structure by examining its average term to maturity  Approximating the bond price change using duration is equivalent to moving along the slope of the bond price-yield curve

6. comm 324 --- W. Suo Slide 6 Example: Calculation of MAC and MOD  Given information A $1,000 par bond with a YTM of 10% has three years to maturity and a 5% coupon rate Currently sells for $875.657

7. comm 324 --- W. Suo Slide 7 Example: Calculation of MAC and MOD  MAC can be calculated using the previous present value calculations TPV of CF Each CFs PV as fraction of Price T weighted by CF 145.4550.051911 × 0.05191 = 0.05191 241.3220.047182 × 0.04718 = 0.09438 3788.8810.90093 × 0.9009 = 2.70270 1.00000MAC = 2.84899 MOD = MAC  (1+y) = 2.84899  1.10 = 2.5899

8. comm 324 --- W. Suo Slide 8 Interest Rate Risk  Interest rate elasticity measures a bond’s price sensitivity to changes in interest rates

9. comm 324 --- W. Suo Slide 9 Example: Evaluating a Bond’s EL  Given information: A bond has a 10% coupon rate and a par of $1,000. Its current price is $1,000 as the YTM is 10% If interest rates were to rise from 10% to 11%, what would the new price be?  The price drops by $17.13 or 1.713%  -$17.13  $1,000 = -0.01713 or –1.713%  The increase in YTM from 10% to 11% is a percentage change of  (0.11 – 0.10)  1.1 = 0.0090909 or 0.9% Results in an EL of –0.01713  0.00909 = -1.90

10. comm 324 --- W. Suo Slide 10 Interest Rate Risk  MAC can also be used to calculate a bond’s elasticity MAC = [(t=1)($90.909  $1,000] + [(t=2)($909.091)  $1,000] = 1.90 years  Interest rate elasticity and MAC are equally good measures of interest rate risk

11. comm 324 --- W. Suo Slide 11  Price change is proportional to duration and not to maturity Duration/Price Relationship  Or, if we denote D * = modified duration

12. comm 324 --- W. Suo Slide 12 Why is duration a key concept?  It’s a simple summary statistic of the effective average maturity of the portfolio;  It measures the sensitivity of bond price change relative to the change in yield  It is an essential tool in immunizing portfolios from interest rate risk;  It is a measure of interest rate risk of a portfolio

13. comm 324 --- W. Suo Slide 13 Contrasting Time Until Maturity and Duration  MAC  T For zeros MAC = T For non-zeros MAC < T Earlier and/or larger CFs result in shorter MAC Macauley durations for a bond with a 6% YTM at various times to maturity Coupon Rate T2%4%6%8% 10.9950.9900.9850.981 24.7564.5584.3934.254 108.8918.1697.6627.286 2014.98112.98011.90411.232 5019.45217.12916.27315.829 10017.56717.23217.12017.064  17.667

14. comm 324 --- W. Suo Slide 14 Contrasting Time Until Maturity and Duration

15. comm 324 --- W. Suo Slide 15 Contrasting Time Until Maturity and Duration

16. comm 324 --- W. Suo Slide 16 Contrasting Time Until Maturity and Duration Duration and convexity for a bond with a 5% coupon at YTM of 10% 2-year bond12-year bond22-year bond PV$913.22$659.32$561.42 MAC1.958.5811 MOD1.777.810 Convexity0.020.210.58 As horizon increases, bond’s MAC, MOD increase.

17. comm 324 --- W. Suo Slide 17 Contrasting Time Until Maturity and Duration  A bond’s duration is inversely related to its coupon rate Duration and convexity for a bond with a 12 year maturity at YTM of 10% Zero coupon 5% coupon10% coupon 15% coupon PV$318.63$659.32$1,000$1,340.68 MAC128.587.506.96 MOD10.917.86.816.33 Convexity0.100.210.200.19

18. comm 324 --- W. Suo Slide 18 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 Rule 5 The duration of a level perpetuity is (1+y)/y Rule 6 The duration of a level annuity is equal to: Rule 7 The duration for a coupon bond is equal to:

19. comm 324 --- W. Suo Slide 19 Duration and Convexity % Price Change Duration Pricing error from convexity Yield

20. comm 324 --- W. Suo Slide 20 Convexity of Two Bonds 0 Change in yield to maturity (%) Percentage change in bond price Bond A Bond B

21. comm 324 --- W. Suo Slide 21 Correction for Convexity Correction for Convexity:

22. comm 324 --- W. Suo Slide 22 Convexity calculation 8% Bond Time Sem. PaymentPV of CF (10%) Weightt(t+1)x C4 14038.095.0395.0790 24036.281.0376.2257 3 4 40 1040 sum 34.553 855.611 964.540.0358.8871 1.000.4299 17.7413 18.4759

23. comm 324 --- W. Suo Slide 23 Convexity calculation (cont.)  Convexity is computed like duration, as a weighted average of the terms (t 2 +t) (rather than t) divided by (1+y) 2  Thus, in the above example, it is equal to 18.4759/1.05 2 = 16.7582 in semester terms.

24. comm 324 --- W. Suo Slide 24 Other Measures  Effective duration  PVBP (also know as PV01, DVBP, DV01) How to calculate it?

25. comm 324 --- W. Suo Slide 25 Duration and Convexity of Callable Bonds 0 Interest Rate Call Price Region of positive convexity Region of negative convexity Price-yield curve is below tangent 5% 10%