1. Copyright 2014 by Diane S. Docking1 Duration & Convexity

2. Learning Objectives Know how to the calculate duration of a security. Know how to calculate the convexity of a security. Understand the economic meaning of duration. Copyright 2014 by Diane S. Docking 2

3. Duration Duration allows for the comparison of securities of different coupons, maturities, etc. Duration measures the weighted average life of an instrument. It equals the average time necessary to recover the initial cost.  E.g..: A bond with 4 years until final maturity with a duration of 3.5 years indicates that an investor would recover the initial cost of the bond in 3.5 years, on average, regardless of intervening interest rate changes. Duration measures of the price sensitivity of a financial asset with fixed cash flows to interest rate changes. 3 Copyright 2014 by Diane S. Docking

4. Macaulay Duration Using annual compounding, Macaulay duration (D) is: where: CF =the interest and/or principal payment that occurs in period t, t =the time period in which the coupon and/or principal payment occurs, i =the current market rate or current market yield on the security Numerator is: PV of Future CFs weighted by period of receipt Denominator is: PV of Future CFs = Current Price Duration is in years or fraction of years 4 Copyright 2014 by Diane S. Docking

5. 5 Duration For interest rate increases, Duration overestimates the price decrease. For interest rate decreases, Duration underestimate s the price increase Change in price predicted by duration yield Price Actual change in price i0i0 i1i1

6. Copyright 2014 by Diane S. Docking 6 Example 1: Calculating Duration: 10-yr, 10% Coupon Bond; r m =10% < 10 years This is current Price = P 0 6,759.02 1,000.00

7. Copyright 2014 by Diane S. Docking 7 Example 2: Calculating Duration: 10-yr, 10% Coupon Bond; r m =20% < 10 years This is current Price = P 0 Verify Price: FV = 1,000 n = 10 yrs. Pmt = $100 i = 20%  PV = 580.75 3,323.01 580.75

8. Copyright 2014 by Diane S. Docking 8 Example 3: Calculating Duration: 10-yr, 20% Coupon Bond; r m =10% < 10 years This is current Price = P 0 Verify Price: FV = 1,000 n = 10 yrs. Pmt = $200 i = 10%  PV = 1,614.46 9,662.61 1,614.46

9. Copyright 2014 by Diane S. Docking 9 Example 4: Calculating Duration: 5-yr, 10% Coupon Bond; r m =10% < 5 years This is current Price = P 0 4,169.87 1,000.00

10. Copyright 2014 by Diane S. Docking 10 Example 5: Calculating Duration: 5-yr, Zero-Coupon Bond; r m =10% This is current Price = P 0 D=maturity Verify Price: FV = 1,000 n = 5yrs. Pmt = 0 i = 10%  PV = 620.92 3,104.61 620.92

11. Copyright 2014 by Diane S. Docking 11 Example 6: Calculating Duration: 10-yr, Zero-Coupon Bond; r m =20% This is current Price = P 0 1615.06 161.506 Verify Price: FV = 1,000 n = 10 yrs. Pmt = 0 i = 20%  PV = 161.51 D=maturity

12. Copyright 2014 by Diane S. Docking 12 Asset Properties and Duration MaturityCouponRmDuration Ex. 110 yrs.10% 6.759 yrs. Ex. 210 yrs.10%20%5.722 yrs. Ex. 310 yrs.20%10%5.985 yrs. Ex. 45 yrs.10% 4.170 yrs. Ex. 55 yrs.0%10%5.000 yrs. Ex. 610 yrs.0%20%10.000 yrs.

13. Copyright 2014 by Diane S. Docking 13 Asset Properties and Duration 1. For bonds with the same coupon rate and the same yield, the bond with the longer maturity will have the________ duration. (Ex.1 vs. Ex.4) 2. For bonds with the same maturity and the same yield, the bond with the lower coupon rate will have the ________ duration. (Ex. 1 vs. Ex. 3; Ex. 2 & 6; Ex. 4 vs. Ex. 5) 3. When interest rates rise, the duration of a coupon bond ________. (Ex. 1 vs. Ex. 2) 4. The lower the initial yield, the _______ the duration for a given bond. (Ex. 1 vs. Ex. 2)

14. Copyright 2014 by Diane S. Docking 14 Key facts about duration 1. The longer a bond’s duration, the its sensitivity to interest rate changes 2. The duration of a______________ bond = bond’s term to maturity 3. The Macaulay duration of any coupon bond is always ______ than the bond’s term to maturity 4. Duration is the duration of a portfolio of securities is the weighted-average of the durations of the individual securities, with the weights equaling the proportion of the portfolio invested in each. 5. The more frequently a security pays interest or principal, the its duration. Duration

15. Copyright 2014 by Diane S. Docking 15 Macaulay Duration Duration – annual interest pmts: Duration – semi-annual interest pmts:

16. Copyright 2014 by Diane S. Docking 16 Modified Duration Modified duration – annual interest pmts: Modified duration – semi-annual interest pmts:

17. Copyright 2014 by Diane S. Docking 17 Duration and Price Sensitivity So, an estimate of the percentage change in the price of a financial asset is: So, an estimate of $ price change in the price of a financial asset is: or

18. Copyright 2014 by Diane S. Docking 18 Duration estimates of price change If annual interest payments: If semi-annual interest payments:

19. Copyright 2014 by Diane S. Docking 19 Duration For interest rate increases, Duration overestimates the price decrease. For interest rate decreases, Duration underestimates the price increase Change in price predicted by duration yield Price Actual change in price

20. Copyright 2014 by Diane S. Docking 20 Problem 1: Duration – Annual Payments Assume today is September 1, 2XX1. Midwest Bank owns the following security: G.E. Corporate bond, 10 ¾%, September 2XX6. Today’s closing price is 124 3 / 32. Assume interest is paid annually. 1. What is the security’s YTM? 2. If market interest rates increase 1%, what will be the security’s price? 3. What is the security’s Macaulay Duration? Modified Duration? 4. What is the dollar change in price for a 1% increase in market rates as estimated by Duration? 5. Compare the actual price change to the estimated price change.

21. Problem 1 Solution: Duration - Annual Payments 1. YTM: PV = 124 3/32 = 124.09375 = $1,240.94 FV = 1,000 Pmt = 107.50 n = 5 yrs. therefore R m = YTM =___________ 2.R m increases 1% to 6.1603%: FV = 1,000 R m = 6.1603% Pmt = 107.50 n = 5 yrs. therefore PV = ___________ Copyright 2014 by Diane S. Docking 21 Makes sense: rates increase, price decreases

22. Copyright 2014 by Diane S. Docking 22 Problem 1 Solution: Duration - Annual Payments 5,231.38 1,240.94 4.2157 1.051603

23. Problem 1 Solution: Duration - Annual Payments (cont.) 4. From Duration: Macaulay or Modified Copyright 2014 by Diane S. Docking 23

24. Problem 1 Solution: Duration - Annual Payments (cont.) 5. Original Price when (Rm = 5.1603%) $1,240.94 Duration est. of price change Duration est. of new Price$1,191.19 vs. $1,192.49 Copyright 2014 by Diane S. Docking 24 Diff = Duration overestimated price decrease.

25. Copyright 2014 by Diane S. Docking 25 Problem 2: Duration – Semi-annual payments Assume today is September 1, 2XX1. Midwest Bank owns the following security: U.S. Treasury bond, 10 ¾%, September 2XX6. Today’s closing price is 124 3 / 32. Assume interest is paid semi-annually. 1. What is the security’s YTM? 2. If market interest rates increase 1%, what will be the security’s price? 3. What is the security’s Macaulay Duration? Modified Duration? 4. What is the dollar change in price for a 1% increase in market rates as estimated by Duration? 5. Compare the actual price change to the estimated price change.

26. Problem 2 Solution: Duration – Semi-annual Payments 1. YTM: PV = 124 3/32 = 124.09375 = $1,240.94 FV = 1,000 Pmt = 107.50/2 = 53.75 n = 5 yrs. x 2 = 10 therefore R m = YTM = 2.6069% semi-annual; 5.2137% annual 2.R m increases 1% to 6.2137% annual/ 2 = 3.1069% FV = 1,000 R m = 3.1069% Pmt = 53.75 n = 10 therefore PV = $1,192.42 Copyright 2014 by Diane S. Docking 26

27. Copyright 2014 by Diane S. Docking 27 Problem 2 Solution: Duration – Semi-annual Payments (cont.)

28. 4. From Duration (in years): Macaulay or Modified Copyright 2014 by Diane S. Docking 28

29. Problem 2 Solution: Duration – Semi-annual Payments (cont.) 5. Original Price when (Rm = 5.21%) $1,240.94 Duration est. of price change Duration est. of new Price$1,191.21 vs. $1,192.42 Copyright 2014 by Diane S. Docking 29 Diff = Duration overestimated price decrease.

30. Copyright 2014 by Diane S. Docking30 Duration For interest rate increases, Duration overestimates the price decrease. For interest rate decreases, Duration underestimates the price increase Change in price predicted by duration yield Price Actual change in price

31. Convexity Taking convexity into account: For interest rate increases, the actual reduction in price will be less than that predicted by duration For interest rate decreases, the actual increase in price will be more than that predicted by duration Change in price predicted by duration yield Price Actual change in price Convexity adds amount back 31Copyright 2014 by Diane S. Docking

32. 32 Convexity Using annual compounding, Convexity (CX) derived from Macaulay duration (D) is: where: CF =the interest and/or principal payment that occurs in period t, t =the time period in which the coupon and/or principal payment occurs, i =the current market rate or market yield on the security We call this D prime = D ’

33. Copyright 2014 by Diane S. Docking33 Convexity Convexity – annual interest pmts: Convexity – semi-annual interest pmts:

34. Copyright 2014 by Diane S. Docking34 Example 1A: Calculating Macaulay Convexity: 10-yr 10% Coupon Bond; r m =10% 63.8790 (1.10) 2

35. Copyright 2014 by Diane S. Docking35 Example 2A: Calculating Macaulay Convexity: 10-yr 10% Coupon Bond; r m =20% 50.5865 (1.20) 2

36. Copyright 2014 by Diane S. Docking36 Example 3A: Calculating Macaulay Convexity: 10-yr 20% Coupon Bond; r m =10% 52.8650 (1.10) 2

37. Copyright 2014 by Diane S. Docking37 Example 4A: Calculating Macaulay Convexity: 5-yr 10% Coupon Bond; r m =10% 23.4357 (1.10) 2

38. Copyright 2014 by Diane S. Docking38 Example 5A: Calculating Macaulay Convexity: 5-yr Zero-Coupon Bond; r m =10% (assume annual payments) 18,627.64 620.92 30.0000 (1.10) 2

39. Copyright 2014 by Diane S. Docking39 Asset Properties and Convexity 1. For bonds with the same coupon rate and the same yield, the the bond with the longer maturity will have the greater convexity. (Ex.1A vs. Ex.4A) 2. For bonds with the same maturity and the same yield, the the bond with the lower coupon rate will have the greater convexity. (Ex. 1A vs. Ex. 3A) 3. When interest rates rise, the convexity of a coupon bond falls. (Ex. 1A vs. Ex. 2A) 4. The greater the initial yield, the less the convexity for a given bond. (Ex. 1A vs. Ex. 2A)

40. Copyright 2014 by Diane S. Docking40 Key facts about convexity 1. Convexity increases with bond maturity 2. Given the same maturity, coupon bonds are _____convex than zero-coupon bonds. Convexity

41. Copyright 2014 by Diane S. Docking41 Price change explained by Convexity: If annual interest payments: If semi-annual interest payments:

42. Copyright 2014 by Diane S. Docking42 Problem 1A: Duration and Convexity – Annual Payments Assume today is September 1, 2XX1. Midwest Bank owns the following security: G.E. Corporate bond, 10 ¾%, September 2XX6. Today’s closing price is 124 3 / 32. Assume interest is paid annually. 1. What is the security’s YTM? 2. If market interest rates increase 1%, what will be the security’s price? 3. What is the security’s Macaulay Duration? Modified Duration? 4. What is the security’s Convexity? 5. What is the dollar change in price for a 1% increase in market rates: a) From Duration? b) From Convexity? 6. Compare the actual price change to the estimated price change.

43. Copyright 2014 by Diane S. Docking43 Problem 1A Solution: Duration & Convexity Annual Payments 3.4. 5,231.38 1,240.94 29,489.86 1,240.94 23.7642 (1.051603) 2

44. Problem 1A Solution: Duration & Convexity - Annual Payments (cont.) 5. a) From Macaulay Duration: 5. b) From Convexity: Copyright 2014 by Diane S. Docking44

45. Problem 1A Solution: Duration & Convexity - Annual Payments (cont.) 6. Original Price when (Rm = 5.1603%) $1,240.94 Duration est. of price change Duration est. of new Price$1,191.19 vs. $1,192.49 Convexity est. of price change not explained by duration + 1.33 Duration + Convexity est. of new Price$1,192.52 vs. $1,192.49 Copyright 2014 by Diane S. Docking45 Diff = Duration overestimated price decrease. Diff =

46. Copyright 2014 by Diane S. Docking46 Problem 2A: Duration and Convexity – Semi-annual payments Assume today is September 1, 2XX1. Midwest Bank owns the following security: U.S. Treasury bond, 10 ¾%, September 2XX6. Today’s closing price is 124 3 / 32. Assume interest is paid semi-annually. 1. What is the security’s YTM? 2. If market interest rates increase 1%, what will be the security’s price? 3. What is the security’s Duration? 4. What is the security’s Convexity? 5. What is the dollar change in price for a 1% increase in market rates: a) From Duration? b) From Convexity? 6. Compare the actual price change to the estimated price change.

47. Copyright 2014 by Diane S. Docking47 Problem 2A Solution: Duration & Convexity Semi-annual Payments 3.4.

48. Problem 2A Solution: Duration & Convexity – Semi- annual Payments (cont.) 5. a) From Macaulay Duration: 5. b) From Convexity (in years): Copyright 2014 by Diane S. Docking48

49. Problem 2A Solution: Duration & Convexity – Semi- annual Payments (cont.) 6. Original Price when (Rm = 5.21%) $1,240.94 Duration est. of price change Duration est. of new Price$1,191.21 vs. $1,192.42 Convexity est. of price change not explained by duration + 1.24 Duration + Convexity est. of new Price$1,192.45 vs. $1,192.42 Copyright 2014 by Diane S. Docking49 Diff = Duration overestimated price decrease. Diff =