# Chapter 9

## Chapter 9

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##### Presentation Transcript

1. Chapter9 Bond Prices and Yields

2. Bond Characteristics • Face or par value • Coupon rate • Coupon payment • Maturity • Yield to maturity

3. Accrued interest and quoted bond prices Accrued Interest = (Annual coupon payment/2)x(days since last coupon payment/days separate coupon payment) Invoice price = quoted price + accrued interest

4. Provisions of Bonds • Secured or unsecured • Call provision • Convertible provision • Put provision (putable bonds) • Floating rate bonds • Sinking funds

5. Bond Pricing T  ParValue C P T t = + B + T + ( 1 r ) T ( 1 r ) = t 1 Bond price = PV of Annuity + PV of lump sum CF PB = Price of the bond Ct = interest or coupon payments T = number of periods to maturity r = semi-annual discount rate or the semi-annual yield to maturity

6. Example: Price of 8%,semiannual coupon payment, 10-yr. with yield at 6% 20 1 1 å P = ´ + ´ 40 1000 t 20 B ( 1 . 03 ) ( 1 . 03 ) = t 1 P = 1 , 148 . 77 B Coupon = 4%*1,000 = 40 (Semiannual) Discount Rate = 3% (Semiannual Maturity = 10 years or 20 periods Par Value = 1,000

7. Exercise in class • A coupon bond which pays interest semi-annually, has a par value of \$1,000, matures in 5 years, and has a yield to maturity of 8%. If the coupon rate is 10%, the intrinsic value of the bond today will be __________. A) \$855.55 B) \$1,000 C) \$1,081 D) \$1,100 2. A coupon bond which pays interest of \$40 annually, has a par value of \$1,000, matures in 5 years, and is selling today at a \$159.71 discount from par value. The actual yield to maturity on this bond is __________. A) 5% B) 6% C) 7% D) 8%

8. Bond Prices and Yields Prices and Yields (required rates of return) have an inverse relationship • When yields get very high the value of the bond will be very low • When yields approach zero, the value of the bond approaches the sum of the cash flows

9. Prices and Yield Price Yield

10. Alternative Measures of Yield • Current Yield • Annual coupon payment/current bond price • Yield to Call • Call price replaces par • Call date replaces maturity • Example: • Suppose the 8% coupon (semiannual payment), 30-year maturity bond sells for \$1,150 and is callable in 10 years at a call price of \$1,100. What is the yield to maturity and yield to call? • Given: PMT: 40; N: 60; FV:1000; PV: -1150  YTM = 6.82% • Given: PMT: 40, N: 20; FV:1100; PV: -1150  YTC = 6.64%

11. Alternative Measures of Yield • Holding Period Yield • Considers actual reinvestment of coupons • Considers any change in price if the bond is held less than its maturity • You purchased a 5-year annual interest coupon bond one year ago. Its coupon interest rate was 6% and its par value was \$1,000. At the time you purchased the bond, the yield to maturity was 4%. If you sold the bond after receiving the first interest payment and the bond's yield to maturity had changed to 3%, your annual total rate of return on holding the bond for that year would have been __________. A) 5.00% B) 5.51% C) 7.61% D) 8.95%

12. Exercise in class • A coupon bond which pays interest of \$50 annually, has a par value of \$1,000, matures in 5 years, and is selling today at an \$84.52 discount from par value. The current yield on this bond is __________. A) 5% B) 5.46% C) 5.94% D) 6.00% 2. A callable bond pays annual interest of \$60, has a par value of \$1,000, matures in 20 years but is callable in 10 years at a price of \$1,100, and has a value today of \$1055.84. The yield to call on this bond is __________. A) 6.00% B) 6.58% C) 7.20% D) 8.00%

13. Premium and Discount Bonds • Premium Bond • Coupon rate exceeds yield to maturity • Bond price will decline to par over its maturity • Discount Bond • Yield to maturity exceeds coupon rate • Bond price will increase to par over its maturity

14. Figure 9.6 Premium and Discount Bonds over Time

15. Default Risk and Ratings • Rating companies • Moody’s Investor Service • Standard & Poor’s • Fitch • Rating Categories • Investment grade • Speculative grade (BBB or BaB below)

16. Figure 9.8 Definitions of Each Bond Rating Class

17. Factors Used by Rating Companies • Coverage ratios • Leverage ratios • Liquidity ratios • Profitability ratios • Cash flow to debt

18. Term Structure of Interest Rates • Relationship between yields to maturity and maturity • Yield curve - a graph of the yields on bonds relative to the number of years to maturity • Usually Treasury Bonds • Have to be similar risk or other factors would be influencing yields

19. Figure 9.10 Yields on Long-Term Bonds

20. Figure 9-11 Yield Curves

21. Theories of Term Structure • Expectations • Long term rates are a function of expected future short term rates • Upward slope means that the market is expecting higher future short term rates • Downward slope means that the market is expecting lower future short term rates • Liquidity Preference • Upward bias over expectations • The observed long-term rate includes a risk premium

22. Problems with Traditional Theories Expectations theory • The term structure is almost always upward sloping, but interest rates have not always risen. • It is often the case that the term structure turns down at very long maturities. Maturity preference theory • The U.S. government borrows much more heavily short term than long term. • Many of the biggest buyers of fixed-income securities, such as pension funds, have a strong preference for long maturities

23. Market segmentation theory • The U.S. government borrows at all maturities. • Many institutional investors, such as mutual funds, are more than willing to move maturities to obtain more favorable rates. • There are bond trading operations that exist just to exploit even very small perceived premiums.

24. Forward Rates Implied in the Yield Curve + + + - n n 1 ( 1 ) ( 1 ) ( 1 ) y y f = - n n 1 n 2 1 ( 1 . 12 ) ( 1 . 11 ) ( 1 . 1301 ) = For example, using a 1-yr and 2-yr rates Longer term rate, y(n) = 12% Shorter term rate, y(n-1) = 11% Forward rate, a one-year rate in one year = 13.01%

25. Exercise in class Consider the following \$1,000 par value zero-coupon bonds: The expected one-year interest rate two years from now should be __________. A) 7.00% B) 8.00% C) 9.00% D) 10.00%

26. Ch10 Managing Bond Portfolios

27. 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

28. 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

29. Bond Pricing Relationships (cont.) • 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

30. Figure 10.1 Change in Bond Price as a Function of YTM

31. 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

32. Duration: Calculation t = + Pr ice ( 1 y ) ] w [CF t t T å = ´ D t w t = t 1 = CF Cash Flow for period t t

33. Duration Calculation

34. Figure 10.3 Duration as a Function of Maturity

35. Duration/Price Relationship Price change is proportional to duration and not to maturity DP/P = -D x [Dy / (1+y)] D* = modified duration D* = D / (1+y) DP/P = - D* x Dy

36. Example 34 A bond pays annual interest. Its coupon rate is 7%. Its value at maturity is \$1,000. It matures in three years. Its yield to maturity is presently 8%. The duration of this bond is __________. A) 2.60 B) 2.73 C) 2.81 D) 3.00 A bond presently has a price of \$1,030. The present yield on the bond is 8.00%. If the yield changes from 8.00% to 8.10%, the price of the bond will go down to \$1,020. The duration of this bond is __________. A) -10.5 B) -8.5 C) 9.7 D) 10.5

37. Uses of Duration • Summary measure of length or effective maturity for a portfolio • Immunization of interest rate risk (passive management) • Net worth immunization • Target date immunization • Measure of price sensitivity for changes in interest rate

38. Exercise 12 • A bank has \$50 million in assets, \$47 million in liabilities and \$3 million in shareholders' equity. If the duration of its liabilities are 1.3 and the bank wants to immunize its net worth against interest rate risk and thus set the duration of equity equal to zero, it should select assets with an average duration of __________.  a   1.22b.  1.50c.  1.60d.   2.00 • An 8%, 30-year bond has a yield-to-maturity of 10% and a modified duration of 8.0 years. If the market yield drops by 15 basis points, there will be a __________ in the bond's price.  a.   1.15% decrease b   1.20% increasec.   1.53% increased.   2.43% decrease

39. Example 322 A bond is presently worth \$1,080.00 and its yield to maturity is 8%. If the yield to maturity goes down to 7.84%, the value of the bond will go to __________ if the duration of the bond is 9. A) \$1,034.88 B) \$1,036.00 C) \$1,094.00 D) \$1,123.60 An 8%, 30-year bond has a yield-to-maturity of 10% and a modified duration of 8.0 years. If the market yield drops by 15 basis points, there will be a __________ in the bond's price. A) 1.15% decrease B) 1.20% increase C) 1.53% increase D) 2.43% decrease create a portfolio with a duration of 4 years, using a 5 year zero-coupon bond and a 3 year 8% annual coupon bond with a yield to maturity of 10%, one would have to invest ________ of the portfolio value in the zero-coupon bond. A) 50% B) 55% C) 60% D) 75%

40. Figure 10.4 Growth of Invested Funds

41. Figure 10.5 Immunization

42. Pricing Error from Convexity Price Pricing Error from Convexity Duration Yield

43. Correction for Convexity Modify the pricing equation: D P 1 2 = - ´ D + ´ ´ D D y Convexity ( y ) 2 P Convexity is Equal to: é ù N 1 ( ) CF å + t 2 t ê ú t 2 t ´ + + P (1 y) ( 1 y ) ë û = t 1 Where: CFt is the cash flow (interest and/or principal) at time t.

44. Figure 10.6 Bond Price Convexity

45. Figure 10.7 Convexity of Two Bonds

46. Active Bond Management: Swapping Strategies • Substitution swap • Intermarket swap • Rate anticipation swap • Pure yield pickup • Tax swap

47. Contingent Immunization • Allow the managers to actively manage until the bond portfolio falls to a threshold level • Once the threshold value is hit the manager must then immunize the portfolio • Active with a floor loss level

48. Figure 10-8 Contingent Immunization

49. Interest Rate Swaps • Interest rate swap basic characteristics • One party pays fixed and receives variable • Other party pays variable and receives fixed • Principal is notional • Growth in market • Started in 1980 • Estimated over \$60 trillion today • Hedging applications