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Chapter 10

Chapter 10. Bond Prices and Yields. Straight Bond . Obligates the issuer of the bond to pay the holder of the bond: A fixed sum of money (principal, par value, or face value) at the bond’s maturity Constant, periodic interest payments (coupons) during the life of the bond (Sometimes)

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Chapter 10

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  1. Chapter 10 Bond Prices and Yields

  2. Straight Bond • Obligates the issuer of the bond to pay the holder of the bond: • A fixed sum of money (principal, par value, or face value) at the bond’s maturity • Constant, periodic interest payments (coupons) during the life of the bond (Sometimes) • Special features may be attached • Convertible bonds • Callable bonds • Putable bonds

  3. Straight Bond Basics • $1,000 face value • Semiannual coupon payments

  4. Straight Bonds • Suppose a straight bond pays a semiannual coupon of $45 and is currently priced at $960. • What is the coupon rate? • What is the current yield?

  5. Straight Bond Prices & Yield to Maturity • Bond Price: • Present value of the bond’s coupon payments • + Present value of the bond’s face value • Yield to maturity (YTM): • The discount rate that equates today’s bond price with the present value of the future cash flows of the bond

  6. Bond Pricing Formula PV of coupons PV of FV Where: C = Annual coupon payment FV = Face value M = Maturity in years YTM = Yield to maturity

  7. Straight Bond PricesCalculator Solution Where: C = Annual coupon payment FV = Face value M = Maturity in years YTM = Yield to maturity N = 2M I/Y = YTM/2 PMT = C/2 FV = 1000 CPT PV

  8. Straight Bond Prices PV of coupons PV of FV For a straight bond with 12 years to maturity, a coupon rate of 6% and a YTM of 8%, what is the current price? Price = $457.41 + $390.12 = $847.53

  9. Calculating a Straight Bond Price Using Excel • Excel function to price straight bonds: =PRICE(“Today”,“Maturity”,Coupon Rate,YTM,100,2,3) • Enter “Today” and “Maturity” in quotes, using mm/dd/yyyy format. • Enter the Coupon Rate and the YTM as a decimal. • The "100" tells Excel to us $100 as the par value. • The "2" tells Excel to use semi-annual coupons. • The "3" tells Excel to use an actual day count with 365 days per year. Note: Excel returns a price per $100 face.

  10. Spreadsheet Analysis

  11. Par, Premium and Discount Bonds Par bonds: Price = par value YTM = coupon rate Premium bonds: Price > par value YTM < coupon rate The longer the term to maturity, the greater the premium over par Discount bonds: Price < par value YTM > coupon rate The longer the term to maturity, the greater the discount from par

  12. Premium Bond Price • 12 years to maturity • 8% coupon rate, paid semiannually • YTM = 6% Price = $457.41 + $390.12 = $1,169.36

  13. Discount Bonds Consider two straight bonds with a coupon rate of 6% and a YTM of 8%. If one bond matures in 6 years and one in 12, what are their current prices?

  14. Premium Bonds Consider two straight bonds with a coupon rate of 8% and a YTM of 6%. If one bond matures in 6 years and one in 12, what are their current prices?

  15. Bond Value ($) vs Years to Maturity Premium CR>YTM 8%>6% YTM= CR M 1,000 CR<YTM 6%<8% Discount 12 6 0

  16. Premium and Discount Bonds • In general, when the coupon rate and YTM are held constant: For premium bonds: the longer the term to maturity, the greater the premium over par value. For discount bonds: the longer the term to maturity, the greater the discount from par value.

  17. Relationships among Yield Measures For premium bonds: coupon rate > current yield > YTM For discount bonds: coupon rate < current yield < YTM For par value bonds: coupon rate = current yield = YTM

  18. A Note on Bond Quotations • If you buy a bond between coupon dates: • You will receive the next coupon payment • You might have to pay taxes on it • You must compensate the seller for any accrued interest.

  19. A Note on Bond Quotations • Clean Price = Flat Price • Bond quoting convention ignores accrued interest. • Clean price = a quoted price net of accrued interest • Dirty Price = Full Price = Invoice Price • The price the buyer actually pays • Includes accrued interest added to the clean price.

  20. Clean vs. Dirty PricesExample • Today is April 1. Suppose you want to buy a bond with a 8% annual coupon payable on January 1 and July 1. • The bond is currently quoted at $1,020 • The Clean price = the quoted price = $1,020 • The Dirty or Invoice price = $1,020 plus (3mo/6mo)*$40 =$1,040

  21. Calculating Yields • Trial and error • Calculator • Spreadsheet

  22. Calculating YieldsTrial & Error A 5% bond with 12 years to maturity is priced at 90% of par ($900). Selling at a discount YTM > 5% Try 6% --- price = $915.32 too high Try 6.5% --- price = $876.34 too low Try 6.25% --- price = $895.56 a little low Actual = 6.1933%

  23. Calculating YieldsCalculator A 5% bond with 12 years to maturity is priced at 90% of par ($900). N = 24 PV = -900 PMT = 25 FV = 1000 CPT I/Y = 3.0966 x 2 = 6.1933%

  24. Calculating YieldsSpreadsheet 5% bond with 12 years to maturity,priced at 90% of par =YIELD(“Now”,”Maturity”,Coupon, Price,100,2,3) “Now” = “06/01/2008” “Maturity” = “06/01/2020” Coupon = .05 Price = 90 (entered as a % of par) 100 redemption value as a % of face value “2” semiannual coupon payments “3” actual day count (365) =YIELD(“06/01/2008”,”06/01/2020”,0.05,90,100,2,3) = 0.06193276

  25. Spreadsheet Analysis

  26. Callable Bonds • Gives the issuer the option to: • Buy back the bond • At a specified call price • Anytime after an initial call protection period. • Most bonds are callable Yield-to-call may be more relevant

  27. Yield to Call Where: C = constant annual coupon CP = Call price of bond T = Time in years to earliest call date YTC = Yield to call

  28. Yield to Call • Suppose a 5% bond, priced at 104% of par with 12 years to maturity is callable in 2 years with a $20 call premium. What is its yield to call? N = 4 # periods to first call date PV = -1040 PMT = 25 FV = 1,020 Face value + call premium CPT I/Y = 1.9368 x 2 = 3.874% Remember: resulting rate = 6 month rate

  29. Interest Rate Risk • Interest Rate Risk = possibility that changes in interest rates will result in losses in the bond’s value • Realized Yield = yield actually earned or “realized” on a bond • Realized yield is almost never exactly equal to the yield to maturity, or promised yield

  30. Interest Rate Risk and Maturity

  31. Malkiel’s Theorems • Bond prices and bond yields move in opposite directions. • For a given change in a bond’s YTM, the longer the term to maturity, the greater the magnitude of the change in the bond’s price. • For a given change in a bond’s YTM, the size of the change in the bond’s price increases at a diminishing rate as the bond’s term ot maturity lengthens.

  32. Malkiel’s Theorems • For a given change in a bond’s YTM, the absolute magnitude of the resulting change in the bond’s price is inversely related to the bond’s coupon rate. • For a given absolute change in a bond’s YTM, the magnitude of the price increase caused by a decrease in yield is greater than the price decrease caused by an increase in yield.

  33. Bond Prices and Yields

  34. Duration • Duration measure the sensitivity of a bond price to changes in bond yields. • Two bonds with the same duration, but not necessarily the same maturity, will have approximately the same price sensitivity to a (small) change in bond yields.

  35. Macaulay Duration % Bond Price ≈ -Duration x (10.5) A bond has a Macaulay Duration = 10 years, its yield increases from 7% to 7.5%. How much will its price change? Duration = 10 Change in YTM = .075-.070 = .005 YTM/2= .035 %Price ≈ -10 x (.005/1.035) = -4.83%

  36. Modified Duration • Some analysts prefer a variation of Macaulay’s Duration, known as Modified Duration. • The relationship between percentage changes in bond prices and changes in bond yields is approximately:

  37. Modified Duration (10.6) (10.7) %Bond Price ≈ -Modified Duration x YTM A bond has a Macaulay duration of 9.2 years and a YTM of 7%. What is it’s modified duration? Modified duration = 9.2/1.035 = 8.89 years

  38. Modified Duration (10.6) (10.7) %Bond Price ≈ -Modified Duration x YTM A bond has a Modified duration of 7.2 years and a YTM of 7%. If the yield increases to 7.5%, what happens to the price? %Price ≈ -7.2 x .005 = -3.6%

  39. Calculating Macaulay’s Duration • Macaulay’s duration values stated in years • Often called a bond’s effective maturity • For a zero-coupon bond: Duration = maturity • For a coupon bond: Duration = a weighted average of individual maturities of all the bond’s separate cash flows, where the weights are proportionate to the present values of each cash flow.

  40. Calculating Macaulay Duration (10.8) Suppose a par value bond has 12 years to maturity and an 8% coupon. What is its duration?

  41. General Macaulay Duration Formula (10.9) Where: CPR = Constant annual coupon rate M = Bond maturity in years YTM = Yield to maturity assuming semiannual coupons

  42. Calculating Macaulay’s Duration • In general, for a bond paying constant semiannual coupons, the formula for Macaulay’s Duration is: • In the formula, C is the annual coupon rate, M is the bond maturity (in years), and YTM is the yield to maturity, assuming semiannual coupons.

  43. Using the General Macaulay Duration Formula • What is the modified duration for a bond that matures in 15 years, has a coupon rate of 5% and a yield to maturity of 6.5%? • Steps: • Calculate Macaulay duration using 10.9 • Convert to Modified duration using 10.6

  44. Using the General Macaulay Duration Formula Bond matures in 15 years Coupon rate = 5% Yieldto maturity = 6.5% • Calculate Macaulay duration using 10.9

  45. Using the General Macaulay Duration Formula Bond matures in 15 years Coupon rate = 5% Yieldto maturity = 6.5% Macaulay Duration = 10.34 years • Convert to Modified duration using 10.6

  46. Calculating Duration Using Excel • Macaulay Duration -- DURATION function • Modified Duration -- MDURATION function =DURATION(“Today”,“Maturity”,Coupon Rate,YTM,2,3)

  47. Calculating Macaulay and Modified Duration

  48. Duration Properties: All Else Equal • The longer a bond’s maturity, the longer its duration. • A bond’s duration increases at a decreasing rate as maturity lengthens. • The higher a bond’s coupon, the shorter is its duration. • A higher yield to maturity implies a shorter duration, and a lower yield to maturity implies a longer duration.

  49. Properties of Duration

  50. Bond Risk Measures based on Duration • Dollar Value of an 01:(10.10) ≈ Modified Duration x Bond Price x 0.0001 = Value of a basis point change • Yield Value of a 32nd:(10.11) ≈ In both cases, the bond price is per $100 face value.

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