1. 6- 1 CHAPTER 6 Bond Valuation Models Key features of Bonds Bond valuation Measuring yield Assessing risk

2. 6- 2 Key Features of a Bond Par value: face amount; paid at maturity. Assume $1,000. Coupon interest rate: stated interest rate. Multiply by par value to get dollar interest payment. Generally fixed.

3. 6- 3 Maturity: years until bond must be repaid. Declines over time. Issue date: date when bond was issued. Call provision Call Protection period Call Premium Default Risk Special Features

4. 6- 4 MINICASE 6 PAGE 246 SIMPLE?

5. 6- 5 Value = +.... ABSOLUTELY FUNDAMENTAL. How can we value assets on the basis of expected future cash flows? CF 1 (1 + r) 1 CF 2 (1 + r) 2 CF n (1 + r) n

6. 6- 6 The discount rate r is the opportunity cost of capital and depends on: riskiness of cash flows. general level of interest rates. Inflation Supply and Demand of Money and Credit Production opportunities Time profile of consumption How is the discount rate determined?

7. 6- 7 Again, the discount rate r is the opportunity cost of capital. i.e. the rate of return that could be earned on alternative investments of similar risk. Remember: r i = r * + IP + LP + MRP + DRP for debt securities. How is the discount rate determined?

8. 6- 8 The coupon payments (an annuity). A lump sum (the maturity, or par, value to be received in the future). Value =Present Value of the interest annuity and the Present value of the maturity value. The cash flows of a bond consist of:

9. 6- 9 010 100 1,000 12 100 Find the value of a 10-year annual coupon bond, when r d = 10%.

10. 6- 10 Enter: Solve for PV = $1,000, or 100% of Par. Computer: Two Ways to Solve Using tables: Value = INT(PVIFA 10%,10 )+ M(PVIF 10%,10 ). Calculator: NI/YRPVPMTFV 10 10 100 1000

11. 6- 11 10 10 100 1000 NI/YR PV PMTFV -1,000 The bond consists of a 10-year, 10% annuity of $100/year plus a $1,000 lump sum at t = 10: $ 614.46 385.54 $1,000.00 PV annuity PV maturity value Value of bond ====== INPUTS OUTPUT

12. 6- 12 020 100 1,000 12 100 Find the value of a similar 20-year annual coupon bond, when r d = 10%.

13. 6- 13 Rule: When the required rate of return (r d ) equals the coupon rate, the bond value (or price) equals the: ?

14. 6- 14 PAR VALUE

15. 6- 15 NI/YRPVPMTFV What would the value of the bonds be if inflation rose causing k d = 13%? Solution: ?

16. 6- 16 10-year, 10% coupon bond, r d =13% NPVFV 10 13 100 1000 Solution: -837.21 or 83.72% of par. I/YRPMT

17. 6- 17 NI/YRPVPMTFV What if it were a 20 year, 10% annual coupon bond, with r d = 13%? 20-year, 10% coupon bond 20 13 100 1000 Solution: -789.25, or 78.925% of Par

18. 6- 18 When r d rises above the coupon rate, bond values fall below par. They sell at a discount: A Discount Bond

19. 6- 19 NI/YRPVPMTFV What would the value of the bonds be if r d = 7%? 10-year bond 10 7 100 1000 Solution: -1210.71, (Price 121.07)

20. 6- 20 NI/YRPVPMTFV What would the value of the bonds be if r d = 7%? 20-year bond 20 7 100 1000 Solution: -1317.82, (Price 131.78)

21. 6- 21 When r d falls below the coupon rate, bond values rise above par. They sell at a premium: A PREMIUM BOND.

22. 6- 22 What would happen over time to the value of a 10 % coupon, 30 bond if (1)interest rates stayed at 10%? (2)If interest rates immediately rose to 13% and stayed there? (3)If interest rates immediately fell to 7.5% and stayed there?

23. 6- 23 If interest rates stay at 10% Value of 30 year bond? $1000. Value of 10 year bond? $1000 Value of 1 year bond? $1000

24. 6- 24 If interest rates rise to 13% and stay there? Value of 30 year bond? $775.13 Value of 10 year bond? $837.21 Value of 1 year bond? $973.45

25. 6- 25 If interest rates fell to 7% and stayed there? Value of 30 year bond? $1372.27 Value of 10 year bond? $1210.71 Value of 1 year bond? $1028.04

26. 6- 26 Value of 30 year, 10% coupon bond over time: 1372 1211 1000 837 775 M r d = 10% r d = 7% r d = 13% 30 20 10 0 Years to Maturity NOTE: Not symmetric

27. 6- 27 Value of 30 year, 10% coupon bond over time: 1372 1211 1000 837 775 M r d = 10% r d = 7% r d = 13% 30 20 10 0 Years to Maturity BULLET-VIC

28. 6- 28 What if 100 year bond? $769.23 if k = 13% $1000 if k = 10% $1428.07 if k = 7% Cf9-23 Cf9-24

29. 6- 29 Summary If r d remains constant: At maturity, the value of any bond must equal its par value. Over time, the value of a premium bond will decrease to its par value. Over time, the value of a discount bond will increase to its par value. A par value bond will stay at its par value.

30. 6- 30 Value of 30 year, 10% coupon bond over time: 1372 1211 1000 837 775 M r d = 10% r d = 7% r d = 13% 30 20 10 0 Years to Maturity ACTUAL BOND PRICE PROFILE

31. 6- 31 What is “yield to maturity”? YTM is the rate of return earned on a bond held to maturity. Also called “promised yield”; or That discount rate which equates PV (Bond’s CF) to its price.

32. 6- 32 What is the YTM on a 10-year, 9% annual coupon, $1,000 par value bond that sells for $887? 90 01910 r d =? 1,000 PV 1. PV 10 PV M 887 Find r d that “works”!

33. 6- 33 First, what does the fact that this is a discount bond tell you about the relationship between r d and the bond’s coupon rate?

34. 6- 34 INPUTS OUTPUT Find r d V B = +... + + INT (1 + r d ) 1 INT (1 + r d ) N M (1 + r d ) N 887 = +... + +. 90 (1 + r d ) 1 90 (1 + r d ) 10 1,000 (1 + r d ) 10 10 -887 90 1000 NI/YR PV PMT FV 10.91 Computer: (e-1)

35. 6- 35 Find YTM if price were $1,134.20. Sells at a premium. Because coupon = 9% > r d = 7.08%, bond’s value > par. Computer: INPUTS OUTPUT 10 -1134.2 90 1000 NI/YR PV PMT FV 7.08

36. 6- 36 If coupon rate > r d, premium. If coupon rate < r d, discount. If coupon rate = r d, par bond. If r d rises (falls), price falls (rises). At maturity, price = par.

37. 6- 37 Definitions Current yield=. Cap gains yld=. = YTM = +. Annual coupon pmt Current price Change in price Beg of period price Exp total return Exp Curr yld Exp cap gains yld

38. 6- 38 Find current yield and capital gains yield for a 9%, 10-year bond when the bond sells for $887 (and YTM = 10.91%.) Current yield= = 0.1015 = 10.15%. $90 $887

39. 6- 39 Remember: YTM = Current yield + Capital gains yield. Therefore, Shortcut Answer: Cap gains yield= YTM - Current yld. = 10.91% - 10.15% = 0.77%. Could also find values in Years 0 and 1, get difference, and divide by value in Year 0. Same answer.

40. 6- 40 Proof: Value of a 9 year, 9% coupon bond if YTM = 10.91? $893.87 (P 1 - P 0 )/ P 0 = ($893.87 - $887)/$887 =.0077 =.77%

41. 6- 41 Current yield = = 7.94%. Capital gains yield = 7.08% - 7.94% = -0.85%. Current yield = = 7.94%. Capital gains yield = 7.08% - 7.94% = -0.85%. $90 $1,134.20 $90 $1,134.20 10-year, 9% coupon bond with price = $1,134.20 (YTM = 7.08%)

42. 6- 42 9 year, 9% coupon bond, YTM = 7.08%; Price = $1124.52 (P 1 - P 0 )/P 0 = 1124.52 - 1134.20 1134.20 = -0.85%

43. 6- 43 What is interest rate (price) risk? Does a 1-yr or 10-yr 10% bond have more price risk? Price risk: Rising r d causes bond’s price to fall. r d 1-year Change 10-year Change 5% $1,048 $1,386 10% 1,000 4.8% 1,000 38.6% 15% 956 4.4% 749 25.1%

44. 6- 44 1-year 10-year kdkd Value 0 500 1000 1500 0%5%10%15%

45. 6- 45 DURATION a measure of price risk Duration = Sum( w i t i ) where w i = PV ( CF i ) / sum (PV ( CF i )) = PV ( CF i ) / Price t i = time to the i th cash flow

46. 6- 46 DURATION e.g. [PV(CF 1 )/P] x 1 + [PV(CF 2 )/P] x 2 + [PV(CF 3 )/P] x 3 +... +[PV(CF n )/P] x n

47. 6- 47 What is the duration of a 1 year 10% coupon bond, r d =.10 0 1 1100

48. 6- 48 What is the duration of a 2 year 10% coupon bond, r d =.10 Cash flows: 0 1 2 100 1100

49. 6- 49 What is the duration of a 2 year 10% coupon bond, r d =.10 Duration = [((100/(1.1))/1000)*1] + [((1100/(1.1) 2 )/1000)*2] = Greater Interest rate risk than a 1 year bond. 1.91

50. 6- 50 DURATION WHAT IS DURATION OF A 2 YEAR ZERO COUPON BOND?

51. 6- 51 What is the duration of a 2 year zero coupon bond? 0 2 [(1000/(1 + r) 2 )/(1000/(1 + r) 2 )] x 2 = 2 1000

52. 6- 52 WHO CARES? Who Cares?

53. 6- 53 WHAT IS REINVESTMENT RATE RISK?

54. 6- 54 What is reinvestment rate risk? The risk that CFs will have to be reinvested in the future at lower rates, reducing income. Illustration: Suppose you just won $500,000 playing the lottery. You’ll invest the money and live off the interest. You buy a 1-year bond at par having a 10% interest rate.

55. 6- 55 Year 1 income = $50,000. At year- end get back $500,000 to reinvest. If rates fall to 3%, income will drop from $50,000 to $15,000. Had you bought 30-year bonds, income would have remained constant.

56. 6- 56 Long-term bonds: High price risk, low reinvestment rate risk. Short-term bonds: Low price risk, high reinvestment rate risk. TANSTAAFL Nothing is riskless!

57. 6- 57 CASE HANDOUT!

58. 6- 58 SEMI-ANNUAL BONDS Return to consideration of 10% coupon bonds (6-9). Virtually all coupon bonds issued in the U.S. have semi-annual coupons.

59. 6- 59 Semiannual Bonds 1.Multiply years by 2 to get periods = 2n. 2.Divide nominal rate by 2 to get periodic rate = r d /2. 3.Divide annual INT by 2 to get PMT = INT/2. INPUTS OUTPUT 2nr d /2 OK INT/2OK NI/YR PV PMT FV

60. 6- 60 Find the value of 10-year, 10% coupon, semiannual bond if r d = 10%; if r d = 13%; if r d = 7%. INPUTS OUTPUT

61. 6- 61 For 10%; 2(10) 10/2 100/2 20 5.0 50 1000 NI/YR PV PMTFV Find the value of 10-year, 10% coupon, semiannual bond if r d = 10%; INPUTS OUTPUT

62. 6- 62 For 10%; 2(10) 10/2 100/2 20 5.0 50 1000 NI/YR PV PMTFV -1000 Find the value of 10-year, 10% coupon, semiannual bond if r d = 10%; INPUTS OUTPUT

63. 6- 63 For 13%; 2(10) 13/2 100/2 20 6.5 50 1000 NI/YR PV PMTFV -834.72 Find the value of 10-year, 10% coupon, semiannual bond if r d = 13%; INPUTS OUTPUT

64. 6- 64 For 7%; 2(10) 7/2 100/2 20 3.5 50 1000 NI/YR PV PMTFV -1213.19 Find the value of 10-year, 10% coupon, semiannual bond if r d = 7%. INPUTS OUTPUT

65. 6- 65 Compare these results with the similar results for Annual Bonds.

66. 6- 66 10% Annual and Semiannual Bonds ANNUAL k d 1 yr 10 yr 30 yr 7% 1211 10% 1000 13% 837 Semiannual k d 1 yr 10 yr 30 yr 7% 1213 10% 1000 13% 835

67. 6- 67 Spreadsheet Functions for Bond Valuation See Ch 06 Mini Case.xls for details. PRICE YIELD

68. 6- 68 You could buy, for $1,000, either a 10%, 10-year, annual payment bond or an equally risky 10%, 10-year semiannual bond. Which would you prefer?

69. 6- 69 You could buy, for $1,000, either a 10%, 10-year, annual payment bond or an equally risky 10%, 10-year semiannual pmt. bond. Which would you prefer? The semiannual bond’s EFF% is: 10.25% > 10% EFF% on annual bond, so buy semiannual bond. EFF%= ( 1 + ) - 1 = ( 1 + ) - 1 = 10.25%. m i Nom m 2.10 2

70. 6- 70 If $1,000 is the proper price for the semiannual bond, what is the proper price for the annual payment bond? Semiannual bond has k Nom = 10%, with EFF% = 10.25%. Should earn same EFF% on annual payment bond, so: 10 10.25 100 1000 NI/YR PV PMT FV -984.80 INPUTS OUTPUT

71. 6- 71 At a price of $984.80, the annual and semi-annual bonds would be in equilibrium, because investors would earn EFF% = 10.25% on either bond.

72. 6- 72 00112233... 88 100100100... 100 What is the cash flow stream of a perpetual bond with an annual coupon of $100?

73. 6- 73 A perpetuity is a cash flow stream of equal payments at equal intervals into infinity. V perpetuity =. PMT r

74. 6- 74 V 10% = = $1000. V 13% = = $769.23. V 7% = = $1428.57. V 10% = = $1000. V 13% = = $769.23. V 7% = = $1428.57. $100 0.10 $100 0.10 $100 0.13 $100 0.13 $100 0.07 $100 0.07

75. 6- 75 YIELD TO CALL

76. 6- 76 A 10-year, 10% semiannual coupon, $1,000 par value bond is selling for $1,135.90 with an 8% yield to maturity. It can be called after 5 years at $1,050. What’s the bond’s nominal yield to call (YTC)? INPUTS OUTPUT

77. 6- 77 0 1 2 3 4 5 0 50 50 50 50 50 50 50 50 50 (50+ 1050) 1 2 3 4 5

78. 6- 78 How does adding a call provision affect a bond? Issuer can refund if rates decline. That helps the issuer but hurts the investor. Therefore, borrowers are willing to pay more, and lenders require more, on callable bonds. Most bonds have a deferred call and a declining call premium. Later

79. 6- 79 A 10-year, 10% semiannual coupon, $1,000 par value bond is selling for $1,135.90 with an 8% yield to maturity. It can be called after 5 years at $1,050. What’s the bond’s nominal yield to call (YTC)? 10 -1135.9 50 1050 N I/YR PV PMT FV 3.765 x 2 = 7.53% INPUTS OUTPUT Question j

80. 6- 80 k Nom = 7.53% is the rate brokers would quote. Could also calculate EFF% to call: EFF% = (1.03765) 2 - 1 = 7.672%. This rate could be compared to monthly mortgages, etc.

81. 6- 81 Bond brokers quote (annual) nominal rates. This is o.k. if you are comparing semi-annual rates, as you would be with bonds. But, if you are comparing returns of instruments having different payment patterns, e.g. MBS’s, you should use EAR.

82. 6- 82 If you bought this bond, would you be more likely to earn YTM or YTC?

83. 6- 83 If you bought this bond, would you be more likely to earn YTM or YTC? Coupon rate = 10% vs. YTC = r d = 7.53%. Could raise money by selling new bonds which pay 7.53%. Could thus replace bonds which pay $100/year with bonds that pay only $75.30/year. Investors should expect a call, hence YTC = 7.5%, not YTM = 8%.

84. 6- 84 n In general, if a bond sells at a premium, then (1) coupon > r d, so (2) a call is likely. n So, expect to earn: YTC on premium bonds. YTM on par & discount bonds.

85. 6- 85 Disney recently issued 100-year bonds with a YTM of 7.5%--this represents the promised return. The expected return was less than 7.5% when the bonds were issued. Why? Bond may be called If issuer defaults, investors receive less than the promised return. Therefore, the expected return on corporate and municipal bonds is less than the promised return.

86. 6- 86 Bond Ratings Provide One Measure of Default Risk Investment GradeJunk Bonds Moody’s AaaAaABaaBaBCaaC S&P AAAAAABBBBBBCCCD

87. 6- 87 What factors affect default risk and bond ratings? Financial performance Debt ratio TIE, FCC ratios Current ratios (More…)

88. 6- 88 Provisions in the bond contract Secured versus unsecured debt Senior versus subordinated debt Guarantee provisions Sinking fund provisions Debt maturity (More…)

89. 6- 89 Other factors Earnings stability Regulatory environment Potential product liability Accounting policies

90. 6- 90 What’s a sinking fund? Provision to pay off a loan over its life rather than all at maturity. Similar to amortization on a term loan. Reduces risk to investor, shortens average maturity. But not good for investors if rates decline after issuance.

91. 6- 91 1.Call x% at par per year for sinking fund purposes. 2.Buy bonds on open market. Company would call (at par) if r d is below the coupon rate and bond sells at a premium. Use open market purchase if r d is above coupon rate and bond sells at a discount. Sinking funds are generally handled in 2 ways

92. 6- 92 Bankruptcy Two main chapters of Federal Bankruptcy Act: Chapter 11, Reorganization Chapter 7, Liquidation Typically, company wants Chapter 11, creditors may prefer Chapter 7.

93. 6- 93 If company can’t meet its obligations, it files under Chapter 11. That stops creditors from foreclosing, taking assets, and shutting down the business. Company has 120 days to file a reorganization plan. Court appoints a “trustee” to supervise reorganization. Management usually stays in control.

94. 6- 94 Company must demonstrate in its reorganization plan that it is “worth more alive than dead.” Otherwise, judge will order liquidation under Chapter 7.

95. 6- 95 If the company is liquidated, here’s the payment priority: 1.Secured creditors from sales of secured assets. 2.Trustee’s costs 3.Wages, subject to limits 4.Taxes 5.Unfunded pension liabilities 6.Unsecured creditors 7.Preferred stock 8.Common stock

96. 6- 96 In a liquidation, unsecured creditors generally get zero. This makes them more willing to participate in reorganization even though their claims are greatly scaled back. Various groups of creditors vote on the reorganization plan. If both the majority of the creditors and the judge approve, company “emerges” from bankruptcy with lower debts, reduced interest charges, and a chance for success.

97. 6- 97 Enough for now on bonds?

98. 6- 98 Week 2s Capital Gains Yield Change in price Price I.e. P 1 – P 0 P 0 Next Period: (P 2 – P 1 )/ P 1