1. Zvi WienerContTimeFin - 6 slide 1 Financial Engineering Interest Rates and Fixed Income Securities Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049

2. Zvi WienerContTimeFin - 6 slide 2 Bonds A bond is a contract, paid up-front that yields a known amount at a known date (maturity). The bond may pay a dividend (coupon) at fixed times during the life. Additional options: callable, puttable, indexed, prepayment options, etc. Credit risk, recovery ratio, rating.

3. Zvi WienerContTimeFin - 6 slide 3 Term Structure of IR time to maturity r short term IR long term IR spot rate

4. Zvi WienerContTimeFin - 6 slide 4 Known IR V - value of a contract. r(t) - short term interest rate. If there is no risk and no coupons then dV = rVdt V(t) = V(T)e -rt if there is a continuous dividend stream dV+cVdt = rVdt

5. Zvi WienerContTimeFin - 6 slide 5 Known IR If r is not constant, but not risky r(t) dV = r(t)Vdt If there is a continuous dividend stream dV+c(t)Vdt = r(t)Vdt

6. Zvi WienerContTimeFin - 6 slide 6 Known IR Assume that there are zero coupon bonds for all possible ttm (time to maturity). Denote the price of these bonds by V(t,T).

7. Zvi WienerContTimeFin - 6 slide 7 Known IR

8. Zvi WienerContTimeFin - 6 slide 8 Yield

9. Zvi WienerContTimeFin - 6 slide 9 Typical yield curves time to maturity yield increasing decreasing humped

10. Zvi WienerContTimeFin - 6 slide 10 Typical yield curves F increasing - the most typical. F decreasing - short rates are high but expected to fall. F humped - short rates are expected to fall soon.

11. Zvi WienerContTimeFin - 6 slide 11 Term Structure Explanations Expectation hypothesis states F 0 =E(P T ) this hypothesis is be true if all market participants were risk neutral.

12. Zvi WienerContTimeFin - 6 slide 12 Term Structure Explanations Normal Backwardation (Keynes), commodities are used by hedgers to reduce risk. In order to induce speculators to take the opposite positions, the producers must offer a higher return. Thus speculators enter the long side and have the expected profit of E(P T ) – F 0 > 0

13. Zvi WienerContTimeFin - 6 slide 13 Term Structure Explanations Contango is similar to the normal backwardation, but the natural hedgers are the purchasers of a commodity, rather than suppliers. Since speculators must be paid for taking risk, the opposite relation holds: E(P T ) – F 0 < 0

14. Zvi WienerContTimeFin - 6 slide 14 8% Coupon Bond Zero Coupon Bond

15. Zvi WienerContTimeFin - 6 slide 15 Duration F. Macaulay (1938) Better measurement than time to maturity. Weighted average of all coupons with the corresponding time to payment. Bond Price = Sum[ CF t /(1+y) t ] suggested weight of each coupon: w t = CF t /(1+y) t /Bond Price What is the sum of all w t ?

16. Zvi WienerContTimeFin - 6 slide 16 Macaulay Duration A weighted sum of times to maturities of each coupon. What is the duration of a zero coupon bond?

17. Zvi WienerContTimeFin - 6 slide 17 Macaulay Duration

18. Zvi WienerContTimeFin - 6 slide 18 Macaulay Duration

19. Zvi WienerContTimeFin - 6 slide 19 Duration Sensitivity to IR changes: F Long term bonds are more sensitive. F Lower coupon bonds are more sensitive. F The sensitivity depends on levels of IR.

20. Zvi WienerContTimeFin - 6 slide 20 Duration The bond price volatility is proportional to the bond’s duration. Thus duration is a natural measure of interest rate risk exposure.

21. Zvi WienerContTimeFin - 6 slide 21 Modified Duration The percentage change in bond price is the product of modified duration and the change in the bond’s yield to maturity.

22. Zvi WienerContTimeFin - 6 slide 22 Comparison of two bonds Coupon bond with duration 1.8853 Price (at 5% for 6m.) is $964.5405 If IR increase by 1bp (to 5.01%), its price will fall to $964.1942, or 0.359% decline. Zero-coupon bond with equal duration must have 1.8853 years to maturity. At 5% semiannual its price is ($1,000/1.05 3.7706 )=$831.9623 If IR increase to 5.01%, the price becomes: ($1,000/1.0501 3.7706 )=$831.66 0.359% decline.

23. Zvi WienerContTimeFin - 6 slide 23 Duration Maturity D 03m6m1yr3yr5yr10yr30yr 15% coupon, YTM = 15% Zero coupon bond

24. Zvi WienerContTimeFin - 6 slide 24 Example A bond with 30-yr to maturity Coupon 8%; paid semiannually YTM = 9% P 0 = $897.26 D = 11.37 Yrs if YTM = 9.1%, what will be the price?

25. Zvi WienerContTimeFin - 6 slide 25 Example A bond with 30-yr to maturity Coupon 8%; paid semiannually YTM = 9% P 0 = $897.26 D = 11.37 Yrs if YTM = 9.1%, what will be the price?  P/P = -  y D*  P = -(  y D*)P = -$9.36 P = $897.26 - $9.36 = $887.90

26. Zvi WienerContTimeFin - 6 slide 26 What Determines Duration? F Duration of a zero-coupon bond equals maturity. F Holding ttm constant, duration is higher when coupons are lower. F Holding other factors constant, duration is higher when ytm is lower. F Duration of a perpetuity is (1+y)/y.

27. Zvi WienerContTimeFin - 6 slide 27 What Determines Duration? F Holding the coupon rate constant, duration not always increases with ttm.

28. Zvi WienerContTimeFin - 6 slide 28 Duration

29. Zvi WienerContTimeFin - 6 slide 29

30. Zvi WienerContTimeFin - 6 slide 30

31. Zvi WienerContTimeFin - 6 slide 31

32. Zvi WienerContTimeFin - 6 slide 32 Duration can be regarded as the discount-rate elasticity of the bond price Modern Approach

33. Zvi WienerContTimeFin - 6 slide 33 Duration can be used to measure the price volatility of a bond: Modern Approach

34. Zvi WienerContTimeFin - 6 slide 34 What are the natural bounds on duration? Can duration be bigger than maturity? Can duration be negative? How to measure duration of a portfolio? Modern Approach

35. Zvi WienerContTimeFin - 6 slide 35 Duration: Modern Approach

36. Zvi WienerContTimeFin - 6 slide 36 Duration of a Portfolio

37. Zvi WienerContTimeFin - 6 slide 37 Duration of a Portfolio

38. Zvi WienerContTimeFin - 6 slide 38 Simon Benninga, Financial Modelling, the MIT press, Cambridge, MA, ISBN 0-262-02437-3, $45 MIT Press tel: 800-356-0343 http://mitpress.mit.edu/book-home.tcl?isbn=0262024373 see also my advanced lecture notes on duration Convexity is a similar measurement but with second derivative. Modern Approach to Duration

39. Zvi WienerContTimeFin - 6 slide 39 F Implementation in Excel F Duration Patterns F Duration of a bond with uneven payments F Calculating YTM for uneven periods F Nonflat term structure and duration F Immunization strategies F Cheapest to deliver option and Duration Financial Modelling by Simon Benninga

40. Zvi WienerContTimeFin - 6 slide 40 Passive Bond Management Passive management takes bond prices as fairly set and seeks to control only the risk of the fixed-income portfolio. F Indexing strategy – attempts to replicate a bond index F Immunization – used to tailor the risk to specific needs (insurance companies, pension funds)

41. Zvi WienerContTimeFin - 6 slide 41 Bond-Index Funds Similar to stock indexing. Major indices: Lehman Brothers, Merill Lynch, Salomon Brothers. Include: government, corporate, mortgage- backed, Yankee bonds (dollar denominated, SEC registered bonds of foreign issuers, sold in the US).

42. Zvi WienerContTimeFin - 6 slide 42 Bond-Index Funds Properties: many issues not all are liquid replacement of maturing issues Tracking error is a good measurement of performance. According to Salomon Bros. With $100M one can track the index within 4bp. tracking error per month.

43. Zvi WienerContTimeFin - 6 slide 43 Cellular approach

44. Zvi WienerContTimeFin - 6 slide 44 Immunization Immunization techniques refer to strategies used by investors to shield their overall financial status from exposure to interest rate fluctuations.

45. Zvi WienerContTimeFin - 6 slide 45 Net Worth Immunization Banks and thrifts have a natural mismatch between assets and liabilities. Liabilities are primarily short-term deposits (low duration), assets are typically loans or mortgages (higher duration). When will banks lose money, when IR increase or decline?

46. Zvi WienerContTimeFin - 6 slide 46 Gap Management ARM are used to reduce duration of bank portfolios. Other derivative securities can be used. Capital requirement on duration (exposure). Basic idea: to match duration of assets and liabilities.

47. Zvi WienerContTimeFin - 6 slide 47 Target Date Immunization Important for pension funds and insurances. Price risk and reinvestment risk. What is the correlation between them?

48. Zvi WienerContTimeFin - 6 slide 48 Target Date Immunization Accumulated value 0 t* t Original plan

49. Zvi WienerContTimeFin - 6 slide 49 Target Date Immunization Accumulated value 0 t* t IR increased at t*

50. Zvi WienerContTimeFin - 6 slide 50 Target Date Immunization Accumulated value 0 t*Dt

51. Zvi WienerContTimeFin - 6 slide 51 Target Date Immunization Accumulated value 0 t*Dt Continuous rebalancing can keep the terminal value unchanged

52. Zvi WienerContTimeFin - 6 slide 52 Good Versus Bad Immunization value 0 8% r Single payment obligation $10,000

53. Zvi WienerContTimeFin - 6 slide 53 Good Versus Bad Immunization value 0 8% r Single payment obligation Good immunizing strategy $10,000

54. Zvi WienerContTimeFin - 6 slide 54 Good Versus Bad Immunization value 0 8% r Single payment obligation Good immunizing strategy $10,000

55. Zvi WienerContTimeFin - 6 slide 55 Good Versus Bad Immunization value 0 8% r Single payment obligation Good immunizing strategy $10,000 Bad immunizing strategy

56. Zvi WienerContTimeFin - 6 slide 56 Standard Immunization Is very useful but is based on the assumption of the flat term structure. Often a higher order immunization is used (convexity, etc.). Another reason for goal oriented mutual funds (retirement, education, housing, medical expenses).

57. Zvi WienerContTimeFin - 6 slide 57 Duration Immunization F Duration protects against small IR changes. F Duration assumes a parallel change in the TS. F Immunization is based on nominal IR. F Immunization is very conservative and is inappropriate for many portfolio managers. F The passage of time changes both duration and horizon date, one need to rebalance. F Duration changes if yields change. F Obtaining bonds for immunization can be difficult.

58. Zvi WienerContTimeFin - 6 slide 58 Cash Flow Matching and Dedication Is a very reasonable strategy, but not always realizable. Uncertainty of payments. Lack of perfect match Saving on transaction fees.

59. Zvi WienerContTimeFin - 6 slide 59 Active Bond Management Mainly speculative approach based on ability to predict IR or credit enhancement or market imperfections (identifying mispriced loans).

60. Zvi WienerContTimeFin - 6 slide 60 Contingent Immunization 0 5 yrt value $10,000 $12,000

61. Zvi WienerContTimeFin - 6 slide 61 Contingent Immunization 0 5 yrt value $10,000 $12,000 Stop boundary

62. Zvi WienerContTimeFin - 6 slide 62 Contingent Immunization 0 5 yrt value $10,000 $12,000 Stop boundary

63. Zvi WienerContTimeFin - 6 slide 63 Contingent Immunization 0 5 yrt value $10,000 $12,000 Stop boundary

64. Zvi WienerContTimeFin - 6 slide 64 Interest Rate Swap One of the major fixed-income tools. Example: 6m LIBOR versus 7% fixed. Exchange of net cash flows. Risk involved: IR risk, default risk (small). Why the default risk on IR swaps is small?

65. Zvi WienerContTimeFin - 6 slide 65 Interest Rate Swap Company ACompany BSwap dealer 6.95% 7.05% LIBOR No need in an actual loan. Can be used as a speculative tool or for hedging.

66. Zvi WienerContTimeFin - 6 slide 66 Interest Rate Swap Can not be priced as an exchange of two loans (old method). Why?

67. Zvi WienerContTimeFin - 6 slide 67 Currency Swap A similar exchange of two loans in different currencies. Subject to a higher default risk, because of the principal. Is useful for international companies to hedge currency risk.

68. Zvi WienerContTimeFin - 6 slide 68 Modeling a Swap A simple fixed versus floating swap. Current fixed rate on a 30 years loan is 7% with semi annual payments for simplicity. Current floating rate is 6%. Notional amount is 1,000. How can we model our future payments?

69. Zvi WienerContTimeFin - 6 slide 69 Modeling a Swap There are two flows of cash. At maturity they cancel each other. The fixed part has payments known in advance. The only uncertainty is with the floating part. We need a simple model of interest rates.

70. Zvi WienerContTimeFin - 6 slide 70 Modeling a Swap 012360 6% Floating IR

71. Zvi WienerContTimeFin - 6 slide 71 Modeling a Swap 012360 6% Floating IR 1101011010

72. Zvi WienerContTimeFin - 6 slide 72 Modeling a Swap 012360 6% Floating IR 1101011010

73. Zvi WienerContTimeFin - 6 slide 73 Modeling a Swap 012360 6% Floating IR 1101011010

74. Zvi WienerContTimeFin - 6 slide 74 Modeling a Swap 012360 6% Floating IR 1101011010

75. Zvi WienerContTimeFin - 6 slide 75 Modeling a Swap 012360 6% Floating IR 1101011010

76. Zvi WienerContTimeFin - 6 slide 76 Modeling a Swap 012360 6% Floating IR Arithmetical BM – all jumps of the same size, direction is defined by the sequence of random variables that you have prepared. 1101011010

77. Zvi WienerContTimeFin - 6 slide 77 Modeling a Swap 012360 6% Floating IR Geometrical BM – for an up jump you multiply the current level by a constant u > 1, for a downward jump you multiply by d < 1. 1101011010

78. Zvi WienerContTimeFin - 6 slide 78 Modeling a Swap 012360 6% Floating IR Geometrical BM – jumps have different sizes but up*down = down*up – an important property! 1101011010

79. Zvi WienerContTimeFin - 6 slide 79 Home Assignment Evaluate the swap with your sequence of random or pseudo-random numbers using both approaches arithmetical and geometrical. Up jumps are 10 bp., and 1.1 Down -10bp., ans 0.9 Your side is fixed, discount at 7% annually. You do not have to submit, but bring it to the class, we will discuss it.

80. Zvi WienerContTimeFin - 6 slide 80 Financial Engineering New securities created: IO (negative duration) PO CMO Swaptions Caps and Caplets Floors Ratchets

81. Zvi WienerContTimeFin - 6 slide 81 Why TS is not flat? Assume that TS is flat, but varies with time. Then the price of a zero coupon bond maturing in time  is e -r . How one can form an arbitrage portfolio? Requirements: zero investment, never losses, sometimes gains.

82. Zvi WienerContTimeFin - 6 slide 82 Why TS is not flat? Take 3 bonds, maturing in 1,2, and 3 years. The current prices are: P 1 = e -r, P 2 = e -2r, P 3 = e -3r. We want to form a portfolio with a one-year bonds, b two-years, c three-years. So the first requirement is ae -r + be -2r + ce -3r =0

83. Zvi WienerContTimeFin - 6 slide 83 Why TS is not flat? So the second requirement is that there are no possible losses Equate duration of long and short sides. -ae -r - 2be -2r - 3ce -3r =0 The two equations can be solved simultaneously. Solution is a zero-investment, zero-loss portfolio - arbitrage.

84. Zvi WienerContTimeFin - 6 slide 84 Why TS is not flat? r price r now

85. Zvi WienerContTimeFin - 6 slide 85 Why TS is not flat? So lve[ { -e -r - 2be -2r - 3ce -3r == 0, -e -r - 2be -2r - 3ce -3r == 0}, {b,c}] r price r now

86. Zvi WienerContTimeFin - 6 slide 86 Term Structure Models Let V(t,T) be the price at time t of an asset paying $1 at time T. Obviously V(T,T) =1. Under the equivalent martingale measure the discounted price is a martingale, so

87. Zvi WienerContTimeFin - 6 slide 87 Term Structure Models

88. Zvi WienerContTimeFin - 6 slide 88 One Factor Models Assume that the short rate is the only factor.

89. Zvi WienerContTimeFin - 6 slide 89 One Factor Models Consider a riskless portfolio consisting of two bonds: V 1, and V 2 (with ttm T 1 and T 2 ). The riskless portfolio can be formed as How to choose  and  so that the portfolio is riskless?

90. Zvi WienerContTimeFin - 6 slide 90 One Factor Models This portfolio is riskless, so it earns the risk free interest.

91. Zvi WienerContTimeFin - 6 slide 91 One Factor Models

92. Zvi WienerContTimeFin - 6 slide 92 One Factor Models

93. Zvi WienerContTimeFin - 6 slide 93 One Factor Models

94. Zvi WienerContTimeFin - 6 slide 94 One Factor Models

95. Zvi WienerContTimeFin - 6 slide 95 One Factor Models

96. Zvi WienerContTimeFin - 6 slide 96 One Factor Models

97. Zvi WienerContTimeFin - 6 slide 97 One Factor Models