1. Fall-02 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html EMBAF Zvi Wiener Based on Chapter 5 in Fabozzi Bond Markets, Analysis and Strategies Factors Affecting Bond Yields and the Term Structure of Interest Rates

2. Zvi WienerFabozzi Ch 5 slide 2 Base Interest Rate Treasury Libor Prime

3. Zvi WienerFabozzi Ch 5 slide 3 Term Structure of Interest Rates Time to maturity r zero 03m6m1yr3yr5yr10yr30yr Yield curve

4. Zvi WienerFabozzi Ch 5 slide 4 http://bond.yahoo.com/rates.html http://www.ratecurve.com/yc2.html

5. Zvi WienerFabozzi Ch 5 slide 5 Yield Curve = Term Structure of IR r maturity Flat Normal Inverted

6. Zvi WienerFabozzi Ch 5 slide 6

7. Zvi WienerFabozzi Ch 5 slide 7 Types of Issuers Governments Agencies Corporate Municipals Others …

8. Zvi WienerFabozzi Ch 5 slide 8 Factors affecting Bond yields and TS Base interest rate - benchmark interest rate Risk Premium - spread Expected liquidity Market forces - Demand and supply

9. Zvi WienerFabozzi Ch 5 slide 9 Taxability of interest qualified municipal bonds are exempts from federal taxes. After tax yield = pretax yield (1- marginal tax rate)

10. Zvi WienerFabozzi Ch 5 slide 10 Do not use yield curve to price bonds PeriodAB 1-9$6$1 10$106$101 They can not be priced by discounting cashflow with the same yield because of different structure of CF. Use spot rates (yield on zero-coupon Treasuries) instead!

11. Zvi WienerFabozzi Ch 5 slide 11 On-the-run Treasury issues Off-the-run Treasury issues Special securities Lending Repos and reverse repos

12. Zvi WienerFabozzi Ch 5 slide 12 30 yr US Treasuries

13. Zvi WienerFabozzi Ch 5 slide 13 10 yr US Treasuries

14. Zvi WienerFabozzi Ch 5 slide 14 Duration and Term Structure of IR

15. Zvi WienerFabozzi Ch 5 slide 15 Partial Duration Key rate duration

16. Zvi WienerFabozzi Ch 5 slide 16 Determinants of the Yield Curve Federal Reserve sets a target level for the fed funds rate - the rate at which depository institutions make uncollaterized overnight loans to one another. Long-term rates reflect expectations of future rates and can be influenced by the outlook for monetary policy.

17. Zvi WienerFabozzi Ch 5 slide 17 Liquidity Bid-offer spread 1-2 cents per $100 face Corporate bonds for example 13 cents High yield bonds 19 cents on-the-run - recently issued in a particular maturity class. With time became off-the-run. Flight to Quality (fall 98) bid-ask 16-25 cents.

18. Zvi WienerFabozzi Ch 5 slide 18 Term Structure of IR If we knew the future IR: 0(Today)8% 110% 211% 311%

19. Zvi WienerFabozzi Ch 5 slide 19 Term Structure of IR If we knew the future IR: 0(Today)8% 110% 211% 311%

20. Zvi WienerFabozzi Ch 5 slide 20 r 1 = 8% r 1 = 10% r 3 = 11% r 4 = 11% Spot rate is the yield to maturity on zero- coupon bonds.

21. Zvi WienerFabozzi Ch 5 slide 21 Future versus Spot Rates r 1 = 8% r 1 = 10% r 3 = 11% r 4 = 11% y 1 = 8% y 2 = 8.995% y 3 = 9.66% y 4 = 9.993%

22. Zvi WienerFabozzi Ch 5 slide 22 Forward Rates Suppose you will need a loan in two years from now for one year. How one can create such a loan today? Go short a three-year zero coupon bond. Go long a two-year zero coupon bond.

23. Zvi WienerFabozzi Ch 5 slide 23 Suppose you will need a loan in two years from now for one year. How one can create such a loan today? Go short a three-year zero coupon bond. Go long a two-year zero coupon bond. +1 0 0-1.3187 -1 0+1.188 0 01 23

24. Zvi WienerFabozzi Ch 5 slide 24 Forward Rates (1 + y n ) n = (1 + y n-1 ) n-1 (1 + f n ) (1 + y n ) n (1 + y n-1 ) n-1 +1 -1.3187 -1 +1.188 01 23

25. Zvi WienerFabozzi Ch 5 slide 25 Forward Rates (1 + y n ) n = (1 + y n-1 ) n-1 (1 + f n ) (1 + y n ) n (1 + y n-1 ) n-1 +1 -1.3187 -1 +1.188 01 23 fnfn

26. Zvi WienerFabozzi Ch 5 slide 26 Forward Rates In other words we can lock now interest rate for a loan which will be taken in future. To specify a forward interest rate one should provide information about today’s date beginning date of the loan end date of the loan

27. Zvi WienerFabozzi Ch 5 slide 27 Forward Rates Buy a two years bond Buy a one year bond and then use the money to buy another bond (the price can be fixed today). (1+r 2 )=(1+r 1 )(1+f 12 )

28. Zvi WienerFabozzi Ch 5 slide 28 Forward Rates (1+r 3 )=(1+r 1 )(1+f 13 )= (1+r 1 )(1+f 12 )(1+f 13 ) Term structure of instantaneous forward rates.

29. Zvi WienerFabozzi Ch 5 slide 29 Forward Rates - Advanced Let P(t,s) be the price at time t of a pure discount bond maturing at time s > t. Then the yield to maturity R(t,T) is the internal rate of return at time t on a bond maturing at t+T. P(t, t+T) = Exp[-R(t,T)*T] Then R(t,T) = - Log[P(t, t+T)]/T

30. Zvi WienerFabozzi Ch 5 slide 30 Forward Rates - Advanced The integral of the forward rates gives the yield to maturity:

31. Zvi WienerFabozzi Ch 5 slide 31 Forward Rates - Advanced The integral of the forward rates gives the yield to maturity: or alternatively

32. Zvi WienerFabozzi Ch 5 slide 32

33. Zvi WienerFabozzi Ch 5 slide 33 FRA Forward Rate Agreement A contract entered at t=0, where the parties (a lender and a borrower) agree to let a certain interest rate R*, act on a prespecified principal, K, over some future time period [S,T]. Assuming continuous compounding we have at time S:-K at time T: Ke R*(T-S) Calculate the FRA rate R* which makes PV=0 hint: it is equal to forward rate

34. Zvi WienerFabozzi Ch 5 slide 34 The Expectations Hypothesis Suggested by Lutz. Forward interest rates is the expected future spot rate. Cox-Ingersoll-Ross have investigated this hypothesis and find that it is not consistent with an economic equilibrium. However it gives often a right direction for expectations.

35. Zvi WienerFabozzi Ch 5 slide 35 Liquidity Preference Hicks (1939) suggested that lenders demand a premium for locking up their money for long period of time. This implies that the term structure will be always upward sloping. The theory ignores the borrowing side of the market.

36. Zvi WienerFabozzi Ch 5 slide 36 Market Segmentation and Preferred Habitat Theories Modigliani and Sutch The market is segmented, investors absolutely prefer one maturity over another. This means that there is no connection between interest rates for different maturities.

37. Zvi WienerFabozzi Ch 5 slide 37 Modern Theories Equilibrium Theories: CIR, BP Non-equilibrium Theories: Dothan, Vasicek, Ho-Lee, Hull-White, HJM Most of them are based on a Brownian Motion as a source of market uncertainty.

38. Zvi WienerFabozzi Ch 5 slide 38 Brownian Motion Time B

39. Zvi WienerFabozzi Ch 5 slide 39 Brownian Motion Starts at the origin Is continuous Is normally distributed at each time Increments are independent Markovian property Technical conditions

40. Zvi WienerFabozzi Ch 5 slide 40 Ch. 5: Questions 2, 3, 10, 13. Home Assignment Chapter 5

41. Zvi WienerFabozzi Ch 5 slide 41 Measuring the Term Structure There are too many data plus some noise. The easiest way to measure the TS is with liquid zero coupon bonds. We obtain a series of points.

42. Zvi WienerFabozzi Ch 5 slide 42 Measuring the Term Structure Time to maturity r zero 03m6m1yr3yr5yr10yr30yr

43. Zvi WienerFabozzi Ch 5 slide 43 First Order Spline Time to maturity r zero 03m6m1yr3yr5yr10yr30yr

44. Zvi WienerFabozzi Ch 5 slide 44 Second Order Spline Time to maturity r zero 03m6m1yr3yr5yr10yr30yr

45. Zvi WienerFabozzi Ch 5 slide 45 Measuring the Term Structure There are too many data plus some noise. The easiest way to measure the TS is with liquid zero coupon bonds. We obtain a series of points. One can connect them with a spline. First order is good for pricing simple bonds. For swaps one need a very high precision.