1. 9-1 Chapter 9 Term Structure of Interest Rates

2. 9-2 Business Cycle Patterns for the Term Structure Yield Maturity Rising Declining

3. 9-3 uThe most common yield curve shape is upward-sloping. uDeclining yield curves occur when interest rates are historically high. uShort-term interest rates are more variable than long-term interest rates. Yield Curves

4. 9-4 Historical evidence indicates two other empirical regularities of the term structure: uFor maturities of six months and less, the yield curve has an upward slope most of the time. uThe prices of long-term bonds are more variable than the prices of short- term bonds.

5. 9-5 Short maturities:[Small][Large] Long maturities:[Huge][Small] The duration effect dominates for long maturities.

6. 9-6 uEach maturity is a separate market. uThere is no substitutability between maturities. uThis could explain any yield curve shape. Segmented Markets Theory

7. 9-7 Increasing Liquidity Premium Rates Maturity Risk Premium 1

8. 9-8 uInvestors prefer the shortest maturity. uInvestors are risk averse. uLonger-term bonds are riskier. uBorrowers are ignored. Assumptions

9. 9-9 What does it mean to be risk averse? uRisk neutral: utility per dollar is constant. uRisk averse: utility per dollar is decreasing. uRisk seeker : utility per dollar is increasing.

10. 9-10 Utility Functions Utility $ Constant utility per $ $1 Increasing utility per $ Risk Seeking $1 Risk Neutral Risk Averse Decreasing utility per $

11. 9-11 Actual utility functions may have risk averse and risk seeking sections. $ Utility

12. 9-12 Money Substitute Theory uThere are many large investors with temporarily excess funds. uIn a short while, these investors will need these funds. uThey will not invest in maturities longer than the date the funds are needed. uThe price of short maturities are driven up by this large demand and their interest rates down.

13. 9-13 A very steep yield curve for short maturities is implied. Maturity y

14. 9-14 The Expectations Hypothesis uThis theory says that forward interest rates are determined by interest rates anticipated to prevail at future dates. uNotation.

15. 9-15 Points in Time 210 R 0,1 f 0,2 f 0,3 f 0,4 34 Rates observed 0

16. 9-16 Points in Time 210 R 0,1 f 0,2 f 0,3 f 0,4 34 Rates observed 0 21 R 1,1 f 1,3 f 1,4 34 1

17. 9-17 Points in Time 210 R 0,1 f 0,2 f 0,3 f 0,4 34 Rates Observed 0 21 R 1,1 f 1,3 f 1,4 34 1 2 R 2,1 f 2,4 34 2 Elapsed Time and Spot and Forward Interest Rates

18. 9-18 Points in Time 210 f 0,2 f 0,3 f 0,4 34 Rates observed 0 21 R 1,1 34 1 2 R 2,1 34 2 Unbiased Expectations Hypothesis: Forward Rates Predict Future Spot Interest Rates R 3,1 34 3 R 0,1

19. 9-19

20. 9-20 Forward Interest Rate Probability R 1, 1 Mean Forward interest rate f 0, 2 is the mean of the one-period spot rate one year from now. f 0, 2 =E(R 1, 1 )

21. 9-21 Forward Interest Rate Probability R 2, 1 Mean Forward interest rate f 0, 3 is the mean of the one-period spot rate two years from now. f 0, 3 =E(R 2, 1 )

22. 9-22 Points in Time 210 Expected future spot rates0.060.060.06 Forward rates f 0,2 = 0.06f 0,3 = 0.06f 0,4 = 0.06 Spot ratesR 0,1 = 0.06R 0,2 = 0.06R 0,3 = 0.06R 0,4 = 0.06 34 Example of a Flat Term Structure

23. 9-23 Points in Time 210 Expected future spot rates0.060.08040.10 Forward rates f 0,2 = 0.06f 0,3 = 0.0804f 0,4 = 0.10 Spot ratesR 0,1 = 0.04R 0,2 = 0.05R 0,3 = 0.06R 0,4 = 0.0699 34 Example of a Rising Term Structure

24. 9-24 Points in Time 210 Expected future spot rates0.070.060.05 Forward rates f 0,2 = 0.07f 0,3 = 0.06f 0,4 = 0.05 Spot rates R 0,1 =.1131R 0,2 =.0913R 0,3 =.0808R 0,4 = 0.073 34 Example of a Declining Term Structure

25. 9-25 Points in Time 210 Expected future spot rates0.070.060.05 Forward rates f 0,2 = 0.07f 0,3 = 0.06f 0,4 = 0.05 Spot ratesR 0,1 = 0.0501R 0,2 = 0.06R 0,3 = 0.06R 0,4 = 0.0575 34 Example of a Humped Term Structure

26. 9-26 Combined Theory f 0,2 = E[R 1,1 ] + L 2. f 0,3 = E[R 2,1 ] + L 3, where L 3 > L 2. General case: f 0,j = E[R j-1,1 ] + L j.

27. 9-27 Points in Time 210 Expected future spot rates0.050.050.05 Liquidity premiumL 2 = 0.01L 3 = 0.015L 4 = 0.02 Forward ratesf 0,2 = 0.06f 0,3 = 0.065f 0,4 =0.07 Spot rates R 0,1 = 0.05R 0,2 = 0.055R 0,3 = 0.058R 0,4 = 0.061 34 Rising Term Structure and Constant Expected Interest Rates

28. 9-28 Points in Time 210 Expected future spot rates0.040.030.02 Liquidity premiumL 2 = 0.01L 3 = 0.015L 4 = 0.02 Forward ratesf 0,2 = 0.05f 0,3 = 0.045f 0,4 = 0.04 Spot ratesR 0,1 = 0.055R 0,2 = 0.052R 0,3 = 0.05R 0,4 = 0.047 34 Rising Liquidity Premiums and a Declining Term Structure

29. 9-29 P 0,n =The price observed at Time 0 of strip (zero-coupon bond) maturing at Time n. P 1,n-1 =The price observed at Time 1 of strip (zero-coupon bond) maturing at Time n. HPR = Holding period return.

30. 9-30

31. 9-31 Arithmetic Approximation: If Unbiased Expectations Hypothesis holds: E[HPR] = R 0,1.

32. 9-32 Two-period Case