Chapter 12

1. Chapter 12 The Term Structure of Interest Rates

2. Chapter Summary • Objective: To explore the pattern of interest rates for different-term assets. • The term structure under certainty • Forward rates • Theories of the term structure • Measuring the term structure

3. Overview of Term Structure of Interest Rates • Relationship between yield to maturity and maturity • Information on expected future short term rates can be implied from yield curve • The yield curve is a graph that displays the relationship between yield and maturity • Three major theories are proposed to explain the observed yield curve

4. Important Terms • Bond yields • Spot rates • Forward rates • Yield curve • Term structure or pure yield curve • Structure of forward rates • Using observed rates to predict future rates

5. Yields Upward Sloping Flat Downward Sloping Maturity Yield Curves

6. Term structure analysis under certainty • Assumption: Future one-period interest rates are known at time zero • Zero-coupon bond prices found from these rates • Yields on these zero-coupon bonds (spot rates) found from these prices • Under certainty, forward rates are equal to future short-term rates

7. Expected Interest Rates in Coming Years (Table 12.1) Expected One-Year Rates in Coming Years YearInterest Rate 0 (today) 8% 1 10% 2 11% 3 11%

8. Pricing of Bonds using Expected Rates PVn = Present Value of $1 in n periods r1 = One-year rate for period 1 r2 = One-year rate for period 2 rn = One-year rate for period n

9. Long-Term Rates and Bond Prices using Expected Rates Time to Maturity Price of Zero* YTM 1 $925.93 8.00% 2 841.75 8.995 3 758.33 9.660 4 683.18 9.993 * $1,000 Par value zero

10. Summary Reminder • Objective: To explore the pattern of interest rates for different-term assets. • The term structure under certainty • Forward rates • Theories of the term structure • Measuring the term structure

11. 3-year investment Alternative 1: Buy and hold a three-year zero 100 131.87 = 100(1+y3)3 1-year investment Alternative 2: Buy a two-year zero and reinvest proceeds in one-year zero 2-year investment 100 118.80 = 100(1+y2)2 131.87 = 118.80(1+r3) Two Alternative Scenarios

12. Forward Rates from Observed Long-Term Rates fn = one-year forward rate for period n yn = yield for a security with a maturity of n

13. Example of Forward Rates using Table 12.2 Numbers 4 yr = 9.993; 3yr = 9.660; fn = ? (1.0993)4 = (1.0966)3 (1+fn) (1.46373)/(1.31870) = (1+fn) fn = .10998 or 11%` Note: this is expected rate that was used in the prior example

14. Downward Sloping Spot Yield Curve Zero-Coupon RatesBond Maturity 12% 1 11.75% 2 11.25% 3 10.00% 4 9.25% 5

15. Forward Rates for Downward Sloping Yield Curve 1yr Forward Rates 1yr [(1.1175)2 / 1.12] - 1 =0.115006 2yrs [(1.1125)3 / (1.1175)2] - 1 =0.102567 3yrs [(1.1)4 / (1.1125)3] - 1 =0.063336 4yrs [(1.0925)5 / (1.1)4] - 1 =0.063008

16. Uncertainty and forward rates • Under certainty investors are indifferent between a short-term bond and a long-term bond sold before maturity, or between one long-term investment and a sequence of rolled-over short-term investments • Under uncertainty the strategy whose return does not depend on an unknown future bond price is less risky

17. Summary Reminder • Objective: To explore the pattern of interest rates for different-term assets. • The term structure under certainty • Forward rates • Theories of the term structure • Measuring the term structure

18. Theories of Term Structure • The Expectations Hypothesis • Liquidity Preference • Upward bias over expectations • Market Segmentation • Preferred Habitat

19. Expectations Theory • Observed long-term rate is a function of today’s short-term rate and expected future short-term rates • Long-term and short-term securities are perfect substitutes • Forward rates that are calculated from the yield on long-term securities are market consensus expected future short-term rates

20. Liquidity Premium Theory • Long-term bonds are more risky • Investors will demand a premium for the risk associated with long-term bonds • Yield curve has an upward bias built into the long-term rates because of the risk premium • Forward rates contain a liquidity premium and are not equal to expected future short-term rates

21. Yields Observed Yield Curve Forward Rates Liquidity Premium Maturity Liquidity Premiums and Yield Curves

22. Yields Observed Yield Curve Forward Rates Liquidity Premium Maturity Liquidity Premiums and Yield Curves

23. Market Segmentation and Preferred Habitat Theories • Short- and long-term bonds are traded in distinct markets • Trading in the distinct segments determines the various rates • Observed rates are not directly influenced by expectations • Preferred Habitat Theory • Modification of market segmentation • Investors will switch out of preferred maturity segments if premiums are adequate

24. What does the record say? • Yield curves are mostly upward-sloping • Liquidity premiums are hard to estimate and may not be constant • Inverted yield curves generally point to declining interest rates • Steeply rising yield curves are generally interpreted as signaling impending rate increases

25. Summary Reminder • Objective: To explore the pattern of interest rates for different-term assets. • The term structure under certainty • Forward rates • Theories of the term structure • Measuring the term structure

26. Measuring the term structure- The bootstrapping method • Derive spot rates from bond yields of varying maturities • Treat each coupon as a mini-zero coupon bond • Use bonds of progressively longer maturities, starting from T-bills • “Clean price” method and “dirty price” method

27. Example: section 12.6, from Figure 11.1 • Observe prices and yields on August 17, 2001; find the spot rate for December 1, 2002 • Observed yields: 3.90%, 4.04% for 6M and 12M, respectively • Observed clean price for bond expiring on December 1, 2002: $1002.29 • Dirty price = clean price + (time elapsed in semesters) x coupon

28. Bootstrapping example (cont.) • Solving, we find y3=4.16% annually

29. Using Spot Rates to price Coupon Bonds • A coupon bond can be viewed as a series of zero coupon bonds • To find the value, each payment is discounted at the zero coupon rate • Once the bond value is found, one can solve for the yield • It’s the reason for which similar maturity and default risk bonds sell at different yields to maturity

30. Sample Bonds Assuming annual compounding

31. Calculation of Price Using Spot Rates (Bond A)

32. Calculation of Price Using Spot Rates (Bond B)

33. Solving for the YTM Bond A • Bond Price = 978.54 • YTM = 6.63% Bond B • Price = 1,047.56 • YTM = 6.61%