# The Term Structure of Interest Rates CHAPTER 13

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1 The Term Structure of Interest Rates CHAPTER 13

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 Interpreting the term structure 13-2

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 Two major theories are proposed to explain the observed yield curve 13-3

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 13-4

5 Yield Curves Yields Upward Sloping Flat Downward Sloping Maturity 13-5

6 Spot Rates, Bond Pricing and the Yield Curve Spot rates can be found by discounting zero-coupon bonds of various maturities For maturities longer than one year zero-coupon bonds are created artificially, by stripping regular coupon bonds and selling each coupon separately Coupon bonds are priced by discounting each cash flow by the appropriate spot rate The pure yield curveis a plot of the spot rates against their respective maturities 13-6

7 Table 13.1 Yields and Prices to Maturity on Zero-Coupon Bonds (\$1,000 Face Value) 13-7

8 Figure 13.2 Two 2-Year Investment Programs 13-8

9 Figure 13.3 Short Rates versus Spot Rates 13-9

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 13-10

11 Forward Rates from Observed Rates (1 + y n n ( 1 + fn) = n 1 (1 + yn 1) f n = one-year forward rate for period n y n = yield for a security with a maturity of n n n 1 ( 1 + y ) = (1 + y ) (1 + f ) n n 1 n ) 13-11

12 Example of Forward Rates using Figure yr = 8.00% 3yr = 7.00% fn =? (1.08) 4 = (1.07) 3 (1+f n ) (1.3605) / (1.2250) = (1+f n ) f n =.1106 or 11.06% Note: this is expected rate that was used in the prior example

13 Downward Sloping Spot Yield Curve Zero-Coupon Rates Bond Maturity 12% % % % %

14 Forward Rates for Downward Sloping Yield Curve 1yr Forward Rates 1yr [(1.1175) 2 / 1.12] -1 = yrs [(1.1125) 3 /(1.1175) 2 ]-1 = yrs [(1.1) 4 /(1.1125) 3 ]-1 = yrs [(1.0925) 5 /(1.1) 4 ]-1 =

15 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 shortterm investments Under uncertainty the strategy whose return does not depend on an unknown future bond price is less risky 13-15

16 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 13-16

17 Theories of Term Structure The Expectations Hypothesis Forward rates are expectations of future short term rates Liquidity Preference Upward bias over expectations 13-17

18 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 13-18

19 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 shortterm rates 13-19

20 Figure 13.4 Yield Curves (with Liquidity Premiums) 13-20

21 Figure 13.4 Yield Curves (with Liquidity Premiums) 13-21

22 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 Interpreting the term structure 13-22

23 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 13-23

24 Figure 13.5 Yield Volatility of Long Term Bonds,

25 Figure 13.6 Yields on Long-Term Versus Short-Term Government Securities: Term Spread,

26 Forward Rates as Forward Contracts Forward rates are also market interest rates We can contract forward loans by buying and selling zero coupon bonds of varying maturities Suppose f 2 is the forward rate between periods 1 and 2 Then you sell 1+f 2 two-year zeros for every one year zero that you buy This is a loan arranged today for period 2 at interest rate f

27 Figure 13.7 Engineering a Synthetic Forward Loan 13-27

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