CHAPTER 15 The Term Structure of Interest Rates McGraw-Hill/Irwin Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
15-2 Overview of Term Structure The yield curve is a graph that displays the relationship between yield and maturity. Information on expected future short term rates can be implied from the yield curve.
15-3 Figure 15.1 Treasury Yield Curves See Treasury.gov Many other interesting links, for example: stockcharts.com
15-4 Bond Pricing Yields on different maturity bonds are not all equal there is a term structure. We need to consider each bond cash flow as a stand-alone zero-coupon bond. The value of the bond should be the sum of the values of its parts. Bond stripping and bond reconstitution offer opportunities for arbitrage.
Table 15.1 Prices and Yields to Maturities on Zero-Coupon Bonds ($1,000 Face Value) 15-5 These prices are in the form: Price CashFlow 1 ytm t t
15-6 Example 15.1 Valuing Coupon Bonds Value a 3 year, 10% coupon bond using discount rates from Table 15.1: Price $100 1.05 $100 1.06 2 $1100 1.07 3 Price = $1082.17 YTM = 6.88% 6.88% is less than the 3-year rate of 7%.
15-7 Two Types of Yield Curves Pure Yield Curve The pure yield curve uses stripped or zero coupon Treasuries. The pure yield curve may differ significantly from the on-the-run yield curve. On-the-run Yield Curve The on-the-run yield curve uses recently issued coupon bonds selling at or near par. The financial press typically publishes onthe-run yield curves.
15-8 Yield Curve Under Certainty Suppose you want to invest for 2 years: Buy and hold a 2-year zero -or- Rollover a series of 1-year bonds Equilibrium (or no arbitrage) requires that both strategies provide the same return. 1+r 1 1+r 2 (1+y 2 ) 2
Figure 15.2 Two 2-Year Investment Programs 15-9 (1+y 2 ) 2 1+r 1 1+r 2
15-10 Yield Curve Under Certainty Buy and hold vs. rollover: 2 1 y r r 1 1 2 1 2 1 y 1 r 1 r 2 1 2 2 1 1+r 1 1+r 2 (1+y 2 ) 2 Next year s 1-year rate (r 2 ) is just enough to make rolling over a series of 1-year bonds equal to investing in the 2- year bond.
15-11 Spot Rates vs. Short Rates Spot rate the rate that prevails today for a given maturity Short rate the rate for a given maturity (e.g. one year) at different points in time. A spot rate is the geometric average of its component short rates. 1 1 2 1 1... 1 n 1 y r r r n n
Short Rates and Yield Curve Slope 15-12 When next year s short rate, r 2, is greater than this year s short rate, r 1, the yield curve slopes up. May indicate market expects rates to rise. When next year s short rate, r 2, is less than this year s short rate, r 1, the yield curve slopes down. May indicate market expects rates to fall.
15-13 Figure 15.3 Short Rates versus Spot Rates
15-14 Forward Rates from Observed Rates ( (1 y ) n 1 f ) n n ( 1 y 1 1 ) n n f n = one-year forward rate for period n y n = yield for a security with a maturity of n (1 n n 1 y ) (1 f ) (1 y ) 1 (1+y n-1 ) n-1 (1+y n ) n n n 1+f n n
15-15 Example 15.4 Forward Rates The forward interest rate is a forecast of a future short rate implied by the market. Example: compute forward rate for year 4: rate for 4-year maturity = 8% rate for 3-year maturity = 7% 4 4 1 3 1 y 1.08 f 4 1.1106 4 3 1 y 1.07 3 f 11. 06% 4
15-16 Interest Rate Uncertainty Suppose that today s rate is 5% and the expected short rate for the following year is E(r 2 ) = 6%. The value of a 2-year zero is: $1000 1.05 1.06 $898.47 The value of a 1-year zero is: $1000 1.05 $952.38
15-17 Interest Rate Uncertainty The investor wants to invest for 1 year. Buy the 2-year bond today and plan to sell it at the end of the first year for $1000/1.06 =$943.40. or: Buy the 1-year bond today and hold to maturity.
15-18 Interest Rate Uncertainty What if next year s interest rate is more (or less) than 6%? The actual return on the 2-year bond is uncertain!
15-19 Interest Rate Uncertainty Investors require a risk premium to hold a longer-term bond. This liquidity premium compensates short-term investors for the uncertainty about future prices.
15-20 Theories of Term Structure Expectations Forward rates come from market consensus Liquidity Preference Upward bias over expectations due to premium the market requires
15-21 Expectations Theory Observed long-term rate is a function of today s short-term rate and expected future short-term rates. 2 ( 1 y ) (1 y )(1 f ) 2 1 2 1 y ) 2 (1 y )(1 E r ) 2 f n = E(r n ) and liquidity premiums are zero. ( 1 2
15-22 Liquidity Premium Theory Long-term bonds carry more risk; therefore, f n generally exceeds E(r n ) The excess of f n over E(r n ) is the liquidity premium The yield curve has an upward bias built into the long-term rates because of the liquidity premium
15-23 Figure 15.4 Yield Curves - A
15-24 Figure 15.4 Yield Curves - B
15-25 Figure 15.4 Yield Curves - C
15-26 Figure 15.4 Yield Curves - D
15-27 Interpreting the Term Structure The yield curve reflects expectations of future interest rates. The forecasts of future rates are clouded by other factors, such as liquidity premiums. An upward sloping curve could indicate: Rates are expected to rise And/or Investors require large liquidity premiums to hold long term bonds.
15-28 Interpreting the Term Structure The yield curve is a good predictor of the business cycle. Long term rates tend to rise in anticipation of economic expansion. Inverted yield curve may indicate that interest rates are expected to fall and signal a recession.
Figure 15.6 Term Spread: Yields on 10-year vs. 90-day Treasury Securities 15-29
15-30 Forward Rates as Forward Contracts In general, forward rates will not equal the eventually realized short rate Still an important consideration when trying to make decisions: Locking in loan rates
Figure 15.7 Engineering a Synthetic Forward Loan 15-31