The Term Structure of Interest Rates

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.
INVESTMENTS | BODIE, KANE, MARCUS
15-13
Figure 15.3 Short Rates versus Spot Rates
INVESTMENTS | BODIE, KANE, MARCUS
15-14
Forward Rates from Observed Rates
15-10
Yield Curve Under Certainty
• Buy and hold vs. rollover:
(1 y2 ) 2 (1 r1 ) x(1 r2 ) 1 y2 (1 r1 ) x(1 r2 )
1 2
• Next year’s 1-year rate (r2) is just enough to make rolling over a series of 1-year bonds equal to investing in the 2-year bond.
INVESTMENTS | BODIE, KANE, MARCUS
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.
• Price = $1082.17 and YTM = 6.88% • 6.88% is less than the 3-year rate of 7%.
INVESTMENTS | BODIE, KANE, MARCUS
15-7
Two Types of Yield Curves
On-the-run Yield Pure Yield Curve Curve • The pure yield curve • The on-the-run yield uses stripped or zero curve uses recently coupon Treasuries. issued coupon bonds • The pure yield curve selling at or near par. may differ significantly • The financial press from the on-the-run typically publishes onyield curve. the-run yield curves.
CHAPTER 15
The Term Structure of Interest Rates
INVESTMENTS | BODIE, KANE, MARCUS
McGraw-Hill/Irwin Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
INVESTMENTS | BODIE, KANE, MARCUS
15-22
Liquidity Premium Theory
• Long-term bonds are more risky; therefore, fn generally exceeds E(rn) • The excess of fn over E(rn) is the liquidity premium.
(1 yn ) n (1 f n ) n 1 (1 yn 1 )
fn = one-year forward rate for period n
yn = yield for a security with a maturity of n
(1 yn ) n (1 yn1 ) n1 (1 f n )
INVESTMENTS | BODIE, KANE, MARCUS
15-15
Example 15.4 Forward Rates
• The forward interest rate is a forecast of a future short rate. • Rate for 4-year maturity = 8%, rate for 3year maturity = 7%.
• We need to consider each bond cash flow as a stand-alone zero-coupon bond. • Bond stripping and bond reconstitution offer opportunities for arbitrage. • The value of the bond should be the sum of the values of its parts.
INVESTMENTS | BODIE, KANE, MARCUS
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!
15-6
Example 15.1 Valuing Coupon Bonds
• Value a 3 year, 10% coupon bond using discount rates from Table 15.1:
$100 $100 $1100 Price 2 1.05 1.06 1.07 3
INVESTMENTS | BODIE, KANE, MARCUS
15-5
Table 15.1 Prices and Yields to Maturities on Zero-Coupon Bonds ($1,000 Face Value)
INVESTMENTS | BODIE, KANE, MARCUS
4 1 y4 1 f4 3 1 y3
1.084 1.1106 3 1.07
f 4 11.06%
INVESTMENTS | BODIE, KANE, MARCUS
15-16
Interest Rate Uncertainty
• Suppose that today’s rate is 5% and the expected short rate for the following year is E(r2) = 6%. The value of a 2-year $1000 zero is:
INVESTMENTS | BODIE, KANE, MARCUS
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 MENTS | BODIE, KANE, MARCUS
15-20
Theories of Term Structure
• Expectations • Liquidity Preference – Upward bias over expectations
INVESTMENTS | BODIE, KANE, MARCUS
• The yield curve has an upward bias built into the long-term rates because of the liquidity premium.
INVESTMENTS | BODIE, KANE, MARCUS
INVESTMENTS | BODIE, KANE, MARCUS
15-12
Short Rates and Yield Curve Slope
• When next year’s short rate, r2 , is greater than this year’s short rate, r1, the yield curve slopes up. – May indicate rates are expected to rise. • When next year’s short rate, r2 , is less than this year’s short rate, r1, the yield curve slopes down. – May indicate rates are expected to fall.
15-21
Expectations Theory
• Observed long-term rate is a function of today’s short-term rate and expected future short-term rates.
• fn = E(rn) and liquidity premiums are zero.
1.051.06
$898.47
• The value of a 1-year zero is:
$1000 $952 .38 1.05
INVESTMENTS | BODIE, KANE, MARCUS
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. – 0r– Buy the 1-year bond today and hold to maturity.
• Equilibrium requires that both strategies provide the same return.
INVESTMENTS | BODIE, KANE, MARCUS
15-9
Figure 15.2 Two 2-Year Investment Programs
INVESTMENTS | BODIE, KANE, MARCUS
INVESTMENTS | BODIE, KANE, MARCUS
15-3
Figure 15.1 Treasury Yield Curves
INVESTMENTS | BODIE, KANE, MARCUS
15-4
Bond Pricing
• Yields on different maturity bonds are not all equal.
INVESTMENTS | BODIE, KANE, MARCUS
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.