7.0term structure of interest rates
TRANSCRIPT
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CHAPTER 15
The Term Structure of Interest
Rates
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The yield curve is a graph that displays therelationship between yield and maturity.
Information on expected future short termrates can be implied from the yield curve.
Overview of Term Structure
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Figure 15.1 Treasury Yield Curves
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Bond Pricing
Yields on different maturity bonds are not allequal.
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 sumof the values of its parts.
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Table 15.1 Prices and Yields to Maturities on
Zero-Coupon Bonds ($1,000 Face Value)
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Example 15.1 Valuing Coupon Bonds
Value a 3 year, 10% coupon bond using
discount rates from Table 15.1:
Price = $1082.17 and YTM = 6.88% 6.88% is less than the 3-year rate of 7%.
32 07.11100$
06.1100$
05.1100$Price
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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-runyield curve.
On-the-run Yield Curve
The on-the-run yield
curve uses recentlyissued coupon bonds
selling at or near par.
The financial press
typically publishes on-the-run yield curves.
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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 requires that both strategies
provide the same return.
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Figure 15.2 Two 2-Year Investment Programs
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Yield Curve Under Certainty
Buy and hold vs. rollover:
Next years 1-year rate (r2) is just enough
to make rolling over a series of 1-year
bonds equal to investing in the 2-yearbond.
22 1 2
1
2
2 1 2
(1 ) (1 ) (1 )
1 (1 ) (1 )
y r x r
y r x r
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Spot Rates vs. Short Rates
Spot ratethe rate that prevails today for a
given maturity
Short ratethe 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.
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Short Rates and
Yield Curve Slope
When next years short
rate, r2, is greater than
this years short rate, r1,
the yield curve slopesup.
May indicate rates
are expected to rise.
When next years short
rate, r2, is less than this
years short rate, r1, the
yield curve slopesdown.
May indicate rates
are expected to fall.
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Figure 15.3 Short Rates versus Spot Rates
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11)1(
)1()1(
nn
nn
ny
yf
fn= one-year forward rate for period n
yn= yield for a security with a maturity of n
)1()1()1( 1
1 n
n
n
n
n fyy
Forward Rates from Observed Rates
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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 3-year
maturity = 7%.
1106.107.1
08.1
1
11
3
4
3
3
4
44
y
yf
%.f 06114
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Interest Rate Uncertainty
Suppose that todays rate is 5% and the
expectedshort rate for the following year is
E(r2)= 6%. The value of a 2-year zero is:
The value of a 1-year zero is:
47.898$
06.105.1
1000$
38.952$05.1
1000$
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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.
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Interest Rate Uncertainty
What if next years interest rate is more (or
less) than 6%?
The actual return on the 2-year bond is
uncertain!
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Interest Rate Uncertainty
Investors require a risk premium to hold alonger-term bond.
This liquidity premium compensates short-term investors for the uncertainty about
future prices.
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Expectations
Liquidity Preference
Upward bias over expectations
Theories of Term Structure
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Expectations Theory
Observed long-term rate is a function of
todays short-term rate and expected
future short-term rates.
fn= E(rn) and liquidity premiums are zero.
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Long-term bonds are more risky; therefore,
fngenerally exceeds E(rn)
The excess offnover E(rn) is the liquiditypremium.
The yield curve has an upward bias builtinto the long-term rates because of the
liquidity premium.
Liquidity Premium Theory
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Figure 15.4 Yield Curves
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Figure 15.4 Yield Curves
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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.
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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.
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Figure 15.6 Term Spread: Yields on 10-year vs.
90-day Treasury Securities
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Forward Rates as Forward Contracts
In general, forward rates will not equal theeventually realized short rate
Still an important consideration when
trying to make decisions : Locking in loan rates
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Figure 15.7 Engineering a Synthetic Forward
Loan