FINC4101
Investment Analysis
Instructor: Dr. Leng Ling
Topic: Term Structure Analysis
1
Learning objectives
1.
2.
3.
4.
5.
6.
7.
Define term structure of interest rates and yield curve.
Explain the purpose of theories of the term structure
Describe the expectations theory and discuss its
implications.
Show how different expectations of future short-term
interest rates can lead to different yield curves.
Use the expectations theory to infer future short-term
interest rates (forward rates).
Describe the liquidity preference theory.
Define the liquidity premium.
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Term structure of interest rates
 Term

structure of interest rates:
Relationship between yields to maturity and
term to maturity across bonds.
 Yield
curve: The graphical representation
of the term structure.
3
Term structure/ yield curve
has 4 different ‘looks’

Term structure/ yield curve can take on
one of the following shapes:
1.
Rising/ upward sloping (most common)
2.
Inverted/ downward sloping/ falling
3.
Hump-shaped, rising and then falling
4.
Flat
4
Term structure/ yield curve
has 4 different ‘looks’
5
Expectations Theory (1)
 Yields
to maturity are determined by
expectations of future short-term interest
rates.
 In other words, the shape of the term
structure/ yield curve depends on the
expected short-term interest rates in the
future.
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Expectations Theory (2)
 Depending
on what investors expect the
future short-term rate to be:
 Higher than current short-term rate, or
 Lower than current short-term rate, or
 Same as current short-term rate
We get different yield curves.
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Expectations Theory (3)
 market
expecting increases in future shortterm interest rates leads to upward sloping
yield curve.
 market expecting decreases in future
short-term interest rates leads to
downward sloping yield curve.
 market expecting no change in future
short-term interest rates leads to flat yield
curve.
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How does the
expectations theory work?
Current 1-year interest rate is 8%.
 Suppose everyone in the market expects that
one year from now, the 1-year interest rate will
rise to 10%.
 How does this expectation determine the current
2-year interest rate?
 All interest rates are quoted on an annual
basis.
 In this situation, investors can choose one of two
possible strategies.

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Investment strategies

For simplicity, assume



Rollover:



Buy 1-year bond now.
When it matures, buy another 1-year bond next year and hold till
it matures.
Buy-and-hold:


All bonds are zero-coupon bonds with $1000 face value.
You can invest in fractional amount of bonds
Buy a 2-year bond now and hold it till it matures in year 2.
The expectations theory says that the two strategies
should produce the same expected total returns.
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Deriving the 2-year interest rate (1)


Suppose you have $1 to invest.
With the rollover strategy, at the end of year 1, you get:
$1 x (1 + current 1-year interest rate)
= 1 x (1.08) = 1.08



Suppose at the end of year 1, the 1-year interest rate is
10% as expected.
You invest $1.08 (total proceeds) in a 1-year bond
promising 10%.
After two years, you get: 1.08 x (1.10) = $1.188
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Deriving the 2-year interest rate (2)
 Let
y2 be the 2-year interest rate.
 According to the expectations theory,
investing $1 in a 2-year bond with an
interest rate of y2 must also produce a
total return of $1.188. That is,
$1 ´ (1 + y 2 )2 = 1.188
1 + y2 =
y2 =
1.188
1.188 - 1 = 0.08995 or 8.995%
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Compare 1-year and 2-year rates
 1-year
interest rate: 8%
 2-year interest rate: 8.995%
Conclusion: longer maturity bonds have
higher interest rates/ yields to maturity.
 The term structure is upward sloping!

* If we observe 1-year rate 8% and 2-year rate 8.995%,
then market expects that, one year from now, the 1year rate will increase from current 8% to 10%.
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Verify for yourself
 If
the market expects future 1-year interest
rate to fall to 6%:
2-year interest rate is 6.995%.
 Yield curve is downward sloping.

 If
market expects future 1-year interest
rate to stay at 8%:
2-year interest rate is 8%.
 Yield curve is flat.

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Inferring expected future rates (1)

Expectations theory says that the expected total returns
from the buy-and-hold strategy and the rollover strategy
are the same.

We can use this assumption to infer expected future
interest rates, which are called “forward rates”.

Forward rate: The inferred short-term interest rate for a
future period that makes the expected total return of a
long-term bond equal to that of rolling over short-term
bonds.
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Inferring expected future rates (2)
 In
general, we obtain the forward rate by
equating the return on an n-period zerocoupon bond with that of an (n – 1)-period
zero-coupon bond rolled over into a oneyear bond in year n:
(1 + yn)n = (1 + yn-1)n-1(1 + fn)
YTM of n-period
zero coupon bond
YTM of (n-1)-period
zero coupon bond
Forward rate for
period n.
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Inferring expected future rates (3)
 Thus,
the forward rate formula is
n
(1 + y n )
fn =
1
n- 1
(1 + y n - 1)
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Applying the forward rate formula
 Two-year
maturity bonds offer yield-tomaturity of 6%, and three-year bonds have
yields of 7%. What is the forward rate for
the third year?
 Verify
that the forward rate is 9.03%
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Problems (1)

The current yield curve for default-free zerocoupon bonds is as follows:
Maturity (Years) YTM

1
10%
2
11
3
12
Assume that the zero-coupon bonds have a face
value of $1000.
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Problems (2)
1.
2.
3.
What are the implied one-year forward rates?
Assume that the pure expectations hypothesis
is correct. If market expectations are accurate,
one year from now what will be the yields to
maturity on one- and two-year zero coupon
bonds?
If you purchase a two-year zero-coupon bond
now, what is the expected total rate of return
over the first year? What if you purchase a
three-year zero-coupon bond?
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Liquidity Preference Theory


Investors prefer to hold short-term bonds because shortterm bonds have more ‘liquidity’ than long-term bonds.

Short-term bonds offer greater price certainty and trade in more
active markets.

Investors like these attributes.
Because long-term bonds are less liquid by comparison,
investors demand a liquidity premium for holding longterm bonds.

Liquidity premium: extra expected return demanded by investors
as compensation for the lower liquidity of long-term bonds.
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Liquidity Preference Theory
The theory says that investors demand a
liquidity premium on long-term bonds.
 Thus, long-term bonds have higher interest rates
than short-term bonds.
 This gives rise to an upward sloping yield curve.
 Even if future short-term interest rates are
expected to remain unchanged, yield curve will
be upward sloping because of the liquidity
premium.

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Liquidity premium
 The
spread between the forward rate of
interest and the expected short-term rate:
Liquidity premium
= forward rate – expected short-term rate
Re-arranging the formula, we get
Forward rate =
expected short-term rate + liquidity premium
So, forward rate > expected short-term rate
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More problems (1)
 The
yield to maturity on one-year zero
coupon bonds is 8%. The yield to maturity
on two-year zero-coupon bonds is 9%.
a) What is the forward rate for the second
year?
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More problems (2)
b) If you believe the expectations theory,
what is your best guess as to the expected
short-term interest rate next year?
c) If you believe the liquidity preference
theory, is your best guess as to next year’s
short-term interest rate higher or lower
than in (b)?
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More problems (3)

1.
2.
3.
4.
Which of the following statements is true?
The expectations theory indicates a flat yield curve if
anticipated future short-term rates exceed current shortterm rates.
The basic conclusion of the expectations theory is that
the long-term rate is equal to the anticipated short-term
rate.
The liquidity preference theory indicates that, all other
things being equal, longer maturities will have higher
yields.
The liquidity preference theory states that a rising yield
curve implies that the market anticipates increases in
interest rates.
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Summary
1.
2.
3.
4.
5.
6.
Term structure of interest rates.
Theories of the term structure try to explain the shape of
the term structure/ yield curve.
Pure Expectations theory and its implications.
Calculate forward rates using observed interest rates of
different maturities.
Liquidity Preference theory and its implications.
Relationship between forward rate, expected short-term
rate and liquidity premium.
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Practice 8
 Chapter
10:
41 (a)(b), 43.
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