FIXED-INCOME SECURITIES
Chapter 3
Term Structure of Interest
Rates: Empirical Properties
and Classical Theories
Outline
•
•
•
•
•
Types of TS
Shapes of the TS
Dynamics of the TS
Stylized Facts
Theories of the TS
Types of Term Structures
• The term structure of interest rates is the series of
interest rates ordered by term-to-maturity at a given
time
• The nature of interest rate determines the nature of
the term structure
– The term structure of yields to maturity
– The term structure of zero coupon rates
– The term structure of forward rates
• TS shapes
–
–
–
–
Quasi-flat
Increasing
Decreasing
Humped
How is the curve today?
Quasi-Flat
US YIELD CURVE AS ON 11/03/99
7.00%
6.50%
par yield
6.00%
5.50%
5.00%
4.50%
4.00%
0
5
10
15
maturity
Quasi-Flat
20
25
30
Increasing
JAPAN YIELD CURVE AS ON 04/27/01
2.50%
2.00%
par yield
1.50%
1.00%
0.50%
0.00%
0
5
10
15
maturity
Increasing
20
25
30
Decreasing
UK YIELD CURVE AS ON 10/19/00
6.00%
par yield
5.50%
5.00%
4.50%
0
5
10
15
maturity
Decreasing (or
inverted)
20
25
30
Humped (1)
EURO YIELD CURVE AS ON 04/04/01
5.50%
par yield
5.00%
4.50%
4.00%
0
5
10
15
20
maturity
Humped
(decreasing then
increasing)
25
30
Humped (2)
US YIELD CURVE AS ON 02/29/00
7.00%
par yield
6.50%
6.00%
5.50%
0
5
10
15
20
maturity
Humped
(increasing then
decreasing)
25
30
Dynamics of the Term Structure
• The term structure of interest rates changes
in response to
– Wide economic shocks
– Market-specific events
• Example
– On 10/31/01, Treasury announces that there will not be any further
issuance of 30 year bonds
– Price of existing 30 year bonds is pushed up (buying pressure)
– 30 year rate is pushed down
Example – US YTM TS
US term structure of government YTM
7
6
YTM
5
0
4
4
J-01
8
12
S-00
16
maturity
M-00
J-00
20
S-99
24
M-99
28
J-99
time
Stylized Facts (1) : Mean Reversion
• Mean reversion: high (low) values tend to be
followed by low (high) values
• Example: 10 Y swap rate versus Dow Chemical
50
45
40
35
30
Dow Chemical en US $
Taux de swap 10 ans en %
25
20
15
10
5
01/01/1999
01/01/1998
01/01/1997
01/01/1996
01/01/1995
01/01/1994
01/01/1993
01/01/1992
01/01/1991
01/01/1990
0
Stylized Facts (2) : Correlation
• Rates with different maturities are
– Positively correlated one another
– Not perfectly correlated though (more than one factor)
– Correlation decreases with difference in maturity
• Example: France (1995-2000)
1M
3M
6M
1Y
2Y
3Y
4Y
5Y
7Y
10Y
1M
3M
6M
1Y
2Y
3Y
4Y
5Y
7Y
10Y
1
0.999
1
0.908 0.914
1
0.546 0.539 0.672
1
0.235 0.224 0.31
0.88
1
0.246 0.239 0.384 0.808 0.929
1
0.209 0.202 0.337 0.742 0.881 0.981
1
0.163 0.154 0.255 0.7
0.859 0.936 0.981
1
0.107 0.097 0.182 0.617 0.792 0.867 0.927 0.97
1
0.073 0.063 0.134 0.549 0.735 0.811 0.871 0.917 0.966
1
Stylized Facts (3)
• The evolution of the interest rate curve can be split
into three standard movements
– Shift movements (changes in level), which account for 70 to 80% of
observed movements on average
– A twist movement (changes in slope), which accounts for 15 to 30% of
observed movements on average
– A butterfly movement (changes in curvature), which accounts for 1 to 5% of
observed movements on average
• That 3 factors account for more than 90% of the
changes in the TS is valid
– Whatever the time period
– Whatever the market
Shift Movements
Upwar -Downward Shift Movements
7
6
yield (in %)
5
4
3
2
1
0
0
5
10
15
maturity
20
25
30
Twist Movements
Flattening - Steepening Twist Movements
8
7
6
yield (in %)
5
4
3
2
1
0
0
5
10
15
maturity
20
25
30
Butterfly Movements
Concave - Convex Butterfly Movements
6
5.5
5
yield (in %)
4.5
4
3.5
3
2.5
2
0
5
10
15
maturity
20
25
30
Theories of the Term Structure
• Studying the TS boils down to wondering about the
preferences of participants' for curve maturities
– Investors
– Borrowers
•
Indeed, if they were indifferent in terms of maturity
– Interest rate curves would be invariably flat
– Notion of TS would be meaningless
• Market participants' preferences can be guided
– By their expectations
– By the nature of their liability or asset
– By the level of the risk premiums they require
Theories of the Term Structure
• Term structure theories attempt to account for the
relationship between interest rates and their residual
maturity
• They fall within the following categories
– Pure expectations
– Pure risk premium
• Liquidity premium
• Preferred habitat
• Market segmentation
• To these main types, we can add
– The biased expectations theory, that combines the first two theories
Theories of the Term Structure
• Remember:
1+R0,t = [(1+ R0,1)(1+ F1,2)(1+ F2,3)…(1+ Ft-1,t)]1/t
• The pure expectations theory postulates that forward
rates exclusively represent future short term rates as
expected by the market
• The pure risk premium theory postulates that forward
rates exclusively represent the risk premium required
by the market to hold longer term bonds
• The market segmentation theory postulates that
– Each of the two main market investor categories is invariably located on a
given curve portion (short, long)
– As a result, short and long curve segments are perfectly impermeable
Pure Expectations
• TS reflects market expectations of future short-term
rates
– An increasing (resp. flat, resp. decreasing) structure means that the market
expects an increase (resp. a stagnation, resp. a decrease) in future shortterm rates
• Example: from a flat curve to an increasing curve
– The current TS is flat at 5%
– Investors expect a 100bp increase in rates within one year
– For simplicity, assume that the short (resp.long) segment of the curve is the
one-year (resp. two-year) maturity
– Then, under these conditions, the interest rate curve will not remain flat but
will be increasing
– Why?
Pure Expectations (Cont’)
• Consider a long-term investor (2-year horizon)
– His objective is maximizing his return on the period
– Either invests in a long 2-year security or invests in a short 1-year security,
then reinvests in one year the proceeds in another 1-year security
• Before interest rates adjust at the 6% level
– First option returns an annual return of 5% over two years
– Second option returns 5% the first year and, according to his expectations,
6% the second year, i.e., 5.5% on average per year over two years
• Investor will thus buy short bonds (one year) rather
than long bonds (two years)
– Similar behavior for the short-term investor (return on 2-y bond after 1 year
is 4.05%=(5+105/1.06-100)/100 < 5% (return on 1 year bond)
– As a result
• The price of the one-year bond will increase (its yield will decrease)
• The price of the two-year bond will decrease (its yield will increase)
– The curve will steepen
Pure Expectations (Cont’)
• In summary, market participants behave collectively
to let the relative appeal of one maturity compared to
the others disappear
• In other words, they neutralize initial preferences for
some curve maturities
• The pure expectations theory has an important
limitation
– Investors behave in accordance with their expectations for the unique
purpose of maximizing their investment return
– They are risk neutral - They do not take into account the fact that their
expectations may be wrong
– The pure risk premium theory includes this contingency
Pure Risk Premium
• Indeed, if forward rates were perfect predictors of
future rates, future bond prices would be known with
certainty
• Unfortunately, it is not the case
– Future interest rates are unknown (re-investment risk)
– Future bond prices are unknown (market risk)
• Example: an investor with a 3 years horizon
– May invest in a 3-year zero coupon bond and holding it until maturity
– May invest in a 5-year zero coupon bond and selling it in 3 years
– May invest in a 10-year zero coupon bond and selling it in 3 years
• What would you prefer?
– Return of the first investment is known ex-ante with certainty
– Not the case for the 2nd and 3rd
– We don’t know the price of these instruments in three years
Pure Risk Premium (con’t)
• However, we know something about their risk
(volatility)
• A bond price risk measured by price volatility
– Tends to increase with maturity (P’(r)>0)
– In a decreasing proportion (P’’(r)<0)
• Assume interest rates increase to 6%
– The long bond price will fall to 5/1.06+ 105/1.062 = $ 98.17
– The short one will fall to 105/1.06 = $99.06.
– Decrease in 2-year bond price nearly twice as big as decrease in 1-year
bond price
• Pure risk premium theory: TS reflects risk premium
required by the market for holding long bonds
• The two versions of this theory differ about the shape
of the risk premium
Pure Risk Premium - Liquidity
• Risk premium increases with maturity in a
decreasing proportion
• Formally
1+Ro,t = [(1+ R0,1) (1+ E(R1,2)+ L2) (1+ E(R2,3)+ L3)… (1+ E(Rt-1,t)+ Lt)]1/t
• Lk is the liquidity (actually risk) premium required by
the market to invest in a bond maturing in k years
0 = L1 < L2 < L3 < ... < Lt
L2 -L1 > L3 -L2 > L4 -L3 > ... > Lt -Lt-1
• Hence, an investor will be interested in holding all
the longer bonds as their return contains a high risk
premium, offsetting their higher volatility
• Implies that a “normal” TS is increasing
Preferred Habitat
• Postulates that risk premium is not uniformly
increasing
• Indeed, investors have a preferred investment
horizon dictated by the nature of their liabilities
(shorter is not always better)
• Nevertheless, depending on bond supply and
demand on specific segments
– Some lenders and borrowers are ready to move away from their preferred
habitat
– Provided that they receive a risk premium that offsets their price or
reinvestment risk aversion
• Thus, all curves shapes can be accounted for
Market Segmentation
• Extreme version of pure risk premium theory
– Investors never move away from preferred habitat (infinite risk premia)
– Commercial banks invest on a short/medium term basis
– Life-insurance companies and pension funds invest on a long term basis
• Shape of the curve determined
– By supply and demand on short and long-term bond markets
– Insurance comp., pension funds are structural buyers of long-term bonds
– Commercial banks' behavior is more volatile: banks prefer to lend money
directly to corporations and individuals than invest in bond securities
• Their demand for short-term bonds is influenced by
business conditions
– During growth periods, sell bonds to meet corporations' and individuals'
demand for loans => relative increase in short-term yields
– During slow-down periods, corporations and individuals pay back their
loans, thus increasing bank funds; then banks invest in short-term bonds
=> relative decrease in short-term yields compared to long-term yields
Biased Expectations Theory and
Stochastic Approach
• All these theories are not mutually exclusive
• Biased expectations theory is an integrated
approach
– Combines pure expectations theory and risk premium theory
– Postulates that TS reflects market expectations of future interest rates as
well as permanent liquidity premia that vary over time
• Thus, all curve shapes can be accounted for
• Stochastic Approach
–
–
–
–
Uncertainty about future interest rates is not implicit in current TS
Difficult to correctly anticipate future interest rates driven by surprise effects
TS modeled as a predictable term plus a stochastic process
This theory represents an alternative to traditional theories generally used
for pricing and hedging contingent claims
Synthesis – Explanation of TS
Shapes
Curve type
Pure expectations
Risk premium
Biased expectations
The market expects a relative
interest rate stability; the risk premium
rises with maturity in a decreasing
proportion.
The market expects a great increase
The risk premium rises
The market expects
Rising curve
in interest rates ; the risk premium
with maturity.
a great increase in
rises with maturity in a decreasing
interest rates.
proportion.
The market expects a great decrease
The liquidity premium
The market expects
Falling curve
in interest rates ; the risk premium
cannot explain it;
a great decrease in
according to the preferred rises with maturity in a decreasing
interest rates.
habitat, the risk premium proportion.
decreases with maturity.
The market expects a decrease followed
The liquidity premium
Humped curve The market expects
or not by an increase in interest rates;
cannot explain it;
first an increase or
decrease in interest rates, according to the preferred the risk premium rises with maturity
habitat, the risk premium in a decreasing proportion.
and then a decrease or
increase in interest rates. increases or decreases
with maturity, and then
decreases or increases
with maturity.
Quasi-flat curve The market expects
a moderate increase
in interest rates.
The risk premium rises
with maturity in a
decreasing proportion.
Market segmentation
Banks have slightly more funds to
invest than insurance companies.
Banks have much more funds to
invest than insurance companies.
Banks have far less funds to
invest than insurance companies.
Banks and insurance companies
have the same amount of funds to
invest; their investment segments
are disjoint.