Term Structure of Interest Rates
Ali Umut Irturk
789139-3
Survey submitted to the Economics Department of the
University of California, Santa Barbara in fulfillment of the requirement for
M.A. Theory of Finance Economics 234B
Spring 2006 Prof. Clement G. Krouse
June 2006 Santa Barbara, California
Keywords: Interest Rates, Yield, Term Structure
Copyright 2006, Ali Umut IRTURK
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Abstract
In this survey, firstly I describe the fundamentals of interest rates and
yield curves. After required background information for the term structure is established,
I move on the main subject of this survey: Term Structure of Interest Rates. We can define
the term structure of interest rates as calculation of the relation between the yields on
default-free securities which only differ in their term to maturity. This relationship has
several determinants, such as interest rates and yield curves, which are always
concerned by economics to establish the term structure. Investors and economists
strongly believe that the shape of the yield curve reflects the conditions for monetary
policy and the market's future expectation for interest rates. In other words, term
structure is important for us because it integrates the market’s anticipations of future
events by offering a complete schedule of interest rates across time. Thus, the
understanding of the explanation of the term structure gives us a way to extract this
information and to predict how changes in the underlying variables will affect the yield
curve.
The novel contributions are: first, that in a stratified context of the fundamentals
provides the basis of term structure of interest rates in the names of interest rates and
yield curves; second, empirical evidences gives the background information for
understanding this important concept; third, important theories of the term structure
which will be important when measuring the term structure; fourth, term structure
models to understand the measurement of the term structure. I believe that this stratified
content of the survey will improve the understanding of the readers about the term
structure of interest rates.
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1. Introduction It is well known that, if identical bonds have different terms to
maturity, consequently their interest rates differ. Term structure of interest
rates is the relationship among yields on financial instruments with identical
tax, risk and liquidity characteristics, however they gives different terms to
maturity. Thus, we can say that the term structure of interest rates refers to
the relationship between bonds of different terms. Here, yield curve is
constructed by plotting the interest rates of bonds against their terms. For
instance, term structure can be defined as the yield curve which is displaying
the relationship between spot rates of zero coupon securities and their term
to maturity. As can be seen, there is a strong connection between interest
rates and yield curve. The term structure of interest rates is a very important
research area for economists. We can ask ourselves that what makes the term
structure of interest rates so important. Because, economists and investors
believe that the shape of the yield curve reflects the market's future
expectation for interest rates and the conditions for monetary policy.
Before moving the concept of term structure of interest rates, we need
to consider some important term structure fundamentals: interest rates and
yield curves.
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1.1 Interest rates
Interest rate is the monthly effective rate payment on borrowed
money. If the person is a creditor, this will be received. It is expressed as the
percentage of the borrowed sum. In modern financial theory, interest rates
and their determinants are probably the most computationally difficult part.
Although it is hard to compute, the interest rates provide very valuable
information to the economists. There are three important reasons to explain
why interest rates are important. Firstly, the modern fixed income market
includes not only bonds but all kinds of derivative securities sensitive to
interest rates. Secondly, interest rates are important in pricing all other
market securities since they are used in time discounting. Lastly, on
corporate level since most investment decisions are based on some
expectations regarding alternative opportunities and the cost of capital—
both depend on the interest rates. In time, there will be changes in interest
rates or differences between interest rates. We know that the interest rates
have some variables which affect the interest rates in time, such as default
risk, tax treatment, marketability, term to maturity, call or put features and
convertibility.
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Best way to understand the interest rate computation is to give a basic
example [5]. This makes the interest rate computation very clear. Suppose a
payment of $1, which will be made with certainty (risk-free interest rate) at
time t. If the market price of $1 paid in time t from now is P0 , then we can
find the interest rate for time t using the simple discount formula,
The interest rate rt in this formula is known as the pure discount interest rate
for time t.
1.2 Yield Curves
If there are some bonds with the same risk, liquidity and tax
characteristics but different maturities, given their maturity we can compare
the differences in their interest rates. The common tool for this analysis is
the yield curves. Yield curves plot the interest rates of same bond with the
same characteristics (except maturity) against their corresponding different
maturity terms. As can be seen, the yield curve is a chart which illustrates
the relationship among yields on bonds that differ only in their term to
maturity.
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There are some different classifications of yield curves. Generally,
there are three different yield curves: upward sloping, downward sloping and
flat. I would like to give different representation of these three curves.
Firstly, upward sloping curve can be defined as ascending sloping or normal
curve. Secondly, downward sloping can be defined as descending.
Upward sloping yield curve. The upward sloping yield curves are most
commonly observed type of yield curves in developed nations;
Figure 1 Upward sloping yield curve
If a yield curve is upward sloped, short term interest rates are below
long term interest rates. In other words, longer term interest rates are usually
higher than shorter term interest rates that why we are saying normal yield
curve. It is thought to reflect the higher inflation-risk premium that investors
demand for longer term bonds. Through most of the post-Great Depression
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era to present the yield curve has been called normal when yields rise as
maturity lengthens, that is, when the slope of the yield curve is positive. This
positive slope reflects investor expectations for the economy to grow in the
future and, importantly, for this growth to be associated with a greater risk
that inflation rises in the future than falls. This expectation for higher
inflation in the future than the present generates both an expectation that the
central bank will tighten monetary policy by raising short term interest rates
in the future to slow economic growth and dampen inflationary pressure and
the need for a risk premium associated with the uncertainty about the future
rate of inflation and the risk this poses to the future value of cash flows.
Investors price these risks into the yield curve by demanding higher yields
for maturities further into the future.
However, normal being associated with a positive slope has not
always been the norm. Through much of the 19th century and early 20th
century the US economy experienced trend growth with persistent deflation,
not inflation. During this period the yield curve was typically inverted,
reflecting the fact that deflation made future cash flows more valuable than
current cash flows. During this period of persistent deflation, a normal yield
curve was negatively sloped.
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Downward sloping yield curve. Downward sloping is sometimes
called inverted yield curve assuming the normal yield curve has the positive
slope.
Figure 2 Downward sloping yield curve
If a yield curve is downward sloped, long run interest rates are below
short term interest rates. This shape is often seen when the market expects
interest rates to fall. Under this abnormal and contradictory situation, long-
term investors will settle for lower yields now if they think the economy will
slow or even decline in the future. An inverted curve may indicate a
worsening economic situation in the future. However, technical factors such
as a flight-to-quality or global economic or currency situations may cause
demand for bonds on the long end of the yield curve causing rates to fall.
This was seen in 1998 during the "Long Term Capital" failure (slight
inversion on part of the curve).
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Flat yield curve.
Figure 3 Flat yield curve
If a yield curve is flat, short term interest rates are equal to long term
interest rates. A small or negligible difference between short and long term
interest rates occurs later in the economic cycle when interest rates increase
due to higher inflation expectations and tighter monetary policy. This is
called a shallow or flat yield curve and higher short term rates reflect less
available money, as monetary policy is tightened, and higher inflation later
in the economic cycle.
Here I would like to consider some other characteristics of yield
curves. I stated above that there are three most usual yield curves. However,
the relationship between an interest rate and the term to maturity of a bond is
usually not linear, so that yield curves can often be classified as humped.
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Figure 4 Humped yield curve
Shape often seen when the market expects that interest rates will first
rise (fall) during a period and fall (rise) during another.
When interest rates change by the same amount for bonds of all terms,
this is called a parallel shift in the yield curve since the shape of the yield
curve stays the same, although interest rates are higher or lower across the
curve. A change in the shape of the yield curve is called a twist and means
that interest rates for bonds of some terms change differently than bond of
other terms. When the difference between long and short term interest rates
is large, the yield curve is said to be steep. This is thought to reflect a loose
monetary policy which means credit and money is readily available in an
economy. This situation usually develops early in the economic cycle when
a country's monetary authorities are trying to stimulate the economy after a
recession or slowdown in economic growth. The low short term interest
rates reflect the easy availability of money and low or declining inflation.
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Higher longer term interest rates reflect investors' fears of future inflation,
recognizing that future monetary policy and economic conditions could be
much different. Tight monetary policy results in short term interest rates
being higher than longer term rates. This occurs as a shortage of money and
credit drives up the cost of short term capital. Longer term rates stay lower,
as investors see an eventual loosening of monetary policy and declining
inflation. This increases the demand for long term bonds which lock in the
higher long term rates.
As can be seen above results, we propose several ideas with looking at
the yield curve. However, there is an important question: what determines
the shape of a yield curve? Someone can give any answer to this question,
but this answer has to incorporate three empirical facts:
a) Interest rates on bonds of different maturities move together
over time.
b) When short term interest rates are low, long run interest rates
tend to be high, such that yield curves are upward sloped and
vice versa.
c) Yield curves are generally almost always upward sloped.
I will consider these empirical facts in Section 3.
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1.3 The Term Structure
As I mentioned above, we can study the term structure visually by
plotting a yield curve at a point in time. The general characteristics are
determined in Section 1.2 as ascending, descending or flat. Besides, Section
4 and 5 are devoted the several theories and methods which explains the
shape of the yield curve.
Here, I want to introduce the usage of term structure mathematically.
Thus, we need a short and incomplete review of bond terminology. Suppose
that a standard bond has the following characteristics [5]:
Bond’s face value which is denoted by D (can be called notional value)
Bond’s coupon payments which are denoted by C. (can be called percentage
of its face value)
Bond’s maturity which is denoted by N, is the date of the last payment that
consists of the face value plus the coupon rate.
Additionally, coupon rate is the percentage rate of the face value paid as
coupons. If we know the term structure of interest rates, the price of a bond
with yearly coupons can be calculated by:
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It is sometimes more convenient for analytic computations to assume that a
bond makes a continuous stream of payments between time 0 and time N.
Consider the payment during the time from t till t + dt, which can be shown
as Ct dt, thus the bond price can be given by:
(continuous compounding)
As a general explanation of the term structure, economic theory [1]
suggests that one important factor explaining the differences in the interest
rates on different securities may be differences in their terms. Thus, the
relationship between the terms of securities and their market rates of interest
is known as the term structure of interest rates. The diagram which is called
yield curve and described above, is used to display the term structure of
interest rates on securities of a particular type at a particular point in time.
In the next Section, I go over empirical evidence of term structure.
Section 3 presents the theories of the term structure, such as expectation
theory, liquidity preference theory and preferred habitat theory. Then, in
Section 4 I show extensively what the models are to compute the term
structure of interest rates. Finally, Section 5 concludes the paper.
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2. Empirical Evidence of Term Structure I would like to introduce the empirical evidence of term structure by
using the model of Campbell R. Harvey [2]. Using this model, we can see
that the slope of the term structure correctly predicted the five cyclical
turnings points over the last 40 years. I have to say that many more
expensive econometric models cannot predict these turning points. This
model is described in the dissertation of Campbell R. Harvey and the
September/October 1989 Financial Analysts Journal article [3]. Basically,
this interest rate based model is very simple. It has only two components:
slope of the term structure (the long term – short term yield spread) and
measure of the average propensity to hedge in the economy.
First of all, we need to learn the link between the term structure and
economic growth to especially understand how to predict business cycle
turning points with the term structure. Basically, if we consider the basic
intuition of the model, we can see that interest rates are ex ante measures and
represents expected future payoffs. Besides, if market rates are set, we can
assume that expectations of future economic growth influence this process.
This basic intuition is described using an example in the paper. If we assume
that investors want to insure their economic well being, thus most of the
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investors would prefer a reasonably stable level of income rather than very
high income in one stage of the business cycle and very low income in
another stage. Besides the investors side, assume that the economy is
currently in a growth stage and the general agreement is for a slowdown or
recession during the next year. Here, this desire to hedge will lead
consumers to purchase a financial instrument which will deliver payoffs in
the slowdown, such as a one year discount bond. At this point, if many
consumers buy the one-year bond, then the price of the security will increase
and the yield to maturity will decrease. Besides, the consumers may sell
their shorter term assets to finance the purchase of the one year bonds. Thus,
this selling pressure will drive down the price of the short term instrument,
besides raise its yield.
After understanding the assumptions, if a recession is expected under
these circumstances, we will see long rates decrease and short rates will
increase. Thus, the yield curve or term structure will become flat or inverted.
As a result, we can say that the shape of the term structure of interest rates
provides a forecast of future economic growth.
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In this research, the historical performance of the model is considered
over U.S. economy. We are looking for recessions which are defined as the
period between an economic trough and peak. Considered recessionary
periods are classified by the National Bureau of Economic Research
(NBER). Record of the term structure [4]:
• Recession 1969 Quarter 4 – 1970 Quarter 4, total GDP decline is .1%
Term structure begins inversion in 1968 Quarter 3 correctly. (Predicts
the recession four quarters in advance.)
• Recession 1973 Quarter 4 – 1975 Quarter 1, total GDP decline is
4.2%
Term structure begins inversion in 1973 Quarter 2 correctly. (Predicts
the recession with a two quarter lead time.)
• Recession 1980 Quarter 1 – 1980 Quarter 3, total GDP decline is 2.6
%.
Term structure begins inversion in 1978 Quarter 4 correctly. (Predicts
the downturn with a five quarter lead.)
• Recession 1981 Quarter 3 – 1982 Quarter 4, total GDP decline is
2.7%.
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Term structure begins inversion in 1980 Quarter 4 correctly. (Predicts
the recession with a four quarter advance signal.)
• Recession 1990 Quarter 3 – 1991 Quarter 1, total GDP decline is
1.8%.
Term structure shows inversion in 1989 Quarter 2 till 1989 Quarter 4.
(Predicts the recession with five quarter lead even though this
inversion is mild when compared with the others.)
Here, I have to indicate that the magnitude of the inversions reveals the
severity of the recession. According to this, we know that in nearly 1995, the
term structure came very close to inverting. Thus, U.S. economy
experienced slower growth, however it was not an evidence of recession.
3. Theories of the Term Structure
We know that in a world of certainty, equilibrium forward rates must
go along with future spot rates. However, if we are considering a world
which is of uncertainty, the analysis becomes much more complex and
difficult. Thus, many researchers considered the certainty of the world
model when they are working on term structure. Then they continue by
examining stochastic generalizations of the certainty equilibrium
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relationship. There are four important and well-known theories of term
structure: Expectations Theory, Liquidity Preference Theory, Market
Segmentation Theory and Preferred Habitat Theory. I will consider them one
by one.
3.1 Expectations Theory (Pure Expectations Theory)
Firstly, I would like to mention that there are various versions of
expectation theory. Generally, these place predominant emphasis on the
expected values of future spot rates or holding-period returns. For better
understanding, the best way is to consider its simplest form. In this situation,
the expectations hypothesis postulates that bonds are priced so that the
implied forward rates are equal to the expected spot rates. Thus, there is a
world of certainty. We can characterize this approach by one of the
following propositions:
a) The return on holding a long-term bond to maturity is equal to
the expected return on repeated investment in a serious of the
short-term bonds.
b) Or we can say that the expected rate of return over the next
holding period is the same for bonds of all maturities.
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In this theory, shape of the yield curve is based on market participants’
expectations of future interest rates. Besides, with the assumption that
arbitrage opportunities will be minimal, we can create the yield curve. For
instance, if an investor has an expectation about what a 1-year interest rate
will be next year, we can calculate the 2-year interest rate by compounding
this year and next years interest rate.
Thus, for example, if people expect that short-term interest rates will
be 20% on average in the next two years, as a result the interest rate on 2-
year bonds will be 20% too.
If we compare these theory characteristics with the empirical facts for
shape of a yield curve, we can see that this theory explains fact 1 and 2. In
Section 1.2, the stated empirical facts were;
a) Interest rates on bonds of different maturities move together
over time.
b) When short term interest rates are low, long run interest rates
tend to be high, such that yield curves are upward sloped and
vice versa.
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As mentioned above, this theory states that if we combine a period of
short term bonds, and compare with the same period of longer term bond,
the total interest earned should be equivalent. This means that the
expectations theory can explain the movements of short-term and long-term
interest rates. Thus, the yield curve can be used to predict future interest
rates. Additionally, we can predict that if the slope of the curve is ascending,
future interest rates will increase and if the slope of the curve is descending,
future interest rates will decrease. These conclusions prove the first and
second empirical facts.
As a result, in this theory investors can be assumed to trade in an
efficient market where they have excellent information and minimal trading
costs. The importance of this theory is that the other theories presume less
efficient markets. However, in this situation the yield curves are usually
upward sloping, normal. On the other hand, short-term interest rates are as
likely to fall as to rise. This is not consistent with the real world. The
expectations theory can not explain the usual upward slope of the yield
which is the third empirical fact.
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3.2 Liquidity Preference Theory
We can say that it is an offshoot of the expectation theory. This theory
concurs with expectations theory in a way that it gives the same importance
to the expected future spot rates, but places more weight on the effects of the
risk preferences of market participants. Liquidity preference theory asserts
that risk aversion will cause forward rates to be systemically greater than
expected spot rates. This amount is usually stated as an amount increasing
with maturity. If we consider this theory in a different way, we can say that
the longer-term interest rates are not only reflect investors’ future
assumptions for the interest rates, but also includes a premium for holding
these longer-term bonds which we state as term premium or liquidity
premium. This premium concept introduces the compensation of investor for
the added risk of having its money tied up for a longer period. This term
premium is the increment required to include to induce investors to hold
long-term securities. Additionally, it includes the uncertainty of the greater
price.
We can summarize as that generally long-term rates have greater risk
and thus investors need greater premiums to give up liquidity. Besides, long-
term rates have greater price variability and less marketability. If we
consider the empirical facts of yield curve, the last comment holds for the
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liquidity premium theory. Because this theory explains an upward sloping
yield curve.
3.3 Market Segmentation Theory (Segmented Markets Theory)
I introduced the premium concept in liquidity preference theory.
However, market segmentation theory introduces a different term premium
concept. In this theory, individuals have strong maturity preferences, thus
bonds of different maturities trade in separate markets. This means that
markets for bonds of different maturities are completely separated and
segmented and cannot substitutable. As a result, the demand and supply of
bonds of particular maturity are little affected by the bonds of neighboring
maturities’ prices and generally determined independently. We can say that
borrowers have particular periods for which they want to borrow and lenders
have particular holding periods in mind.
Investors have to decide whether they need short-term or long-term
instruments. In this situation, we know that investors prefer their portfolio to
be liquid. Thus, they will prefer short-term instruments to long-term
instruments. This results so that short-term instruments will receive higher
demand in the market. This higher demand to the short-term instruments will
cause higher prices and lower yield. Here, we can see that one of our
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empirical facts is true. Because here, short-term yield is lower than long-
term yield. Besides, this explains us the normal yield curve. However, this
theory cannot explain that yields of different terms move together, because
the supply and demand of the two markets are independent. This theory
explains only the empirical fact 3.
In this theory, term premiums do not need to be positive and an
increasing function of maturity.
3.4 Preferred Habitat Theory
The preferred habitat theory is similar to market segmentation theory
that researchers use some arguments similar to the market segmentation
theory. However, they recognize its limitations and combine it with aspects
of the other theories. We can say that preferred habitat theory is the
combination of the market segmentation theory and expectations theory,
because investors care both expected returns and maturity. Additionally,
investors have different investment horizons and to buy bonds with
maturities outside their habitat, they need a meaningful premium. Thus, this
theory allows market participants to trade outside of their preferred maturity
if adequately compensated for the additional risk. But, we have to remember
that investors prefer short-term to long-term bonds and never prefer a long-
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term bond if this offers the same expected return as a series of short-term
bond. Here, short-term investors are more prevalent in the fixed-income
market, thus longer-term rates tend to be higher than short-term rates.
Preferred habitat theory intended for a plausible rationale for term
premiums which is not restricted in sign or monotonicity, rather than as a
necessary causal explanation. We can predict the future like the following
examples:
a) If the yield curve slopes slightly upward, investors predict
interest rates to stay about the same.
b) If the yield curve slopes sharply upward, short-term rates are
predicted to rise.
c) If the yield curve slopes flat, short-term rates are predicted to
fall slightly.
d) If the yield curve slopes downward, the investors predict a
sharp decline in interest rates.
Here, I completed the definitions of the term structure theories. We
see that economics are interested in term structure theories, because of many
reasons. If we state some of these reasons:
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a) The accuracy of the predictions of different term structure theories
is relatively easy to evaluate, when the actual rate term structure of
interest rates is easy to observe. Here, I stated the theories which
are based on assumptions and principles that have applications in
other branches of economic theory, such as expectations theory.
b) Term structure theories explain the ways in which changes in
short-term interest rates affect the levels of long-term interest rates.
Economic theory states that monetary policy may have a direct
effect on short-term interest rates, but little, if any, direct effect on
longer-term rates.
c) Term structure may provide information about the expectations of
participants in financial markets. Thus, these expectations are of
considerable interest to forecasters and policy-makers. Many
economists believe that the people best able to forecast events in a
market are in fact the participants in that market.
At last we face this question: Which Theory is Right?
We can say that Preferred Habitat Theory is the most consistent theory to the
day-to-day changes in the term structure. However, if we consider the long-
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run, expectations of future interest rates and liquidity premiums become
important components of the position and shape of the yield curve.
4. Term Structure Models 4.1 Merton, 1973, dr = bdt + σdB
Capital asset pricing model is the one of the most important
developments in capital market theory [6]. It is known as the Sharpe-
Lintner-Mossin mean-variance equilibrium model of exchange. Even though
this model is considered among many papers, it is criticized too much. For
example, one of these reasons is that the assumption that investors choose
their portfolios according to Markowitz mean-variance criterion. However,
the model is still used because it is an equilibrium model which provides a
strong specification of the relationship among asset yields that is easily
interpreted, and the empirical evidence suggests that it does explain a
significant fraction of the variation in asset returns.
Robert C Merton developed an equilibrium model of the capital market
which is an intertemporal consumer-investor behavior based model. This
model
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a) has the simplicity and empirical tractability of the capital asset
pricing model;
b) is consistent with expected utility maximization and the limited
liability of assets;
c) provides a specification of the relationship among yields that is
more consistent with empirical evidence.
Unfortunately, the assumptions, for example: principally
homogeneous expectations, which it holds in common with the above
mentioned classical model, make the new model subject to same criticisms.
4.2 Vasicek, 1977, dr = α(γ - r)dt + σdB
Vasicek [7] gives an explicit characterization of the term structure of
interest rates in an efficient market. The model is widely used for pricing the
bonds. Additionally, it uses the Ornstein-Uhlenbeck process to compute the
spot interest rate. This model is a one-factor model which means that rates
depend on the spot interest rate. Thus, the spot rate defines the whole term
structure.
Besides the general characteristics, I want to state the main advantage
and disadvantage of the model. It has the advantage that it can be used to
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value all interest-rate-contingent claims in a consistent way. Its main
disadvantage is that it involves several unobservable parameters and do not
provide a perfect fit to the initial term structure of interest rates.
4.3 Cox, Ingersoll, Ross (CIR), 1985, dr = α(γ- r)dt + σrdB
The researchers [8] developed an intertemporal general equilibrium
asset pricing model. We know that the effective concepts when determining
the bond prices are risk aversion, investment alternatives, anticipations and
preferences about the timing of consumption.
The researchers considered the problem of determining the term
structure as being a problem in general equilibrium theory, and their
approach contains elements of all of the previous theories. Anticipations of
future events are important, as are risk preferences and the characteristics of
other investment alternatives. Also, individuals can have specific
preferences about the timing of their consumption, and thus have, in that
sense, a preferred habitat. Thus, their model permits detailed predictions
about how changes in a wide range of underlying variables will affect the
term structure.
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We can say that the model developed by Cox, Ingersoll and Ross has
the same main advantage and disadvantage which are stated above in
Vasicek model.
4.4 Ho, Lee, 1986, dr = Ө (t )dt + σdB
Ho et. al. [10] proposes an alternative approach to pricing models. The
approach is taking the term structure as given, and deriving the feasible
subsequent term structure movements. These movements must satisfy
certain constraints to ensure that they are consistent with an equilibrium
framework. Specifically, the movements cannot permit arbitrage profit
opportunities. They called these interest rate movements arbitrage-free rate
movements (AR). When the AR movements are determined, the interest rate
contingent claims are then priced by the arbitrage methodology which is
used in CIR. Therefore, their model is a relative pricing model in the sense
that they price the contingent claims relative to the observed term structure;
however, they do not endogenize the term structure as the CIR model do.
Thus, Ho and Lee pioneered a new approach by showing how an
interest rate model can be designed so that it is automatically consistent with
any specified initial term structure.
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4.5 Hull, White (extended Vasicek), 1990, dr = (Ө(t ) - βr )dt + σdB
Hull, White (extended CIR), 1990, dr = (Ө(t ) - βr )dt + σrd
The researchers [11] showed that the one-state-variable interest-rate
models of Vasicek (1977) and Cox, Ingersoll, and Ross (1985) can be
extended so that they are consistent with both the current term structure of
interest rates and either the current volatilities of all spot interest rates or the
current volatilities of all forward interest rates. The extended Vasicek model
is shown to be very tractable analytically. The article compares option prices
obtained using the extended Vasicek model with those obtained using a
number of other models.
Besides, the researchers present two one-state variable models of the
short-term interest rate. Both are consistent with both the current term
structure of interest rates and the current volatilities of all interest rates. In
addition, the volatility of the short-term interest rate can be a function of
time. The user of the models can specify either the current volatilities of spot
interest rates (which will be referred to as the term structure of spot rate
volatilities) or the current volatilities of forward interest rates (which will be
referred to as the term structure of forward rate volatilities). The first model
is an extension of Vasicek. The second model is an extension of Cox,
Ingersoll, and Ross.
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The main contribution of this paper is to show how the process
followed by the short-term interest rate in the two models can be deduced
from the term structure of interest rates and the term structure of spot or
forward interest-rate volatilities. The parameters of the process can be
determined analytically in the case of the extended Vasicek model, and
numerically in the case of the extended Cox, Ingersoll, and Ross (CIR)
model. Once the short-term interest rate process has been obtained, either
model can be used to value any interest-rate contingent claim. European
bond options can be valued analytically when the extended Vasicek model is
used.
4.6 Black, Karasinski, 1991, d log r = (Ө(t )- βlog r )dt+σdB
Black et. al. [12] describe a one-factor model for bond and option
pricing that is based on the short-term interest rate and that allows the target
rate, mean reversion and local volatility to vary deterministically through
time. For any horizon, the distribution of possible short rates is lognormal,
so the rate neither falls below zero nor reflects off a barrier at zero. A model
like this allows one to match the yield curve, the volatility curve and the cap
curve. Surprisingly, adding to future local volatility lowers the volatility
curve.
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A conventional binary tree with probabilities of 0.5 but variable time
spacing is used to value bonds and options. When the inputs are constant, the
slope of the yield curve starts out positive and ends up negative, while its
curvature shifts from negative to positive. Even when mean reversion is
zero, the volatility curve has a negative slope.
The researchers presented a one-factor model of bond prices, bond
yields, and related options. The single factor that is the source of all
uncertainty is the short-term interest rate. They assumed no taxes or
transaction costs, no default risk and no extra costs for borrowing bonds.
They also assumed that all security prices are perfectly correlated in
continuous time.
Here, before moving to the Health-Jarrow-Mortons’ approach [14], I
would like to consider what we learned so far. Generally, the term structure
models which are prior to Health-Jarrow and Morton were finite
dimensional Markovian models. In Markovian models, the interest rate
economy is determined by the spot rate and besides, but not necessarily, one
or two additional state variables. This enabled the use of standard arbitrage
arguments, along the lines of Black and Scholes (1973) and Merton (1973),
33
to derive the PDE for the bond and bond option prices which, in turn,
enabled the application of well-developed techniques from the theory of
PDEs to obtain analytic solutions, and numerical solutions in cases where
this was not possible. The progenitors of this approach could be regarded as
Vasicek (1977) and Brennan and Schwartz (1979). After Vasicek’s model is
established, many of the interest rate models are proposed using this model.
In Vasicek’s model, the spot rate was assumed as a mean reverting process
with constant volatility and constant mean reversion level. As I stated
before, the common tool used in these models was the no-arbitrage
arguments of Black-Scholes and Merton, which produced the pricing partial
differential equation for the bond, and bond option, prices in a systematic
manner. Well developed techniques from the theory of partial differential
equations were then applied to solve, either analytically or numerically,
these pricing equations.
These early models are useful because they have analytic solutions.
However, the calibration of model parameters to observed market data is a
non-trivial task. Especially, many models cannot be calibrated consistently
to the initial yield curve. Additionally, the relationship between the model
parameters and the market observed variables are not always clear and we
34
cannot always incorporate observer market features, such as humped
volatility curve, into these models.
The quantity driving this class of models was the instantaneous spot
rate of interest, and, since the spot rate is a non-traded quantity, these models
usually involved the market price of interest rate risk. And as the market
price of risk is an unobservable quantity, assumptions then had to be made,
often based on mathematical convenience rather than economic
considerations, so as to obtain a pricing PDE that enabled the application of
various solution techniques.
4.7 Heath, Jarrow, Morton (1992)
By contrast, the Heath, Jarrow and Morton model [14] provides us a
very general interest rate framework which is capable of incorporating most
of the market observed features. This model takes as the quantities driving
the model the continuum of instantaneous forward rates, which are directly
related to the prices of traded bonds. Furthermore, the HJM models are
automatically calibrated to the initial yield curve, and the connection
between the model parameters and the market variables often emerge from
the theory. They used techniques from stochastic calculus to construct a very
general framework for the evolution of interest rates that had the useful
35
feature that the model is naturally calibrated to the currently observed yield
curve.
Although the HJM model is widely accepted as the most general and
consistent framework under which to study interest rate derivatives, the
added complexity and the absence of efficient numerical techniques under
the general HJM framework saw the earlier models retain their popularity,
particularly among practitioners.[13] The main drawback of the HJM model
is that these models are non-Markovian in general, and as a result, the
techniques from the theory of PDEs no longer applicable to these models.
For the general HJM model, Monte Carlo simulation, which can often be
time consuming, is the only method of solution. To overcome these
problems, many researchers, such as Carverhill (1994), Ritchken and
Sankarasubramanian (1995) have considered ways of transforming the HJM
models to Markovian systems. In these transformed systems, the desirable
properties of the earlier Markovian models and the HJM framework coexist,
and provide useful settings under which to study interest rate derivatives.
However, with the rapid advances in computer technology, HJM models are
becoming increasingly practical, and various forms of the model are
36
currently being adopted by practitioners for the pricing and hedging of
interest rate derivatives.
The main inputs into the HJM framework are the forward rate
volatility processes, and it was shown that the Cox-Ingersoll-Ross model
was a special case of the general 1-factor HJM framework, corresponding to
a particular choice of the volatility process. In the standard HJM model, the
uncertainty in the interest rate market is represented by Wiener processes
which drive the forward rate process. Consequently, all other processes in
the interest rate market, such as forward rate volatilities, are also driven by
the same Wiener processes. Because of the characteristics of these
processes, the standard HJM model does not incorporate additional
independent sources of stochastic volatility.
With powerful computers and mathematical techniques, investors and
academics are constantly striving to build models which explain the shape of
the yield curve and hopefully provide insight into the future direction of
interest rates. This has given rise to "yield curve" strategies which are
employed by bond managers to add value to their portfolios.
37
5. Conclusions In this survey, I investigated the term structure of interest rates with
providing the fundamentals, theories and models. Firstly, this survey started
with the fundamentals of the interest rates, yield curves and term structure.
Secondly, section 2 established the empirical evidence of the term structure
concept. In this section, the historical performance of Campbell R. Harvey’s
model is considered over U.S. economy. We looked for recessions which are
defined as the period between an economic trough and peak. As a result of
this section, we can say that the shape of the term structure of interest rates
provides a forecast of future economic growth depending on the interest
rates movement. In section 3, we examined the theories of the term structure
of interest rates, such as Expectations Theory, Liquidity Preference Theory,
Market Segmentation Theory and Preferred Habitat Theory. I stated the
general definitions of these theories. Additionally, I looked at why these
theories are important for the researchers. At last, I gave some comments
about which can be the most suitable model for today. In section 4, the
models of the term structure are deeply investigated. This is an important
and emerging subject in term structure measurement research. These models
help the economists to compute the term structure of interest rates.
38
After this wide area of survey, I would like to conclude this survey
with stating three important points. First one is the roles of term structure of
interest rates in making of the monetary policy. Second one is the some
comments about tactical policy decisions. And at last, I would like to state
some comments about the pitfalls in using the term structure [15].
Firstly, it can be predicted by anyone that the term structure of interest
rates play an important role in the making of monetary policy.
Here, I state several results, such as [15]:
a) Long rates show the extent to which a central bank has reached
price stability.
b) Significant bond rate movements affect the timing and
magnitude of monetary policy actions.
c) The ability of bond rates to forecast changes in inflation trends
is not especially good.
d) The influence of policy actions on longer-term rates can be
quite inconsistent.
e) The degree of restraint transmitted by policy is difficult to
manage in a transition between high and low inflation regimes.
39
f) If the low inflation is secure, the effect of policy on the
economy becomes more predictable.
Term structure can be used to make tactical policy decisions. Thus, I
need to state the below comments;
When there is a need for policy to preempt a rise in inflation, thus
inflation expectations puts a premium on the long bond rate as an indicator
of credibility for low inflation.
a) Policy leverage on long rates is regime dependent and will
vary with a central bank’s commitment to its credibility for
low inflation and price stability.
b) Policy generally follows long rates because of two reasons.
Firstly, long rates embody expectations for future short rate
policy actions and secondly long rate movements signal
changing inflation expectations that may precipitate a policy
reaction.
c) Bond market vigilantes do not make central banks irrelevant.
d) The yield curve can be used effectively to distinguish policy
actions from policy impulses for to state how much policy is
in the pipeline.
40
Lastly, I would like to state the most important part of the survey:
pitfalls in using the term structure. There are some serious pitfalls in
measuring the inflation risk in the outlook for the economy with using bond
rates. And another pitfall is to measure the degree to which a series of short-
term interest rate policy actions will be carried to the economy through
longer-term interest rates. Here, I would like to consider these situations
from Marvin Goodfriends’ paper.
In Goodfriends’ paper [15], firstly he considers the “Bond Rate
Forecasting Failures.” He states that long bond is a good indicator to
understand the central bank’s commitment to low inflation. The reason is
that the significant bond rate movements of the long bonds which attract the
central bankers. As another important reason, an ongoing inflation trend is
always reflected in higher bond rates. And we know that the term structure
does contain information for forecasting cyclical swings in inflation.
However, after these important comments, he asks this important question:
“Which bond rates actually have proven to be good forecasters of future
inflation trends?” And states that, if researchers or economists are trying
foresee the changes in the trend of inflation, bond rates have not done as
41
well. As an example, we cannot see the big jump in trend inflation which is
occurred in the late 1960s and early 1970s from the U.S. bond rates. In this
situation, bond rates did not move up as a perfect indicator.
Then, M. Goodfriend gives another important example about this
important pitfall. He considers the U.S. 30-year bond rate. This rate was
roughly in the same 8 percent range in early 1992 and 1977, and inflation
was 3 percentage points lower in 1992 than in early 1977. Suppose that a
real long-term interest rate of around 3 percent, the long-term expected rate
of inflation would have been about 5 percent in both years. Apparently,
investors perceived the 6 percent inflation rate which was temporarily high
in early 1977, and they perceived the 3 percent inflation rate in 1992 which
was temporarily low. However, he states that the five years beginning in
1977 saw the worst inflation of the period, and thus inflation has fallen by a
percentage point or more since 1992. Besides these facts, the U.S. long rate
rose to around 14 percent in the summer of 1984 seems incredible when
trend inflation since then has remained around 4 percent or less. Using these
facts, we can understand that bond rates are not very good predictors of
changes in inflation trends before.
42
He considers the second pitfall as “Policy Actions and Long Rates.”
Consider these two different situations:
Firstly, The Fed moved short-term rates up by about 3 percentage
points between the spring of 1988 and the spring of 1989, here the 30-year
bond rate increased relatively little, and as a result, the yield curve was
inverted. Again in a different time period, the Fed again moved short rates
up by 3 percentage points between February 1994 and February 1995.
However, in this case, the long rate moved up from a trough of less than 6
percent in October 1993 to peak at over 8 percent in November of 1994, and
thus the yield curve did not invert. If we consider these two policy tightening
episodes in terms of magnitude, we see that the magnitudes are similar and
not far removed in time. Additionally, inflation rose modestly in the late
1980s and then held steady at around 3 percent between 1994-1995 periods.
Thus, here we see that the behavior of the long rate differed significantly in
the two periods. As a reason, we can say that the effect of a policy tightening
on long rates has different effects because of the circumstances. These can
be underlying factors, for example: the state of the business cycle and the
nation’s commitment to low inflation.
43
After the first question, he states another one: “Even if these two
episodes can be seen as reflecting similar correlations between the bond rate
and the short rate, is there any reason to expect the correlation to be stable in
the future?” The answer for this question is no. In the low-inflation 1950 and
1969 period, long rates are varied relatively little with short rates. Then
inflation expectations were anchored securely, and the range in which the
Fed varied short rates to stabilize the economy was smaller in this period
than it was in the 1970s, ’80s, and ’90s. Here, if the Fed succeeds in getting
full credibility for low inflation in these years, short and long rates should
once again co-vary which was in the earlier period. Thus, the late 1980s and
mid-1990s can be seen as a transition period because short and long rates
continued to exhibit the kind of covariation that observed in the period of
high inflation.
He considers the third pitfall as “Direct Policy Leverage on the Long
Real Rate.” Monetary policy transmission can be viewed as running from
short-term real interest rates which are managed by central banks to the
longer-term real rates which influence aggregate demand. Goodfriend states
that there are two major pitfalls to overcome in estimating such direct policy
leverage on the long real rate. First pitfall is the difference between policy
44
actions and policy impulses. These must be distinguished. Because interest
rate policy actions would not be expected to affect to longer-term rates
much, for to measure the effect of policy on longer-term rates, one must
construct and use a sequence of policy impulses. Second pitfall, it is
reasonable to predict the effect of a nominal short rate policy impulse on the
nominal long-term rate in real terms when the current inflation is stable and
inflation expectations are well-anchored. Those conditions were only
satisfied in the 1950s and early ’60s and never completely satisfied since
then. Only when actual inflation has been well-behaved in the 1990s, the
relatively large movements in long bond rates showed that inflation
expectations were still not firmly anchored.
After these important points, he considers simple evidence on the
leverage that short rate policy actions exert on long rates. In the late 1970s,
Cook and Hahn (1989) found that a 100-basis-point increase in the Fed’s
nominal federal funds rate target increased the nominal 30-year rate by 13
basis points on average. They used a narrow day or two time window in their
calculations. Then, Goodfriend considered this situation and after two rough
calculations in his 1993 paper, he suggests a larger 25-basis-point effect on
the 30-year rate per 100-basis-point short rate policy action in 1979 and
45
1980. Here, when these policy actions were taken, if we assume that both
inflation expectations and actual inflation were relatively unchanging on
average, thus we can interpret these estimates of policy leverage in real
terms.
46
VI – References
1- Russell, Steven. Understanding the Term Structure of Interest Rates:
The Expecations Theory. Federal Reserve Bank of St. Louis Review,
July-August 1992.
2- Campbell R. Harvey. Recovering Expectations of Consumption
Growth from an Equilibrium Model of the Term Structure of Interest
Rates, University of Chicago, December 1986
3- Campbell R. Harvey. Forecasting Economic Growth with the Bond
and Stock Markets, Financial Analysts Journal September/October,
(1989): 38-45
4- Campbell R. Harvey. Predicting Business Cycle Turning Points with
the Term Structure, WWWFinance. 1995
5- Simon Benninga and Zvi Wiener. Term Structure of Interest Rates.
Mathematica in Education and Research Vol. 7 No. 2 1998
6- Merton, Robert C. (1973). An Intertemporal Capital Asset Pricing
Model. Econometrica 41 (September), pp. 867-886
7- Vasicek, Oldrich A. (1977). An Equilibrium Characterization of the
Term Structure. Journal of Financial Economics 5 (November), pp.
177-88
47
8- Cox, John C., J. E. Ingersoll, and S. A. Ross (1985). A Theory of the
Term Structure of Interest Rates. Econometrica, 53, 385-408
9- Sick, Gordon. A Theory of the Term Structure of Interest Rates by
John C. Cox, Jonathan E. Ingersoll, Jr. and Stephen A. Ross.
Haskayne, School of Business. University of Calgary.
10- Ho, T. S. Y. and S.-B Lee (1986). Term Structure Movements and
Pricing Interest Rate Contingent Claims. Journal of Finance 41, 1011-
1029
11- Hull, J. and A. White (1990). Pricing Interest-Rate-Derivative
Securities. The Review of Financial Studies 3, 573-92
12- Black, F. and P.Karasinski (1991) Bond and Option Pricing when
Short Rates are Lognormal. Financial Analysts Journal (July -
August), 52-59
13- Carl CHIARELLA and OH Kang KWON. Classes of Interest Rate
Models Under the HJM Framework. September 16,1999.
14- David Heath, Robert Jarrow, Andrew Morton. Bond Pricing and the
Term Structure of Interest Rates: A New Methodology for Contingent
Claims Valuation. Econometrica, Vol. 60, No. 1 (Jan., 1992) , pp. 77-
105.
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15- Goodfriend, Marvin. Using the Term Structure of Interest Rates for
Monetary Policy. Federal Reserve Bank of Richmond Economic
Quarterly Volume 84/3 Summer 1998
16- Poole, William. Understanding the Term Structure of Interest Rates.
President, Federal Reserve Bank of St. Louis. Speech.
