Term Structure of Interest Rates

Transcription

1 Term Structure of Interest Rates Ali Umut Irturk 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

2 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. 2

3 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. 3

4 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. 4

5 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 P 0, then we can find the interest rate for time t using the simple discount formula, The interest rate r t 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. 5

6 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 6

7 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. 7

8 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, longterm 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). 8

9 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. 9

10 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. 10

11 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. 11

12 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: 12

13 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. 13

14 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 14

15 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. 15

16 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 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 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 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 Quarter 4, total GDP decline is 2.7%. 16

17 Term structure begins inversion in 1980 Quarter 4 correctly. (Predicts the recession with a four quarter advance signal.) Recession 1990 Quarter 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 17

18 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. 18

19 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. 19

20 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. 20

21 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, longterm rates have greater price variability and less marketability. If we consider the empirical facts of yield curve, the last comment holds for the 21

22 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 22

23 empirical facts is true. Because here, short-term yield is lower than longterm 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- 23

24 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: 24

25 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- 25

26 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 26

27 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 27

28 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. 28

29 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. 29

30 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. 30

31 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. 31

32 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), 32

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 33

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 34

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 35

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. 36

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. 37

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. 38

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. 39

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 shortterm 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 40

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 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 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. 41

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 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 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. 42

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 43