Term Structure Theories

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  eBook - ePub

  The Creators of Inside Money

  A New Monetary Theory

  • D. Gareth Thomas(Author)
  • 2018(Publication Date)
  • Palgrave Macmillan

    (Publisher)

  The term structure of interest rates, however, recognises a common link between them all in the form of expectations of the future, whether long or short, which determines the holding of various financial assets of maturity as well as influencing the determination of aggregate demand and supply within the real economy. The term structure, therefore, is important for Central Bank ’s policymakers. If these monetary instruments affect short-term rates of interest in the first instance, which leads to the determination of long-term rates of interest, which drives capital and consumption expenditure, then analysing the term structure is crucial for understanding the transmission mechanism of monetary policy (Fender 2012). 1 Nevertheless, it is also relevant for many households in terms of the portfolio choice of assets. Suppose a family requires expenditure on private school fees in ten years’ time and decides to save now. There are a number of options. They could save by investing into a ten-year bond. Alternatively, they could purchase a short-term bill and then take the earnings into another bond each time it matures, until the ten years are up. Clearly, the important components determining the choice will be the expected return (or cost) and the risk involved, embodied in the term structure. Therefore, the analysis must consider the various theoretical models put forward in the literature to explain the relationship between interest rates on bonds (or bills) of differing maturity, although the hypothesis can applied to other assets as diverse as housing and the mortgage rate. The foremost theory of the term structure of interest rates is the so-called expectations hypothesis, which focuses on the rôle of expectations of future short-term interest rates in the determination of prices and yields on longer-term bills (or bonds). There a number of ways in which the theory in the literature differs in terms of the length of the bills (or bonds) included in the analysis

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  Theory Of Valuation (2nd Edition)

  • Sudipto Bhattacharya, George Michael Constantinides(Authors)
  • 2005(Publication Date)
  • World Scientific

    (Publisher)

  Econometrica, Vol. 53, No. 2 (March, 1985) A THEORY OF THE TERM STRUCTURE OF INTEREST RATES' BY JOHN C. COX, JONATHAN E. INGERSOLL, JR., AND STEPHEN A. Ross This paper uses an intertemporal general equilibrium asset pricing model to study the term structure of interest rates. In this model, anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices. Many of the factors traditionally mentioned as influencing the term structure are thus included in a way which is fully consistent with maximizing behavior and rational expectations. The model leads to specific formulas for bond prices which are well suited for empirical testing. 1. INTRODUCTION THE TERM STRUCTURE of interest rates measures the relationship among the yields on default-free securities that differ only in their term to maturity. The determinants of this relationship have long been a topic of concern for economists. By offering a complete schedule of interest rates across time, the term structure embodies the market's anticipations of future events. An 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. In a world of certainty, equilibrium forward rates must coincide with future spot rates, but when uncertainty about future rates is introduced the analysis becomes much more complex. By and large, previous theories of the term structure have taken the certainty model as their starting point and have proceeded by examining stochastic generalizations of the certainty equilibrium relationships. The literature in the area is voluminous, and a comprehensive survey would warrant a paper in itself. It is common, however, to identify much of the previous work in the area as belonging to one of four strands of thought. First, there are various versions of the expectations hypothesis.

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  Fixed Income Analysis

  • Barbara S. Petitt(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  Solution to 5: A is correct. Although the local expectations theory predicts that the short-run return for all bonds will be equal to the risk-free rate, most of the evidence refutes that claim. Returns from long-dated bonds are generally higher than those from short-dated bonds, even over relatively short investment horizons. This market evidence is consistent with the risk–expected return trade-off that is central to finance and the uncertainty surrounding future spot rates.

  5. Modern Term Structure Models

  Modern term structure models provide quantitatively precise descriptions of how interest rates evolve. A model provides a sometimes simplified description of a real-world phenomenon on the basis of a set of assumptions; models are often used to solve particular problems. These assumptions cannot be completely accurate in depicting the real world, but instead, the assumptions are made to explain real-world phenomena sufficiently well to solve the problem at hand.

  Interest rate models attempt to capture the statistical properties of interest rate movements. The detailed description of these models depends on mathematical and statistical knowledge well outside the scope of the investment generalist’s technical preparation. Yet, these models are very important in the valuation of complex fixed-income instruments and bond derivatives. Thus, we provide a broad overview of these models in this chapter. Equations for the models and worked examples are given for readers who are interested.

  5.1. Equilibrium Term Structure Models

  Equilibrium term structure models are models that seek to describe the dynamics of the term structure using fundamental economic variables that are assumed to affect interest rates. In the modeling process, restrictions are imposed that allow for the derivation of equilibrium prices for bonds and interest rate options. These models require the specification of a drift term (explained later) and the assumption of a functional form for interest rate volatility. The best-known equilibrium models are the

  Cox–Ingersoll–Ross model

  10 and the

  Vasicek model

  ,11 which are discussed in the next two sections.

  Equilibrium term structure models share several characteristics:

  • They are one-factor or multifactor models. One-factor models assume that a single observable factor (sometimes called a state variable) drives all yield curve movements. Both the Vasicek and CIR models assume a single factor, the short-term interest rate, r

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  The Economics of Financial Markets

  • Roy E. Bailey(Author)
  • 2005(Publication Date)
  • Cambridge University Press

    (Publisher)

  See the introduction to volume I of Ross (2000) and the references cited there. 328 The economics of financial markets opportunities, and (b) if the model of the term structure were correct (i.e. an acceptably close approximation to the actual evolution of bond prices). The predictions can then be compared with realized bond prices in order to inform investment strategies (i.e. to provide signals about which bonds to buy or sell) or to provide price quotations in negotiations about OTC contracts. Alternatively, from a more disinterested academic perspective, the predictions can be tested using methods similar to those described in chapter 9. Here the goal is to identify which theories of economic behaviour are more (or less) consistent with the observed patterns of bond prices and their rates of return. 13.7 Summary 1. Studies of the term structure of interest rates seek to reveal the relationship among the yields on bonds with different times to maturity. Commonly, the term structure is expressed by a yield curve, which plots yields as a function of the number of years remaining before redemption of the bonds. 2. Because the time to maturity is not the only dimension across which bonds differ, it is necessary to control for other characteristics of bonds, in particular their coupon payments. The most straightforward approach is to construct yield curves for zero-coupon bonds (bonds that promise to make a single payment at maturity). 3. Another attribute of some bonds is that the promised payments are adjusted for changes in the price level. From the observed prices (yields) of these index-linked bonds, it is possible to estimate real yield curves that express yields to maturity adjusted for future price level changes as a function of time to maturity. It then becomes possible to extract estimates of expected inflation rates, for various periods in the future, from market prices (yields) for real and nominal bonds.

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  The Theory of Interest

  • Friedrich A. Lutz, Friedrich Lutz(Authors)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  Part FourThe Term Structure of Interest Rates

  Our presentation of the various theories of the term structure of interest rates will be organized not by author but, rather, according to the type of the theory. We begin with the best known one according to which the long-term interest rate is an average of the expected short-term rates. By bringing in the required qualifications, we can subsume under this theory a very large part of the contributions to the problem of the term structure of interest rates. In Chapter 18 we shall deal with two further attempts to construct theories of the rate structure, and in Chapter 19 we shall consider the question of the “basic rate”, – i.e., which of the many interest rates the theory of interest should primarily determine so as to give the whole structure a peg. We shall omit a discussion of the valuable attempts to verify some of the theories, even though we have followed them with much interest and were pleased to see that one author believes he is able to refute the most widely accepted theory by appealing to the empirical evidence1 , while another thinks that the facts support it2 and a third claims that the same empirical material that was used by the first supports rather than refutes this theory.3

  1 J. M. Culbertson, ‘The Term Structure of Interest Rates’, Quarterly Journal of Economics, 71 (1957) 485-517.

  2 D. Meiselman, The Term Structure of Interest Rates (Englewood Cliffs, N. J., 1962).

  3 J. B. Michaelsen, The Term Structure of Interest Rates: Comment’, Quarterly Journal of Economics, 77 (1963) 166-174.

  Passage contains an image

  CHAPTER 17The Expectation Theory of the Term Structure of Interest Rates

  Already Irving Fisher, as we saw, viewed the long-term rate as an average of the short-term rates which are expected to prevail during the period for which a long-term loan is granted. He was the first to develop what we here call the “expectation theory” of the term structure of interest rates.4 In expounding this theory, let us first assume perfect foresight and neglect the costs which arise if an investor shifts from short maturities to longer ones or vice versa. In this case competition will equalize the yield on all investments undertaken for equal periods, regardless of what form they take – i.e., whether we are dealing with a short-term investment repeated several times, or with a single long-term investment, or with several short-term investments succeeded by a longer-term one, and so on. If the condition of equal yields on all possible configurations of investments over the same time span is not fulfilled, profitable arbitrage operations can and will be carried out with the result that yields will be equalized. The sum into which a dollar grows if invested for n years at the long-term rate

  Rn

  must then be equal to the sum into which it would grow if it were invested several times in succession at the short-term rates r 1 , r 2 ... r n

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  Handbook of Economic Expectations

  • Ruediger Bachmann, Giorgio Topa, Wilbert van der Klaauw(Authors)
  • 2022(Publication Date)
  • Academic Press

    (Publisher)

  9 All three estimates share a similar broad pattern – a fairly stable natural rate around 2% up until the financial crisis followed by a marked decline.

  17.3 Expectations and the term structure of interest rates

  Monetary policy affects the aggregate economy primarily via the term structure of interest rates. While central banks have tight control over short-term rates, the efficacy of monetary policy depends on the ability to affect longer-maturity interest rates which drive the saving and investment decisions of households and firms. Standard macroeconomic models assume that the transmission mechanism of monetary policy is given by the expectations hypothesis: yields on longer-term government bonds reflect the average short rate that investors expect to prevail over the life of the bond.

  We now use the fitted term structure of expectations to evaluate how well movements in future short rate expectations explain long-term yields. We do this by decomposing observed government bond yields into an expectations hypothesis component and a residual component which we interpret as a measure of the subjective term premium perceived by professional forecasters. Because our survey-based term premiums represent the residual between yields and expected short rates, we can remain agnostic about what specifically they represent. For example, they might reflect shifts in investor risk attitudes leading to time variation in expected excess bond returns, differences between the expectations of the marginal investor and consensus expectations, or frictions in the bond market which prevent the elimination of arbitrage opportunities. Hereafter we will refer to this measure as the “term premium” for simplicity.

  17.3.1 Decomposing the term structure of interest rates

  We obtain zero coupon U.S. Treasury yields from the Gurkaynak et al. (2007) dataset available on the Board of Governors of the Federal Reserve's research data page.10 The sample period is March 1983–December 2019. Let

  y t

  ( n )

  be the continuously compounded yield on an n-month discount bond and

  i t

  the risk-free nominal short rate at time t. To separate longer-term from short-term expectations, we conduct our analyses in terms of forward rates, defined as the current yield of an n-month bond maturing in

  n + m

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  Pricing and Hedging Interest and Credit Risk Sensitive Instruments

  • Frank Skinner(Author)
  • 2004(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  The end result of this process is inconclusive. However, some recent progress has been made concerning extracting information contained in the term structure of interest rates and using this information to make forecasts of future economic conditions. This work is mildly support-ive of the expectations hypothesis implying that while there may be risk premiums embedded in the sovereign term structure of interest rates, it appears that at least in part the term structure of interest rates is set in accordance with investors’ expectations concerning future possible economic conditions. More interestingly, it looks as though we can extract this information and use it to make forecasts. Fama (1984), Mishkin (1988) and Hardouvelis (1988) all find that forward rates can predict the future direction of short-term interest rates. In other words, while the expectations theory may not be able to predict the level of future rates of interest, there appears to be some validity to the expectations hypothesis since changes in the forward rate appear to have some forecasting ability to predict the direction in future rates of interest. Papageorgiou and Skinner (2002) build upon this work by showing that changes in the forward rate are able to predict directional changes in interest rate one month ahead with rather more than 60% success. Estrella and Hardouvelis (1991) find that increases in the slope of the term structure foreshadow improvements in real economic activ-ity while Estrella and Mishkin (1998) find that decreases in the slope of the term structure can indicate an increased likelihood of future recessions. Also, Estrella and Mishkin (1997) find that increases in the slope of the term structure are associated with increases in inflation as long as five years ahead. The idea here is that the monetary authority (e.g. European Cent-ral Bank in the Eurozone, the Fed in the US) attempts to adjust interest rates to control inflation and maintain sustainable economic growth.

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  Fixed Income Securities

  Valuation, Risk, and Risk Management

  • Pietro Veronesi(Author)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  18.3.3 Conclusion The model presented in this section is simple, but it does provide some economic intuition to the term structure models that we have seen in previous chapters, and that we will see in future chapters as well. This model has numerous drawbacks, including the fact that it does not fit the yield curves very well. For instance, while Panel B of Figure 18.5 shows the term spread for various levels of risk aversions, and the spread appears reasonable, the yield curves corresponding to this panel are not reasonables: For h = 70 we obtain too low (in fact negative) interest rates, while for h = 140 we obtain way too high yields. A smaller variation in h produces reasonable yield curves, but not enough variation in term spreads, and thus on the market price of risk. The problem with this model is its simplicity, in fact, and indeed this model can be generalized in multiple ways to provide both more reasonable yield curves as well as more interesting dynamics for bond prices and the market price of risk. For instance, we assumed that the expected GDP growth g is constant, which it isn’t in the data. Similarly, the level of risk aversion h may be assumed time varying, which induces additional interest dynamics: For instance, we know that a high h implies a higher term spread and a higher market price of risk. This implies that a time variation in h may lead to the empirical observation that a higher slope of the term structure implies a higher expected return, as documented in Section 7.3 in Chapter 7. But the model can be useful to interpret the yield curve. For instance, according to the model, if we observe a yield curve that is relatively steep, this can be due to any of the following three reasons: 1. Market participants expected high future inflation, and thus high future spot rates.

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  Fixed Income Analysis

  • (Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  Chapter 7 The Term Structure and Interest Rate Dynamics 359 they interpreted as level, steepness, and curvature. The level movement refers to an upward or downward shift in the yield curve. The steepness movement refers to a non-parallel shift in the yield curve when either short-term rates change more than long-term rates or long-term rates change more than short-term rates. The curvature movement is a reference to movement in three segments of the yield curve: The short-term and long-term segments rise while the middle-term segment falls, or vice versa. Exhibit 11 illustrates these factors. EXHIBIT 11 Primary Yield Curve Factors: Level, Slope, and Curvature Yield (%) Term Term Slope Curvature Level In practice, the level movement factor explains most of the total changes in swap and bond market yields. This factor may be interpreted as a reflection of parallel yield curve moves in which rates move in the same direction and by a similar order of magnitude. The steepness factor addresses the shape of the curve, with short-term yields typically moving more than long-term yields. These changes take place over time and therefore explain less of the total variance in rates than the level factor. Finally, the third factor, curvature, tends to have a nega- tive impact on intermediate yields and a positive impact on short- and long-term yields. This variable explaining the “twist” in the yield curve has the smallest impact of the three. 8. THE MATURITY STRUCTURE OF YIELD CURVE VOLATILITIES AND MANAGING YIELD CURVE RISKS • explain the maturity structure of yield volatilities and their effect on price volatility 8.1. Yield Volatility Quantifying interest rate volatilities is important for fixed income managers for at least two reasons. First, most fixed-income instruments and derivatives have embedded options. Option values, and hence the values of the fixed-income instrument, crucially depend on the level of interest rate volatilities.

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  Statistics for Finance

  • Erik Lindström, Henrik Madsen, Jan Nygaard Nielsen(Authors)
  • 2018(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Here the statistical term estimation of the term structure will be used throughout. In Section 14.11.3, the Extended Kalman filtering technique from Chapter 14 will be used, where the spot interest rate model describes the underlying process and the solution of the bond pricing equation is the measurement equation. Thus the method enables us to estimate both parameters and implied interest rates directly from observed bond prices. 11.6.1 Polynomial methods In practice it is fairly common to assume that the yield curve may be approximated by a polynomial in T of order s, i.e., Y (T) = α − 1 + α 1 T + α 2 T 2 + … + α s T s. ⁢ (11.152) This is a reasonably general formulation. A number of estimation methods exist and have been implemented in statistical packages, which we shall not discuss here. A program package called RIO, which is based on cubic splines, has been developed at the Aarhus School of Business. This package is also used today in a number of financial institutions, because it allows the modeller to split the term structure into a number of segments. The package can also calculate other types of information. 11.6.2 Decay functions Decay functions are very useful for term structure estimation if the term structure should converge to a constant interest rate for T → ∞. The simplest possible decay function is Y (T) = α 0 + α 1 exp (− α 2 T). ⁢ (11.153) In the next section, we discuss in some detail an extension of this model. 11.6.3 Nelson-Siegel method The relation between the price of a zero-coupon bond, P (t, T), and the instantaneous forward rate, F (t, T), is given by (11.109), i.e., P (t, T) = exp (− ∫ t T f (t, s) d s). ⁢ (11.154) The yield curve follows from (11.13),. i.e., Y (t, T) = − log [ P (t, T) ] T − t = 1 T − t ∫ t T f (t, s) d s. ⁢ (11.155) Today, at t = 0, we may observe the yield of a number of bonds with different maturities

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