Estimating the Short Rate from the Term Structures in the Vasicek Model

The fact that long term interest rates have been higher on average than short rates in the twentieth century has often been interpreted in the term structure literature as evidence of the existence of positive or term premiums. The purpose of this paper is to point out that an average differential between long and short rates does not necessarily represent a differential between returns realized by holders of long versus short term bonds. In particular, it is shown that if short term rates are positively autocorrelated, interest rate differentials overstate differentials in realized rates of return. We conclude that liquidity or term premiums properly estimated from the Durand yield curve data are not consistent with the liquidity preference theory. LONG TERM INTEREST RATES have been higher on average than short term rates during the twentieth century. The average differential over the period 1900-1958 between long term interest rates and spot one year rates in the Durand yield curve data increases monotonically with term to about half a percentage point at forty years. The interpretation given to these differentials in the term structure literature is that they are the additional rate of return earned on average by capital invested in long term bonds. The purpose of this paper is to point out that an average differential between long and short term interest rates does not necessarily represent a differential between realized rates of return. This is because the realized increment to capital invested in a sequence of short term bonds depends on the ex post product of uncertain future spot rates. The expected value of this ex post product will in general differ from the product of expected future spot rates. We show that if short term rates are positively autocorrelated, the interest rate differentials overstate differentials in realized rates of return. We shall refer to differentials in realized rates of return as term premiums. The proper interpretation of interest rate differentials is important to the assessment of the evidence for various theories of the term structure. The liquidity preference theory of J. R. Hicks [4, pp. 144-147] predicts that a term premium will be earned by capital invested in long term bonds because their holders require compensation for risk of capital fluctuation. Forward interest rates will exceed one period spot rates on average by the amount of premium which rises monotonically with horizon. The positive and monotonic average differentials between forward and spot one year rates in the Durand data have been

In order to simplify the analysis, whether of the determination of the rate of interest, as in Chapter 10, or of the demand for assets within the context of a portfolio adjustment model, it is necessary to reduce the size of the model to manageable proportions. This is often achieved by aggregation over assets which are close substitutes, where relative variations in supply will have little impact on prices. It is exactly such issues, whether long-dated bonds are close substitutes for short bonds or whether variations in the relative supply of bonds will affect the pattern of bond prices, that lie at the heart of arguments about the determination of the term structure of interest rates. It is, perhaps, unfortunate that analysis of the term structure often begins in a certainty context since then all assets are virtually perfect substitutes by definition, though the mechanics of the relationship between present bond rates, implicit forward rates, and present and forward bond prices can be set out more easily in such circumstances.If investors were risk-neutral and held similar expectations, then the introduction of uncertainty would not alter the term structure greatly. Both assumptions are, however, implausible. It is, therefore, necessary to take account of risk aversion among investors with differing holding periods and expectations. The attempt to do so by adding a constant liquidity premium into an equation otherwise based on risk-neutral behaviour (i.e., assuming that the sole objective is to maximise the mean expected return) is unsatisfactory. The risk involved in adding a bond of differing maturity to the portfolio is not theoretically a constant but depends on several factors, such as the expected variance and covariances of the bond, the investor's holding period, etc. The analytical considerations are reviewed in Section 1.If investors exhibit risk aversion, short bonds should not be very close substitutes for long bonds; some economists, in particular Culbertson, have gone so far as to argue on such grounds that the market will be segmented. On the other hand, bonds of adjacent maturities are likely to be very close substitutes, though less so at the short end of the market. The interesting question then is how quickly, if at all, does the degree of substitution fade as the distance along the maturity spectrum increases.Such a question can be answered only by examining the data. In Section 2, a number of the earlier empirical studies, undertaken in the 1960s and the first half of the 1970s, are first surveyed. Much of the increased interest in this subject then can be traced to Meiselman's reformulation of the expectations hypothesis, in terms of systematic revisions to expected future short-term rates in the light of errors in forecasting the current short rate. Such studies usually relied on obtaining observations of implicit forward rates from calculated yield curves and the tests of their hypotheses could have been affected by the method used to construct such curves. The studies undertaken by Modigliani, with Sutch and Shiller, got away from the use of constructed yield curves by regressing actual long rates on past and present short rates. Almost all these early empirical exercises produced quite good statistical fits. There seemed at that time, circa 1975, no doubt that expectations played a major role in determining the term structure, though exactly how these expectations were formed and revised remained uncertain. More surprising, in view of the prevalence of risk aversion, was the general failure of such empirical studies to reveal any clear signs of supply factors influencing the pattern of rates, though a number of reasons can be adduced in partial explanation for this.In the course of the 1970s, the rational expectations theory was applied to the analysis of asset price determination. This theory posited that, in an efficient market, prices should incorporate all available public knowledge. This approach occasioned some more rigorous tests than had the earlier studies of the role of expectations in determining the term structure; as is discussed subsequently in Section 3, research undertaken in the late 1970s and early 1980s revealed two major features of the term structure that appeared inconsistent with the rational expectations, efficient market hypothesis. First, given the auto-correlative, time series properties of short rates, long rates appeared to under-react to changes in short rates. Second, given the historical time path of short rates (and company dividends), long rates (and share prices) seemed excessively volatile. These two findings, which appear at first glance contradictory, can be explained either if long rates (and share prices) are also influenced by other long-lasting factors (e.g., changes in the preferences and attitudes to risk of investors), or if there was good reason at times for investors to believe that the future time path of short rates (and dividends) would diverge markedly from those that did historically occur. Both these latter hypotheses may have some explanatory power. Even so, this research leaves the expectations theory with relatively little role in the determination of long term rates. We seem to know less now than we had believed that we knew in the early 1970s.

We present a pricing kernel that summarizes well the main features of the dynamics of interest rates and risk in postwar U.S. data and use it to uncover how the pricing kernel has moved with the short rate in this data. Our findings imply that standard monetary models miss an essential link between the central bank instrument and the economic activity that monetary policy is intended to affect, and thus we call for a new approach to monetary policy analysis. We sketch a new approach using an economic model based on our pricing kernel. The model incorporates the key relationships between policy and risk movements in an unconventional way: the central bank's policy changes are viewed as primarily intended to compensate for exogenous business cycle fluctuations in risk that threaten to push inflation off target. This model, while an improvement on standard models, is considered just a starting point for their revision. It leads to critical questions that researchers need to answer as they continue to revise their approach to monetary policy analysis. Copyright © 2009 by the National Bureau of Economic Research.

Background and statement of the problem: Previous studies on Australian data provide inconclusive results on expectation hypothesis. Moreover, it is argued that Australian long-term rate is influenced by US long term rate, which signals decoupling of short rate from long rate. This decoupling, if true, makes monetary policy transmission ineffective. Accordingly, the paper examines expectation hypothesis with Australian data. Research methodology and data: Daily, weekly, monthly and quarterly data on 30-day, 90-day and 180-day dealers bill and 5-year and 10-year Australian government bond rate over the period from 1976 to 2016 are sourced from Datastream. Unit root test with structural break is applied to check the stationarity of data. Threshold co-integration approach is followed to test the expectation hypothesis of interest rate. Principal component analysis is used to identify common global factors affecting long rates in developed countries. Finally, Gewek’s instantaneous feedback method is applied to examine contemporaneous movement of long rate in developed countries. Research findings: Empirical analysis shows that all interest rate series are nonstationary at level; however, stationary at their first differences, which are suitable for co-integration analysis. Threshold co-integration results show that short rate and long rate in Australia are co-integrated, that is, expectation hypothesis holds. We also find evidence of threshold effect in some cases. Using principal component and instantaneous feedback method we find that long term bond yields in developed markets are driven by some unobserved global factors and the USA long rate has no causal influence on other developed markets’ long rate. We conclude that in Australia short rate is not de-coupling from long rate and hence effectiveness of monetary policy is well maintained. Policy implications: Expectation hypothesis is essential for the effective transmission of monetary policy. The hypothesis, if does not hold, causes interruption in monetary policy transmission. We find that expectation hypothesis holds for Australian data; hence RBA’s monetary policy transmission should not foresee any interruption. Moreover, bond contains premium for expected inflation, so it can be used as an indicator of RBA’s commitment for low inflation. Besides, financial intermediaries can effectively use the link between short and long rate in setting their interest on long-term loans based on short term interest rates.

In this paper we present a number of general solutions in the application of portfolio analysis to the problem of the demand for money when the wealth-holder is able to choose intertemporally among term differentiated bonds and money balances. This is a topic which has received considerable attention over the years although until very recently there existed neither sound theoretical results nor well-developed empirical work for a demand-for-money problem in which a complete term structure is included. The bulk of the literature--both theoretical and empirical as described by Laidler [9]--has phrased the question in terms of which interest rate (a short rate or a long rate) belongs in the demand for money rather than coming to grips directly with a problem in which both figure. Recently, though, Friedman [5] has developed some theoretical results for a term structure-demand for money problem using an inventory-theoretic paradigm and has obtained a priori results on the demand for money for the slope of the term structure and the level of interest rates. Our purpose in the present paper is to examine the term structure/demand for money problem from the point of view of the mean-variance model, in order to provide a portfolio rationale for the appearance of the term structure in the demand for money and to provide a model which lends itself to the determination of comparative equilibrium results. The model also can be used as the basis of empirical work. The demand-for-money-portfolio literature we are building on is essentially that initiated by J. Tobin [10], while a general term-structure-portfolio paper was developed by G. Bierwag and M. Grove [2], and more recently by G. Bierwag [3] and J. Bernstein and D. Fisher [1]. Tobin and his successors were interested in generalizing the demand for money of J. M. Keynes [8] in such a way as to provide a rigorous rationale for the negative relation

On biases in tests of the expectations hypothesis of the term structure of interest rates

This paper presents a unifying theory for valuing contingent claims under a stochastic term of interest rates. The methodology, based on the equivalent martingale measure technique, takes as given an initial forward rate curve and a family of potential stochastic processeE for its subsequent movements. A no arbitrage condition restricts this family of processes yielding valuation formulae for interest rate sensitive contingent claims which do not explicitly depend on the market prices of risk. Examples are provided to illustrate the key results. IN RELATION TO the term of interest rates, arbitrage pricing theory has two purposes. The first, is to price all zero coupon (default free) bonds of varying maturities from a finite number of economic fundamentals, called state variables. The second, is to price all interest rate sensitive contingent claims, taking as given the prices of the zero coupon bonds. This paper presents a general theory and a unifying framework for understanding arbitrage pricing theory in this context, of which all existing arbitrage pricing models are special cases (in particular, Vasicek (1977), Brennan and Schwartz (1979), Langetieg (1980), Ball and Torous (1983), Ho and Lee (1986), Schaefer and Schwartz (1987), and Artzner and Delbaen (1988)). The primary contribution of this paper, however, is a new methodology for solving the second problem, i.e., the pricing of interest rate sensitive contingent claims given the prices of all zero coupon bonds. The methodology is new because (i) it imposes its stochastic directly on the evolution of the forward rate curve, (ii) it does not require an inversion of the term structure to eliminate the market prices of risk from contingent claim values, and (iii) it has a stochastic spot rate process with multiple stochastic factors influencing the term structure. The model can be used to consistently price (and hedge) all contingent claims (American or European) on the term structure, and it is derived from necessary and (more importantly) sufficient conditions for the absence of arbitrage. The arbitrage pricing models of Vasicek (1977), Brennan and Schwartz (1979), Langetieg (1980), and Artzner and Delbaen (1988) all require an IFormerly titled Bond Pricing and the Term Structure of Interest Rates: A New Methodology.

Recent events have highlighted the potential importance of nonlinear efiects of flscal variables (notably debt and deflcits) on interest rates: While in times when government solvency is not a concern the standard crowding-out efiects are of moderate magnitude, in times when default risk becomes an issue the interest rate efiects can become very large. This paper provides new evidence on the magnitude of these efiects. For the case when default risk is not a concern, it uses an arbitrage-free term structure model to estimate the dynamic efiects of flscal policy shocks on interest rates along the entire maturity spectrum. For the case when default risk becomes a concern (thereby violating a central assumption of the term structure model), I present evidence based on EMU government bond spread regressions on time-varying efiects of national flscal policies on spreads as well as the time-varying sensitivity of yield spreads to international risk aversion as a function of the state of flscal policy. JEL classiflcation: E6, H6.

The integration of emerging economies with developed economies has changed the behaviour of interest rates and exchange rate fluctuation. The current study tries to analyse the implication of expectation hypothesis (EH) and term structures of interest rates between India and US. Using vector auto regressive estimates, the study tries to test the dynamic interdependence of interest rates on exchange rate fluctuation. Further, the study estimates Granger causality tests and Impulse Response Functions to test the behaviour of interest rate movements for a period of nineteen years ranging from June 1996 to June 2015.The empirical results of the study show evidence in line with the existence of EH in the case of emerging market. Nevertheless, in the case of advanced economies we do not find any evidence for EH. The findings revealed that the spread between long and short rate of India is influenced by short-term interest rates and past values of Indian spread. This implies that the fluctuations in the long rate over the short rate evidenced the strong presence of EH as far as emerging economy is concerned.To the best of our knowledge, this is the first study in Indian market, which tests the role of EH in interest rate fluctuations along with exchange rate. Since majority of the studies on term structure of interest rates focus on developed markets, the present study is an attempt to test the causal relationship between developed and developing economies.

Recent empirical research shows that a reasonable characterization of federal-funds-rate targeting behavior is that the change in the target rate depends on the maturity structure of interest rates and exhibits little dependence on lagged target rates. See, for example, Cochrane and Piazzesi (2002). The result echoes the policy rule used by (1994) to rationalize the empirical failure of the `expectations hypothesis' applied to the term- structure of interest rates. That is, rather than forward rates acting as unbiased predictors of future short rates, the historical evidence suggests that the correlation between forward rates and future short rates is surprisingly low. showed that a desire by the monetary authority to adjust short rates in response to exogenous shocks to the term premiums imbedded in long rates (i.e. yield-curve smoothing), along with a desire for smoothing interest rates across time, can generate term structures that account for the puzzling regression results of Fama and Bliss (1987). also clearly pointed out that this reduced-form approach to the policy rule, although naturally forward looking, needed to be studied further in the context of other response functions such as the now standard Taylor (1993) rule. We explore both the robustness of McCallum's result to endogenous models of the term premium and also its connections to the Taylor Rule. We model the term premium endogenously using two different models in the class of affine term structure models studied in Duffie and Kan (1996): a stochastic volatility model and a stochastic price-of- risk model. We then solve for equilibrium term structures in environments in which interest rate targeting follows a rule such as the one suggested by (i.e., the McCallum Rule). We demonstrate that McCallum's original result generalizes in a natural way to this broader class of models. To understand the connection to the Taylor Rule, we then consider two structural macroeconomic models which have reduced forms that correspond to the two affine models and provide a macroeconomic interpretation of abstract state variables (as in Ang and Piazzesi (2003)). Moreover, such structural models allow us to interpret the parameters of the term-structure model in terms of the parameters governing preferences, technologies, and policy rules. We show how a monetary policy rule will manifest itself in the equilibrium asset-pricing kernel and, hence, the equilibrium term structure. We then show how this policy can be implemented with an interest-rate targeting rule. This provides us with a set of restrictions under which the Taylor and Rules are equivalent in the sense if implementing the same monetary policy. We conclude with some numerical examples that explore the quantitative link between these two models of monetary policy.

The term structure of interest rates, i.e., yield curve, has long been of interest to monetary policymakers and their advisers. The transmission of monetary is conventionally viewed as running from shortterm interest rates managed by central banks to longer-term rates that influence aggregate demand. A central bank's leverage over longer-term rates comes from fact that market determines these as average expected level of short rates over relevant horizon (abstracting from a term premium and default risk). Working in other direction, long bond rate contains a premium for expected and, thus, serves as an indicator of credibility of a central bank's commitment to low inflation.1 Different theoretical perspectives support two above-mentioned uses of term structure for monetary policy: John Hicks's (1939) expectations theory of term structure supports first, and Irving Fisher's (1896) decomposition of nominal bond rates into expected and an expected real return supports second.2 The two views are compatible in principle, although reconciling them creates difficulties of interpretation in practice. For example, does a steepening yield curve indicate a loss of confidence in central bank's commitment to low inflation, or does it indicate that markets expect tighter in form of a higher path of short rates pursued by central bank? The yield curve contains information of use to monetary policymakers, but it needs to be interpreted in light of judicious subsidiary identifying conditions, together with other data on economy. Some circumstances lend themselves to clearer interpretations than others, and there are many pitfalls. Whether or not one regards longer-term interest rates as economic indicators or as part of transmission mechanism for policy, or both, term structure plays a potentially important role in policymaking process. In spite of its complexity, term structure cannot be ignored. This article addresses some issues involved in using term structure to conduct monetary policy. I begin by discussing long bond rate as an indicator of expectations. Second, I comment on role that bond rates have played in recent U.S. monetary history. Third, I explain how information in yield curve can be used to overcome what I call policy in pipeline problem. Fourth, I review recent empirical evidence supporting two theoretical views underlying our understanding of term structure. I explain how peso problems associated with inflation scares in bond market may help to account for a serious empirical failure of expectations theory of term structure. I also discuss evidence supporting view that significant movements in long-term interest rates are largely driven by expected inflation. Finally, I point out some pitfalls of using term structure to make tactical decisions. 1. PURSUING LOW INFLATION The Chairman of Federal Reserve Board, Alan Greenspan, supports a longrun goal for price stability such that the expected rate of change of general level of prices ceases to be a factor in individual and business decisionmaking.3 The long bond rate is particularly well suited to help a central bank assess degree to which it has achieved low defined in that way. One could compare behavior of yield on a long-term nominal bond to its behavior in a past period in which was very low and public was reasonably confident that it would stay low. For instance, in United States 30-year nominal bond rate ranged from around 3 percent to a little over 4 percent from mid-1950s until mid-1960s, a period in which averaged around 1 to 2 percent, and presumably, long-term expectations were no more than that.4 One would think that if Federal Reserve (the Fed) were to achieve full credibility for low inflation, long bond rate would once again move down to 3 to 4 percent range. …

A Fresh View on the Ho-Lee Model of the Term Structure from a Stochastic Discounting Perspective

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. These place predominant emphasis on the expected values of future spot rates or holdingperiod returns. In its simplest form, the expectations hypothesis postulates that bonds are priced so that the implied forward rates are equal to the expected spot rates. Generally, this approach is characterized by 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 series of the short-term bonds, or (b) the expected rate of return over the next holding period is the same for bonds of all maturities. The liquidity preference hypothesis, advanced by Hicks [16], concurs with the importance of expected future spot rates, but places more weight on the effects of the risk preferences of market participants. It asserts that risk aversion will cause forward rates to be systematically greater than expected spot rates, usually

The Term Structure of Interest Rates in a DSGE Model

The space of probability distributions on a given sample space possesses natural geometric properties. For example, in the case of a smooth parametric family of probability distributions on the real line, the parameter space has a Riemannian structure induced by the embedding of the family into the Hilbert space of square-integrable functions, and is characterised by the Fisher-Rao metric. In the nonparametric case the relevant geometry is determined by the spherical distance function of Bhattacharyya. In the context of term structure modelling, we show that minus the derivative of the discount function with respect to the maturity date gives rise to a probability density. This follows as a consequence of the positivity of interest rates. Therefore, by mapping the density functions associated with a given family of term structures to Hilbert space, the resulting metrical geometry can be used to analyse the relationship of yield curves to one another. We show that the general arbitrage-free yield curve dynamics can be represented as a process taking values in the convex space of smooth density functions on the positive real line. It follows that the theory of interest rate dynamics can be represented by a class of processes in Hilbert space. We also derive the dynamics for the central moments associated with the distribution determined by the yield curve.