Continuous-time Term Structure Models Research Articles

Macroeconomic Implications of Term Structures of Interest Rates Under Stochastic Differential Utility with Non-Unitary EIS

This paper proposes a continuous-time term-structure model under stochastic differential utility with non-unitary elasticity of intertemporal substitution (EIS, henceforth) in a representative-agent endowment economy with mean-reverting expectations on real output growth and inflation. Using this model, we make clear structural relationships among a term structure of real and nominal interest rates, utility form and underlying economic factors (in particular, inflation expectation). Notably, we show that, if (1) the EIS is less than one, (2) the agent is comparatively more risk-averse relative to time-separable utility, (3) short-term interest rates are pro-cyclical, and (4) the rate of expected inflation is negatively correlated with the rate of real output growth and its expected rate, then a nominal yield curve can have a low instantaneous riskless rate and an upward slope.

Changing Times: The Pricing Problem in Non-Linear Models

Finding conditional moments and derivative prices is a common application in continuous-time financial economics, but these quantities are known in closed-form only for a few specific models. Recent research identifies a large class of models for which solutions to such problems have convergent power series, allowing approximation even when not known in closed-form. However, such power series may converge slowly or not at all for long time horizons, limiting their practical use. We develop the method of time transformation, in which the variable representing time is replaced by a non-linear function of itself. With appropriate choice of the time transformation, power series often converge for much longer time horizons, and also much faster, sometimes uniformly for all time horizons. For applications such as bond pricing, in which the time-to-maturity may be many years, rapid convergence is very important for practical application. The ability to approximate solutions accurately and in closed-form simplifies the estimation of non-affine continuous-time term structure models, since the bond pricing problem must be solved for many different parameter vectors during a typical estimation procedure. We show through several examples that the series are easy to derive, and, using term structure models for which bond prices are known explicitly, also show that the series are extremely accurate over a wide range of interest rate levels for arbitrarily long maturities; in some cases, they are many orders of magnitude more accurate than series constructed without time transformations. Other potential applications include pricing of callable bonds and credit derivatives.

Term Structure of Interest Rates Under Recursive Preferences in Continuous Time

This paper proposes a testable continuous-time term-structure model with recursive utility to investigate structural relationships between the real economy and the term structure of real and nominal interest rates. In a representative-agent model with recursive utility and mean-reverting expectations on real output growth and inflation, this paper shows that, if (1) real short-term interest rates are high during economic booms and (2) the agent is comparatively risk-averse (less risk-averse) relative to time-separable utility, then a real yield curve slopes down (slopes up, respectively). Additionally, for the comparatively risk-averse agent, if (3) expected inflation is negatively correlated with the real output and its expected growth, then a nominal yield curve can slope up, regardless of the slope of the real yield curve.

A joint model for the term structure of interest rates and the macroeconomy

AbstractWe present and estimate a continuous time term structure model that incorporates observable macroeconomic variables and latent variables with a clear macroeconomic interpretation. Our model is able to accurately describe the joint dynamics for US macroeconomic variables and the yield curve. However, the observable variables do not explain the long end of the term structure. Central tendencies of these macroeconomic variables do a much better job in this respect. These unobservable factors also play an important role in the description of the interest rate policy rule. Both observable and non‐observable factors determine the risk premia and hence bond excess holding returns. Copyright © 2006 John Wiley & Sons, Ltd.

Evaluating Government Bond Fund Performance with Stochastic Discount Factors

This article shows how to evaluate the performance of managed portfolios using stochastic discount factors (SDFs) from continuous-time term structure models. These models imply empirical factors that include time averages of the underlying state variables. The approach addresses a performance measurement bias, described by Goetzmann, Ingersoll, and Ivkovic (2000) and Ferson and Khang (2002), arising because fund managers may trade within the return measurement interval or hold positions in replicable options. The empirical factors contribute explanatory power in factor model regressions and reduce model pricing errors. We illustrate the approach on US government bond funds during 1986–2000.

Evaluating Fixed Income Fund Performance with Stochastic Discount Factors

We evaluate the performance of fixed income mutual funds using stochastic discount factors from continuous-time term structure models. Time-aggregation of the models for discrete returns generates additional empirical factors, and these factors contribute significant explanatory power to empirical the models. We provide the first conditional performance evaluation for US fixed income mutual funds, conditioning on a variety of discrete ex ante characterizations of the state of the term structure and the economy. During 1985-1999 fixed income funds returned less on average than passive benchmarks that don't pay expenses, but not in all economic states. Fixed income funds typically do poorly when short term interest rates or industrial capacity utilization rates are high, and offer higher returns when quality-related credit spreads are high. We find more heterogeneity across fund styles than across characteristics-based fund groups. Mortgage funds under perform a GNMA index in all economic states. These excess returns are reduced, and typically become insignificant, when we adjust for risk using the stochastic discount factors.

An Empirical Examination of Term Structure Models with Regime Shifts

We examine several continuous-time term structure models in which the short rate is subject both to continuous changes and to discrete shifts. Several regime-switching term structure models are developed, with regime-dependence in various combinations of their drift and diffusion parameters. We examine their predictive power. Our empirical analysis suggests that it is important to attempt to specify the switching model correctly: Badly parameterized switching models may not be an improvement (in terms of pricing) over models which do not allow for regime switching, even when there are clear breaks in the data.

Term Structure Estimation in Low-Frequency Transaction Markets: A Kalman Filter Approach with Incomplete Panel-Data

There are two issues that are of central importance in term structure analysis. One is the modeling and estimation of the current term structure of spot rates. The second is the modeling and estimation of the dynamics of the term structure. These two issues have been addressed independently in the literature. The methods that have been proposed assume a sufficiently complete price data set and are generally implemented separately. However, when the methods are applied to markets with sparse bond price, results are unsatisfactory. We develop a method for jointly estimating the current term structure and its dynamics for markets with low-frequency transactions. We propose solving both issues by using a dynamic term structure model estimated from incomplete panel data. To achieve this, we modify the standard Kalman filter approach to deal with the missing-observation problem. In this way, we can use historic price data in a dynamic model to estimate the current term structure. With this approach we are able to obtain an estimate of the current term structure even for days with an arbitrary low number of price observations. The proposed methodology can be applied to a broad class of continuous-time term-structure models with any number of stochastic factors. To show the implementation of the approach, we estimate a three-factor generalized-Vasicek model using Chilean government bond price data. The approach, however, may be used in any market with low-frequency transactions, a common characteristic of many emerging markets.

Comparing forecasting ability of parametric and non-parametric methods: an application with Canadian monthly interest rates

The primary objective of this article is to compare the forecasting ability of some recent parametric and non-parametric estimation methods by using monthly Canadian interest rate data between 1964:1–1999:1. The two-factor continuous time term structure model of Brennan and Schwartz was estimated where the first factor represents the short rate and the second factor the long rate using the continuous time estimation procedures developed by Bergstrom. The interest rates using the multivariate GARCH model developed by Engle and Kroner, and two non-parametric estimation methods namely, non-parametric kernel smoothing and the artificial neural networks was modelled. For the short-term rates, it has been found that, the Bergstrom's method and the artificial neural networks model have marginally better forecasting performance than that of the linear benchmark. For the long-term rates, none of the methods produced better forecasting precision than that of the benchmark.

Evaluating Government Bond Fund Performance with Stochastic Discount Factors

This paper evaluates the performance of government bond mutual funds with stochastic discount factors from continuous-time term structure models. The approach addresses the interim trading bias described by Goetzmann, Ingersoll and Ivkovic (2000) and Ferson and Khang (2002). It replaces the ad hoc selection of empirical factors and instruments with variables prescribed by theory. Time-aggregation of the models for discrete returns generates empirical factors that contribute explanatory power to the models, and may be useful in other settings. We provide the first conditional performance evaluation for US fixed income mutual funds. During 1986-2000 most government bond funds returned less on average than passive benchmarks that don't pay expenses, but not for all states of the term structure. The abnormal returns are reduced, and with a few interesting exceptions become insignificant, when we adjust for risk using the stochastic discount factors from term structure models.

Nonparametric Estimation of Multifactor Continuous Time Interest Rate Models

In this paper we study the finite sample properties of the nonparametric method developed by Stanton and later extended by Boudoukh, et al. for the estimation of the drifts and diffusions of multifactor continuous-time term-structure models. Monte Carlo simulations from a known parametric model are employed to calculate the performance of the estimator. The paper focuses on the issue of optimal bandwidth selection. The results suggest that, for persistent data-generating processes exhibiting stochastic volatility, such as interest rate data, a bandwidth function that varies over the surface of the data is optimal. The paper also presents a computationally intensive bandwidth-selection procedure that uses dynamic graphics, combining the computational power of the machine with the pattern-recognition abilities of the human brain. The Monte Carlo simulations require the numeric solution of a system of stochastic differential equations. The paper also presents a nonparametric test for the validity of the solutions. This test is useful in other estimation algorithms, such as the efficient method of moments, where numeric solutions of stochastic differential equations are required. The test is also useful as a tool for understanding how the length of the time step used in the numeric solution of the stochastic differential solutions affects the accuracy of the solution.

Forecasting U.K. and U.S. interest rates using continuous time term structure models

In this paper we compare the forecasting performance of different one factor interest rate models commonly used in the financial markets. In particular we estimate the general CKLS model and the special cases of Merton, Vasicek, CIRSR, Dothan, GBM, Brennan-Schwartz, CIRVR, and CEV models using the approach of Nowman (1997). The models are estimated using weekly Euro-currency data for the U.K. and U.S. over a range of maturities. We find the forecasting performance varies across models and markets.

Continuous-time term structure models: Forward measure approach

The problem of term structure of interest rates modelling is considered in a continuous-time framework. The emphasis is on the bond prices, forward bond prices and so-called LIBOR rates, rather than on the instantaneous continuously compounded rates as in most traditional models. Forward and spot probability measures are introduced in this general set-up. Two conditions of no-arbitrage between bonds and cash are examined. A process of savings account implied by an arbitrage-free family of bond prices is identified by means of a multiplicative decomposition of semimartingales. The uniqueness of an implied savings account is established under fairly general conditions. The notion of a family of forward processes is introduced, and the existence of an associated arbitrage-free family of bond prices is examined. A straightforward construction of a lognormal model of forward LIBOR rates, based on the backward induction, is presented.

Maximum Likelihood Estimation of Non-Linear Continuous-Time Term-Structure Models

This paper presents an exact maximum likelihood method which is designed for estimating multi-dimensional time-homogeneous diffusion processes, using discretely sampled observations. The essence of this method is to evaluate the transition density by means of simulating the diffusion processes. The advantages of this approach are that we do not have to derive the distribution for the diffusion process described by the stochastic differential equation, and that we can use all the standard results from the likelihood theory. We apply this exact maximum likelihood method to consider a class of dynamic one factor continuous-time interest rate models all nested within the CKLS-model (Chan, Karolyi, Longstaff and Sanders 1992), which will enable us to compare the different models. The general superiority of this method to existing methods (e.g. QML, GMM) is documented through Monte Carlo studies. We also show that the method can be applied with great benefit to multi-factor continuous-time term-structure models, and we look at a non-linear special case of the Brennan & Schwartz model (1982). Finally, empirical results from the U.S. market will be presented. NOTE: This is a revised version of the paper Estimation of a Dynamic One Factor Continuous-Time Term-Structure Model.

A new class of duration measures

This paper presents a definition of duration for continuous time term structure models based on proportional bond price changes for instantaneous changes in spot rates. We apply this definition to derive a duration measure for Heath et al.'s ( Econometrica, 1992, 60, 77–105) dynamic term structure model.