A Classification of Two-Factor Affine Diffusion Term Structure Models

Dai and Singleton (2000) introduced a typology of affine diffusionmodels when the domain of admissible values of the factors isan intersection of half planes and under some additional constraintson the parameters. This condition on the domain and the additionalsufficient constraints are restrictive and can considerablydiminish the practical interest of affine models. In this articlewe successfully address the research agenda sketched by Duffie,Filipovic, Schachermayer (2003, section 12.2, p. 1042). A systematicinvestigation is performed and our article provides a completetypology in the two-factor case, without prior restrictionson the domain and on the parameters.     

Beyond Single-Factor Affine Term Structure Models

This article proposes a new approach to testing for the hypothesisof a single priced risk factor driving the term structure ofinterest rates. The method does not rely on any parametric specificationof the state variable dynamics or the market price of risk.It simply exploits the constraint imposed by the no-arbitragecondition on instantaneous expected bond returns. In order toachieve our goal, we develop a Kolmogorov-Smirnov test and applyit to data on Treasury bills and bonds for both the United Statesand Spain. We find that the single risk factor hypothesis cannotbe rejected for either dataset.     

A Two-Factor Model for Low Interest Rate Regimes

This paper derives a two-factor model for the term structure of interest rates that segments the yield curve in a natural way. The first factor involves modelling a non-negative short rate process that primarily determines the early part of the yield curve and is obtained as a truncated Gaussian short rate. The second factor mainly influences the later part of the yield curve via the market index. The market index proxies the growth optimal portfolio (GOP) and is modelled as a squared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, this study focuses on the difficult but important case where the short rate stays close to zero for a prolonged period of time. For the proposed model, an equivalent risk neutral martingale measure is neither possible nor required. Hence we use the benchmark approach where the GOP is chosen as numeraire. Fair derivative prices are then calculated via conditional expectations under the real world probability measure. Using this methodology we derive pricing functions for zero coupon bonds and options on zero coupon bonds. The proposed model naturally generates yield curve shapes commonly observed in the market. More importantly, the model replicates the key features of the interest rate cap market for economies with low interest rate regimes. In particular, the implied volatility term structure displays a consistent downward slope from extremely high levels of volatility together with a distinct negative skew. 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: G10, G13     

Valuing Fixed-Income Options and Mortgage-Backed Securities with Alternative Term Structure Models

To value mortgage-backed securities and options on fixed-income securities, it is necessary to make assumptions regarding the term structure of interest rates. We assume that the multi-factor fixed parameter term structure model accurately represents the actual term structure of interest rates, and that the values of mortgage-backed securities and discount bond options derived from such a term structure model are correct. Differences in the prices of interest rate derivative securities based on single-factor term structure models are therefore due to pricing bias resulting from the term structure model. The price biases that result from the use of single-factor models are compared and attributed to differences in the underlying models and implications for the selection of alternative term structure models are considered.     

基于Nelson—Siegel模型的我国利率期限结构的构建

利率问题一直都是经济金融研究中最基础、最核心的问题。利率可以反映出资金的供求状况,并受到物价水平、经济周期和预期等的影响。本文基于中国银行间债券市场的交易数据,利用基于贝叶斯推断的马尔科夫链蒙特卡罗模拟（MCMC）方法估计Hautsch＆Ou（2008）提出的动态的Nelson—Siegel模型,以构建我国的利率期限结构模型。     

A Binomial Basis for the Cox, Ingersoll and Ross Model of the Term Structure of Interest Rates

The technical demands of the Cox, Ingersoll and Ross (1985a and 1985b) papers are such that they can only be mastered by those who have a good understanding of some deep mathematics and statistical concepts, including the techniques of continuous time stochastic calculus and the measure theory upon which it is based, the Kuhn-Tucker theory surrounding non-linear optimisation techniques as well as variational methods founded on solutions of non-linear differential equations. Hence, our purpose here is to formalise both investor preferences and the supply side which underscores the Cox, Ingersoll and Ross (1985b) 'square root' model of the term structure of interest rates in terms of some simple binomial filtration processes, thereby avoiding most of the intricate technical detail contained in the original papers. These procedures not only allow for a more focused evaluation of the model's underlying strengths and weaknesses but also provide a framework for assessing some of the strategies which the model makes available for hedging exposure against adverse interest rate movements.     

Are Indexed Bonds Really Inflation Proof? A Model of Real and Nominal Term Structures when Money has Real Effects

This paper studies the general behavior of the nominal and real term structures of interest rates in a general equilibrium framework. A central bank is introduced in the model as an agent facing a tradeoff between inflation and output and choosing a monetary policy variable. Prices and output are jointly determined in our model endogenously. Two multi-factor nominal and real term structure models are given as examples to illustrate the general model. In our economies, inflation indexed bonds are not completely inflation proof, but are still subject to the influence of inflation uncertainties. The models offer us an empirical framework that can be studied with indexed bond data and nominal bond data together in a single estimation.     

最优多因子仿射利率期限结构模型选择

在利用NS模型估计出市场即期利率的基础上,采用卡尔曼滤波方法对多因子Vasieck和CIR模型进行参数估计,最后运用蒙特卡罗模拟方法对交易所国债价格进行模拟,并与实际价格进行比较,进而确定了符合我们国债市场的最优多因子仿射利率期限结构模型。研究结果表明：多因子CIR模型对数据的拟合效果及对国债价格模拟效果要明显优于多因子Vasicek模型;对于多因子CIR模型而言,因子个数增加并没有提高模型的价格模拟效果;两因子CIR模型具有最优的国债价格模拟效果。     

Gaussian Estimation and Forecasting of Multi-Factor Term Structure Models with an Application to Japan and the United Kingdom

In this paper we extend the exact discrete model of Bergstrom (1966) first used in empirical finance by Brennan and Schwartz (1979) to estimate their two-factor term structure model to estimate other two-factor term structure models using the recent assumption in Nowman (1997) for single factor models. Following Nowman (1997) we use the exact Gaussian estimation methods of Bergstrom (1983–1986, 1990) to estimate two-factor CKLS, Vasicek and CIR models. We estimate the models using monthly UK and Japanese interest rate data and our results indicate that the estimation method works well in practice.     

利率期限结构理论研究综述

本文主要对利率期限结构的理论研究做综述,以20世纪70年代初和90年代末为分界线,70年代以前称为传统的利率期限结构,主要以描述性研究为主;70年代以后称为现代利率期限结构,主要以随机模型研究为主;从20世纪90年代末,开始了两极分化发展。本文分为三个部分：第一部分对20世纪70年代之前传统利率期限结构的描述性理论作了概括;第二部分是现代利率期限结构的定量模型,包括均衡模型和无套利模型;第三部分则主要介绍20世纪90年代末以来的一些最新研究进展,包括市场模型和宏观金融模型等。     

The Term Structure of Interest Rates: Bounded or Falling?

This short paper resolves an apparent contradiction between Feldman's (1989) and Riedel's (2000) equilibrium models of the term structure of interest rates under incomplete information. Feldman (1989) showed that in an incomplete information version of Cox, Ingersoll, and Ross (1985), where the stochastic productivity factors are unobservable, equilibrium term structures are ``interior' and bounded. Interestingly, Riedel (2000) showed that an incomplete information version of Lucas (1978), with an unobservable constant growth rate, induces a ``corner' unbounded equilibrium term structure: it decreases to negative infinity. This paper defines constant and stochastic asymptotic moments, clarifies the apparent conflict between Feldman's and Riedel's equilibria, and discusses implications. Because productivity and growth rates are not directly observable in the real world, the question we answer is of particular relevance.     

Multi-Factor Cox-Ingersoll-Ross Models of the Term Structure: Estimates and Tests from a Kalman Filter Model

This paper presents a method for estimating multi-factor versions of the Cox-Ingersoll-Ross (1985b) model of the term structure of interest rates. The fixed parameters in one, two, and three factor models are estimated by applying an approximate maximum likelihood estimator in a state-space model using data for the U.S. treasury market. A nonlinear Kalman filter is used to estimate the unobservable factors. Multi-factor models are necessary to characterize the changing shape of the yield curve over time, and the statistical tests support the case for two and three factor models. A three factor model would be able to incorporate random variation in short term interest rates, long term rates, and interest rate volatility.     

Estimating and Testing Exponential-Affine Term Structure Models by Kalman Filter

This paper proposes a unified state-space formulation for parameter estimation of exponential-affine term structure models. The proposed method uses an approximate linear Kalman filter which only requires specifying the conditional mean and variance of the system in an approximate sense. The method allows for measurement errors in the observed yields to maturity, and can simultaneously deal with many yields on bonds with different maturities. An empirical analysis of two special cases of this general class of model is carried out: the Gaussian case (Vasicek 1977) and the non-Gaussian case (Cox Ingersoll and Ross 1985 and Chen and Scott 1992). Our test results indicate a strong rejection of these two cases. A Monte Carlo study indicates that the procedure is reliable for moderate sample sizes.     

Design and Estimation of Quadratic Term Structure Models

We consider the design and estimation of quadratic term structuremodels. We start with a list of stylized facts on interest ratesand interest rate derivatives, classified into three layers: (1)general statistical properties, (2) forecasting relations, and (3)conditional dynamics. We then investigate the implications of eachlayer of property on model design and strive to establish amapping between evidence and model structures. We calibrate atwo-factor model that approximates these three layers ofproperties well, and show that a flexible specification for themarket price of risk is important in capturing the stylizedevidence in forecasting relations while factor interactions areindispensable in generating the hump-shaped dynamics of bondyields.     

Forecasting the Term Structure of South African Government Bond Yields

This article uses the parsimonious dynamic Nelson–Siegel model to fit the yields of South African government bonds. We find that the dynamic Nelson–Siegel model has good fitting abilities for all maturities. We further forecast the term structure by seven different dynamic Nelson–Siegel models with time series models. We find that the DNS–VAR–GARCH model is useful for forecasting the short-term rates, the DNS–VAR best predicts the medium-term rates, and the DNS–RW best predicts the long-term rates. In addition, the dynamic Nelson–Siegel models provide better forecasts of yield data than a random walk model, especially for the 12-month forecasting horizons.     

The Term Structure of Implicit Discount Rates in Security Valuation

Abstract:   A reformulation of the residual income model is used to generate estimates of discount rates implicit in UK security prices. The terminal value of the infinite valuation model is incorporated into the coefficient on current earnings. By varying the length of the forecast horizon, different combinations of implicit discount rates are revealed that allow the estimation of time‐variant costs of equity. Results indicate no specific pattern of discount rates, thus revealing neither myopia on short‐term earnings nor excessive optimism on long(er)‐term earnings. Surprisingly, there is weak evidence that if any myopia exists, it is concentrated in larger and lower price‐earnings firms.     

A New Stochastic Duration Based on the Vasicek and CIR Term Structure Theories

The stochastic duration based on the Vasicek and CIR models is theoretically superior to Macaulay's duration. However, empirical tests on bond immunization performance have so far failed to show its superiority. Within the one-factor framework, this paper proposes to use a longer zero-curve yield instead of the original instantaneous interest rate as a proxy for the relevant risk source(s). We prove that the new duration becomes larger, increasing with bond maturity, than the original duration. Bond immunization using Belgian data shows that the new duration definitely beats the original duration and can in some cases outperform Macaulay's duration.     

A Model of the Convenience Yields in On-the-Run Treasuries

The convenience yield differential between on- and off-the-run Treasury securities with identical maturities has two components. A non-cyclical component may arise due to the higher illiquidity of off-the-run bonds. Also, trading in the market for the next issue often causes cyclical shortages of the on-the-runs. When this occurs, owners of the on-the-run bond can earn riskless profits by borrowing at a special repo rate while lending at the prevailing risk free market rate. This second component of the convenience yield, induced by the auction, is cyclical. We first show that special repo rates and the convenience yield are jointly cyclical over the auction cycle. The patterns are statistically significant and pervasive. Repo specials are highest around the announcement day and disappear by the issue day. The off- minus on-the-run yield spread is highest at the beginning of the cycle and collapses near its end, consistent with a decreasing present value of profits over a decreasing horizon. Second, we develop a first no-arbitrage continuous-time model, with both interest and special repo rates stochastic, that prices the on-the-run bonds that command this convenience yield. A simple implementation of the model can generate yields consistent with the evidence.     

中国金融市场通胀预期——基于利率期限结构的量度

通胀预期量度在以通胀预期为导向的货币政策中的意义重大。本文利用卡尔曼滤波法将离散时间两因子无套利广义高斯仿射模型运用于我国银行间债券市场,第一次从中国国债收益率曲线中分解出金融市场的中长期通胀预期L。将L与居民通胀预期和经济学家通胀预期比较,发现从事前看,L优于经济学家通胀预期,稍逊于居民通胀预期;从事后看,L优于居民通胀预期,稍逊于经济学家通胀预期。综合看,L作为金融市场形成的、高频的、反映中长期通胀的预期指数,对货币政策制定具有现实的参考意义。