Term Structure Estimation Research Articles

On estimability of parsimonious term structure models: an experiment with the Nelson–Siegel specification

This study addresses operational issues in estimation of parsimonious term structure models. When using price errors, objective function in term structure estimation is a nonlinear function of the model parameters. This necessarily entails using numerical optimization techniques for estimation, which brings to fore the issue of (sensitivity of final results to) the choice of initialization of the optimization routine. This study assesses the sensitivity of the final objective function value and the final parameter vector to the choice of the ‘initial guess’ during the estimation of the popular Nelson–Siegel model. It turns out that there exist regions in the shape of the objective function where a slight change in (seemingly reasonable) initial vector takes one far from optimum. Choice of the (range of) ‘best’ starting vector turns out to be an empirical matter. Grid search is recommended. One must first get to a subset of initial values that results in the objective function value near a minimum and then assess the sensitivity of the final parameter vector to those relevant (subset of) initial values. The study illustrates the process using a typical trading day's data.

Term Structure Estimation with Survey Data on Interest Rate Forecasts

AbstractThe estimation of dynamic no-arbitrage term structure models with a flexible specification of the market price of risk is beset by severe small-sample problems arising from the highly persistent nature of interest rates. We propose using survey forecasts of a short-term interest rate as an additional input to the estimation to overcome the problem. To illustrate the methodology, we estimate the 3-factor affine-Gaussian model with U.S. Treasury yields data and demonstrate that incorporating information from survey forecasts mitigates the small-sample problem. The model thus estimated for the 1990–2003 sample generates a stable and sensible estimate of the expected path of the short rate, reproduces the well-known stylized patterns in the expectations hypothesis tests, and captures some of the short-run variations in the survey forecast of the changes in longer-term interest rates.

Price informativeness and predictability: how liquidity can help

Information asymmetry and liquidity concentration has been widely discussed in literatures. This study shows how liquidity influences not only forecasting performances of term structure estimation, but also information transmission and price adjustment across markets. Our analysis helps understanding how extreme market movements affect one another. This study examines, and provides a rationale for incorporating, liquidity in estimating term structure. Forecasting performance can be greatly enhanced when conditioning on trading liquidity. It reduces information asymmetry in the sense of Easley and O’Hara (2004) and Burlacu et al. (2007). We adopt a time series forecasting model following Diebold and Li (2006) to compare behaviour of forecasted price errors. Our findings indicate that forecasted price errors in markets with less depth would influence those with more. Information asymmetry induces volatile trading first and then price adjustment is transmitted to another market due to insufficient market depth. Cross-market price adjustment could be as much as 21 bps on average. Compared with previous studies, our results establish a valid reason to condition on liquidity when forecasting prices.

Linearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing

We propose to use the linearity-generating framework to accommodate the evidence of unspanned stochastic volatility: Variations in implied volatilities on interest-rate options such as caps and swaptions are independent of the variations on the interest rate term structure. Under this framework, bond valuation depends only on the transition dynamics of interest-rate factors, but not on their volatilities. Thus, interest-rate volatility is truly unspanned. Furthermore, this framework allows tractable pricing of options on any bond portfolios, including both caps and swaptions. This feat is not possible under existing exponential-affine or quadratic frameworks. Finally, the framework allows sequential estimation of the interest-rate term structure and the interest-rate option implied volatility surface, thus facilitating joint empirical analysis. Within this framework, we perform specification analysis on interest-rate factor transition dynamics and its relation to the interest-rate term structure; we also analyze the interest-rate volatility dynamics and its impact on interest-rate option pricing. We estimate several specifications for the transition dynamics to ten years worth of U.S. dollar LIBOR and swap rates across 15 maturities. We also estimate several interest-rate volatility dynamics specifications using ten years of swaption implied volatilities across a matrix of ten option maturities and seven swap tenors. The estimation results show that the volatility dynamics dictate the option implied volatility variation along the option maturity dimension, whereas the interest-rate transition dynamics dictate the implied volatility variation along the underlying swap maturity dimension.

The Term Structure of Implied Costs of Equity Capital

We model and estimate the term structure of implied costs of equity capital (and implied risk premia) at the rm level for the years 1996-2012, using inter alia synthetic futures prices derived from forward looking option contracts. Empirical tests reject the assumption that implied rm-level costs of equity are constant over different horizons. Instead, we find that the term structure of implied costs of equity are upward sloping and concave for most years and industries. Interestingly, we also fi nd that the term structure of implied costs of equity ‡flattened during the 2008-2009 crisis period. We validate the firm-level term structure estimates by reference to future realized equity returns. Consistent with our estimates being valid costs of equity proxies, we find a robust association between our estimates and future equity returns. Both cross-sectional and time-series asset pricing tests indicate that our time-varying implied costs of equity capital are positively and significantly associated with future equity returns in contrast to traditional static implied and factor model costs of equity which are not associated with future equity returns. We also derive a long-term implied cost of equity capital from the term structure pattern that outperforms standard implied cost of capital estimates. We further show that the term structure model, in contrast to other models, predicts risk premia for the month of earnings announcements consistent with recent literature.

A Binomial Zero Coupon Model of Expected Bond Returns Term Structure

We propose, analyze, and implement a model for the estimation of zero-coupon expected bond returns (EBR), expanding the model developed by Afik and Benninga (AB, 2010). The central idea of AB is to use the matrix of rating transitions to estimate a Markov model of the EBR. Unlike AB, this paper presents a tree like model that can be used to aggregate market data for each rating category and extract its zero-coupon term structure of EBR and its yield risk premium components, including a certainty equivalence premium which is related to the systematic risk of the CAPM. We use corporate bond transaction data from the United States and a rating agency transition matrix to extract the term structure of EBR and its components and the term structure of risk-neutral default probabilities. We then apply our model and its results to the estimation of CDS premium term structure and implement it on our US bond market data set. We believe this model can be extended and applied to a variety of research and practical purposes. A few early examples are included in our forthcoming papers on implied recovery rates, abnormal returns and premia, and market implied bond rating.

Zero-Coupon Yield Curve Estimation with the Packagetermstrc

Since zero-coupon rates are rarely directly observable, they have to be estimated from market data. In this paper we review several widely-used parametric term structure estimation methods. We propose a weighted constrained optimization procedure with analytical gradients and a globally optimal start parameter search algorithm. Moreover, we introduce the <b>R</b> package <b>termstrc</b>, which offers a wide range of functions for term structure estimation based on static and dynamic coupon bond and yield data sets. It provides extensive summary statistics and plots to compare the results of the different estimation methods. We illustrate the application of the package through practical examples using market data from European government bonds and yields.

Challenges in Empirical Estimation of Term Structure in India

Subsequent to reform measures initiated by the government during the 1990s signalling the end of the era of government-determined interest rates, the Indian bond market has moved to a phase where market forces have been driving the interest rates putting forward at the same time challenges in the estimation of term structure of interest rates. There are problems in implementation of term structure estimation models in every market, even developed ones, yet due to problems largely attributed to lack of liquidity and efficient price discovery in the Indian bond market, term structure estimation is quite a challenging task. This article is a part of a larger study focusing on estimation and prediction of term structure in Indian bond market and goes deep into issues specific to the Indian bond market related to the challenges faced by researchers and others in term structure estimation.

Term Structure Estimation and Highly Persistent Processes in a Bayesian Context

Term Structure Estimation and Highly Persistent Processes in a Bayesian Context

Bayesian inference for the hazard term structure with functional predictors using Bayesian predictive information criteria

A Bayesian method for estimation of a hazard term structure is presented in a functional data analysis framework. The hazard terms structure is designed to include the effects of changes in economic conditions, as well as trends in stock prices and accounting variables from financial statements. The hazard function contains time-varying parameters that are modelled using splines. To estimate the model parameters, a Markov-chain Monte Carlo sampling algorithm is developed. The Bayesian predictive information criterion is employed to assess the default predictive power of the estimated model. The method is then applied to a Japanese firm’s default data listed on the Japanese Stock Exchange. The results demonstrate that the proposed method performs well.

Term‐structure estimation in markets with infrequent trading

AbstractThere are two issues that are of central importance in term‐structure analysis. One is the modelling and estimation of the current term structure of spot rates. The second is the modelling 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, there are serious problems when these methods are applied to markets with sparse bond prices.We develop a method for jointly estimating the current term‐structure and its dynamics for markets with infrequent trading. 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 infrequent trading, a common characteristic of many emerging markets. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Term Structure Estimation with Survey Data on Interest Rate Forecasts

The estimation of dynamic no-arbitrage term structure models with a flexible specification of the market price of risk is beset by a severe small-sample problem arising from the highly persistent nature of interest rates. We propose using survey forecasts of a short-term interest rate as an additional input to the estimation to overcome the problem. The three-factor pure-Gaussian model thus estimated with the U.S. Treasury term structure for the 1990-2003 period generates a stable estimate of the expected path of the short rate, reproduces the well-known stylized patterns in the expectations hypothesis tests, and captures some of the short-run variations in the survey forecast of the changes in longer-term interest rates.

Flexible Term Structure Estimation: Which Method is Preferred?

We show that the recently developed non-parametric procedure for fitting the term structure of interest rates developed by Linton, Mammen, Nielsen, and Tanggaard (J Econ 105(1):185–223, 2001) overall performs notably better than the highly flexible McCulloch (J Finon 30:811–830, 1975) cubic spline and Fama and Bliss (Am Econ Rev 77:680–692, 1987) bootstrap methods. However, if interest is limited to the Treasury-bill region alone then the Fama–Bliss method demonstrates superior performance. We further show, via simulation, that using the estimated short rate from the Linton–Mammen–Nielsen–Tanggaard procedure as a proxy for the short rate has higher precision then the commonly used proxies of the one and three month Treasury-bill rates. It is demonstrated that this precision is important when using proxies to estimate the stochastic process governing the evolution of the short rate

Term Structure Estimation with Survey Data on Interest Rate Forecasts

The estimation of dynamic no-arbitrage term structure models with a flexible specification of the market price of risk is beset by a severe small-sample problem arising from the highly persistent nature of interest rates. We propose using survey forecasts of a short-term interest rate as additional input to the estimation to overcome the problem. The three-factor pure-Gaussian model thus estimated with the U.S. Treasury term structure for the 1990-2003 period generates a stable estimate of the expected path of the short rate, reproduces the well-known stylized patterns in the expectations hypothesis tests, and captures some of the short-run variations in the survey forecast of the changes in longer-term interest rates.

Currency dependence of corporate credit spreads

Many pricing and risk management models need credit spread curves as an input. In the corporate bond market the estimation of credit spread curves is not trivial. Most issuers have only too few bonds outstanding and frequently these bonds are denominated in different currencies. To ensure a sufficient number of bonds for the estimation procedure in many cases bonds in different currencies have to be used which implies that the estimation procedure has to take into account potential currency effects. Under the hypothesis of zero correlation between the default variables and the exchange rates deflated by the relevant money market accounts we show using a rather general pricing framework that credit spreads are expected to be equal across different currencies. This paper analyses these effects and presents a new model which allows to estimate a credit spread curve for a single issuer with bonds in different currencies. This new model is based on the multi-curve estimation approach which allows a parsimonious joint estimation of a risk free term structure and the credit spread curve of the issuer. We reject the hypothesis of zero correlation between credit and exchange rate risk and present empirical evidence that there are significant differences of issuer specific credit spreads across different currencies in a representative sample of international corporate bonds. Moreover, this implies that dollar related credit spread curves cannot be used without special care for pricing defaultable claims denominated in other currencies.

Term Structure Estimation in Illiquid Government Bond Markets

With increasing liquidity of the Indian sovereign debt market since 1997, it has become possible to estimate the term structure in India. However, the market is characterised by several frictions that cause individual securities to be priced differently from the ‘average’ pricing in the market. In such a scenario, traditional estimation procedures like ordinary least squares using various functional forms do not perform well. In this paper, we find that mean absolute deviation is a better estimation procedure in illiquid markets than the ordinary least square. We further discover a novel liquidity weighted objective function for parameter estimation. We model the liquidity function using the exponential and hyperbolic tangent functions and suggest the most robust model for estimating term structures in India.

Parsimonious Estimation of Credit Spreads

The traditional method of credit spread estimation based on subtracting independently estimated risk-free and risky term structures of interest rates can yield unrealistically shaped and often irregular credit spread curves. A more parsimonious joint estimation of the risk-free term structure and the credit spread might serve be a good alternative, but it is hard to decide whether a seemingly irregular shape of the credit spread curve is an economic result or is merely an artifact of the estimation model. An empirical examination of European Monetary Union government bond data shows that traditional estimation models with different functional forms produce differing irregularities in the credit spread curves, while joint estimation procedures result in well-behaved and consistent curves that are less volatile. The more parsimonious joint estimation procedures have virtually the same explanatory power as the traditional methods. A simple linear joint cubic splines specification performs surprisingly well compared to a more comprehensive non-linear model.

On the consistency of ratings and bond market yields

We study the consistency of the credit-risk orderings implicit in ratings and bond market yields. By analyzing errors in term structure estimates for bonds with particular ratings, we show that for significant periods, a quarter of some categories of high credit quality bonds are rated in a manner that is inconsistent with their pricing. Adjusting for economic determinants of spreads (tax, liquidity and risk premiums) and allowing for the dynamic adjustment of ratings and spreads largely eliminates the inconsistencies, however.

Constant Maturity Yield Estimates for India: How well do they Depict the Term Structure?

Central banks over the world report the base interest rates of their economy in terms of constant maturity yield curves. We introduce estimation of constant maturity yields for the Indian economy and through a rigorous comparison with zero-coupon yield estimates using econometric techniques, show that even in economies with underdeveloped and imperfect bond markets, it is possible to generate qualitatively accurate term structure estimates, based on secondary market yields.

Estimating the Interest Rate Term Structure of Corporate Debt With a Semiparametric Penalized Spline Model

This article provides a new methodology for estimating the term structure of corporate debt using a semiparametric penalized spline model. The method is applied to a case study of AT&T bonds. Typically, very few data are available on individual corporate bond prices, too little to find a nonparametric estimate of term structure from these bonds alone. This problem is solved by “borrowing strength” from Treasury bond data. More specifically, we combine a nonparametric model for the term structure of Treasury bonds with a parametric component for the credit spread. Our methodology generalizes the work of Fisher, Nychka, and Zervos in several ways. First, their model was developed for Treasury bonds only and cannot be applied directly to corporate bonds. Second, we more fully investigate the problem of choosing the smoothing parameter, a problem that is complicated because the forward rate is the derivative-log{D(t)}, where the discount function D is the function fit to the data. In our case study, estimation of the derivative requires substantially more smoothing than selected by generalized cross-validation (GCV). Another problem for smoothing parameter selection is possible correlation of the errors. We compare three methods of choosing the penalty parameter: generalized cross-validation (GCV), the residual spatial autocorrelation (RSA) method of Ellner and Seifu, and an extension of Ruppert's empirical bias bandwidth selection (EBBS) to splines. Third, we provide approximate sampling distributions based on asymptotics for the Treasury forward rate and the bootstrap for corporate bonds. Confidence bands and tests of interesting hypotheses, for example, about the functional form of the credit spreads, are also discussed.