Zero-coupon Rates Research Articles

Model-free nonparametric bounds for zero-coupon interest rates in bond markets without the no arbitrage principle

ABSTRACT Real-world market quotes of bonds can sometimes contradict the no arbitrage principle within the discounted cash flow model. This makes some kinds of no-arbitrage-based analyses impossible, e.g., inferring the bounds for the term structure of interest rates. Instead of using a more sophisticated model to account for pricing peculiarities, we propose a replacement for the no arbitrage principle. While being theoretically plausible, it does not get contradicted by the data – thus, we can infer the same kind of term structure bounds from it, even given the contradictions with the no arbitrage principle. Using a dataset of Russian sovereign bond quotes, we show that the no arbitrage inconsistency problem arises in 10–20% of cases and that the proposed approach successfully overcomes it.

Fuel up with OATmeals! The case of the French nominal yield curve

We construct the French nominal yield curve using Svensson33 methodology and all available public data of French nominal government debt securities—Obligations Assimilables du Trésor (OATs). Our sample period starts in October 1987 and ends in April 2018. We find that the functioning of the French sovereign bond market has improved dramatically following the onset of the euro area and has been functioning reasonably well since then, with the exceptions of the Global Financial Crisis period and the European sovereign crisis period. We also find that, the French nominal on-the-run securities have, on average, a negligible liquidity premium, in sharp contrast to the U.S. nominal Treasury market, where such a premium is sizable. On average, the level and the slope of the French zero-coupon rates have been decreasing since the Global Financial Crisis.

GENERAL ANALYSIS OF LONG-TERM INTEREST RATES

We introduce here the idea of a long-term swap rate, characterized as the fair rate of an overnight indexed swap (OIS) with infinitely many exchanges. Furthermore, we analyze the relationship between the long-term swap rate, the long-term yield, (F. Biagini, A. Gnoatto & M. Härtel (2018) Affine HJM Framework on [Formula: see text] and long-term yield, Applied Mathematics and Optimization 77 (3), 405–441, F. Biagini & M. Härtel (2014) Behavior of long-term yields in a lévy term structure, International Journal of Theoretical and Applied Finance 17 (3), 1–24, N. El Karoui, A. Frachot & H. Geman (1997) A note on the behavior of long zero coupon rates in a no arbitrage framework. Working Paper. Available at Researchgate: https://www.researchgate.net/publication/5066730) , and the long-term simple rate (D. C. Brody & L. P. Hughston (2016) Social discounting and the long rate of interest, Mathematical Finance 28 (1), 306–334) as long-term discounting rate. Finally, we investigate the existence of these long-term rates in two-term structure methodologies, the Flesaker–Hughston model and the linear-rational model. A numerical example illustrates how our results can be used to estimate the nonoptional component of a CoCo bond.

Investigating the expectation hypothesis and the risk premium dynamics: new evidence for Brazil

We re-examine the validity of the expectation hypothesis (EH) of the term structure for the Brazilian fixed income market, using data from January 2000 to June 2017. Furthermore, we investigated the out-of-sample predictability of bond excess returns by means of common factors extracted from a cross-section of Brazilian macro-variables and zero-coupon interest rates. The EH is rejected throughout the term structure examined on the basis of the statistical tests across the entire maturity spectrum considered. Our results confirm previous findings, mostly obtained for developed markets, that a linear combination of forward rates and macroeconomic factors can explain a substantial portion of movements in bonds excess returns, contributing novel and up-to-date evidence from a large and dynamic emerging bond market, such as Brazil. Furthermore, we find that the factor extracted from a large panel of macroeconomic variables generates significant gains in forecasting bond excess returns relative to yield curve information.

Zero-coupon interest rates: Evaluating three alternative datasets

The zero-coupon yield curve is a common input for most financial purposes. We consider three popular yield curve datasets and explore the extent to which the decision as to what dataset to use for a particular application may have an impact on the results. Many term structure papers evaluate alternative models for estimating zero coupon bonds based on their ability to replicate bond prices. However, in this paper we take a step forward by analyzing the consequences of using these alternative datasets in estimates of other moments and variables such as interest rate volatilities or the resulting forward rates and their correlations. After finding significant differences, we also explore the existence of volatility spillover effects among these three datasets. Finally, we illustrate the relevance of the choice of one particular dataset by examining the differences that may arise when testing the expectations hypothesis. In the conclusions, we provide guidance to end users in selecting a particular dataset.

Risk Framework Analysis in the Management of Sovereign Debt: The Argentine Case

The main objective of this paper is to develop a practical approach to Argentina’s sovereign risk management. Through Contingent Claim Analysis (CCA), Gape, Gray, Lim and Xiao (2008)[1] developed a sovereign risk framework whereby we can construct a marked to market sovereign balance sheet and obtain a set of credit risk indicators that can help policy-makers: set thresholds for foreign reserves, design risk mitigation strategies and select best policy options. The main contribution is that instead of using a conventional index such as GBI-EM1 in order to estimate the volatility of domestic currency liabilities, we use 24 sovereign domestic currency bonds to construct an interest rate covariance matrix. That is, an interest rate sensitive sovereign portfolio, whose risk factor variations2 are represented by a vector of the portfolio PV01 (present value of a basis point change) with respect to each interest rate of the zero-coupon yield curve. Since zero-coupon rates are rarely directly observable, we must estimate them from market data. In this paper we implemented a widely-used parametric term structure estimation method called Nelson and Siegel. For Argentina we generated two yield curves, i.e., sets of fixed maturity interest rates determined by Badlar and CER.

Flight-to-Quality in the Euro Area: Convenience Yield in German Government Bond Rates

German government bonds are very popular among investors because of their liquidity and safety, and they are also widely used in repo transactions as collateral. During the recent financial crisis the excess demand for German securities pushed their yields below the theoretical risk-free rate. The convenience yield, the deviation of German rates from the riskless rate, is a useful measure of the flight-to-quality/liquidity behaviour of investors. In this paper, I propose a Gaussian affine term structure model for German zero-coupon yields that allows the estimation of the convenience yield as one the pricing factors. I show that the model can be relatively easily estimated, with only a little use of numerical techniques. The estimated factor can be interpreted as convenience yield, and the model provides a good fit for both German zero-coupon yields and Eurepo rates.

On the Level-Slope-Curvature Effect in Yield Curves and Eventual Total Positivity

Principal components analysis has become widely used in a variety of fields. In finance and, more specifically, in the theory of interest rate derivative modeling, its use has been pioneered by Litterman and Scheinkman [J. Fixed Income, 1 (1991), pp. 54--61]. Their key finding was that a few components explain most of the variance of treasury zero-coupon rates and that the first three eigenvectors represent level, slope, and curvature (LSC) changes on the curve. This result has been, since then, observed in various markets. Over the years, there have been several attempts at modeling correlation matrices displaying the observed effects as well as trying to understand what properties of those matrices are responsible for them. Using recent results of the theory of total positiveness [O. Kushel, Matrices with Totally Positive Powers and Their Generalizations, 2014], we characterize these matrices and, as an application, we shed light on the critique to the methodology raised by Lekkos [J. Derivatives, 8 (2000), pp. 72--83].

A yield-only model for the term structure of interest rates

AbstractThis paper develops a term structure model for the UK nominal, real and implied inflation spot zero-coupon rates simultaneously. We start with fitting a descriptive yield curve model proposed by Cairns (1998) to fill the missing values for certain given days at certain maturities in the yield curve data provided by the Bank of England. We compare four different fixed ‘exponential rate’ parameter sets and decide the set of parameters which fits the data best. With the chosen set of parameters we fit the Cairns model to the daily values of the term structures. By applying principal component analysis on the hybrid data (Bank of England data and fitted spot rates for the missing values) we find three principal components, which can be described as ‘level’, ‘slope’ and ‘curvature’, for each of these series. We explore the relation between these principal components to construct a ‘yield-only’ model for actuarial applications. Main contribution of this paper is that the models developed in the paper enable the practitioners to forecast three term structures simultaneously and it also provides the forecast for whole term structures rather than just short and long end of the yield curves.

Libor Market Model with OIS-Discounting: Extension, Clarification and Valuation

Since the Financial Crisis in 2009, interest rate derivative markets have witnessed a dramatic change in the valuation, especially with respect to cash-flow discounting, forward curve building und counterpart risks. The application of OIS discount curve to value collateralised interest derivatives results in a major change in valuation models since the pricing numeraires are directly connected with discount factors. To capture the effect of OIS discounting, an additional stochastic process for overnight rates is required. In this paper, I extend the classical Libor Market Model (LMM) by adding a short rate process for overnight rates, and then unify OIS zero-coupon bonds and tenor forward rates under a unique OIS forward measure, for example, the OIS terminal measure. Within the OIS-LMM, I could clarify the market practices and models in valuating secured derivatives where OIS adjustments in observable forward rates are identified as a result of the separation of discount curve and forward curve in the new multi-curve world. The suggested OIS-LMM is extended parsimoniously so that the existing pre-crisis LMM can be easily modified to capture OIS discounting.

Term Structure Modelling by Using Nelson-Siegel Model

Zero coupon rates are not observable in the market for a range of maturities. Therefore, an estimation methodology is required to derive the zero coupon yield curves from observable data. If we deal with approximations of empirical data to create yield curves it is necessary to choose suitable mathematical functions. We use parametric model of Nelson and Siegel. The current mathematical apparatus employed for this kind of approximation is outlined. This theoretical background is applied to an estimation of the zero-coupon yield curve derived from the Czech coupon bond market. There are many methodologies and each can provide surprisingly different results. Nevertheless, each seeks to provide an estimation that fit the data well while maintaining an easily interpretable form. On an initial test data sample we have not faced any problems, reported elsewhere, of not having found the global optimum or having found multiple local minima.

An Analysis of the Ultra Long-Term Yields

The discounting of very long-term cash-flows is crucial for the valuation of long-term investment projects. In this paper, we analyze the market prices of US government bonds with very long-term time-to-maturity, and emphasize some statistical specificities of very long-term zero-coupon rates, that standard Gaussian affine term structure models do not account for. In addition, we describe and estimate three Gaussian Nelson-Siegel affine term structure models, and highlight the model characteristics, which are necessary to match the dynamics of very long-term interest rates.

The Information Content of the French and German Government Bond Yield Curves: Why Such Differences?

In this paper, we evaluate the information content of the yield curve as regards future interest rates and inflation in France and Germany. An original data set of long-term zero-coupon interest rates for French and German government bonds was constructed for the period 1980-97. Empirical evidence shows that the German yield curve has a significant information content about the future average change in short-term rates and the future path of inflation. The information content of the French yield curve is much more limited and is only relevant for the average change in short-term rates. We show that the difference between the results obtained for both countries mainly stems from lower variability in German risk premia than in French risk premia.

GENERALIZATION OF THE DYBVIG-INGERSOLL-ROSS THEOREM AND ASYMPTOTIC MINIMALITY

The long-term limit of zero-coupon rates with respect to the maturity does not always exist. In this case we use the limit superior and prove corresponding versions of the Dybvig–Ingersoll–Ross theorem, which says that long-term spot and forward rates can never fall in an arbitrage-free model. Extensions of popular interest rate models needing this generalization are presented. In addition, we discuss several definitions of arbitrage, prove asymptotic minimality of the limit superior of the spot rates, and illustrate our results by several continuous-time short-rate models.

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.

Arbitrage with Fixed Costs and Interest Rate Models

AbstractWe study securities market models with fixed costs. We first characterize the absence of arbitrage opportunities and provide fair pricing rules. We then apply these results to extend some popular interest rate and option pricing models that present arbitrage opportunities in the absence of fixed costs. In particular, we prove that the quite striking result obtained by Dybvig, Ingersoll, and Ross (1996), which asserts that under the assumption of absence of arbitrage long zero-coupon rates can never fall, is no longer true in models with fixed costs, even arbitrarily small fixed costs. For instance, models in which the long-term rate follows a diffusion process are arbitrage-free in the presence of fixed costs (including arbitrarily small fixed costs). We also rationalize models with partially absorbing or reflecting barriers on the price processes. We propose a version of the Cox, Ingersoll, and Ross (1985) model which, consistent with Longstaff (1992), produces yield curves with realistic humps, but does not assume an absorbing barrier for the short-term rate. This is made possible by the presence of (even arbitrarily small) fixed costs.

The information content of the German term structure regarding inflation

The paper investigates the information content of the German term structure regarding inflation, defined as the ability of the yield curve's slope to predict future changes in inflation rates. The empirical tests show that the German yield curve is informative in that sense, especially in its middle segment between three and eight years. A new robustness test is considered. Besides the specification in terms of yields-to-maturity, which has traditionally been employed by the Bundesbank and by previous empirical research, zero-coupon rates (are considered) estimated using the Svensson (IMF Working Paper No. 114, 1994) approach. This takes account of the fact that tests of the expectations hypothesis are in fact tests of joint hypotheses, among them, that the yield curve specification used gives an unbiased picture of the relevant information and that the specific formulation of the expectations hypothesis is valid. Despite the considerable differences between the two yield curve specifications, the results...

Deriving Zero-Coupon Rates: Alternatives to Orthodoxy

The bootstrapping method for deriving zero-coupon rates from par yield curves implicitly assumes a specific forecast of prospective interest rate changes. In the event that market conditions reflect this standard methodology but, at the same time, individual practitioners reject the implicit forecast, contrarian opportunities will be present. Additionally, to the extent that zero-coupon discount rates are based on unacceptable assumptions, any subsequent present valuation calculation using these rates would be suspect. We propose two methodologies whereby any explicit interest rate forecast may be used for determining zero-coupon interest rates. “Backsliding” calculates zero-coupon returns by assuming interim interest cash flows are always rolled over into the longest maturity coupon-bearing instruments available; “frontsliding” requires rolling these cash flows over into the shortest maturity vehicles. The choice of methodology should be driven by rate forecasts, as well as whether the objective is for f...