Coupon Bonds Research Articles

A Continuous-Time Model of the Term Structure of Interest Rates with Fiscal-Monetary Policy Interactions

We study the term structure implications of the fiscal theory of price level determination. We introduce the intertemporal budget constraint of the government in a general equilibrium model in continuous time. Fiscal policy is set according to a simple rule whereby taxes react proportionally to real debt. We show how to solve for the prices of real and nominal zero coupon bonds.

Bond Implied CDS Spread and CDS-Bond Basis

We derive a simple formula for calculating the CDS spread implied by the bond market price. Using no-arbitrage argument, the formula expresses the bond implied CDS spread as the sum of bond price, bond coupon and Libor zero curve weighted by risky annuities. We show that the bond implied CDS spread is consistent with the standard CDS pricing model if the survival probabilities and recovery are consistent with the bond price.

Revisiting the Expectations Hypothesis of the Term Structure of Interest Rates

The expectations hypothesis of the term structure has been decisively rejected by a large empirical literature that spans several decades. In this paper, using a newly constructed dataset of synthetic zero coupon bond yields, we show that evidence against the expectations hypothesis became very much weaker following the widespread acceptance of its empirical failure to describe the behavior of interest rates in the early 1990s. Indeed, in the period 1991-2004, the expectations hypothesis cannot be rejected for most bond maturities. These results are consistent with the idea that asset pricing anomalies tend to disappear once they are widely recognized.

Superseding Newton with a Superior Yield Algorithm

Determining the yield to maturity of a coupon bond with more than four coupon periods is a two-step process. The first step uses an approximation formula to obtain a first approximation of the true yield. The second step uses an algorithm to advance the first approximation closer to the bond's true yield. Newton's Method is the algorithm used in applications such as Microsoft's Excel YIELD function. This paper evaluates some commonly used approximation formulae before demonstrating a solution algorithm that generally outperforms Newton's Method.

Linearized Nelson-Siegel and Svensson Models for the Estimation of Spot Interest Rates

Linearized versions of the Nelson-Siegel and Svensson models for estimating spot yield curves from samples of coupon bonds are developed and analyzed. It is shown how these models can be made linear in the level, slope and curvature parameters and how prior information about these parameters can be incorporated in the estimation procedure. The performance of the linearized models are assessed in a Monte Carlo setting and with a sample of U.S. government bonds. The results reveal that the linearized models compare favorably to the original models in terms of precision, computing effort and prevalence of local optima.

The Slope of Credit Spread Curves

One way to test a corporate bond pricing model is to examine its predictions for the shape of credit yield spread curves. However, existing empirical studies along this line are known to be problematic because of not controlling properly for the credit quality of bonds (an exception is Helwege and Turner [1999] who study the slope of credit spread curves for high yield coupon bonds). In this article we examine credit spread curves for both zero-coupon and coupon bonds and for both investment grade and high yield bonds. We find that the term structure of credit spreads is usually upward sloping, regardless of credit quality or coupon. <b>TOPICS:</b>Fixed income and structured finance, statistical methods, credit risk management

Nonlinear Filtering in Affine Term Structure Models: Evidence from the Term Structure of Swap Rates

When the relationship between observed fixed-income securities and the latent state variables in dynamic term structure models is nonlinear, existing studies usually linearize this relationship because nonlinear filtering is computationally demanding. We propose the use of the unscented Kalman filter to allow for nonlinearities. To illustrate its advantages, we analyze the cross section of swap rates, which are relatively simple nonlinear instruments. An extensive Monte Carlo experiment demonstrates that the unscented Kalman filter generates much smaller swap rate prediction errors and also smaller errors in parameter estimates. Estimation using swap data indicates large differences between the state variables obtained using the unscented Kalman filter and the more conventional extended Kalman filter. Our findings suggest that the unscented Kalman filter may prove to be a good approach for a number of other problems in fixed income pricing with nonlinear relationships between the state vector and the observations, such as the estimation of term structure models using interest rate derivatives or coupon bonds, and the estimation of quadratic term structure models.

Trading strategies based on term structure model residuals

The term structure of interest rates is an important input for basically every pricing model and is mostly calibrated on coupon bond prices. Therefore, the estimated interest rates should accurately explain the market prices of these bonds. However, nearly all empirical papers on interest rate estimation, e.g. Svensson, L.E.O. 1994. Estimating and interpreting forward interest rates: Sweden 1992–1994, IMF Working Paper, International Monetary Fund, report significant pricing errors in their sample. So an important question is what drives these pricing errors of the bonds. One simple explanation would be different tax treatment or different liquidity, but most papers on this research topic, e.g. Elton, E., and T.C. Green. 1998. Tax and liquidity effects in pricing government bonds. Journal of Finance 53: 1533–62, cannot fully explain the observed pricing errors. Therefore, these errors must be at least partially caused by either model misspecification or by the deviation of particular bond prices from general market conditions, i.e. mispricing revealing insufficient market efficiency. We provide empirical evidence for the German government bond market that risk-adjusted trading strategies based on bond pricing errors can yield about 15 basis points p.a. abnormal return compared to benchmark portfolios. Furthermore, the abnormal returns are continuously achieved over the whole time period and not randomly on a few days and show a relation to changes in the level and the curvature of the term structure of interest rates. Therefore, pricing errors contain economic information about deviations of bond prices from general market conditions and are not exclusively caused by model misspecification and/or differences in liquidity and tax treatment of individual bonds.

The subprime crisis

The paper revises our previous USD 300 bn estimate for mortgage related losses to a range of USD 350-420 bn. In doing this the paper explicitly rejects the previous approach based on implied defaults from ABX pricing, because these prices are affected by illiquidity and extreme volatility; they will likely lead to misleading estimates of losses. Instead it builds a proper default model approach and allows for recovery of collateral via house sales over time. The paper separates out the losses due to commercial banks in the US, and goes on to look at the implied deleveraging required to meet capital standards. It could take 6-12 months for banks to offset losses via earnings alone, depending on Fed rate cuts and the dividend policy of banks. Since even more capital than this is required if banks were to expand their balance sheets, the paper looks at possibilities for capital injections from groups like sovereign wealth funds; and it also looks at a novel plan for the use of public money with an RTC-style approach and the issue of zero coupon bonds. Finally the paper looks at the issues of moral hazard, the likely size of the impact in Europe and Asia and non-bank corporate leverage.

Quantum finance Hamiltonian for coupon bond European and barrier options

Coupon bond European and barrier options are financial derivatives that can be analyzed in the Hamiltonian formulation of quantum finance. Forward interest rates are modeled as a two-dimensional quantum field theory and its Hamiltonian and state space is defined. European and barrier options are realized as transition amplitudes of the time integrated Hamiltonian operator. The double barrier option for a financial instrument is "knocked out" (terminated with zero value) if the price of the underlying instrument exceeds or falls below preset limits; the barrier option is realized by imposing boundary conditions on the eigenfunctions of the forward interest rates' Hamiltonian. The price of the European coupon bond option and the zero coupon bond barrier option are calculated. It is shown that, is general, the constraint function for a coupon bond barrier option can -- to a good approximation -- be linearized. A calculation using an overcomplete set of eigenfunctions yields an approximate price for the coupon bond barrier option, which is given in the form of an integral of a factor that results from the barrier condition times another factor that arises from the payoff function.

Term Structure Estimation

The term structure of interest rates gives the relationship between the yield on an investment and the term to maturity of the investment. Since the term structure is typically measured using default-free, continuously-compounded, annualized zero-coupon yields, it is not directly observable from the published coupon bond prices and yields. This paper focuses on how to estimate the default-free term structure of interest rates from bond data using three methods: the bootstrapping method, the McCulloch cubic-spline method, and the Nelson and Siegel method. Nelson and Siegel method is shown to be more robust than the other two methods. The results of this paper can be implemented using user-friendly Excel spreadsheets.

Pricing and Evaluating a Bond Portfolio Using a Regime Switching Markov Model

We apply a regime switching Markov chain model to determine bond prices. Our work builds upon the work of Thomas, Allen & Morkel-Kingsbury (2002) and Jarrow, Lando & Turnbull (1997). The interest rate process and credit rating migration process are considered. Our aim is to study the price evolution of a portfolio of defaultable bonds. We are interested in determining the density process to compute the Value at Risk (VaR) and Conditional Value at Risk (CVaR) a year ahead. Whereas Thomas et al. describe a model under the pricing measure only, our model takes into consideration both the physical and pricing measures. We also describe the whole (risk free) forward rate curve, with appropriate conditions for the absence of arbitrage, while Thomas et al. describe the spot rate process. Most bond models make use of zero coupon bond prices. By using bond stripping we are able to discover zero coupon bond prices for bonds of difierent ratings. We use a mathematical programming approach to strip coupons and minimise squared error, subject to no arbitrage constraints.

Spot and Forward Interest Rates

This note examines how spot and forward interest rates relate to bond prices and to each other. After defining spot and forward rates, the note shows how to estimate spot rates from data on either zero-coupon bonds or coupon bonds. It also shows how to express these stated rates for different compounding assumptions. The note pays particular attention to understanding how arbitrage forces in markets establish the links between forward and spot rates and how to infer one set of rates from the other. The note deepens understanding of bond pricing, assuming that the reader is already familiar with the basics of present-value techniques. It is a useful springboard for topics (e.g., swaps) that require familiarity with spot and forward rates.

On the Need for a New Approach to Analyzing Monetary Policy

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.

Robust Recovery Risk Hedging: Only the First Moment Matters

Credit derivatives are subject to at least two sources of risk: the default time and the recovery payment. This paper examines the impact of modeling the recovery payment on hedging strategies in a reduced-form model as well as a structural model. We show that all hedging approaches based on a quadratic criterion do only depend on the expected recovery payment at default and not the whole shape of the recovery payment distribution if the underlying hedging instrument (say, a defaultable zero coupon bond) jumps to or reaches a pre-specified value when the credit event occurs. This justifies assuming a \emph{certain} recovery rate conditional on default time and interest rate level. Hence, this result allows a simplified modeling of credit risk.

Konstrukce výnosové křivky pomocí vládních dluhopisů v České republice

The paper deals with yield curve construction methods using coupon bonds in Czech bond market. Generally, there are more possibilities how to approach this problem: bootstraping, splines, parametric functions. Due to the lack of tradable public bonds and due to the fact that existing bonds do not pay coupons at the same date of the year, traditional bootstraping method could not be applied under Czech market conditions. It seemed appropriate to use parametrical solutions to the yield curve issue and minimise the sum of squares of differences between market and theoretical prices. There were presented three function types which arrived to similar results in the paper. The authors also used Svensson parametric function to demonstrate the possible use of parametric yield curve construction. It was shown that, after duration adjustment, it can indicate shift in market expectations regarding future short term interest rate moves, and thus regarding future monetary policy, pretty well.

On the Mathematics of Duration Crossovers

In this paper, we present, mathematically, the conditions under which longer maturity, lower coupon bonds have shorter durations than shorter maturity, higher coupon bonds. We consider differences in the timing and size of coupon payments, yields-to-maturity, and maturity in the calculation of duration.

A fast algorithm for numerical solutions to Fortet's equation

A fast algorithm for computation of default times of multiple firms in a structural model is presented. The algorithm uses a multivariate extension of the Fortet's equation and the structure of Toeplitz matrices to significantly improve the computation time. In a financial market consisting of M ⪢ 1 firms and N discretization points in every dimension the algorithm uses O ( n log n · M · M ! · N M ( M - 1 ) / 2 ) operations, where n is the number of discretization points in the time domain. The algorithm is applied to firm survival probability computation and zero coupon bond pricing.

Modelling the Fair Value of Annuities Contracts: The Impact of Interest Rate Risk and Mortality Risk

The purpose of this paper is to analyze the problem of the fair valuation of annuities contracts. The market consistent valuation of these products requires a pricing framework which includes the two main sources of risk affecting the value of the annuity, i.e. interest rate risk and mortality risk. As the IASB has not set any specific guidelines as to which models are the most appropriate for these risks, in this note we consider a range of different models calibrated with historical data. We calculate the fair value of the annuity as a portfolio of zero coupon bonds, each with maturity set equal to the date of the annuity payments; the weights in the portfolio are given by the survival probabilities. Moreover, we focus on the additional information provided by stochastic simulations in order to define a suitable risk margin. The nature of the risk margin is one of the main key issues concerning the IASB and Solvency project.

Insider trading in an equilibrium model with default: a passage from reduced-form to structural modelling

We study, in the framework of Back [Rev. Financial Stud. 5(3), 387–409 (1992)], an equilibrium model for the pricing of a defaultable zero coupon bond issued by a firm. The market consists of a risk-neutral informed agent, noise traders, and a market maker who sets the price using the total order. When the insider does not trade, the default time possesses a default intensity in the market’s view as in reduced-form credit risk models. However, we show that, in equilibrium, the modelling becomes structural in the sense that the default time becomes the first time that some continuous observation process falls below a certain barrier. Interestingly, the firm value is still not observable. We also establish the no expected trade theorem that the insider’s trades are inconspicuous.