Caio Almeida

Decomposing and Simulating the Movements of Term Structures of Interest Rates in Emerging Eurobond Markets

CRISTIANO AUGUSTO COELHO FERNANDES is with Pontificia Universidade Cat6lica of Rio de Janeiro. e movements of a term structure of interest rates are usually assumed to be driven by three orthogonal factors: parallel shfts, changes in slope, and changes in curvature (see Litterman and Scheinkman [ 19911). Principal components analysis has traditionally been used to obtain the most important factors explaining the dynamics of a term structure (see Mardia, Kent, and Bibby [1992]). The objective is to obtain those orthogonal linear combinations that best explain the covariance matrix of the term structure of interest rates. The approach is simple. Given a data base of term structures, select a set of maturities, estimate the covariance matrix of the yields for this set of maturities and, finally, apply principal components analysis to this covariance matrix to obtain the most important orthogonal factors. Identitjring and simulating the most important factors driving the movements of a term structure of interest rates is important for portfolio managers, risk managers, and derivatives analysts. Fixed-income securities present dfferent exposures to the orthogonal factors driving the movements of a term structure of interest rates. For example, while short-term bonds are basically sensitive to parallel shifts, long-term bonds are more sensitive to parallel shifts and changes in slope. All three orthogonal factors are important when long-term bonds with embedded options are analyzed. A fixed-income portfolio manager should be able to simulate different movements in a term structure to vahdate an investment strategy (see Carino et al. [1994]). A risk manager should be able to estimate the impact of dfferent risk factors on a fixed-income portfolio to suggest optimal hedges (Singh [1997]). A r

Assessing Misspecified Asset Pricing Models with Empirical Likelihood Estimators

Hansen and Jagannathan (1997) compare misspecified asset pricing models based on least-square projections on a family of admissible stochastic discount factors. We extend their fundamental contribution by considering Minimum Discrepancy (MD) projections where misspecification is measured by convex functions that can explicitly take into account higher moments of asset returns. The MD problems are solved on dual spaces with the interpretation of optimal portfolio problems based on HARA utility functions, producing a family of estimators that captures the least-square problem as a particular case. We use our proposed methodology to test the Consumption Asset Pricing Model and illustrate, under several different discrepancy functions and regions of the parametric space, the relation between the parametric proxy model, and the closest admissible SDF. On the estimation problem, not surprisingly, all MD estimators clearly reject the CCAPM model. However, some of these estimators lead to admissible SDFs that are very distinct from the one implied by the least-square solution. By their pricing implications, this rich set of optimal MD SDFs represent useful tools to diagnose missing factors in asset pricing models.

Does Curvature Enhance Forecasting

In this paper, we analyze the importance of curvature term structure movements on forecasts of interest rates. An extension of the exponential three-factor Diebold and Li (2006) model is proposed, where a fourth factor captures a second type of curvature. The new factor increases model ability to generate volatility and to capture nonlinearities in the yield curve, leading to a significant improvement of forecasting ability. The model is tested against the original Diebold and Li model and some other benchmarks. Based on a forecasting experiment with Brazilian fixed income data, it obtains significantly lower bias and root mean square errors for most examined maturities, and under three different forecasting horizons. Robustness tests based on two sub-sample analyses partially confirm the favorable results.

Forecasting Bond Yields with Segmented Term Structure Models

Recent empirical analysis of interest rate markets documents that bond demand and supply directly affect yield curve movements and bond risk premium. Motivated by those findings we propose a parametric interest rate model that allows for segmentation and local shocks in the term structure. We split the yield curve in segments presenting their own local movements that are globally interconnected by smoothing conditions. Two classes of segmented exponential models are derived and compared to successful term structure models based on a sequence of out-of-sample forecasting exercises. Adopting U.S. interest rates data available from 1985 to 2008, the segmented models present overall better forecasting performance suggesting that local shocks might indeed be important determinants of yield curve dynamics.

AFFINE PROCESSES, ARBITRAGE-FREE TERM STRUCTURES OF LEGENDRE POLYNOMIALS, AND OPTION PRICING

Multivariate Affine term structure models have been increasingly used for pricing derivatives in fixed income markets. In these models, uncertainty of the term structure is driven by a state vector, while the short rate is an affine function of this vector. The model is characterized by a specific form for the stochastic differential equation (SDE) for the evolution of the state vector. This SDE presents restrictions on its drift term which rule out arbitrages in the market. In this paper we solve the following inverse problem: Suppose the term structure of interest rates is modelled by a linear combination of Legendre polynomials with random coefficients. Is there any SDE for these coefficients which rules out arbitrages? This problem is of particular empirical interest because the Legendre model is an example of factor model with clear interpretation for each factor, in which regards movements of the term structure. Moreover, the Affine structure of the Legendre model implies knowledge of its conditional characteristic function. From the econometric perspective, we propose arbitrage-free Legendre models to describe the evolution of the term structure. From the pricing perspective, we follow Duffie et al. [22] in exploring their conditional characteristic functions to obtain a computational tractable method to price fixed income derivatives.

Term structure movements implicit in Asian option prices

In this paper we implement dynamic term structure models that adopt bonds and Asian options in the estimation process. The goal is to analyse the pricing and hedging implications of term structure movements when options are (or are not) included in the estimation process. We investigate how options affect the shape, risk premium and hedging structure of the dynamic factors. We find that the inclusion of options affects the loadings of the slope and curvature factors, and considerably changes the risk premium and hedging structure of all dynamic factors.

TIME-VARYING RISK PREMIA IN EMERGING MARKETS: EXPLANATION BY A MULTI-FACTOR AFFINE TERM STRUCTURE MODEL

From the empirical viewpoint, the Expectation Hypothesis Theory (EHT) of the term structure of interest rates has been extensively tested and rejected for US term structure data. Dai and Singleton [6] show that under the settings of Affine term structure models it is possible that one matches both the historical term structure dynamics and capture an important stylized fact that have contradicted the EHT: Time-varying risk premia. In emerging markets, economic conditions tend to be much less stable than in developed markets. For this reason, if risk premia is dynamic in such markets, intuition would suggest that it is more volatile than in developed markets, implying a stronger statistical rejection of the EHT. In this paper, we verify the robustness of Dai and Singletons results under these more extreme market conditions. We estimate an arbitrage free Affine Gaussian model for the term structure of swaps in the Brazilian market. We propose an extensive empirical analysis which consists on: defining the optimal number of factors to be used in the model, estimating the model, giving interpretation to the state variables in terms of risk factors, and studying the model implied risk premia. In the end, we propose an application for risk management of interest rates futures portfolios.

A GENERALIZATION OF PRINCIPAL COMPONENT ANALYSIS FOR NON-OBSERVABLE TERM STRUCTURES IN EMERGING MARKETS

Principal Component Analysis (PCA) has been traditionally used for identifying the most important factors driving term structures of interest rates movements. Once one maps the term structure dynamics, it can be used in many applications. For instance, portfolio allocation, Asset/Liability models, and risk management, are some of its possible uses. This approach presents very simple implementation algorithm, whenever a time series of the term structure is disposable. Nevertheless, in markets where there is no database for discount bond yields available, this approach cannot be applied. In this article, we exploit properties of an orthogonal decomposition of the term structure to sequentially estimate along time, term structures of interest rates in emerging markets. The methodology, named Legendre Dynamic Model (LDM), consists in building the dynamics of the term structure by using Legendre Polynomials to drive its movements. We propose applying LDM to obtain time series for term structures of interest rates and to study their behavior through the behavior of the Legendre Coefficients levels and first differences properly normalized (Legendre factors). Under the hypothesis of stationarity and serial independence of the Legendre factors, we show that there is asymptotic equivalence between LDM and PCA, concluding that LDM captures PCA as a particular case. As a numerical example, we apply our technique to Brazilian Brady and Global Bond Markets, briefly study the time series characteristics of their term structures, and identify the intensity of the most important basic movements of these term structures.

Nonparametric Tail Risk, Stock Returns and the Macroeconomy

This paper introduces a new tail risk measure based on the risk-neutral excess expected shortfall of a cross-section of stock returns. We propose a novel way to risk neutralize the returns without relying on option price information. Empirically, we illustrate our methodology by estimating a tail risk measure over a long historical period based on a set of size and book-to-market portfolios. We find that a risk premium is associated with long-short strategies with portfolio sorts based on tail risk sensitivities of individual securities. Our tail risk index also provides meaningful information about future market returns and aggregate macroeconomic conditions. Results are robust to the cross-sectional information selected to compute the tail risk measure.

Credit Spread Arbitrage in Emerging Eurobond Markets

In the corporate emerging Eurobond fixed-income market there are two main sources of credit risk: sovereign risk and the relative credit quality of issuers of the eurobonds. This article presents a model to estimate, in a one-step procedure, both the term structure of interest rates and the credit spread function of a diversified international portfolio of Eurobonds with different credit ratings. The estimated term structures can be used to analyze credit spread arbitrage opportunities in Eurobond markets. Numerical examples in the Argentinean, Brazilian, and Mexican Eurobond markets illustrate the practical use of the methodology.