تجزیه و تحلیل ریسک نرخ بهره: نوسانات تصادفی در ساختار دوره بازدهی اوراق قرضه دولتی

We propose a Nelson–Siegel type interest rate term structure model where the underlying yield factors follow autoregressive processes with stochastic volatility. The factor volatilities parsimoniously capture risk inherent to the term structure and are associated with the time-varying uncertainty of the yield curve’s level, slope and curvature. Estimating the model based on US government bond yields applying Markov chain Monte Carlo techniques we find that the factor volatilities follow highly persistent processes. We show that yield factors and factor volatilities are closely related to macroeconomic state variables as well as the conditional variances thereof.

Much research in financial economics has been devoted to the modeling and forecasting of interest rates and the term structure thereof. However, only a few approaches explicitly account for time-varying interest rate risk. A potential reason is the typically high (cross-sectional) dimension of the term structure whose multivariate volatility is cumbersome to model. Consequently, most studies focus on the riskiness of (selective) interest rates for given maturities or study aggregated volatility measures based on a common component as recently suggested by Koopman et al. (2010) or based on bond market portfolios in the spirit of Engle et al., 1990 and Engle and Ng, 1993. In this paper, we propose capturing the riskiness inherent to the term structure of interest rates by an extended Nelson–Siegel (1987) term structure model where the underlying yield curve factors reveal stochastic volatility. We see this approach as a parsimonious alternative to a model where the individual time series of yields themselves reveal (high-dimensional) time-varying volatility. Modeling stochastic volatility directly in the Nelson–Siegel factors reduces the dimension of the stochastic volatility process to three and allows capturing time-varying uncertainty associated with the yield curve’s level, slope and curvature. Accordingly, the so-called ‘level volatility’ reflects the volatility with respect to the overall level of yields, whereas the ‘slope volatility’ captures the time-varying riskiness in the spread between short-term and long-term yields. Correspondingly, the ‘curvature volatility’ is associated with the risk due to changes in the term structure curvature. This paper contributes to the recent empirical literature on the modeling of interest rate dynamics. Classical equilibrium models, such as, e.g., Vasicek, 1977, Cox et al., 1985, Duffie and Kan, 1996, Dai and Singleton, 2002 and Duffee, 2002 and no-arbitrage approaches in the line of, e.g., Hull and White, 1990 and Heath et al., 1992 describe the evolution of short rates in terms of underlying risk factors. They typically use affine structures which allow constructing the expected yields at other maturities given assumptions about the underlying dynamics of risk factors and risk premia. These models – typically combined with stochastic volatility processes (e.g., Heston, 1993) – are workhorses for the pricing of bonds and interest rate derivatives. However, though these approaches are powerful in explaining the term structure across different maturities at a single point in time, they are not very successful in forecasting interest rates and the term structure thereof. In fact, in both empirical research as well as financial practice, the latter task is dominantly addressed using factor models. A well-known approach in this area is the Nelson and Siegel (1987) exponential components framework which is neither an equilibrium nor a no-arbitrage model but can be heuristically motivated by the expectations hypothesis of the term structure. In this setting, the term structure is captured by three factors which are associated with the yield curve’s level, slope and curvature. In a related approach, Litterman and Scheinkman, 1991 propose such factors as the first three principal components based on the bond return covariance matrix. Diebold and Li (2006) propose a simple dynamic implementation of the Nelson and Siegel (1987) model and employ it to model and to predict the yield curve. Diebold et al. (2006) extend this approach by including macroeconomic variables while Koopman et al. (2010) allow for time-varying loadings and include a common volatility component. Motivated by the lacking empirical evidence on the role of term structure volatility, we aim at filling this gap in the literature and address the following three major research questions: (i) To which extent do yield curve factors reveal time-varying volatility? (ii) Are factor volatilities correlated with macroeconomic fundamentals? (iii) Are there dynamic interdependencies between macroeconomic variables, their conditional variances and term structure risk? We represent the Nelson–Siegel model in a state space form, where both the (unobservable) yield factors and their stochastic volatility processes are treated as latent factors following autoregressive processes. We estimate the model using Markov chain Monte Carlo (MCMC) methods based on monthly unsmoothed Fama–Bliss zero yields from 1964 to 2003. In a second step, we use the estimated yield curve factors and volatility factors as components of a VAR model including macroeconomic variables, such as capacity utilization, industrial production, inflation, unemployment, GDP growth, as well as the federal funds rate, among others. Based on our empirical study, we can summarize the following main findings: (i) We find strong evidence for time-varying term structure risk with the yield curve volatilities following persistent dynamics. (ii) Both yield factors and factor volatilities are strongly correlated with macroeconomic fundamentals, such as capacity utilization, inflation, unemployment rates, federal funds rates and GDP growth. (iii) Prediction error variance decompositions show evidence for significant long-run effects of macroeconomic state variables and their conditional variances on term structure movements and associated interest rate risk. Finally, in a forecasting exercise, we illustrate the effect of stochastic volatility in yield factors on the precision of yield forecasts. The remainder of the paper is structured as follows. In Section 2, we describe the dynamic Nelson and Siegel (1987) model as put forward by Diebold and Li (2006) and discuss the proposed extension allowing for stochastic volatility processes in the yield factors. Section 3 presents the data and illustrates the estimation of the model using MCMC techniques. In Section 4, we investigate the dynamic interdependencies between yield factors, factor volatilities and macroeconomic variables. Section 5 shows the model’s in-sample and out-of-sample forecasting ability. Finally, Section 6 concludes.

We propose a dynamic Nelson–Siegel type yield curve factor model, where the underlying factors reveal stochastic volatility. By estimating the model using MCMC techniques we extract both the Nelson–Siegel factors as well as their volatility components and relate them to underlying macroeconomic variables. This approach allows us to link the approaches by Diebold and Li, 2006 and Diebold et al., 2006 on factor-based term structure modeling with the GARCH-in-Mean models by Engle et al., 1990 and Engle and Ng, 1993 capturing interest rate risk premia. We summarize the following main findings: (i) Term structure risk as represented by the volatility of yield curve factors is time-varying and highly persistent. (ii) Yield term factors and – to an even larger extent – factor volatilities are correlated with key macroeconomic variables reflecting capacity utilization, unemployment, GDP growth, inflation and monetary policy. (iii) Macroeconomic variables have more long-run predictability for term structure volatilities than for the term structure itself. Capacity utilization, inflation, GDP growth and unemployment rates are important long-term predictors for risk inherent to the level and slope of the yield curve. (iv) Conversely, yield factors have significant forecasting ability for capacity utilization, federal funds rates, unemployment and M1 growth but have only negligible impacts on the volatilities thereof. These results indicate that risk inherent to the shape of the yield curve is relevant and seems to be effectively captured by a stochastic volatility component in the curvature factor.