Monetary Policy and the Yield Curve Antulio N. Bomfim∗ Federal Reserve Board First Draft: November 8, 2002 This Draft: February 13, 2003 Abstract This paper examines the empirical properties of a two-factor affine model of the term structure of interest rates, estimated with LIBOR and interest rate swap data from 1989 through 2001. Despite its rel- ative simplicity, the model fits the interest rate data remarkably well, both across time and maturity, and identifies changes in the current and expected stance of monetary policy as primary movers of the yield curve. JEL Classification: G12, E43 Keywords: Affine models, expectations hypothesis, latent factors ∗Board of Governors of the Federal Reserve System, Washington, DC 20551; E-mail: abomfi[email protected]; Fax: (202) 452-2301, Tel.: (202) 736-5619. I am grateful to Brian Sack and Hao Zhou for helpful comments, to Joseph Rosenberg for excellent research assistance, and to seminar participants at the Federal Reserve Board for their insights. The opinions expressed in this paper are not necessarily shared by the Board of Governors of the Federal Reserve System or any other members of its staff. 1 Introduction A growing literature has suggested that the entire yield curve moves primarily in response to changes in a small number of potentially unobservable common factors, but the factors themselves are typically described based on their effects on the “level, slope, and curvature” of the yield curve, rather than in terms of the financial and economic fundamentals that drive market interest rates—Dai and Singleton (2000) and Litterman and Scheinkman (1991).1 In this paper, I discuss and test empirically an economic interpretation for an otherwise standard no-arbitrage multifactor affine framework, very much in the tradition of Balduzzi, Das, and Foresi (1998) and Babbs and Nowman (1999). The results suggest that actual and expected changes in monetary policy account for a remarkably large share of the variation in bond yields in the United States across the whole maturity spectrum. The proposition that monetary policy plays an important role in the determination of market interest rates is hardly new. Indeed, in laying the groundwork for the two-factor affine model used in the analysis, I briefly review, in Sections 2 and 3, a well-known form of the expectations hypothesis of the yield curve, which postulates that yields on longer-term bonds reflect market participants’ expectations of future values of short-term yields, which are themselves largely set by monetary policy. What is new and noteworthy among the results reported in this paper is the extent to which estimates 1Recent work by Ang and Piazzesi (2002) and Piazzesi (2001) deviates somewhat from this norm and is related to this paper to the extent that at least some of the common factors discussed in those papers are given an explicit economic interpretation. For instance, Ang and Piazzesi propose a multifactor framework that incorporates both observable (macroe- conomic) factors and latent factors, although the latter are not fully accounted for by identifiable fundamentals. 1 of the two latent factors that drive the yield curve in the model can be mapped into commonly used measures of the current and expected stance of monetary policy. Based on the estimation approach described in Section 4, I report, in Section 5, that while the first factor tracks observed short-term interest rates very well, the second moves nearly one-to-one with medium- term monetary policy expectations, as inferred by eurodollar futures rates. The close relationship between the second factor and policy expectations is especially remarkable given that futures rates were not used in the estimation of the model. In Section 6, I discuss the main conclusions of the analysis and point to potential directions for further research. 2 Background and Motivation Let Z(t; T )denotethetime-t price of a zero-coupon bond that matures at time T with no risk of default. Assuming, without loss of generality, that the bond has a face value of $1, its continuously compounded yield-to-maturity, R(t; T ), is such that: Z(t; T )=e−R(t;T )(T −t) (1) The continuously compounded time-t forward rate for the future period [T1;T2]canbeshowntobe log(Z(t; T2)) − log(Z(t; T1)) R(t; T1;T2)=− (2) T2 − T1 2 and thus the instantaneous forward rate with maturity T —the time-t forward rate for the infinitesimal future period [T;T + dt]—is written as @log(Z(t; T )) f(t; T )=− (3) @T The instantaneous riskless spot rate, also referred to in this paper as the short rate, is given by: r(t)=f(t; t)(4) Note that by integrating both sides of (3) and combining the result with (1) we obtain R T f(t; S)dS R(t; T )= t (5) T − t which gives the familiar result that the time-t yield on a longer-term zero- coupon bond can be thought of as the average of time-t forward rates pre- vailing over the life of the bond. Intuition and financial reasoning tell us that forward rates should embody expectations of future spot rates (plus a potentially non-zero risk premium). Motivated by this insight, the modeling approach adopted in this paper starts with a model for the evolution of the short rate, which, in turn leads to the specification of the common factors behind yield curve fluctuations. Before turning to the specifics of the modeling framework, however, we will take a closer look at equation (5), which should help us relate the model presented here to other work in the term structure literature. In addition, I shall argue that equation (5) is a useful tool for adding an economic interpretation to the model. 3 2.1 The Expectations Hypothesis and a Role for Mon- etary Policy Equation (5) serves as a basis for a well-known form of the expectations hypothesis of the term structure of interest rates. This version of the hy- pothesis is based on the assumption that the instantaneous forward rate f(t; S) is equal to the expectation of the corresponding future spot rate plus a time-invariant risk premium φ(S − t), f(t; S)=Et[r(S)] + φ(S − t)(6) where Et[:] denotes an expectation conditioned on time-t information, com- puted under the objective probability measure.2 Combining equations (5) and (6), we obtain R T Et[r(S)]dS R(t; T )= t +Φ(T − t)(7) T − t which is the so-called yield-to-maturity form of the expectations hypothesis— see, e.g., Cox, Ingersoll, and Ross (1981)—augmented with a time-invariant risk premium. Equation (7) says that yields on longer-term zero-coupon bonds represent market participants’ expectations of the average level of short-term interest rates over the maturity of the bond plus a risk premium Ê T φ(S−t)dS − ≡ t Φ(T t) T −t . Equation (7) provides a direct link between longer-term interest rates and 2In its \pure" version, this form of the expectations hypothesis assumes that φ(S −t)= 0. In this paper, I work with the weaker form of the hypothesis, which allows for non-zero risk premiums. 4 expectations of future short-term interest rates. A common approach in the economics literature on the term structure of interest rates, which I shall refer to as the “structural approach,” is to assume that short rates are primarily determined by monetary policy and then to propose an equation—often re- ferred to as a “monetary policy reaction function”—relating movements in the short rate to changes in key macroeconomic variables that monetary pol- icymakers are assumed to care about. Thus, using model-based projections for such macro variables, which are effectively taken to be the factors behind movements in the yield curve, one can derive the model-implied forecasts of the short rate and assess the model’s ability to capture the main sources of variation in longer-term yields.3 In this paper, I follow what I shall call a “market-based” approach to estimating (7). Rather than trying to model the behavior of the short rate by assuming a specific reaction function for monetary policymakers, and attempting to forecast the macro variables that enter into that function, I rely almost exclusively on observed interest rates to infer the short rate expectations and associated risk premiums in the minds of financial market participants. Indeed, a key idea behind this paper is to impose a minimum set of assumptions on the evolution of the short rate and to let the whole configuration of market interest rates tell us as much as possible about market participants’ attitudes toward risk and their expectations of where the short rate (and potentially the yield curve) is headed. 3This approach is similar in spirit to the work of Evans and Marshall (1998). 5 3 The Modeling Framework Consider the following model for the evolution of the short rate, r(t), dr(t)=k[θ(t) − r(t)]dt + vdWr(t)(8) dθ(t)=α[β − θ(t)]dt + ηdWθ(t)(9) r(t) is assumed to error-correct toward its time-varying central tendency θ(t) with a mean reversion coefficient k. θ(t) is also assumed to follow a mean-reverting process, with α denoting its speed of mean reversion and β its long-run value. dWr(t)anddWθ(t) are uncorrelated stochastic shocks (infinitesimal increments to standard Brownian motions) hitting the short rate and its central tendency, respectively, and v and η are their volatilities. In terms of the economics of the model, θ(t) can be thought of as where market participants think the monetary authority will take the short rate in the future, an implicit notion of a policy reaction function, and β is the steady-state value of the short rate.