Inflation Bets or Deflation Hedges

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Ináation Bets or Deáation Hedges? The Changing Risks of Nominal Bonds John Y. Campbell, Adi Sunderam, and Luis M. Viceira 1 First draft: June 2007 This version: March 21, 2011 1 Campbell: Department of Economics, Littauer Center, Harvard University, Cambridge MA 02138, USA, and NBER. Email john_campbell@harvard.edu. Sunderam: Harvard Business School, Boston MA 02163. Email asunderam@hbs.edu. Viceira: Harvard Business School, Boston MA 02163 and NBER. Email lviceira@hbs.edu. We acknowledge the extraordinarily able research assistance of Johnny Kang. We are grateful to Geert Bekaert, Andrea Buraschi, Jesus Fernandez-Villaverde, Wayne Ferson, Javier Gil-Bazo, Pablo Guerron, John Heaton, Ravi Jagannathan, Jon Lewellen, Monika Piazzesi, Pedro Santa-Clara, George Tauchen, and seminar participants at the 2009 Annual Meeting of the American Finance Association, Bank of England, European Group of Risk and Insurance Economists 2008 Meeting, Sixth Annual Empirical Asset Pricing Retreat at the University of Amsterdam Business School, Harvard Business School Finance Unit Research Retreat, Imperial College, Marshall School of Business, NBER Fall 2008 Asset Pricing Meeting, Norges Bank, Society for Economic Dynamics 2008 Meeting, Stockholm School of Economics, Tilburg University, Tuck Business School, and Universidad Carlos III in Madrid for helpful comments and suggestions. This material is based upon work supported by the National Science Foundation under Grant No. 0214061 to Campbell, and by Harvard Business School Research Funding.Abstract The covariance between US Treasury bond returns and stock returns has moved considerably over time. While it was slightly positive on average in the period 1953ñ 2009, it was unusually high in the early 1980ís and negative in the 2000ís, particularly in the downturns of 2001ñ2 and 2008ñ9. This paper speciÖes and estimates a model in which the nominal term structure of interest rates is driven by four state variables: the real interest rate, temporary and permanent components of expected ináation, and the ìnominal-real covarianceî of ináation and the real interest rate with the real economy. The last of these state variables enables the model to Öt the changing covariance of bond and stock returns. Log bond yields and term premia are quadratic in these state variables, with term premia determined by the nominal-real covariance. The concavity of the yield curveó the level of intermediate-term bond yields, relative to the average of short- and long-term bond yieldsó is a good proxy for the level of term premia. The nominal-real covariance has declined since the early 1980ís, driving down term premia.1 Introduction Are nominal government bonds risky investments, which investors must be rewarded to hold? Or are they safe investments, whose price movements are either inconsequential or even beneÖcial to investors as hedges against other risks? US Treasury bonds performed well as hedges during the Önancial crisis of 2008ñ9, but the opposite was true in the early 1980ís. The purpose of this paper is to explore such changes over time in the risks of nominal government bonds. To understand the phenomenon of interest, consider Figure 1, an update of a similar Ögure in Viceira (2010). The Ögure shows the history of the realized beta (regression coe¢ cient) of 10-year nominal zero-coupon Treasury bonds on an aggregate stock index, calculated using a rolling three-month window of daily data. This beta can also be called the ìrealized CAPM betaî, as its forecast value would be used to calculate the risk premium on Treasury bonds in the Capital Asset Pricing Model (CAPM) that is often used to price individual stocks. Figure 1 displays considerable high-frequency variation, much of which is attributable to noise in the realized beta. But it also shows interesting low-frequency movements, with values close to zero in the mid-1960ís and mid-1970ís, much higher values averaging around 0.4 in the 1980ís, a spike in the mid-1990ís, and negative average values in the 2000ís. During the two downturns of 2001ñ3 and 2008ñ9, the average realized beta of Treasury bonds was about -0.2. These movements are large enough to cause substantial changes in the Treasury bond risk premium implied by the CAPM. Nominal bond returns respond both to expected ináation and to real interest rates. A natural question is whether the pattern shown in Figure 1 is due to the changing beta of ináation with the stock market, or of real interest rates with the stock market. Figure 2 summarizes the comovement of ináation shocks with stock returns, using a rolling three-year window of quarterly data and a Örst-order quarterly vector autoregression for ináation, stock returns, and the three-month Treasury bill yield to calculate ináation shocks. Because high ináation is associated with high bond yields and low bond returns, the Ögure shows the beta of realized deáation shocks (the negative of ináation shocks) which should move in the same manner as the bond return beta reported in Figure 1. Indeed, Figure 2 shows a similar history for the deáation beta as for the nominal bond beta. Real interest rates also play a role in changing nominal bond risks. In the period 1since 1997, when long-term Treasury ináation-protected securities (TIPS) were Örst issued, Campbell, Shiller, and Viceira (2009) report that TIPS have had a predominantly negative beta with stocks. Like the nominal bond beta, the TIPS beta was particularly negative in the downturns of 2001ñ3 and 2008ñ9. Thus not only the stock-market covariances of nominal bond returns, but also the covariances of two proximate drivers of those returns, ináation and real interest rates, change over time. In the CAPM, assetsí risk premia are fully explained by their covariances with the aggregate stock market. Other modern asset pricing models allow for other ináuences on risk premia, but still generally imply that stock-market covariances have considerable explanatory power for risk premia. Time-variation in the stock-market covariances of bonds should then be associated with variation in bond risk premia, and therefore in the typical shape of the Treasury yield curve. Yet the enormous literature on Treasury bond prices has paid little attention to this phenomenon. This paper begins to Öll this gap in the literature. We make three contributions. First, we write down a simple term structure model that captures time-variation in the covariances of ináation and real interest rates, and therefore of nominal bond returns, with the real economy and the stock market. Importantly, the model allows these covariances, and the associated risk premia, to change sign. It also incorporates more traditional ináuences on nominal bond prices, speciÖcally, real interest rates and both transitory and temporary components of expected ináation. Second, we estimate the parameters of the model using postwar quarterly US time series for nominal and ináation-indexed bond yields, stock returns, realized and forecast ináation, and realized second moments of bond and stock returns calculated from daily data within each quarter. The use of realized second moments, unusual in the term structure literature, forces our model to Öt the phenomenon of interest. Third, we use the estimated model to describe how the changing stock-market covariance of bonds should have altered bond risk premia and the shape of the Treasury yield curve. The organization of the paper is as follows. Section 2 reviews the related literature. Section 3 presents our model of the real and nominal term structures of interest rates. Section 4 describes our estimation method and presents parameter estimates and historical Ötted values for the unobservable state variables of the model. Section 5 discusses the implications of the model for the shape of the yield curve and the movements of risk premia on nominal bonds. Section 6 concludes. An Appendix to this paper available online (Campbell, Sunderam, and Viceira 2010) presents details of the model solution and additional empirical results. 22 Literature Review Nominal bond risks can be measured in a number of ways. A straightforward approach is to measure the covariance of nominal bond returns with some measure of the marginal utility of investors. According to the Capital Asset Pricing Model (CAPM), for example, marginal utility can be summarized by the level of aggregate wealth. It follows that the risk of bonds can be measured by the covariance of bond returns with returns on the market portfolio, often proxied by a broad stock index. Alternatively, one can measure the risk premium on nominal bonds, either from average realized excess bond returns or from variables that predict excess bond returns such as the yield spread (Shiller, Campbell, and Schoenholtz 1983, Fama and Bliss 1987, Campbell and Shiller 1991) or a more general linear combination of forward rates (Stambaugh 1988, Cochrane and Piazzesi 2005). If the risk premium is large, then presumably investors regard bonds as risky. This approach can be combined with the Örst one by estimating an empirical multifactor model that describes the cross-section of both stock and bond returns (Fama and French 1993). These approaches are appealingly direct. However, the answers they give depend sensitively on the sample period that is used. The covariance of nominal bond returns with stock returns, in particular, is extremely unstable over time and even switches sign (Li 2002, Guidolin and Timmermann 2006, Christiansen and Ranaldo 2007, David and Veronesi 2009, Baele, Bekaert, and Inghelbrecht 2010, Viceira 2010). The average level of the nominal yield spread is also unstable over time as pointed out by Fama (2006) among others. An intriguing fact is that the movements in the average yield spread seem to line up to some degree with the movements in the CAPM beta of bonds. The average yield spread, like the CAPM beta of bonds, was lower in the 1960ís and 1970ís than in the 1980ís and 1990ís. Viceira (2010) shows that both the short-term nominal interest rate and the yield spread forecast the CAPM beta of bonds over the period 1962ñ2007. On the other hand, during the 2000ís the CAPM beta of bonds was unusually low while the yield spread was fairly high on average. Another way to measure the risks of nominal bonds is to decompose their returns into several components arising from di§erent underlying shocks. Nominal bond returns are driven by movements in real interest rates, ináation expectations, and the risk premium on nominal bonds over short-term bills. Several papers, including Barsky (1989), Shiller and Beltratti (1992), and Campbell and Ammer (1993) have estimated the covariances of these components with stock returns, assuming the 3covariances to be constant over time. The literature on a¢ ne term structure models also proceeds by modelling state variables that drive interest rates and estimating prices of risk for each one. Many papers in this literature allow the volatilities and risk prices of the state variables to change over time, and some allow risk prices and hence risk premia to change sign. 2 Several recent a¢ ne term structure models, including Dai and Singleton (2002) and Sangvinatsos and Wachter (2005), are highly successful at Ötting the moments of nominal bond yields and returns. Some papers have also modelled stock and bond prices jointly, but no existing models allow bond-stock covariances to change sign. 3 The contributions of our paper are Örst, to write down a simple term structure model that allows for bond-stock covariances that can move over time and change sign, and second, to confront this model with historical US data. The purpose of the model is to Öt new facts about bond returns in relation to the stock market, not to improve on the ability of a¢ ne term structure models to Öt bond market data considered in isolation. Our introduction of a time-varying covariance between state variables and the stochastic discount factor, which can switch sign, means that we cannot write log bond yields as a¢ ne functions of macroeconomic state variables; our model, like those of Beaglehole and Tenney (1991), Constantinides (1992), Ahn, Dittmar and Gallant (2002), and Realdon (2006), is linear-quadratic. 4 To solve our model, we use a general result on the expected value of the exponential of a non-central chi-squared 2Dai and Singleton (2002), Bekaert, Engstrom, and Grenadier (2005), Sangvinatsos and Wachter (2005), Wachter (2006), Buraschi and Jiltsov (2007), and Bekaert, Engstrom, and Xing (2009) specify term structure models in which risk aversion varies over time, ináuencing the shape of the yield curve. These papers take care to remain in the essentially a¢ ne class described by Du§ee (2002). 3 Bekaert et al. (2005) and other recent authors including Mamaysky (2002) and díAddona and Kind (2006) extend a¢ ne term structure models to price stocks as well as bonds. Bansal and Shaliastovich (2010), Eraker (2008), and Hasseltoft (2008) price both stocks and bonds in the longrun risks framework of Bansal and Yaron (2004). Piazzesi and Schneider (2006) and Rudebusch and Wu (2007) build a¢ ne models of the nominal term structure in which a reduction of ináation uncertainty drives down the risk premia on nominal bonds towards the lower risk premia on ináationindexed bonds. Similarly, Backus and Wright (2007) argue that declining uncertainty about ináation explains the low yields on nominal Treasury bonds in the mid-2000ís. 4Du¢ e and Kan (1996) point out that linear-quadratic models can often be rewritten as a¢ ne models if we allow the state variables to be bond yields rather than macroeconomic fundamentals. Buraschi, Cieslak, and Trojani (2008) also expand the state space to obtain an a¢ ne model in which correlations can switch sign. 4distribution which we take from the Appendix to Campbell, Chan, and Viceira (2003). To estimate the model, we use a nonlinear Öltering technique, the unscented Kalman Ölter, proposed by Julier and Uhlmann (1997), reviewed by Wan and van der Merwe (2001), and recently applied in Önance by Binsbergen and Koijen (2008). 3 A Quadratic Bond Pricing Model We now present a term structure model that allows for time variation in the covariances between real interest rates, ináation, and the real economy. In the model, both real and nominal bond yields are linear-quadratic functions of the vector of state variables and, consistent with the empirical evidence, the conditional volatilities and covariances of excess returns on real and nominal assets are time varying. 3.1 The SDF and the real term structure We start by assuming that the log of the real stochastic discount factor (SDF), mt+1 = log (Mt+1), follows the process: mt+1 = xt + 2 m 2 + "m;t+1; (1) whose drift xt follows an AR(1) process subject to a heteroskedastic shock t "x;t+1 and a homoskedastic shock "X;t+1: xt+1 = x (1 x ) + xxt + t "x;t+1 + "X;t+1: (2) The innovations "m;t+1, "x;t+1, and "X;t+1 are normally distributed, with zero means and constant variance-covariance matrix. We allow these shocks to be cross-correlated and adopt the notation 2 i to describe the variance of shock "i , and ij to describe the covariance between shock "i and shock "j . To reduce the complexity of the equations that follow, we assume that the shocks to xt are orthogonal to each other; that is, xX = 0. The state variable xt is the short-term log real interest rate. The price of a single-period zero-coupon real bond satisÖes P1;t = Et [exp fmt+1g] ;so that its yield 5