Inflation risk premia and the expectations hypothesis☆

An enormous literature discusses the estimation and properties of dynamic models of the term structure of interest rates. A significant part of this literature focuses on reduced-form affine models.¹ In this paper, first we study the link between these models and the real business cycle literature. We then quantify the properties of the inflation risk premium and study the ability of a general equilibrium model to explain deviations from the expectations hypothesis of interest rates.

The first generation of (completely) affine models makes three assumptions to derive implications about the nominal yield curve. First, the spot interest rate is an affine function of a set of mean-reverting state variables with constant or square-root local volatility. Second, the price of risk is a constant multiple of the local interest rate volatility. Third, inflation is neutral so that the Fisher relation between nominal and real interest rates holds.

Empirical studies of this class of models have exposed several limitations. With regards to the second assumption, Duffee (2002) shows that the restriction on the market price of risk implies bond returns and Sharpe ratios that are too high with respect to the empirical evidence. Dai and Singleton (2000) show that this same assumption makes affine models unable to explain the extent of the deviation from the expectations hypothesis of interest rates. They state that “a three factor CIR-style [Cox, Ingersoll, and Ross] model is wholly incapable of matching linear projection yield (LPY ) coefficients. We attribute this model failure to the constraint in CIR-style models that risk premiums are proportional to factor volatilities”. With regards to the third assumption, there is mounting evidence against the Fisher neutrality assumption. Benninga and Protopapadakis (1983), Fama (1990), and Boudoukh (1993) find that the inflation rate is negatively related to the real interest rate in terms of both realized changes and expected values. Moreover, real returns on nominal bonds decline when inflation increases (Fama, 1976b, Fama, 1990, Fama and Gibbons, 1982).²

Partially in response to these limitations, recent studies have explored more flexible models. Duffee (2002) and Dai and Singleton (2000) estimate reduced-form (essentially) affine models in which the price of risk is specified as a more general (ad hoc) function of the state variables. They identify and discuss the specific features that improve the empirical performance of this class of models.

In this paper, instead of assuming these features as exogenous, we develop and estimate a structural model in which some of these features arise in equilibrium. We explore a monetary version of a real business cycle production economy in which, in equilibrium, the term structure of interest rates has the following properties. First, although the state variables follow affine stochastic processes, the market price of risk is not a constant multiple of the local volatility of interest rates. Second, the inflation risk premium is positive and time varying, i.e., the Fisher hypothesis does not hold. Third, the previous two features make the term structure deviate from the expectations hypothesis of interest rates. Thus, the model can potentially match the empirical LPY coefficients.

The structural model allows us to identify the underlying nominal and real factors and to address a number of economic questions. However, this approach has its shortcomings. First, it shares the known limitations of real business cycle models (Cooley and Hansen, 1995). Second, while the market price of risk is state-dependent, it is not as flexible as the one advocated in the reduced-form approach by Duffee (2002). We study the extent to which the model can describe the dynamics of the term structure despite these limitations. Our structural model is classical in many respects: time-separable preferences, a representative agent, diffusive information, and a constant-returns-to-scale production function. Its main distinguishing features are as follows.

First, we assume a nominal fiscal system. This feature generates the departure from the Fisher hypothesis. In classical monetary real business cycle models, inflation and money are not neutral because if agents anticipate an increase in inflation, they substitute away from activities that use cash in favor of activities that do not. We explore a different and arguably more important channel of monetary nonneutrality. When the fiscal system is not indexed to the general price level, i.e., when taxes and fiscal incentives are calculated on nominal historical values, the inflation rate affects the after-tax real return on capital. This, in turn, affects ex ante decisions on the optimal allocation of (real) resources and therefore also affects both asset prices and risk premia (see Feldstein et al., 1978, Feldstein, 1980, Fisher and Modigliani, 1978). Examples of the nominal nature of the fiscal system include depreciation, capital gains, and interest payments on debt.³ When the fiscal system is based on nominal historical values, inflation is a risk factor with asset pricing implications. Given the dimension of the taxable base and tax rates, the fiscal system is of first-order importance.

Second, monetary policy is endogenous. Similar in spirit to a Taylor (1993) policy rule, the money supply consists of a constant long-term rate plus a term that depends on the gap between the current levels of inflation and output and their long-term targets. This feature allows us to distinguish between exogenous monetary shocks and monetary injections motivated by real shocks.

Third, the real marginal productivity of capital follows a square-root process with stochastic drift. The state variables affecting the drift follow square-root processes with correlated Brownian motions. This setup can capture differential effects of the state variables on the marginal productivity of capital. Under certain parameter configurations, the state variables can independently affect either the instantaneous return on capital or its local variance. This allows us to model separately the uncertainty (unexpected innovations) in the marginal productivity of capital and the volatility of expected innovations in the marginal productivity of capital. We show the importance of this feature to generating a state-dependent market price of risk.

We begin by fully characterizing the stochastic equilibrium of the model. We then estimate the structural parameters using U.S. Treasury bond data and study a number of economic implications. We address several questions. First, can a classical monetary business cycle model generate an affine term structure with a price of risk sufficiently flexible to address Duffee's critique? We show a model that generates an endogenous equilibrium market price of risk that is not a constant multiple of the interest rate volatility. The price of risk is state-dependent and can explain the conditional volatility of interest rates better than a traditional CIR model.

Second, what is the size of the U.S. inflation risk premium? We find that over the last 40 years, the average one-month inflation risk premium has been 15 basis points. However, the average ten-year inflation risk premium has been 70 basis points. The term structure of the inflation risk premium is sharply upward sloping, with the long-term inflation risk premium about four times larger than the short-term premium. The size of the long-term inflation risk premium is a large component of the yield spread between nominal and real bonds. Moreover, the inflation risk premium shows time variation over the business cycle, from 20 to 140 basis points.

Third, can a structural model explain the size of the deviation from the expectations hypothesis (EH) of interest rates? What structural reasons drive such deviations? Our monetary model generates a highly time-varying forward risk premium. The extent of time-variation is sufficient to reject the EH. We analytically solve for the model-implied Campbell-Shiller (1991) regression coefficients. We find that they are not statistically different from those obtained by Campbell and Shiller using empirical data. We then decompose the total risk premium into two components. The first is generated by monetary shocks and the second by real shocks. We find that the monetary factor accounts for 43% of the time variation of the risk premium.

This paper draws on contributions from several streams of literature. In addition to the reduced-form affine term structure literature, discussed earlier, our asset pricing model is closely related to monetary versions of real business cycle models.⁴ As such, our model inherits many of their advantages and disadvantages. For instance, in the absence of frictions, time-separable real business cycle models find it difficult to replicate output growth persistence (Cooley and Hansen, 1995) and the equity premium. For this reason, classical real business cycle models have been generalized to incorporate nominal rigidities, more realistic financial intermediation mechanisms, and nonseparable preferences. Blanchard and Kiyotaki (1987) study a static economy with both wage and price stickiness. Chari et al. (1996) consider a more general real business cycle model with price stickiness. Erceg (1997), Erceg et al. (2000), and Huang and Liu (1999) analyze the effects of exogenous monetary policy shocks in a model with wage contracts à la Taylor. Christiano et al. (2001) consider a model with staggered wage contracts and variable capital utilization. They show that these nominal rigidities can account for observed inertia in inflation and persistence in output. Although they do not derive explicit implications for the inflation risk premium, it is reasonable to expect that these nominal rigidities would increase the inflation risk premium, implied by models with no frictions, by distorting the optimal allocation of capital and increasing the persistence of monetary and technological shocks.

Cooley and Nam, 1998, Dai and Singleton, 2000 study a different monetary channel. They incorporate a debt-contracting problem with costly verification into a standard real business cycle model with limited participation. Financial intermediaries are initially uninformed and must pay a price to obtain information. This feature affects loan intermediation and amplifies the response of capital to the money supply shock.⁵

Our work is also related to Bakshi and Chen (1996), who study a monetary economy in which positive monetary holdings are supported in equilibrium. They derive closed-form solutions for the nominal term structure of interest rates in a model that abstracts from tax distortions. The main differences in our model are: (a) the monetary policy is endogenous (the nominal money supply is allowed to change in response to deviations from monetary and real targets); (b) we introduce taxes in the model and allow for an imperfect indexation mechanism to nominal shocks; and (c) we let the investment opportunity set be affected by inflation innovations so that there exists a (time-varying) risk premium on the inflation rate. The equilibrium process of the general price level is affected by both supply and demand factors. Our results are also related to Evans (1998), who uses data on U.K. index-linked bonds to estimate the real term structure and the risk premium, and to Pennacchi (1991), who studies a generalized Vasiček economy with constant volatility in which the time series for the expected inflation is obtained from survey data as opposed to being estimated. An important advantage of our model is to allow the risk premia to be time varying.⁶ Other related papers include Longstaff and Schwartz (1992), Marshall, 1992, Masulis and DeAngelo, 1980, the two-factor Cox et al. (1985b) model, Constantinides (1992), and Duffie and Kan (1996). However, none of these papers explicitly account for the risk premium on the inflation rate.

The paper is organized as follows. Section 2 sets up the model and characterizes the equilibrium nominal and real term structures of interest rates. Section 3 describes empirical testable restrictions and the econometric method. Section 4 describes the dataset. Section 5 summarizes the results of estimating and testing the structural model. Section 6 illustrates the properties of the inflation risk premium. Section 7 explores the extent to which the model can explain deviations from the expectations hypothesis. Section 8 discusses the implications of the model in terms of conditional second moments. Section 9 studies the tradeoff between fitting asset prices and other macroeconomic variables. Section 10 concludes.