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A Consumption-Based Model of the Term Structure of Interest Rates

AbstractThis paper proposes a consumption-based model that can account for many features of the nominal term structure of interest rates. The driving force behind the model is a time-varying price of risk generated by external habit. Nominal bonds depend on past consumption growth through habit and on expected inflation. When calibrated data on consumption, inflation, and the average level of bond yields, the model produces realistic volatility of bond yields and can explain key aspects of the expectations puzzle documented by Campbell and Shiller (1991) and Fama and Bliss (1987). When Actual consumption and inflation data are fed into the model, the model is shown to account for many of the short and long-run fluctuations in the short-term interest rate and the yield spread. At the same time, the model captures the high equity premium and excess stock market volatility.

Summary (4 min read)

Introduction

  • The negative correlation between surplus consumption and the riskfree rate leads to positive risk premia on real bonds, and an upward sloping yield curve.
  • Expected inflation is calibrated purely to match inflation data.
  • Like these models, the model proposed here assumes that the agent evaluates today’s consumption relative to a reference point that increases with past consumption.

1 Model

  • This section describes the model assumed in this paper.
  • Section 1.1 describes the assumptions for preferences, Section 1.2 describes the assumptions on the price level.
  • Section 1.3 describes the solution method, and Section 1.4 discusses consequences for risk premia on real and nominal bonds.

1.1 Preferences

  • The sensitivity function λ(st) will be described below.
  • In the model of Campbell and Cochrane (1999), the mechanism in (10) does not create timevarying risk premia on bonds for the simple reason that bond returns are constant, and equal to the riskfree rate at all maturities.

1.2 Inflation

  • For simplicity, the authors follow Boudoukh (1993) and Cox, Ingersoll, and Ross (1985), and model inflation as an exogenous process.
  • The correlation between inflation, Zt and consumption can be modeled in a parsimonious way by writing the consumption growth shock vt+1 as vt+1 = σc²t+1.
  • This structure allows for an arbitrary number of state variables and cross-correlations.
  • Multiple lags may be accommodated by increasing the dimension of Zt. 5Harvey (1989) provides direct evidence that the the risk-return tradeoff varies counter-cyclically.
  • 6Since an earlier version of this paper circulated, Buraschi and Jiltsov (2003) study a related model that puts the money supply directly in the utility function.

1.3 Model Solution

  • This section calculates the prices of long-term bonds and stocks.
  • To compute prices on nominal bonds, techniques from affine bond pricing7 are combined with numerical methods.
  • Introducing affine bond pricing techniques improves the efficiency of the calculation and provides insight into the workings of the model.

Bond Prices

  • This paper solves for prices of both real bonds (bonds whose payment is fixed in terms of units of the consumption good) and nominal bonds (bonds whose payoff is fixed in terms of units of the price level).
  • This implies the boundary condition: P0,t = 1.
  • For this problem, numerical integration is superior to calculating the expectation by Monte Carlo.
  • Equation (14) indicates that, unlike real bond prices, nominal bond prices are functions of the state variable Zt as well as st.
  • These formulas can also be used to gain insight into the workings of the model, as explained in Section 1.4.

Aggregate Wealth

  • The market portfolio is equivalent to aggregate wealth, and the dividend equals aggregate consumption.
  • The price-consumption ratio and the return on the market can be calculated using methods similar to those above, with a small but important modification.
  • Because these assets pay no coupons, they have the same recursive pricing relation as bonds (16).
  • Of course the prices are different, and this is because there is a different boundary condition: P e0,t = Ct. 1.
  • This formula can be solved recursively using one-dimensional quadrature.

1.4 Implications for bond risk premia

  • Of interest is the risk premium on the nominal riskfree asset.
  • If σπσc < 0, the one-period nominal bond has a positive risk premium relative to the one-period real bond.
  • Intuitively, this is because σπσ ′ c < 0 implies that inflation and consumption growth are negatively correlated.
  • In general, there is no closed form expression for nominal or real bond prices with maturity greater than one period.
  • These can be determined in some special cases, as described below.

Special cases

  • As long as expected inflation varies, the nominal riskfree rate also varies.
  • These risk premia vary with st, and it is again not possible to solve for bond prices in closed form.
  • Then inflation risk is not priced, and the same reasoning as above shows that P $n,t = exp{−nrf} exp{An +BnZt}.
  • Thus risk premia on nominal bonds are zero except for a constant Jensen’s inequality term.

2 Estimation

  • The results of the previous section suggest that the process assumed for expected inflation will be an important determinant of yields and returns on nominal bonds.
  • This is equivalent to assuming that realized inflation follows an ARMA(1,1) process.
  • Equations (26)–(28) imply an exact likelihood function.
  • The left column reports the parameter estimate, the right column reports the standard error.

3 Implications for Asset Returns

  • This section describes the implications of the model for returns on bonds and stocks.
  • Section 3.1 describes the calibration of the parameters, and the data used to calculate moments of nominal bonds for comparison.
  • Section 3.2 characterizes the price-dividend ratio and the yield spread on real and nominal bonds as functions of the underlying state variables st and expected inflation.
  • Section 3.3 evaluates the model by simulating 100,000 quarters of returns on stocks and nominal and real bonds and compares the simulated moments implied by the model to those on stocks and nominal bonds in the data.
  • Lastly, Section 3.4 shows the implications of the model for the time series of the short-term interest rate and the yield spread, and examines the properties of implied bond risk premia using the technique proposed by Dai and Singleton (2002).

3.1 Calibration

  • The processes for consumption and inflation are calibrated using the estimation of Section 2, while the preference parameters are calibrated using bond and stock returns.9.
  • Then σc and σπ can be found by taking the Cholesky decomposition of the right hand side of (29).
  • Boudoukh fits consumption and inflation parameters to consumption and inflation data, and preference parameters to bond returns.
  • This implies that when the nominal riskfree rate in the model is evaluated at s̄, it equals the yield on the three-month bond.
  • The simulation results in Section 3.3 show that the difference is small.

3.2 Characterizing the Solution

  • As shown in Figure 3, the price-dividend ratio increases with surplus consumption St. As the pricedividend ratio is often taken to be a measure of the business cycle (e.g. Lettau and Ludvingson (2001)), this confirms the intuition that St is a procyclical variable.
  • 10A potential concern with this regression is the relatively high degree of persistence in the surplus consumption ratio.
  • 16 Figure 4 plots the yields on nominal and real bonds for maturities of three months and ten years.
  • Both nominal and real yields decrease with St, but the long yields are more sensitive to St than the short yields.
  • Both long and short-term yields are increasing in expected inflation.

3.3 Simulation

  • To evaluate the predictions of the model for asset returns, 100,000 quarters of data are simulated.
  • Prices of the claim on aggregate consumption , of real, and nominal bonds are calculated numerically, using the method described in Section 1.3.

Returns on the Aggregate Market

  • Table 3 shows the implications of this model for equity returns.
  • The implications of the present model for equity returns are nearly identical to those of Campbell and Cochrane (1999).
  • The model fits the mean and standard deviation of equity returns, even though it was calibrated only to match the ratio.
  • The persistence φ is chosen so that the model fits the correlation of the price-dividend ratio by construction.
  • In addition, results available from the author show that price-dividend ratios have the ability to predict excess returns on equities, just as in the data (Campbell and Shiller (1988), Fama and French (1989)), and that declines in the price-dividend ratio predict higher volatility (Black (1976), Schwert (1989), Nelson (1991)).

Bond Returns

  • Table 4 shows the implications of the model for means and standard deviations of real and nominal bond yields.
  • The model produces average nominal yields that are very similar to those in the data for bonds between maturities of 3 months and 5 years.
  • The previous discussion shows that interest rate risk leads both real and nominal bonds to have positive risk premia.
  • This section shows that risk premia are indeed time-varying, and explains why.
  • 17 While the model succeeds in fitting the pattern of the coefficients in the data, the magnitude of the difference between the slope coefficients and one is smaller in the model than in the data.

3.4 Implications for the Time Series

  • The previous section shows the implications of the model for the population values of aggregate market moments, bond yields, and Campbell and Shiller (1991) regression coefficients.
  • Zt, it is possible to calculate the model’s implications for nominal yields.
  • The argument in Section 3.1 shows that this series is equal to Zt. 20For the 3-month nominal yield, (23) is an approximate closed-form expression.
  • 23 the higher frequency movements in the 70s, and overall, the correlation between the yield spread implied by the model and that in the data is .40.
  • 24 Figure 9 plots the coefficients βRn from the regression (36), along with the coefficients βn from (34) found in the data.

4 Conclusion

  • This paper offers a theory of the nominal term structure based on the preferences of a representative agent.
  • Nevertheless, the implied volatility of yields is close to the sample estimates of nominal yield volatility in the data.
  • This suggests that surplus consumption, which, along with expected inflation drives changes in yields in the model, is a determinant of yields in the data.
  • The second test is whether, when the Campbell-Shiller regressions are adjusted by risk premia on bonds implied by the model, the slope coefficients are closer to unity.
  • In summary, the model is able to capture many of the properties of moments of bond returns in the data, and explain much of the time series variation in short and long-term bond yields.

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The Rodney L. White Center for Financial Research
A Consumption-Based Model of the
Term Structure of Interest Rates
Jessica A. Wachter
27-04

A Consumption-Based Model of the Term Structure
of Interest Rates
Jessica A. Wachter
University of Pennsylvania and NBER
July 9, 2004
I thank Andrew Ang, Ravi Bansal, Michael Brandt, Geert Bekaert, John Campbell, John Cochrane,
Francisco Gomes, Vassil Konstantinov, Martin Lettau, Anthony Lynch, David Marshall, Lasse Pederson,
Andre Perold, Ken Singleton, Christopher Telmer, Jeremy Stein, Matt Richardson, Stephen Ross, Robert
Whitelaw, Yihong Xia, seminar participants at the 2004 Western Finance Association meeting in Vancouver,
the 2003 Society of Economic Dynamics meeting in Paris, and the 2001 NBER Asset Pricing meeting in
Los Angeles, the the NYU Macro lunch, the New York Federal Reserve, Washington University, and the
Wharton School. I thank Lehman Brothers for financial support.
Address: The Wharton School, University of Pennsylvania, 3620 Locust Walk, Philadelphia, PA 19104;
Tel: (215) 898-7634; Email: jwachter@wharton.upenn.edu; http://finance.wharton.upenn.edu/˜ jwachter/

A Consumption-Based Model of the Term Structure
of Interest Rates
Abstract
This paper proposes a consumption-based model that can account for many features of the
nominal term structure of interest rates. The driving force behind the model is a time-varying
price of risk generated by external habit. Nominal bonds depend on past consumption growth
through habit and on expected inflation. When calibrated to data on consumption, inflation, and
the average level of bond yields, the model produces realistic volatility of bond yields and can
explain key aspects of the expectations puzzle documented by Campbell and Shiller (1991) and
Fama and Bliss (1987). When actual consumption and inflation data are fed into the model, the
model is shown to account for many of the short and long-run fluctuations in the short-term interest
rate and the yield spread. At the same time, the model captures the high equity premium and
excess stock market volatility.

Citations
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Journal ArticleDOI
Abstract: Are there important cyclical fluctuations in bond market premiums and, if so, with what macroeconomic aggregates do these premiums vary? We use the methodology of dynamic factor analysis for large datasets to investigate possible empirical linkages between forecastable variation in excess bond returns and macroeconomic fundamentals. We find that “real” and “inflation” factors have important forecasting power for future excess returns on U.S. government bonds, above and beyond the predictive power contained in forward rates and yield spreads. This behavior is ruled out by commonly employed affine term structure models where theforecastability ofbondreturns andbondyields is completely summarized by the cross-section of yields or forward rates. An important implication of these findings is that the cyclical behavior of estimated risk premia in both returns and long-term yields depends importantly on whether the information in macroeconomic factors is included in forecasts of excess bond returns. Without the macro factors, risk premia appear virtually acyclical, whereas with the estimated factors risk premia have a marked countercyclical component, consistent with theories that imply investors must be compensated for risks associated with macroeconomic activity. (JEL E0, E4, G10, G12)

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Cites background or methods from "A Consumption-Based Model of the Te..."

  • ...For example, the real and in ation factors we study may be reasonable proxies for the consumption and in ations shocks that enter models of time-varying risk premia like those of Campbell and Cochrane (1999), Brandt and Wang (2003) and Wachter (2006)....

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  • ...For example, Campbell and Cochrane (1999) and Wachter (2006) study models in which risk aversion varies over the business cycle and is low in good times when the economy is growing quickly....

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  • ...Wachter (2006) adapts the Campbell-Cochrane habit model to examine the nominal term structure of interest rates, and shows that bond risk premia (as well as equity premia) should vary with the slow-moving consumption habit....

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Abstract: This paper incorporates a time-varying intensity of disasters in the Rietz-Barro hypothesis that risk premia result from the possibility of rare, large disasters. During a disaster, an asset’s fundamental value falls by a time-varying amount. This in turn generates time-varying risk premia and thus volatile asset prices and return predictability. Using the recent technique of linearity-generating processes (Gabaix 2007), the model is tractable, and all prices are exactly solved in closed form. In the “variable rare disasters” framework, the following empirical regularities can be understood qualitatively: (i) equity premium puzzle (ii) risk-free rate-puzzle (iii) excess volatility puzzle (iv) predictability of aggregate stock market returns with price-dividend ratios (v) value premium (vi) often greater explanatory power of characteristics than covariances for asset returns (vii) upward sloping nominal yield curve (viiii) a steep yield curve predicts high bond excess returns and a fall in long term rates (ix) corporate bond spread puzzle (x) high price of deep out-of-the-money puts. I also provide a calibration in which those puzzles can be understood quantitatively as well. The fear of disaster can be interpreted literally, or can be viewed as a tractable way to model time-varying risk-aversion or investor sentiment. (JEL: E43, E44, G12)

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Cites background from "A Consumption-Based Model of the Te..."

  • ...In the models of Bekaert, Engstrom, and Grenadier (2005) and Wachter (2006), increases in risk aversion unambiguously increase equity and bond premiums, but their effect on interest rates is actually ambiguous....

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Abstract: This paper quanties how variation in real economic activity and ination in the U.S. inuenced the market prices of level, slope, and curvature risks in U.S. Treasury markets. To accomplish this we develop a novel arbitrage-free DTSM in which macroeconomic risks{ in particular, real output and ination risks{ impact bond investment decisions separately from information about the shape of the yield curve. Estimates of our preferred macro-DTSM over the twenty-three year period from 1985 through 2007 reveal that unspanned macro risks explained a substantial proportion of the variation in forward terms premiums. Unspanned macro risks accounted for nearly 90% of the conditional variation in short-dated forward term premiums, with unspanned real economic growth being the key driving factor. Over horizons beyond three years, these eects were entirely attributable to unspanned ination. Using our model, we also reassess some of Chairman Bernanke’s remarks on the interplay between term premiums, the shape of the yield curve, and macroeconomic activity.

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Abstract: We show that bond risk-premia rise with uncertainty about expected inflation and fall with uncertainty about expected growth; the magnitude of return predictability using these two uncertainty measures is similar to that by multiple yields. Motivated by this evidence, we develop and estimate a long-run risks model with time-varying volatilities of expected growth and inflation. The model simultaneously accounts for bond return predictability and violations of uncovered interest parity in currency markets. We find that preference for early resolution of uncertainty, time-varying volatilities, and non-neutral effects of inflation on growth are important to account for these aspects of asset markets.

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References
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Journal ArticleDOI
Abstract: This paper introduces an ARCH model (exponential ARCH) that (1) allows correlation between returns and volatility innovations (an important feature of stock market volatility changes), (2) eliminates the need for inequality constraints on parameters, and (3) allows for a straightforward interpretation of the "persistence" of shocks to volatility. In the above respects, it is an improvement over the widely-used GARCH model. The model is applied to study volatility changes and the risk premium on the CRSP Value-Weighted Market Index from 1962 to 1987. Copyright 1991 by The Econometric Society.

9,393 citations


Additional excerpts

  • ...In addition, results available from the author show that price–dividend ratios have the ability to predict excess returns on equities, just as in the data (Campbell and Shiller, 1988; Fama and French, 1989), and that declines in the price–dividend ratio predict higher volatility (Black, 1976; Schwert, 1989; Nelson, 1991)....

    [...]

  • ...…from the author show that price–dividend ratios have the ability to predict excess returns on equities, just as in the data (Campbell and Shiller, 1988; Fama and French, 1989), and that declines in the price–dividend ratio predict higher volatility (Black, 1976; Schwert, 1989; Nelson, 1991)....

    [...]


Journal ArticleDOI
Abstract: This paper uses an intertemporal general equilibrium asset pricing model to study the term structure of interest rates. In this model, anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices. Many of the factors traditionally mentioned as influencing the term structure are thus included in a way which is fully consistent with maximizing behavior and rational expectations. The model leads to specific formulas for bond prices which are well suited for empirical testing. 1. INTRODUCTION THE TERM STRUCTURE of interest rates measures the relationship among the yields on default-free securities that differ only in their term to maturity. The determinants of this relationship have long been a topic of concern for economists. By offering a complete schedule of interest rates across time, the term structure embodies the market's anticipations of future events. An explanation of the term structure gives us a way to extract this information and to predict how changes in the underlying variables will affect the yield curve. In a world of certainty, equilibrium forward rates must coincide with future spot rates, but when uncertainty about future rates is introduced the analysis becomes much more complex. By and large, previous theories of the term structure have taken the certainty model as their starting point and have proceeded by examining stochastic generalizations of the certainty equilibrium relationships. The literature in the area is voluminous, and a comprehensive survey would warrant a paper in itself. It is common, however, to identify much of the previous work in the area as belonging to one of four strands of thought. First, there are various versions of the expectations hypothesis. These place predominant emphasis on the expected values of future spot rates or holdingperiod returns. In its simplest form, the expectations hypothesis postulates that bonds are priced so that the implied forward rates are equal to the expected spot rates. Generally, this approach is characterized by the following propositions: (a) the return on holding a long-term bond to maturity is equal to the expected return on repeated investment in a series of the short-term bonds, or (b) the expected rate of return over the next holding period is the same for bonds of all maturities. The liquidity preference hypothesis, advanced by Hicks [16], concurs with the importance of expected future spot rates, but places more weight on the effects of the risk preferences of market participants. It asserts that risk aversion will cause forward rates to be systematically greater than expected spot rates, usually

6,763 citations


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Abstract: Restrictions that a class of general equilibrium models place upon the average returns of equity and Treasury bills are found to be strongly violated by the U.S. data in the 1889–1978 period. This result is robust to model specification and measurement problems. We conclude that, most likely, an equilibrium model which is not an Arrow-Debreu economy will be the one that simultaneously rationalizes both historically observed large average equity return and the small average risk-free return.

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"A Consumption-Based Model of the Te..." refers methods in this paper

  • ...Thus, the model can fit the equity premium puzzle of Mehra and Prescott (1985)....

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Abstract: Expected returns on common stocks and long-term bonds contain a term or maturity premium that has a clear business-cycle pattern (low near peaks, high near troughs). Expected returns also contain a risk premium that is related to longer-term aspects of business conditions. The variation through time in this premium is stronger for low-grade bonds than for high-grade bonds and stronger for stocks than for bonds. The general message is that expected returns are lower when economic conditions are strong and higher when conditions are weak.

3,953 citations


Additional excerpts

  • ...In addition, results available from the author show that price–dividend ratios have the ability to predict excess returns on equities, just as in the data (Campbell and Shiller, 1988; Fama and French, 1989), and that declines in the price–dividend ratio predict higher volatility (Black, 1976; Schwert, 1989; Nelson, 1991)....

    [...]

  • ...…from the author show that price–dividend ratios have the ability to predict excess returns on equities, just as in the data (Campbell and Shiller, 1988; Fama and French, 1989), and that declines in the price–dividend ratio predict higher volatility (Black, 1976; Schwert, 1989; Nelson, 1991)....

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Abstract: We present a consumption†based model that explains a wide variety of dynamic asset pricing phenomena, including the procyclical variation of stock prices, the long†horizon predictability of excess stock returns, and the countercyclical variation of stock market volatility. The model captures much of the history of stock prices from consumption data. It explains the short†and long†run equity premium puzzles despite a low and constant risk†free rate. The results are essentially the same whether we model stocks as a claim to the consumption stream or as a claim to volatile dividends poorly corelated with consumption. The model is driven by an independently and identically distributed consumption growth process and adds a slow †moving external habit to the standard power utility function. These features generate slow countercyclical variation in risk premia. The model posits a fundamentally novel description of risk premia. Investors fear stocks primarily because they do poorly in recessions unrelated to the risks of long†run average consumption growth.

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Q1. What are the contributions in "A consumption-based model of the term structure of interest rates" ?

This paper proposes a consumption-based model that can account for many features of the nominal term structure of interest rates.