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

TL;DR: This paper proposed a consumption-based model that can account for many features of the nominal term structure of interest rates, such as a time-varying price of risk generated by external habit.

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 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

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.

Figures

Table 4: Moments of Bond Yields

Table 3: Statistics for the Aggregate Market

Figure 9: Adjusted long-Rate Regressions. Lines with plus signs represent coefficients β from the regression

Figure 3: The Price-Dividend Ratio as a Function of Surplus Consumption St. The price-dividend ratio is the ratio of aggregate wealth to aggregate consumption (divided by four to annualize).

Table 2: Utility Parameters

Figure 2: Past Consumption Growth and Interest Rate. This figure plots the history of average past (inflation-adjusted) consumption growth ∑40 j=0 φ j∆ct−j and the continuously compounded rate of return on the 90-day Treasury bill, adjusted for inflation. The parameter φ = 0.95. Variables are de-meaned and standardized.

Figure 1: Expected and Realized Inflation. The dotted line plots quarterly changes in log CPI. The solid line plots expected inflation, conditional on past realized inflation, implied by the estimation of Section 2 and the maximum likelihood estimates given in Table 1.

Figure 6: Long-Rate Regressions. Coefficients βn from the regression

Figure 5: Nominal continuously compounded bond yields as a function of surplus consumption and expected inflation. Solid lines denote the 3-month yield, dotted lines the 10-year yield. Yields are plotted for expected inflation at its unconditional mean, at two unconditional standard deviations below the unconditional mean (upside-down triangles), and at two unconditional standard deviations above the unconditional mean (upright triangles). Yields are in annual terms. The unconditional standard deviation is calculated as ΣΣ′/(1− Φ2).

Figure 7: Time series of the 3-month yield in the data and predicted by the model. The solid line plots the time series of the nominal 3-month yield in quarterly data. The dashed line plots the implied time series when quarterly data on consumption and the price level is fed into the model. Expected inflation is taken to be its mean conditional on past inflation data, given the maximum likelihood estimates in Table 1. Using (2), surplus consumption is generated from actual consumption. Both series are de-meaned.

Table 1: Parameter Estimates

Figure 8: Time series of the yield spread on the five-year bond in the data and predicted by the model. The yield spread is the difference in yields on the five-year nominal bond and the threemonth bond. The solid line plots the time series of the yield spread on between bonds in the data. The dashed line plots the implied time series when quarterly data on consumption and the price level is fed into the model. Expected inflation is taken to be its mean conditional on past inflation data, given the maximum likelihood estimates in Table 1. Using (2), surplus consumption is generated from actual consumption. Both series are de-meaned.

Figure 4: Continuously compounded yields on real and nominal bonds as a function of surplus consumption. Dotted lines denote yields on ten-year nominal and real bonds; solid lines denote yields on three-month nominal and real bonds. Open circles denote nominal bonds; closed circles denote real bonds. For the nominal yields, expected inflation is set equal to its unconditional mean of 1% per quarter. Yields are in annual terms.

Full text

"2-:)67-8=3*)227=0:%2-%"2-:)67-8=3*)227=0:%2-%
',30%60=311327 ',30%60=311327
-2%2')%4)67 #,%6832%'908=)7)%6',

3279148-32%7)(3()03*8,)!)61 869'896)3*28)6)783279148-32%7)(3()03*8,)!)61 869'896)3*28)6)78
%8)7%8)7
)77-'%#%',8)6
"2-:)67-8=3*)227=0:%2-%
3003;8,-7%2(%((-8-32%0;36/7%8,88476)437-836=94)22)(9*2')$4%4)67
%683*8,)-2%2')311327%2(8,)-2%2')%2(-2%2'-%0%2%+)1)28311327
)'311)2()(-8%8-32)'311)2()(-8%8-32
#%',8)63279148-32%7)(3()03*8,)!)61 869'896)3*28)6)78%8)7
3962%03*
-2%2'-%0'3231-'7


,884(<(3-36+..@2)'3
!,-74%4)6-74378)(%8 ',30%60=311327,88476)437-836=94)22)(9*2')$4%4)67
36136)-2*361%8-3240)%7)'328%'86)437-836=43&3<94)22)(9

3279148-32%7)(3()03*8,)!)61 869'896)3*28)6)78%8)73279148-32%7)(3()03*8,)!)61 869'896)3*28)6)78%8)7
&786%'8&786%'8
!,-74%4)6463437)7%'3279148-32&%7)(13()08,%8%''39287*361%2=*)%896)73*8,)231-2%08)61
7869'896)3*-28)6)786%8)7!,)(6-:-2+*36')&),-2(8,)13()0-7%8-1):%6=-2+46-')3*6-7/+)2)6%8)(&=
)<8)62%0,%&-831-2%0&32(7()4)2(324%78'3279148-32+63;8,8,639+,,%&-8%2(32)<4)'8)(
-2A%8-32#,)2'%0-&6%8)(83(%8%32'3279148-32-2A%8-32%2(8,)%++6)+%8)1%6/)88,)13()0
463(9')76)%0-78-'1)%27%2(:30%8-0-8-)73*&32(=-)0(7%2(%''39287*368,))<4)'8%8-32749>>0)!,)
13()0%073'%4896)78,),-+,)59-8=46)1-91%2()<')77783'/1%6/)8:30%8-0-8=
-7'-40-2)7-7'-40-2)7
-2%2')?-2%2')%2(-2%2'-%0%2%+)1)28
!,-7.3962%0%68-'0)-7%:%-0%&0)%8 ',30%60=311327,88476)437-836=94)22)(9*2')$4%4)67

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.

Frequently asked questions

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.