Term Premium Dynamics and the Taylor Rule Michael Gallmeyer (Texas A&M) Francisco Palomino (Michigan) Burton Hollifield (Carnegie Mellon) Stanley Zin (Carnegie Mellon) Bank of Canada Conference on Fixed Income Markets
Motivation Empirical success of NA affine term-structure models. Essentially vs. completely affine: Essentially more flexible. Limited economic interpretations of these models. Ideal to link back to the macroeconomy. Identify the latent state variables. Macro aggregates. Monetary policy state variables. Determine the pricing kernel through g.e. restrictions. Model monetary authority setting a short-term nominal rate, i (1) t imposes additional restrictions. = f (macro variables), Term Premium Dynamics and the Taylor Rule 2/21
Questions Can we provide an economic interpretation in conjunction with an interest rate policy rule to an essentially affine model? What can we learn about term premiums when inflation is determined by an interest rate policy rule? Is monetary policy an important source of long-term interest rate variability? Can we learn about policy regimes from long-term rates? Term Premium Dynamics and the Taylor Rule 3/21
Approach and Findings Endowment economy with: preference shocks, an interest rate policy rule to pin down inflation, Leads to an essentially affine equilibrium model for yields. The interest rate rule helps capture an upward-sloping yield curve, volatile long-term yields, & macroeconomic dynamics. Recent features of interest rates are consistent with a more aggressive response to inflation in monetary policy. Term Premium Dynamics and the Taylor Rule 4/21
Related literature Wachter (2006) - Campbell-Cochrane habits. Exogenous inflation. Piazzesi & Schneider (2006) - Recursive utility & learning. Exogenous inflation. Inflation is bad news for consumption. Buraschi & Jiltsov (2007) - Campbell-Cochrane habits. Money supply determines inflation. Gallmeyer, Hollifield, & Zin (2005) & Palomino (2007) - New Keynesian macro model with an affine term structure. Inflation determined by monetary policy & firms staggered price setting. Gallmeyer et al. (2007) - Recursive utility & stochastic volatility. An interest-rate policy rule determines inflation. Endogenous negative correlation between inflation & consumption. Term Premium Dynamics and the Taylor Rule 5/21
Nominal Yields Across Maturity 9 Mean Nominal Yields 8 7 1970:1 to 1989:4 1970:1 to 2005:4 % 6 1990:1 to 2005:4 5 4 0 1 2 3 4 5 6 7 8 9 10 Maturity (Years) 3 Std. Dev. Nominal Yields 2.5 1970:1 to 2005:4 % 2 1970:1 to 1989:4 1.5 1990:1 to 2005:4 1 0 1 2 3 4 5 6 7 8 9 10 Maturity (Years) Term Premium Dynamics and the Taylor Rule 6/21
Completely vs. Essentially Affine Models Completely affine pricing kernel: log M t+1 = Γ 0 + Γ 1 s t + λσ(s t ) 1/2 ε t+1. Essentially affine pricing kernel: log M t+1 = Γ 0 + Γ 1 s t + 1 2 λ(s t) Σλ(s t ) + λ(s t ) Σ 1/2 ε t+1 with λ(s t ) = λ 0 + λ 1 s t. Interest rates: (n) ni e t = E t [M t+n ] i (n) t = 1 [ An + B ] n s t. n Term Premium Dynamics and the Taylor Rule 7/21
Long Rate Volatility in Essentially Affine Models 1 σ(i (n) t ) σ(i (1) t ) = 1 n 1 Φ n λ 1 Φ λ, Φ λ = [Φ Σλ 1 ]. 0.8 0.6 0.4 Φ λ =0.1 Φ λ =0.5 Φ λ =0.9 Φ λ =0.99 0.2 0 3 mths 5 yrs 10 yrs Maturity Φ: Autocorrelation of state variables. λ 1: Price-of-risk sensitivity to state variables. Term Premium Dynamics and the Taylor Rule 8/21
Essentially Affine Economic Model - Real Part [ Utility: E t=0 e δt C 1 γ t 1 γ Q t ]. Consumption Growth (c log C): c t+1 = (1 φ c )θ c + φ c c t + σ c ε c,t+1. Preference Shock (q log Q): q t+1 = 1 2 (η c c t + η ν ν t ) 2 σ 2 c + (η c c t + η ν ν t ) σ c ε c,t+1. Essentially Affine Pricing Kernel: log M t+1 = δ + γ c t+1 q t+1. Term Premium Dynamics and the Taylor Rule 9/21
Essentially Affine Economic Model - Nominal Nominal Pricing Kernel: log(m $ t+1 ) = log(m t+1) π t+1 Exogenous inflation - a benchmark: π t+1 = (1 φ π )θ π +φ π π t +σ π ε π,t+1, ε π,t+1 other shocks. i (n) t = A $ n + B $ n,c c t + B $ n,νν t + B $ n,ππ t. Endogenous inflation via a Taylor Rule. Term Premium Dynamics and the Taylor Rule 10/21
Economic Model - Endogenous Inflation via Taylor Rule Monetary policy sets the 1-period nominal yield: i t = ī + ı c c t + ı π π t + u t with the monetary policy shock given by u t = φ u u t 1 + σ u ε u,t. π t must simultaneously satisfy: 1 the Taylor Rule, 2 the NA bond pricing equation. Term Premium Dynamics and the Taylor Rule 11/21
Equilibrium Inflation Process: Guess and Verify i t {}}{ ī + ı c c t + ı π ( π + π c c t + π ν ν t + π u u t ) +u }{{} t guess for π t log M $ t+1 {}}{ = log E t [exp{ log M t+1 ( π + π c c t+1 + π ν ν t+1 + π u u t+1 )}] }{{} guess for π t+1 π c = γ(φ c σ 2 c η c ) ı c ı π (φ c σ 2 c η c ), π ν = (γ + π c)σ 2 c η ν ı π φ ν, π u = 1 ı π φ u. i (n) t = A $ n + B $ n,c c t + B $ n,νν t + B $ n,uu t. Term Premium Dynamics and the Taylor Rule 12/21
Prices of Risk Shocks: ε = (ε c, ε ν, ε u or ε π ). Real λ(s t ) = (γ + η c c t + η ν ν t, 0, 0). Nominal - exogenous π λ $ (s t ) = λ(s t ) + (0, 0, 1). Nominal - endogenous π t = π + π c c t + π ν ν t + π u u t λ $ (s t ) = λ(s t ) + (π c, π ν, π u ). Term Premium Dynamics and the Taylor Rule 13/21
Inflation & Term Premiums Driven by Monetary Policy E[i t r t ] =... + E[cov t (log M t+1, π t+1 )], where E[cov t (log M t+1, π t+1 )] = π c (γ + η c θ c )σ 2 c E[i (2) t i t ] =... + 1 2 E[cov t(log M $ t,t+1, i t+1 )], where E[cov t (log M $ t,t+1, i t+1 )] = (γ+π c )(γ+π c +η c θ c )(φ c η c σ 2 c )σ 2 c + (-) Term. π c = γ(φc σ2 c ηc ) ıc ı π (φ c σc 2 ηc ) < 0 if A weak response to inflation or A strong response to consumption growth. An upward sloping nominal curve is driven by π c. Term Premium Dynamics and the Taylor Rule 14/21
Calibration Calibrate the exogenous & endogenous inflation models to quarterly U.S. data (1971:3 to 2005:4). Zero coupon yields (3 months - 10 years). Per capita consumption of nondurables & services. Inflation from methodology in Piazzesi & Schneider (2006). Both models calibrated to share the same real dynamics. Term Premium Dynamics and the Taylor Rule 15/21
Calibration - Fitted Policy Rule Parameters Policy rule responds positively to consumption and inflation. Endogenous corr( c t, π t ) < 0. Highly persistent policy shock captures long bond volatility. Term Premium Dynamics and the Taylor Rule 16/21
Calibration - Fitted Preference Parameters Habit η c < 0: Upward-sloping yield curve, Countercyclical price of risk. Taste shock v t : Short rate volatility through η v, Intermediate maturity volatilities through φ v. No external habit model interpretation though: Affine-class restriction invokes tensions on parameters to achieve upward sloping yield curves. Model does not deliver countercyclical real yields. Model requires a taste shock to fit volatilities. Term Premium Dynamics and the Taylor Rule 17/21
Nominal Yield Curve Nominal Yield Curve Highly autocorrelated policy shocks explain long rate volatility. % Panel A: Interest Rates Avg. Level 9 8.5 8 7.5 7 Exogenous π 6.5 Endogenous π % 4 3.5 3 2.5 2 1.5 1 Panel B: Interest Rates Volatility % 0.3 0.25 0.2 0.15 0.1 0.05 6 3 mth 5 yrs 10 yrs Maturity : 1971-2005 0.5 3 mth 5 yrs 10 yrs Maturity 3 0 m Term Premium Dynamics and the Taylor Rule 18/21
Two Policy Experiments Increase the reaction coefficients to (1) inflation & (2) consumption growth to match the average short-term rate (1987-2005). Baseline: i t = 0.007 + 0.79 c t + 1.68π t + u t. ı π : i t = 0.007 + 0.79 c t + 2.14π t + u t. ı c : i t = 0.007 + 1.07 c t + 1.68π t + u t. Term Premium Dynamics and the Taylor Rule 19/21
Two Policy TwoExperiments Policy Experiments Increase the Increase reaction thecoefficients reaction coefficients to (1) inflation to (1)& inflation (2) and (2) consumption consumption growth togrowth match the to match average theshort-term average short-term rate rate (1987-2005). (1987-2005). Baseline: Baseline: i t = 0.007 i t + = 0.79 c 0.007 t + 0.79 c 1.68π t t + u1.68π t. t + u t. ı π : ı π : i t = 0.007 i t + = 0.79 c 0.007 + t 0.79 c 2.14π t t + + u 2.14π t. t + u t. ı c : ı c : i i t = 0.007 t = 0.007 + 1.07 c + 1.07 c t 1.68π t + 1.68π t + u t. t + u t. Panel A: Interest Rates Avg. Level 8 Panel B: Interest Rates Volatility 0.25 Panel C: Avg. Term P 7.5 7 6.5 3 2.5 0.2 0.15 % 6 Baseline 2 5.5 i π 5 i c 1.5 4.5 3 mth 5 yrs 10 yrs 3 mth 5 yrs 10 yrs Maturity Maturity % % 0.1 0.05 0 3 mth 5 yrs Maturity : 1971-2005, : : 1971-2005, 1987-2005 : 1987-2005 Term Premium Dynamics Term Premium and the Dynamics Taylor Rule and the Taylor Rule 19/21
Policy Experiment Changes in the dynamics of inflation are consistent with a more aggressive reaction to inflation. Data Policy Experiment (1971-2005) (1987-2005) Baseline ı π ı c E [ c t] 4 (%) 1.98 1.83 1.98 1.98 1.98 E [π t] 4 (%) 4.46 2.95 4.42 2.71 3.11 σ ( c t) 4 (%) 1.74 1.35 1.74 1.74 1.74 σ (π t) 4 (%) 2.66 1.26 2.69 1.80 2.67 corr ( c t, c t 1) 0.41 0.28 0.41 0.41 0.41 corr (π t, π t 1) 0.84 0.54 0.85 0.70 0.90 corr ( c t, π t) -0.33-0.17-0.18-0.17-0.41 Term Premium Dynamics and the Taylor Rule 20/21
Conclusions A policy rule aids a consumption-based bond pricing model. Highly autocorrelated policy shocks needed. Negative correlation between inflation & real activity. Term structure information can help identify the policy regime. Future Work: Role of endogenous inflation a general N.A. affine model. The monetary policy rule still tractable in the exact discrete-time affine setting of Dai, Le, & Singleton (2006). Jointly capture real & nominal term structures. Source of the policy shock? Inflation & the real side of the economy? Term Premium Dynamics and the Taylor Rule 21/21