Decomposing the Yield Curve

Transcription

1 Decomposing the Yield Curve John H. Cochrane and Monika Piazzesi January 1, 1 Cochrane/Piazzesi () Yield Curve January 1, 1 1 / 1

2 Objective and motivation Yield curve: expected interest rates or risk premiums? Yield: Forward: y (n) t = 1 n E t f (n) t = E t (y (1) t+n y (1) t + y (1) t y (1) t+n 1 + rpy (n) t (n) 1 ) + rpf t Returns : E t (r (n) t+1 ) = y (1) t + rpr (n) t Current risk premium or expected future premium? Term structure of risk premiums? rpy (n) t = 1 n he t rx (n) t+1 + E t rx t+ (n 1)!A ne model with a lot of attention to risk premiums E t rx () i t+n 1 Cochrane/Piazzesi () Yield Curve January 1, 1 / 1

3 A ne model structure Factors, e.g.: X t = x t level t slope t curve t Real factor dynamics: X t+1 = µ + φx t + v t+1 ; E v t+1 v t+1 = V A ne model: f (n) t = E t (y (1) t+n 1 ) = () + δ 1 φn 1 X t φ is easy to t, pure cross section. Need φ for forecasts, premiums Market prices λ: φ φ V λ 1 Market prices λ: E t (rx t+1 ) = () + cov(rx t+1, vt+1 ) (λ + λ 1 X t ) λ (x ) 3 λ t = (λ + λ 1 X t ) = λ (level) λ (x,x ) 1 λ (x,l) 3 1 x t λ (slope) 7 + λ (l,x ) 1 7 level t 5 5 slope t λ (curve) curve t Two issues: 1) How does λ t vary over time (columns)? ) Covariance with which shocks generates a premium (rows)? Can we simplify estimation of unknown parameters, please? Cochrane/Piazzesi () Yield Curve January 1, 1 3 /

4 CP Review rx (n) t+1 = a n + b 1 y (1) t + b f () t b 5 f (5) t + ε (n) t+1 Unrestricted Restricted 5 3!Single-factor model for expected excess returns E t rx (n) t+1 = b n γ f t = bn x t Paper: Eigenvalue decompose covariance matrix of expected returns; x t = γ f t is the dominant (>99%) eigenvector. 5 3 Cochrane/Piazzesi () Yield Curve January 1, 1 / 1

5 CP to a ne model Data: E t rx (n) t+1 = b n γ f t = bn x t Model: E t (rx t+1 ) = () + cov(rx t+1, v t+1 ) (λ + λ 1 X t ) All variation through time in market prices of risk is carried by x t λ (x ) 3 λ (x,x ) 3 λ t = (λ + λ 1 X t ) = λ (level) 1 λ (slope) 7 + λ (l,x ) 1 5 λ (s,x ) 7 λ (curve) 1 5 λ (c,x ) 1 x t level t slope t curve t Cochrane/Piazzesi () Yield Curve January 1, 1 5 / 1

6 E t rx (n) t+1 = () + cov(rx (n) t+1, v t+1 ) (λ + λ 1 X t ) b n x t = cov(rx (n) t+1, v x t+1 )λ(x,x ) 1 x t + cov(rx (n) t+1, v l t+1 )λ(l,x ) 1 x t +... cov(r,level) q r cov(r,x) cov(r,curve) cov(r,slope) Maturity, years Market prices of risk correspond entirely to covariance with the level shock. You can estimate λ 1 with a cross-sectional regression Cochrane/Piazzesi () Yield Curve January 1, 1 / 1

7 Market price of risk summary Market price of risk only varies over time in response to one state variable, x t, and not to level, slope and curvature. Risk premium is only earned in return for exposure to term-structure level shocks vt+1 l. The premium for x, slope, curvature risk is zero. Dramatic simpli cation. Two parameters to estimate, and cross-sectional regression method to do so! φ = φ + V λ 1 E t (rx t+1 ) = () + cov(rx t+1, v t+1 )λ t λ t = λ + λ 1 X 3 t λ t = λ l λ 1l x t level t slope t curve t Cochrane/Piazzesi () Yield Curve January 1, 1 7 / 1

8 Estimation 1 Find risk-neutral φ to t cross section f (n) t = () + δ 1 φn 1 X t (+ε t ) N T min () + δ fφ 1 g φn 1 X t f (n) t n=1 t=1 No forecasting information in risk-neutral transition matrix φ. As usual, very close t. Use cross-sectional regression estimate λ to nd real φ φ = φ + V λ 1 φ = φ + V λ 1l Zero restrictions mean that all but one column of φ is estimated from the cross-section alone! Cochrane/Piazzesi () Yield Curve January 1, 1 8 /

9 Transition Matrix Estimates x level slope curve Risk-neutral: φ x level slope curve Actual: φ x level slope curve The risk-neutral φ from the cross-section = a lot of information about the true φ!.98 does not change. Near unit-root estimation problems are solved. The root is identi ed from the cross section. Cochrane/Piazzesi () Yield Curve January 1, 1 9 / 1

10 Percent Percent Percent Percent True dynamics φ. x is not an AR(1). Slope, curve! x. Can expect future risk premium without current; term-structure of risk premiums... Resp. to x shock Return forecast x level slope curve Resp. to level shock Years Years 1. Resp. to slope shock Resp. to curve shock Years Years Cochrane/Piazzesi () Yield Curve January 1, 1 1 / 1

11 Percent Percent Percent Percent Term structure of risk premiums..3 Resp. to x shock Return forecast Current forwards Expected yield Er factor x.5..3 Resp. to level shock Time and maturity, years. 8 1 Time and maturity, years. Resp. to slope shock Resp. to curve shock Time and maturity, years 8 1 Time and maturity, years Cochrane/Piazzesi () Yield Curve January 1, 1 11 / 1

12 18 5 year forward decomposition, Risk neutral model 1 Yr Yield 5 Yr fwd Expected Y (1) Cochrane/Piazzesi () Yield Curve January 1, 1 1 / 1

13 18 5 year forward decomposition, Return forecast model 1 Yr Yield 5 Yr fwd Expected Y (1) Cochrane/Piazzesi () Yield Curve January 1, 1 13 / 1

14 18 1 year forward decomposition, Risk neutral model 1 Yr Yield 1 Yr fwd Expected Y (1) Cochrane/Piazzesi () Yield Curve January 1, 1 1 / 1

15 18 1 year forward decomposition, Return forecast model 1 Yr Yield 1 Yr fwd Expected Y (1) Cochrane/Piazzesi () Yield Curve January 1, 1 15 / 1

16 18 1 year forward decomposition, half lambda model 1 Yr Yield 1 Yr fwd Expected Y (1) Cochrane/Piazzesi () Yield Curve January 1, 1 1 / 1