Vayanos and Vila, A Preferred-Habitat Model of the Term Structure of Interest Rates December 4, 2007
Overview Term-structure model in which investers with preferences for specific maturities and arbitrageurs Solutions are in affine forms Results Bond risk premia are negatively related to short rates and positively to the term-structure slope Forward rates under-react to expected short rates Despite the presence of two factors, the first principal component explains about 90% of movement
Model Continuum of zero-coupon bonds in zero net supply P t,τ the time-t price of the bond with maturity τ R t,τ = log(pt,τ ) τ yield of the bond Short rate is exogenous dr t = κ r (r r t )dt + σ r db r,t Investors Each investor demands only a specific maturity Demand function of the clientele τ y t,τ = α(τ)τ(r t,τ β t,τ ) Intercept β t,τ = β t
Arbitrageurs Trade a bond portfolio across maturities Denoting time t wealth by W ( t dw t = W t ) T 0 x t,τ dτ r t dt + T o x t,τ dpt,τ P dτ tt,τ Arbitragues solve a standard mean-variance problem max [ E t (dw t ) a 2 Var t(dw t ) ] Maximization over instantaneous mean and variance.
One-Factor MOdel β t = β constant. Short rate r t is the only factor Equilibrium [Ar (τ)rt+c(τ)] P t,τ = e dp t,τ P t,τ = µ t,τ dt A r (τ)σ r db r,t dw [ t = W t r t + ] [ T ] 0 x T t,τ (µ t,τ r t )dτ dt 0 x t,τ A r (τ)dτ σ r db r,t From the FOC, µ t,τ r t = A r (τ)λ r λ r aσr 2 T 0 x t,τ A r (τ)dτ And market clearing, x t,τ = y t,τ Risk neutral parameters κ r = κ r + aσr 2 T 0 α(τ)a r (τ) 2 dτ > κ r Reverse carry case r r + (β r)z β+z c κ r r increasing in β
Term-Structure Movement Forward rates f t,τ τ,τ = log ( P t,τ P t,τ τ ) τ f t,τ = log(pt,τ ) τ Proposition 2 : Effect of Short-Rate Expectations 0 < ft,τ r t < Et(rt+τ ) r t ft,τ rt Et (r t+τ ) rt is decreasing in τ : Under-reaction of forward rates f t,τ = E t(r t+τ ) + [E t (r t+τ ) E t(r t+τ )] σ 2 r /2A r (τ) 2 Proposition 3 : Effect of Investor Demand 0 < ft,τ β < 1 increasing in τ f t,τ β Longer maturities are harder to arbitrage
Risk-Premia and Predictability Fama-Bliss regression 1 τ log ( Pt+ τ,τ P t,τ ) R t, τ = α + γ p (f t,τ τ,τ R t, τ ) + ɛ τ = 1 year and τ = 2, 3, 4, 5 years EH predicts γ p to be zero Found that γ p is positive and the standard deviation of predicted risk premia is large (about 1-1.5%) The model predicts the positive relationship between risk premia and the term-structure slope When r t is low the term structure is upward sloping Arbitrageurs are long bonds and borrow short rates because they see r t will rise and expected future rates will be low The trade should have positive risk premium
Risk-Premia and Predictability Negative relationship between risk premia and the short rate ( ) 1 τ log Pt+ τ,τ P t,τ R t, τ = α s + γ s R t, τ + ɛ Model predicts that γ s = (κ r κ r )A r (τ) < 0 when τ 0 λ r = κ r r κ r r r t(κ r κ r ) Price of risk λ r is affine in r t and changes the sign Campbell and Shiller τ R t+ τ,τ τ R t,τ = α + γ r τ τ (R t,τ R t, τ ) + ɛ EH predicts that γ = 1. But CS find that γ r is smaller than one
Two Factor Model Demand parameter β follows an OU process dβ t = κ β (β β t ) + σ β db β,t The equilibrium is similar, but solved numerically
Effect of Short-Rate Expectations
Effect of Investor Demand Figure: Russian yield spreads.
Principal Component Analysis
Risk Premia and Predictability - FB
Discussion Nice equilibrium model Affine term structure Explains empirical puzzles Larger effect of demand on longer maturities - how much of it is mechanical? Extensions Demand function y t,τ = α(τ)τ(r t,τ β t,τ ) Different demand shocks across maturities