Lecture 5: Asset Pricing Model with Habit Formation Simon Gilchrist Boston Univerity and NBER EC 745 Fall, 2013
Habit model: Assume: ∞ � β t u ( c t , h t ) U = E t =0 , with u given, for instance, by the formula u ( c, h ) = ( c − θh ) 1 − σ , 1 − σ where θ > 0 is a parameter and h t is the habit level. The habit level h t satisfies a law of motion, e.g. it is a function of past consumption choices: ∞ � (1 − δ ) j − 1 c t − j h t = (1 − δ ) h t − 1 + c t − 1 = j =1 .
Comments: External habits (keeping up with the Joneses) – agents take h as exogenous. Internal habits – agents recognize the effect of current consumption on future habits. In addition to asset pricing, habits are used in DSGE models to deliver hump-shaped business cycle dynamics. Microfoundations are weak: very little empirical evidence and unclear how such a model aggregates.
Volatility If agent’s utility is v ( c, h ) = u ( c − h ) instead of u ( c ) , and h grows over time so that its distance to c is always rather small, then a given percentage change in c generates a larger percentage change in c − h : ∆( c − h ) = ∆ c c − h > ∆ c c c . c − h c This is just a “leverage” effect coming from the “subsistence level” h . Hence for a given volatility of c , we certainly get more volatility of marginal utility of consumption ∂u ∂c . This will allow us to become closer to the Hansen-Jagannathan bounds: marginal utility of consumption is volatile, which is what you need for the equity premium puzzle.
Time-varying risk aversion: When agents’ consumption becomes closer to the habit level h , they fear further negative shocks since their utility is concave. The curvature of the utility function, i.e. the “index of relative risk aversion”, is I R ( c ) = − cv ′′ ( c ) = − cu ′′ ( c − h ) , v ′ ( c ) u ′ ( c − h ) which is time-varying. When u is CRRA, u ( c ) = c 1 − γ 1 − γ . Direct calculation yields c I R ( c ) = γ c − h, As c → h , I R ( c ) → ∞ . Hence “time-varying risk aversion”, and hence time-varying risk premia.
Multiplicative habits Utility: β t ( C t /X t ) 1 − γ − 1 ∞ � E 0 , 1 − γ t =0 Assume habit is external then � − γ � X t � γ − 1 � C t +1 M t +1 = C t X t +1 Assume one lag for simplicity: X t = C κ t − 1 then � κ ( γ − 1) � − γ � C t − 1 � C t +1 M t +1 = C t C t
Implications: Assume joint-log normality of consumption growth and asset returns then the risk free rate is: r f,t +1 = − log β − γ 2 σ 2 c 2 + γE t ∆ c t +1 − κ ( γ − 1) ∆ c t The equity premium is unchanged relative to the CRRA model: E t ( r t +1 − r f,t +1 ) + σ r 2 = γσ c On average: r f,t +1 = − log β − γ 2 σ 2 c 2 + ( γ (1 − κ ) + κ ) g so increasing γ will not have as large an impact on reducing r f when 0 < k < 0 Increasing γ for 0 < κ < 1 will also raise the variability of the interest rate in the short-run – this will tend to be counterfactual. Overall difficult to argue this model solves the risk-free rate puzzle for very high γ .
Campbell-Cochrane: Utility: β t ( C t − X t ) 1 − γ − 1 ∞ � E 0 , 1 − γ t =0 where γ denotes the risk-aversion coefficient, X t the external habit level and C t consumption. Assume that consumption growth is i.i.d and log-normal: ∆ c t +1 = g + u t +1 , where u t +1 ∼ i.i.d. N (0 , σ 2 ) .
Surplus consumption ratio: Define the surplus consumption ratio S t ≡ ( C t − X t ) /C t . Note that 0 < S t < 1 . Assume that s t = log( S t ) is related to consumption through the following heteroskedastic AR(1) process: s t +1 = (1 − φ ) s + φs t + λ ( s t )(∆ c t +1 − g ) . This is a generalization of a standard AR(1), i.e. X t = (1 − δ ) X t − 1 + C t − 1 . The function λ ( . ) introduces a nonlinearity, which will prove important.
Pricing kernel: External habits: The habit is assumed here to depend only on aggregate, not on individual, consumption. Thus, the inter-temporal marginal rate of substitution is here: β U c ( C t +1 , X t +1 ) M t +1 = U c ( C t, X t ) � − γ � S t +1 C t +1 = β S t C t βe − γ [ g +( φ − 1)( s t − s )+(1+ λ ( s t ))(∆ c t +1 − g ) ] . = In contrast, with internal habits, the consumer is forward-looking and realizes that increasing C today will result in a higher habit in the future. In this case the SDF is more complicated: j =1 β j U x ( t + 1 + j ) ∂X t +1+ j U c ( t + 1) + � ∞ ∂C t +1 M t +1 = β . j =1 β j U x ( t + j ) ∂X t + j U c ( t ) + � ∞ ∂C t
Sensitivity function: Campbell and Cochrane use the following sensitivity function: λ ( s t ) = 1 � 1 − 2( s t − s ) − 1 , when s ≤ s max , 0 elsewhere , S where S and s max are respectively the steady-state and upper bound of the surplus-consumption ratio, which we set as: γ � S = σ 1 − φ, and 2 s max = s + 1 − S . 2 These two values for S and s max are one possible choice, which they justify to make the habit locally predetermined.
Risk free rate: This sensitivity function allows them to have a constant risk-free interest rate. To see this, note that the risk-free rate is t +1 = − log β + γg − γ (1 − φ )( s t − s ) − γ 2 σ 2 (1 + λ ( s t )) 2 . r f 2 Two effects where s t appears: intertemporal substitution and precautionary savings. When s t is low, households have a low IES which drives the risk free rate up. High risk aversion induces more precautionary savings which drives the risk free rate down. CC offset these two effects by picking λ such that γ (1 − φ )( s t − s ) + γ 2 σ 2 (1 + λ ( s t )) 2 = constant. 2
Alternative specifications By picking a different λ , you can have a risk-free rate which depends, say, linearly on the state variable s t , i.e. r f t +1 = A − Bs t . Extensions of the CC model to study the yield curve (Wachter, JFE) or the forward premium puzzle (Verdelhan, JF) consider the case B < 0 and B > 0 , modifying slightly the CC model. Depending on the value of the structural parameters, the model implies constant, pro- or counter-cyclical interest rates.
Key mechanism Time-varying local risk-aversion coefficient: γ t = − CU CC = γ . U C S t Counter-cyclical market price of risk. Start from: E t ( R e � t +1 ) � ≤ σ t ( M t +1 ) � � � SR t = E t ( M t +1 ) = MPR t , � � σ t ( R e t +1 ) � with an equality for assets that perfectly correlated with the SDF. In this model the market price of risk is: MPR t = γσ (1 + λ ( s t )) = γσ � 1 − 2( s t − s ) . S At the steady-state, SR = γσ/S , but the market price of risk is countercyclical, and hence so is the Sharpe ratio.
Proof of result on MPR: In this model, the SDF is conditionally log-normally distributed, hence we can apply the general formula: � e V ar t (log M t +1 ) − 1 ≃ σ t (log M t +1 ) SR t = Proof: use the log-normal formula E ( X ) + 1 � � E (exp( X )) = exp 2 V ar ( X ) and compute E t ( M t +1 ) = e E t (log M t +1 )+ 1 2 V ar t (log M t +1 ) , and E t ( M 2 t +1 ) − [ E t ( M t +1 )] 2 , V ar t ( M t +1 ) = e 2 E t (log M t +1 )+2 V ar t (log M t +1 ) = e 2 E t (log M t +1 )+ V ar t (log M t +1 ) , − hence the result.
Solving the model Solving numerically this model is complicated, because of the important nonlinearities. The aggregate market is represented as a claim to the future consumption stream. Let P t denote the ex-dividend price of this claim. Then, E t [ M t +1 R t +1 ] = 1 implies that in equilibrium P t satisfies: � P t +1 + C t +1 � E t M t +1 = 1 P t � C t +1 P t � � P t +1 � = E t M t +1 + 1 C t C t +1 C t � � 1 − γ � P t +1 �� � − γ � C t +1 � S t +1 = E t β + 1 S t C t C t +1
Solution: The state variable is s t . We solve for a fixed point, i.e. P t C t = h ( s t ) with � � � − γ � C t +1 � 1 − γ � S t +1 h ( s t ) = E u t +1 β ( h ( s t +1 ) + 1) , S t C t with ∆ c t +1 = g + u t +1 , s t +1 = (1 − φ ) s + φs t + λ ( s t ) u t +1 , u t +1 ∼ i.i.d.N (0 , σ 2 ) .
Fixed point equation This implies the following fixed point equation in h ( s )) : � ∞ h ( s ) = β exp ( f ( s, u )) d Φ( u ) . −∞ where f ( u, s ) = − γ ((1 − φ ) ( s − s ) + λ ( s ) u ) + (1 − γ ) g + (1 − γ ) u ( h ((1 − φ ) s + φs + λ ( s ) u ) + 1) To compute this fixed point we need to evaluate the integral on the right hand side.
Recipe: Set a grid for s, { s 1 , s 2 , ..., s N } ; Take a guess for h , i.e. { h ( s 1 ) , h ( s 2 ) , ..., h ( s N ) } ; For each value of s in the grid, compute the RHS of the fixed-point equation numerically. We need to interpolate h to find its value at the point (1 − φ ) s + φs + λ ( s ) u since it lies outside the grid. To compute the integral numerically, one way is to use a � k ( u ) du = � i ω i k ( u i ) for some points u i . quadrature, e.g. This gives a new guess for h using the fixed-point equation. We keep iterating on this equation until convergence. Once we know h , we have the P-D ratio and the rest can be computed simply, as in Mehra-Prescott. (See Wachter, Finance research letters, for a note detailing the computation of that model.)
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