Lecture 5: Asset Pricing Model with Habit Formation Simon Gilchrist - - PowerPoint PPT Presentation

Lecture 5: Asset Pricing Model with Habit Formation

Simon Gilchrist Boston Univerity and NBER

EC 745

Fall, 2013

Habit model:

Assume: U = E

∞

• t=0

βtu (ct, ht) ,with u given, for instance, by the formula u(c, h) = (c − θh)1−σ 1 − σ , where θ > 0 is a parameter and ht is the habit level. The habit level ht satisfies a law of motion, e.g. it is a function of past consumption choices: ht = (1 − δ)ht−1 + ct−1 =

∞

• j=1

(1 − δ)j−1ct−j .

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 − h = ∆c c c c − h > ∆c 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 IR(c) = −cv′′(c) v′(c) = −cu′′(c − h) u′(c − h) , which is time-varying. When u is CRRA, u(c) = c1−γ

1−γ . Direct calculation yields

IR(c) = γ c c − h, As c → h, IR(c) → ∞. Hence “time-varying risk aversion”, and hence time-varying risk premia.

Multiplicative habits

Utility: E0

∞

• t=0

βt (Ct/Xt)1−γ − 1 1 − γ , Assume habit is external then Mt+1 = Ct+1 Ct −γ Xt Xt+1 γ−1 Assume one lag for simplicity: Xt = Cκ

t−1

then Mt+1 = Ct+1 Ct −γ Ct−1 Ct κ(γ−1)

Implications:

Assume joint-log normality of consumption growth and asset returns then the risk free rate is: rf,t+1 = − log β − γ2 σ2

c

2 + γEt∆ct+1 − κ (γ − 1) ∆ct The equity premium is unchanged relative to the CRRA model: Et (rt+1 − rf,t+1) + σr 2 = γσc On average: rf,t+1 = − log β − γ2 σ2

c

2 + (γ (1 − κ) + κ) g so increasing γ will not have as large an impact on reducing rf 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: E0

∞

• t=0

βt (Ct − Xt)1−γ − 1 1 − γ , where γ denotes the risk-aversion coefficient, Xt the external habit level and Ct consumption. Assume that consumption growth is i.i.d and log-normal: ∆ct+1 = g + ut+1, where ut+1 ∼ i.i.d. N(0, σ2).

Surplus consumption ratio:

Define the surplus consumption ratio St ≡ (Ct − Xt)/Ct. Note that 0 < St < 1. Assume that st = log(St) is related to consumption through the following heteroskedastic AR(1) process: st+1 = (1 − φ)s + φst + λ(st)(∆ct+1 − g). This is a generalization of a standard AR(1), i.e. Xt = (1 − δ)Xt−1 + Ct−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: Mt+1 = β Uc(Ct+1,Xt+1) Uc(Ct,Xt) = β St+1 St Ct+1 Ct −γ = βe−γ[g+(φ−1)(st−s)+(1+λ(st))(∆ct+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: Mt+1 = β Uc(t + 1) + ∞

j=1 βjUx(t + 1 + j) ∂Xt+1+j ∂Ct+1

Uc(t) + ∞

j=1 βjUx(t + j) ∂Xt+j ∂Ct

.

Sensitivity function:

Campbell and Cochrane use the following sensitivity function: λ(st) = 1 S

• 1 − 2(st − s) − 1, when s ≤ smax, 0 elsewhere,

where S and smax are respectively the steady-state and upper bound of the surplus-consumption ratio, which we set as: S = σ

• γ

1 − φ, and smax = s + 1 − S

2

2 . These two values for S and smax 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 rf

t+1 = − log β + γg − γ(1 − φ)(st − s) − γ2σ2

2 (1 + λ(st))2 . Two effects where st appears: intertemporal substitution and precautionary savings. When st 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)(st − s) + γ2σ2 2 (1 + λ(st))2 = constant.

Alternative specifications

By picking a different λ, you can have a risk-free rate which depends, say, linearly on the state variable st, i.e. rf

t+1 = A − Bst.

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: γt = −CUCC UC = γ St . Counter-cyclical market price of risk. Start from: SRt =

• Et(Re

t+1)

σt(Re

t+1)

• ≤ σt(Mt+1)

Et(Mt+1) = MPRt, with an equality for assets that perfectly correlated with the SDF. In this model the market price of risk is: MPRt = γσ (1 + λ(st)) = γσ S

• 1 − 2(st − 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: SRt =

• eV art(log Mt+1) − 1 ≃ σt(log Mt+1)

Proof: use the log-normal formula E(exp(X)) = exp

• E(X) + 1

2V ar(X)

• and compute

Et(Mt+1) = eEt(log Mt+1)+ 1

2 V art(log Mt+1),

and V art(Mt+1) = Et(M2

t+1) − [Et(Mt+1)]2,

= e2Et(log Mt+1)+2V art(log Mt+1) − e2Et(log Mt+1)+V art(log Mt+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 Pt denote the ex-dividend price of this

• claim. Then, Et[Mt+1Rt+1] = 1 implies that in equilibrium Pt

satisfies: Et

• Mt+1

Pt+1 + Ct+1 Pt

• = 1

Pt Ct = Et

• Mt+1

Pt+1 Ct+1 + 1 Ct+1 Ct

• =

Et

• β

St+1 St −γ Ct+1 Ct 1−γ Pt+1 Ct+1 + 1

Solution:

The state variable is st. We solve for a fixed point, i.e. Pt

Ct = h(st) with

h(st) = Eut+1

• β

St+1 St −γ Ct+1 Ct 1−γ (h(st+1) + 1)

• ,

with ∆ct+1 = g + ut+1, st+1 = (1 − φ)s + φst + λ(st)ut+1, ut+1 ∼ i.i.d.N(0, σ2).

Fixed point equation

This implies the following fixed point equation in h(s)): h(s) = β ∞

−∞

exp(f(s, u))dΦ(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, {s1, s2, ..., sN} ; Take a guess for h, i.e. {h (s1) , h (s2) , ..., h (sN)} ; 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

• utside the grid.

To compute the integral numerically, one way is to use a quadrature, e.g.

• k(u)du =

i ωik(ui) for some points ui.

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.)

Basic results:

The model matches the level of the riskless rate and the equity return, and their volatilities.

no variation in expected cash flows, no variation in the risk-free rate, but a lot of variation in risk premia.

Intuition: Risk premia are higher after a sequence of some bad shocks, which push C close to X, hence a very low s.

Suggests that risk premia should be highest in the depth of a recession.

Covariance of returns with st is the key source of risk here.

Additional results:

Volatility of returns is higher in bad times. The model also matches the time-series predictability evidence: dividend growth is not predictable, but returns are predictable, and the volatility of the P-D ratio is accounted for by this latter term (“discount rate news”). Long-run equity premium: because of mean-reversion in stock prices, excess returns on stocks at long horizons are even more puzzling than the standard one-period ahead puzzle. CC note that if the state variable is stationary, the long-run standard deviation of the SDF will not depend on the current state. Key point: in their model, S−γ is not stationary – variance is growing with horizon!