Lecture 6: Recursive Preferences Simon Gilchrist Boston Univerity - - PowerPoint PPT Presentation

Lecture 6: Recursive Preferences

Simon Gilchrist Boston Univerity and NBER

EC 745

Fall, 2013

Basics

Epstein and Zin (1989 JPE, 1991 Ecta) following work by Kreps and Porteus introduced a class of preferences which allow to break the link between risk aversion and intertemporal substitution. These preferences have proved very useful in applied work in asset pricing, portfolio choice, and are becoming more prevalent in macroeconomics.

Value function:

To understand the formulation, recall the standard expected utility time-separable preferences are defined as Vt = Et

∞

• s=0

βs−tu(ct+s), We can also define them recursively as Vt = u(ct) + βEtVt+1,

• r equivalently:

Vt = (1 − β)u(ct) + βEt (Vt+1) .

EZ Preferences

EZ preferences generalize this: they are defined recursively over current (known) consumption and a certainty equivalent Rt (Vt+1) of tomorrow’s utility Vt+1 : Vt = F (ct, Rt (Vt+1)) , where Rt(Vt+1) = G−1 (EtG(Vt+1)) , with F and G increasing and concave, and F is homogeneous of degree one. Note that Rt(Vt+1) = Vt+1 if there is no uncertainty on Vt+1. The more concave G is, and the more uncertain Vt+1 is, the lower is Rt(Vt+1).

Functional forms

Most of the literature considers simple functional forms for F and G: ρ > 0 : F(c, z) =

• (1 − β)c1−ρ + βz1−ρ

1 1−ρ ,

α > 0 : G(x) = x1−α 1 − α. For example: Vt =

• (1 − β)c1−ρ

t

+ β

• EtV 1−α

t+1

1−ρ

1−α

• 1

1−ρ

.

Limits

Limits: ρ = 1 : F(c, z) = c1−βzβ. α = 1 : G(x) = log x. Hence α > 0 : Rt (Vt+1) = Et

• V 1−α

t+1

• 1

1−α ,

α = 1 : Rt(Vt+1) = exp (Et log (Vt+1)) .

Proof

Define f(x) F(c, z) = cf(x) where x = z/c and f(x) =

• 1 − β + βx1−ρ

1 1−ρ

So f′ (x) f (x) = βx−ρ 1 − β + βx1−ρ and lim

ρ→1

f′ (x) f (x) = β X . Since f continuous this implies lim

ρ→1 f (x) = Xβ

(note this is simply the proof that a CES function converges to a Cobb-Douglas as ρ → 1).

Risk Aversion vs IES

In general α is the relative risk aversion coefficient for static gambles and ρ is the inverse of the intertemporal elasticity of substitution for deterministic variations. Suppose consumption today is c today and consumption tomorrow is uncertain: {cL, c, c, ....} or {cH, c, c, ....} , each has prob 1

2.

Utility today: V = F

• c, G−1

1 2G(VL) + 1 2G(VH)

• where VL = F(cL, c) and VH = F(cH, c).

Curvature of G determines how adverse you are to the uncertainty.

If G is linear you only care about the expected value. If not, this is the same as the definition of a certainty equivalent: G( V ) = 1

2G(VL) + 1 2G(VH).

Special Case: Deterministic consumption

If consumption is deterministic: we have the usual standard time-separable expected discounted utility with discount factor β and IES = 1

ρ , risk aversion α = ρ.

Proof: If no uncertainty, then Rt (Vt+1) = Vt+1 and Vt = F (ct, Vt+1) . With a CES functional form for F, we recover CRRA preferences: Vt =

• (1 − β)c1−ρ

t

+ βV 1−ρ

t+1

• 1

1−ρ

Wt = (1 − β)c1−ρ

t

+ βWt+1 = (1 − β)

∞

• j=0

βjc1−ρ

t+j ,

where Wt = V 1−ρ

t

.

Special Case: α = ρ

Similarly, if α = ρ, then the formula Vt =

• (1 − β)c1−ρ

t

+ β

• EtV 1−α

t+1

1−ρ

1−α

• 1

1−ρ

simplifies to V 1−ρ

t

= (1 − β)c1−ρ

t

+ β

• EtV 1−α

t+1

• Define Wt = V 1−ρ

t

, we have Wt = (1 − β)c1−ρ

t

+ βEt (Wt+1) , i.e. expected utility.

Simple example with two lotteries:

Lotteries:

lottery A pays in each period t = 1, 2, ... ch or cl, the probability is 1

2 and the outcome is iid across period;

lottery B pays starting at t = 1 either ch at all future dates for sure, or cl at all future date for sure; there is a single draw at time t = 1.

With expected utility, you are indifferent between these lotteries, but with EZ lottery B is prefered iff α > ρ. In general, early resolution of uncertainty is preferred if and only if α > ρ i.e. risk aversion >

1 IES . This is another way to

motivate these preferences, since early resolution seems intuitively preferable.

Resolution of uncertainty

For lottery A, the utility once you know your consumption is either ch, or cl, since Vh = F(ch, Vh) =

• (1 − β)c1−ρ

h

+ βV 1−ρ

h

• 1

1−ρ .

The certainty equivalent before playing the lottery is G−1 1 2G (ch) + 1 2G (cl)

• =

1 2c1−α

h

+ 1 2c1−α

l

• 1

1−α

. For lottery B, the values satisfy W 1−ρ

h

= (1 − β)c1−ρ

h

+ β 1 2W 1−α

h

+ 1 2W 1−α

l

1−ρ

1−α

, W 1−ρ

l

= (1 − β)c1−ρ

l

+ β 1 2W 1−α

h

+ 1 2W 1−α

l

1−ρ

1−α

,

Resolution of uncertainty

We want to compare G−1 1

2G (Wh) + 1 2G (Wl)

• to

G−1 1

2G (ch) + 1 2G (cl)

• .

Note that the function x → x

1−ρ 1−α is concave if 1 − ρ < 1 − α,

i.e. ρ > α, and convex otherwise. As a result, if ρ > α, 1 2W 1−α

h

+ 1 2W 1−α

l

1−ρ

1−α

≥ 1 2

• W 1−α

h

1−ρ

1−α + 1

2

• W 1−α

l

1−ρ

1−α

= 1 2W 1−ρ

h

+ 1 2W 1−ρ

l

Also W 1−ρ

h

≥ (1 − β)c1−ρ

h

+ β 1 2W 1−ρ

h

+ 1 2W 1−ρ

l

• W 1−ρ

l

≥ (1 − β)c1−ρ

l

+ β 1 2W 1−ρ

h

+ 1 2W 1−ρ

l

Continued

These results imply that if ρ > α then W 1−ρ

h

+ W 1−ρ

l

2 ≥ c1−ρ

h

+ c1−ρ

l

2 . in which case the certainty equivalent of lottery A is higher than the certainty equivalent of lottery B and agents prefer late to early resolution of uncertainty. Technically, EZ is an extension of EU which relaxes the independence axiom. Recall the independence axion is: if x y, then for any z,α : αx + (1 − α)z αy + (1 − α)z. With EZ, “Intertemporal composition of risk matters”: we cannot reduce compound lotteries.

Euler’s Thereom:

We have Vt =

• (1 − β) C1−ρ

t

+ βRt (Vt+1)1−ρ

1 1−ρ

where Rt (Vt+1) =

• Et
• V 1−α

t+1

• 1

1−α

Since Vt is homogenous of degree one, Euler’s thereom implies Vt = MCtCt + EtMVt+1Vt+1

Euler equation:

Taking derivatives: MCt = ∂Vt ∂Ct = (1 − β) V ρ

t C−ρ t

and MVt+1 = ∂Vt ∂Rt (Vt+1) ∂Rt (Vt+1) ∂Vt+1 where ∂Vt ∂Rt (Vt+1) = V ρ

t βRt (Vt+1)−ρ

and ∂Rt (Vt+1) ∂Vt+1 = Rt (Vt+1)α V −α

t+1

This implies MVt+1 = βV ρ

t Rt (Vt+1)α−ρ V −α t+1

IES

Define the intertemporal marginal rate of substitution as St,t+1 = MVt+1MCt+1 MCt = β Ct+1 Ct −ρ Vt+1 Rt (Vt+1) ρ−α The first term is familiar. The second term is next period’s value relative to its certainty equivalent. If ρ = α or there is no uncertainty so that Vt+1 = Rt (Vt+1) this term equals unity.

Household wealth:

Start with the value function: Vt = MCtCt + EtMVt+1Vt+1 Divide by MCt : Vt MCt = Ct + Et MVt+1MCt+1 MCt

• Vt+1

MCt+1 Define Wt = Vt MCt then Wt = Ct + EtSt,t+1Wt+1 is the present-discounted value of wealth.

The return on wealth

Define the cum-dividend return on wealth: Rm,t+1 = Wt+1 Wt − Ct Note that Wt+1 = Vt+1 MCt+1 = V 1−ρ

t+1 Cρ t+1

1 − β Hence Rm,t+1 = V 1−ρ

t+1 Cρ t+1

V 1−ρ

t

Cρ

t − Ct

= Ct+1 Ct ρ V 1−ρ

t+1

V 1−ρ

t

− (1 − β)C1−ρ

t

• Now use fact that

V 1−ρ

t

= (1 − β) C1−ρ

t

+ βRt(Vt+1)1−ρ to obtain Rm,t+1 =

• β

Ct+1 Ct −ρ Rt (Vt+1) Vt+1 1−ρ−1

Certainty Equivalent

Use this equation to solve for the value function relative to the certainty equivalent: R−1

m,t+1

=

• β

Ct+1 Ct −ρ Rt (Vt+1) Vt+1 1−ρ Vt+1 Rt (Vt+1) =

• βRm,t+1

Ct+1 Ct −ρ1/(1−ρ) Comment: we can use this to directly evaluate the cost of uncertain returns and consumption.

SDF:

From above: St,t+1 = β Ct+1 Ct −ρ Vt+1 Rt (Vt+1) ρ−α = βθRθ−1

m,t+1

Ct+1 Ct

− θ

ψ

where θ = 1 − α 1 − ρ and ψ = 1/ρ Note if ρ = α we have St,t+1 = β Ct+1 Ct −ρ Now take logs log St,t+1 = θ log β − θ ψ∆ct+1 − (1 − θ) rm,t+1

Expected returns

The return on the ith asset satisfies: EtSt,t+1Ri

t,t+1 = 1

Taking logs: log

• EtRi

t,t+1

Rf

t+1

• =

−cov(log St,t+1, log Ri

t,t+1)

= θ ψ (cov (∆ct+1, ri,t+1)) + (1 − θ) cov(rm,t+1, ri,t+1) Epstein-Zin is a linear combination of the CAPM and the CCAPM model.

Market return:

For the market return we have log ERm Rf

• = θ

ψcov (∆c, rm) + (1 − θ) σ2

m

We can write as: rm + σm 2 = rf + θ ψcov (∆c, rm) + (1 − θ) σ2

m

Special case: ρ = 1

In this case Rm,t+1 =

• β

Ct+1 Ct −1−1 and σ∆c = σm This implies that: log ERm Rf

• = σ2

m

Risk free rate:

We have: log Rf

t+1

= log Et exp(− log St,t+1) In logs: rf

t+1

= −θ log β + θ ψEt∆ct+1 + (1 − θ) rm − θ ψ 2 σ2

∆c

2 − (1 − θ)2 σ2

m

2 − θ (1 − θ) ψ cov(∆c, rm) Substitute in for the market return to obtain: (1 − θ) rm = (1 − θ) rf − (1 − θ) σm 2 + + (1 − θ)2 σm + (1 − θ) θ ψ cov (∆c, rm)

Risk free rate:

Simplify rf

t = − log β + 1

ψEt∆ct+1 − θ ψ2 σ2

∆c

2 − (1 − θ) σ2

m

2 Again if ρ = α so θ = 1 we have the standard risk-free rate equation. If α > ρ then θ < 1 and the volatility from the market return reduces the real interest rate.

Iid Consumption

Let ∆Ct+1 = g + σcεt+1 Let vt = Vt

Ct and write value function as

vt =  1 − β + βEt

• v1−α

t+1

Ct+1 Ct 1−α 1−ρ

1−α

 

1 1−ρ

Since consumption is iid v is constant.

SDF with iid consumption

With vt = v St,t+1 = β Ct+1 Ct −ρ Vt+1 Rt (Vt+1) ρ−α = β Ct+1 Ct −α     1 Et

• Ct+1

Ct

1−α    

−(1−θ)

Take logs: log St,t+1 = log β − α∆ct+1 + (1 − θ) log Et exp((1 − α)∆ct+1) = log β − α∆ct+1 + (α − ρ) g + (1 − θ) (1 − α)2 σc 2

Risk free rate with iid consumption

Risk free rate: rf = − log EtSt,t+1 = −

• Et log St,t+1 + σ2

s

2

• =

− log β + ρg −

• (1 − θ) (1 − α) + α2 σ2

c

2 If ρ = α this is the standard expression rf = − log β + ρg − ρ2 σ2

c

2

Price-Dividend ratio

Log dividend price ratio: Conjecture a constant q q = EtSt,t+1 Ct+1 Ct (1 + q) where log St,t+1 + ∆ct+1 = log β + (1 − α) ∆ct+1 + (1 − θ) log Et exp((1 − α)∆ct+1) = log β + (1 − α) ∆ct+1 + (α − ρ) g + (1 − θ) (1 − α)2 σc 2

Gordon-Growth Formula

So the price-dividend ratio satsifies: log q 1 + q = log β + (1 − ρ)g − (1 − α)2 θσ2

c

2 = −rf +

• g + σ2

c

2

• − ασ2

c

where the term in brackets is expected consumption growth log(EtCt+1/Ct). Hence this is a risk-adjusted Gordon growth formula.

Risk Premium

The risk premium on a consumption claim is then log EtRt+1 = log Et q + 1 q Ct+1 Ct so that rm + σm 2 − rf = ασ2

c

Consumption-Wealth Ratio:

Start with the identity: Wt+1 = Rm,t+1 (Wt − Ct) to obtain the log-linear equation: ∆wt+1 = rm,t+1 + k +

• 1 − 1

ρ

• (ct − wt)

where ρ = 1 − exp(c − w). Rearrange to obtain (1 − ρ) (ct − wt) = ρrm,t+1 − ρ∆wt+1 + ρk = ρrm,t+1 + ρ [∆ (ct+1 − wt+1) − ∆ct+1] + ρk Present value relationship: ct − wt = ρ (rm,t+1 − ∆ct+1) + ρ (ct+1 − wt+1) + ρk =

∞

• s=1

ρs [rm,t+s − ∆ct+s] + ρ 1 − ρk

Present value expression

Now combine the risk free and market rate Euler equations to

• btain:

rm,t+s − ∆ct+s = (1 − ψ) rm,t+s − µm where µm is a constant that depends on conditional covariances etc.. ct − wt = (1 − ψ) Et

∞

• s=1

ρsrm,t+s + ρ (κ − µm) 1 − ρ The consumption-wealth ratio is an increasing function of expected future returns if the IES < 1. Note, we started with an identity and combined it with the Euler equation for safe vs risky returns for a given IES. Thus these expressions are general and do not depend specifically on EZ preferences.

Unexpected changes in consumption

Now use ct+1 − Etct+1 = Wt+1 − EtWt+1 + (1 − ψ) (Et+1 − Et)

∞

• s=1

ρsrm,t+s+1 = rm,t+1 − Etrm,t+1 + (1 − ψ) (Et+1 − Et)

∞

• s=1

ρsrm,t+s+1 Unexpected returns increase consumption growth. Unexpected future returns increase current consumption growth if the IES < 1.

Some comments

If returns are not forecastable, the consumption-wealth ratio is a constant. In this case, consumption volatility equals the volatility of wealth, or equivalently the market return. In the data this is obviously not true – hence returns must be predictable.

Asset pricing implications:

We can now compute covt (ri,t+1, ∆ct+1) = σic = σim + (1 − ψ) σih where σih is the covariance of ri,t+1 with the surprise in future market returns: σih = cov(ri,t+1, (Et+1 − Et)

∞

• s=1

ρsrm,t+s+1)

Epstein-Zin preferences and Risk Premiums:

Using EZ preferences, the risk premium is: Etri,t+1 − rf,t+1 + σ2

i

2 = θσic ψ + (1 − θ) σim The risk premium for asset i depends on its covariance between current returns and its covariance with news about future market returns: Etri,t+1 − rf,t+1 + σ2

i

2 = ασim + (α − 1) σih where α is the coefficient of relative risk aversion. Note we don’t need to know the IES or consumption growth to price risk in this framework.

EZ preferences and the Equity premium puzzle:

For EZ preferences we can now write the risk premium on the market return as: . Etrm,t+1 − rf,t+1 + σ2

m

2 = ασ2

m + (α − 1) σmh

If returns are unforecastable, σih = 0. Given σim = 0.17 we need α = 2 to obtain a risk premium of 6% So we succeed in matching the risk premium with low relative risk aversion but fail

• n the fact that the consumption-wealth ratio will be a constant,

and consumption volatility should equal wealth volatility. If there is mean reversion and future returns are negatively correlated with current returns then σmh < 0 we would need a higher α. Since mean-reversion is difficult to determine, the estimate could be substantially higher.

Predictable consumption growth:

Consumption growth: ∆ct+1 = g + xt + ut xt = φxt−1 + vt Again use Campbell-Shiller decomposition: rt+1 = ρqt+1 − qt + ∆ct+1 where qt = Pt/Ct the price of a consumption claim and ρ = q 1 + q < 1 Solving forward qt = Et

∞

• s=1

ρs [∆ct+s+1 − rt+s+1]

Price-dividend ratio

Euler equation ∆ct+s = ψrt+s where ψ is the IES so that [∆ct+s+1 − rt+s+1] =

• 1 − 1

ψ

• ∆ct+s+1

The price-dividend ratio satisifies qt =

• ρφ

1 − ρφ 1 − 1 ψ

• xt

Implications

If the IES > 1 then an increase in current consumption growth causes an increase in the price-dividend ratio. Intuition: A persistent increase in consumption growth provides news about future cash flows and discount rates that go in

• pposite directions.

If the IES is high, interest rates don’t need to move very much in response to the change in consumption growth. The cash flow effect dominates.

Comments

We need a high φ to get large volatility in the price-dividend ratio. But this comes from predictable dividend growth not from time-varying returns (the risk free rate is moving but the risk premium is not).

Time-varying volatility:

Now add time-varying volatility: xt = φxt−1 + σtut σt = (1 − γ)σ + γσt−1 + vt We then have a solution of the form qt =

• ρφ

1 − ρφ 1 − 1 ψ

• xt + aσ2

t

where a > 0 if IES > 0 and risk-aversion > 0. We will also get time-varying risk premia – “discount rate news” that offsets the “cash flow” news of the consumption growth shock. In other words, we need time-varying volatility to match the equity premium combined with persistent movements in consumption growth to match the volatility of the price dividend ratio.

Calibration and empirical implementation

Calibration:

IES = 1.5, α = 10 Very high persistence and volatility for shocks to volatility and peristent consumption-growth process.

Issues to think about:

IES > 1 is controversial. Difficult to estimate long-run risk.