Lecture 6: Recursive Preferences

Transcription

1 Lecture 6: Recursive Preferences Simon Gilchrist Boston Univerity and NBER EC 745 Fall, 2013

2 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.

3 Value function: To understand the formulation, recall the standard expected utility time-separable preferences are defined as V t = E t β s t u(c t+s ), s=0 We can also define them recursively as or equivalently: V t = u(c t ) + βe t V t+1, V t = (1 β)u(c t ) + βe t (V t+1 ).

4 EZ Preferences EZ preferences generalize this: they are defined recursively over current (known) consumption and a certainty equivalent R t (V t+1 ) of tomorrow s utility V t+1 : where V t = F (c t, R t (V t+1 )), R t (V t+1 ) = G 1 (E t G(V t+1 )), with F and G increasing and concave, and F is homogeneous of degree one. Note that R t (V t+1 ) = V t+1 if there is no uncertainty on V t+1. The more concave G is, and the more uncertain V t+1 is, the lower is R t (V t+1 ).

5 Functional forms Most of the literature considers simple functional forms for F and G: For example: ρ > 0 : F (c, z) = ( (1 β)c 1 ρ + βz 1 ρ) 1 1 ρ, α > 0 : G(x) = x1 α 1 α. V t = ( (1 β)c 1 ρ t + β ( E t V 1 α t+1 ) 1 ) 1 ρ 1 ρ 1 α.

6 Limits Limits: ρ = 1 : F (c, z) = c 1 β z β. α = 1 : G(x) = log x. Hence α > 0 : R t (V t+1 ) = E t ( V 1 α t+1 ) 1 1 α, α = 1 : R t (V t+1 ) = exp (E t log (V t+1 )).

7 Proof Define f(x) where x = z/c and F (c, z) = cf(x) So and f(x) = ( 1 β + βx 1 ρ) 1 1 ρ f (x) f (x) = Since f continuous this implies βx ρ 1 β + βx 1 ρ f (x) lim ρ 1 f (x) = β X. lim f (x) = Xβ ρ 1 (note this is simply the proof that a CES function converges to a Cobb-Douglas as ρ 1).

8 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: {c L, c, c,...} or {c H, c, c,...}, each has prob 1 2. Utility today: V = F ( ( 1 c, G 1 2 G(V L) + 1 )) 2 G(V H) where V L = F (c L, c) and V H = F (c H, 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 2 G(V L) G(V H).

9 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 R t (V t+1 ) = V t+1 and V t = F (c t, V t+1 ). With a CES functional form for F, we recover CRRA preferences: V t = ( (1 β)c 1 ρ t ) + βv 1 ρ 1 1 ρ t+1 W t = (1 β)c 1 ρ t + βw t+1 = (1 β) where W t = V 1 ρ t. j=0 β j c 1 ρ t+j,

10 Special Case: α = ρ Similarly, if α = ρ, then the formula V t = ( (1 β)c 1 ρ t + β ( E t V 1 α t+1 ) 1 ) 1 ρ 1 ρ 1 α simplifies to V 1 ρ t = (1 β)c 1 ρ t + β ( E t Vt+1 1 α ) Define W t = V 1 ρ t, we have i.e. expected utility. W t = (1 β)c 1 ρ t + βe t (W t+1 ),

11 Simple example with two lotteries: Lotteries: lottery A pays in each period t = 1, 2,... c h or c l, the probability is 1 2 and the outcome is iid across period; lottery B pays starting at t = 1 either c h at all future dates for sure, or c l 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.

12 Resolution of uncertainty For lottery A, the utility once you know your consumption is either c h, or c l, since V h = F (c h, V h ) = ( (1 β)c 1 ρ h ) + βv 1 ρ 1 1 ρ h. The certainty equivalent before playing the lottery is ( 1 G 1 2 G (c h) + 1 ) ( 1 2 G (c l) = 2 c1 α h + 1 ) 1 1 α 2 c1 α l. For lottery B, the values satisfy W 1 ρ h W 1 ρ l = (1 β)c 1 ρ h = (1 β)c 1 ρ l ( 1 + β 2 W 1 α h + 1 ) 1 ρ 2 W 1 α 1 α l, ( 1 + β 2 W 1 α h + 1 ) 1 ρ 2 W 1 α 1 α l,

13 Resolution of uncertainty We want to compare G 1 ( 1 2 G (W h) G (W l) ) to G 1 ( 1 2 G (c h) G (c l) ). 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 2 W 1 α h + 1 ) 1 ρ 2 W 1 α 1 α l 1 2 = 1 2 W 1 ρ h ( ) 1 ρ W 1 α 1 α h W 1 ρ l ( ) 1 ρ W 1 α 1 α l Also W 1 ρ h (1 β)c 1 ρ h ( 1 + β W 1 ρ l (1 β)c 1 ρ l + β 2 W 1 ρ h + 1 ) 2 W 1 ρ l ) ( 1 2 W 1 ρ h W 1 ρ l

14 Continued These results imply that if ρ > α then W 1 ρ h + W 1 ρ l 2 c1 ρ h + c 1 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.

15 Euler s Thereom: We have where V t = ( (1 β) C 1 ρ t + βr t (V t+1 ) 1 ρ) 1 1 ρ R t (V t+1 ) = ( ( )) 1 E t V 1 α 1 α t+1 Since V t is homogenous of degree one, Euler s thereom implies V t = MC t C t + E t MV t+1 V t+1

16 Euler equation: Taking derivatives: MC t = V t = (1 β) V ρ t C C ρ t t and where and This implies MV t+1 = V t R t (V t+1 ) R t (V t+1 ) V t+1 V t R t (V t+1 ) = V ρ t βr t (V t+1 ) ρ R t (V t+1 ) V t+1 = R t (V t+1 ) α V α t+1 MV t+1 = βv ρ t R t (V t+1 ) α ρ V α t+1

17 IES Define the intertemporal marginal rate of substitution as S t,t+1 = MV t+1mc t+1 MC t ( ) ρ ( Ct+1 Vt+1 = β R t (V t+1 ) C t ) 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 V t+1 = R t (V t+1 ) this term equals unity.

18 Household wealth: Start with the value function: Divide by MC t : Define then V t = MC t C t + E t MV t+1 V t+1 V t MC t = C t + E t ( MVt+1 MC t+1 MC t V t W t = MC t W t = C t + E t S t,t+1 W t+1 is the present-discounted value of wealth. ) Vt+1 MC t+1

19 The return on wealth Define the cum-dividend return on wealth: Note that Hence R m,t+1 = Now use fact that to obtain R m,t+1 = W t+1 W t C t t+1 Cρ t+1 W t+1 = V t+1 = V 1 ρ MC t+1 1 β V 1 ρ t+1 Cρ t+1 V 1 ρ t V 1 ρ t R m,t+1 = C ρ t C t = ( Ct+1 C t = (1 β) C 1 ρ t [ β ( Ct+1 C t ) ρ ( V 1 ρ t + βr t (V t+1 ) 1 ρ V 1 ρ t+1 (1 β)c 1 ρ t ) ρ ( ) ] Rt (V t+1 ) 1 ρ 1 V t+1 )

20 Certainty Equivalent Use this equation to solve for the value function relative to the certainty equivalent: [ ( ) ρ ( ) ] Rm,t+1 1 Ct+1 Rt (V t+1 ) 1 ρ = β V t+1 R t (V t+1 ) = ( C t βr m,t+1 ( Ct+1 C t V t+1 ) ρ ) 1/(1 ρ) Comment: we can use this to directly evaluate the cost of uncertain returns and consumption.

21 SDF: From above: S t,t+1 = β ( Ct+1 C t ) ρ ( Vt+1 ) ρ α R t (V t+1 ) where Note if ρ = α we have Now take logs = β θ Rm,t+1 θ 1 C θ t+1 ψ C t θ = 1 α 1 ρ S t,t+1 = β and ψ = 1/ρ ( ) ρ Ct+1 C t log S t,t+1 = θ log β θ ψ c t+1 (1 θ) r m,t+1

22 Expected returns The return on the ith asset satisfies: E t S t,t+1 R i t,t+1 = 1 Taking logs: ( ) Et Rt,t+1 i log R f t+1 = cov(log S t,t+1, log R i t,t+1) = θ ψ (cov ( c t+1, r i,t+1 )) + (1 θ) cov(r m,t+1, r i,t+1 ) Epstein-Zin is a linear combination of the CAPM and the CCAPM model.

23 Market return: For the market return we have ( ) ERm log = θ ψ cov ( c, r m) + (1 θ) σm 2 We can write as: R f r m + σ m 2 = r f + θ ψ cov ( c, r m) + (1 θ) σ 2 m

24 Special case: ρ = 1 In this case ( R m,t+1 = β ( Ct+1 C t ) 1 ) 1 and This implies that: σ c = σ m ( ) ERm log = σm 2 R f

25 Risk free rate: We have: log R f t+1 = log E t exp( log S t,t+1 ) In logs: r f t+1 = θ log β + θ ψ E t c t+1 + (1 θ) r m ( ) θ 2 σ c 2 ψ 2 (1 σ 2 θ)2 m 2 θ (1 θ) cov( c, r m ) ψ Substitute in for the market return to obtain: (1 θ) r m = (1 θ) r f (1 θ) σ m (1 θ) 2 (1 θ) θ σ m + cov ( c, r m ) ψ

26 Risk free rate: Simplify r f t = log β + 1 ψ E t c t+1 θ σ c 2 ψ 2 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.

27 Iid Consumption Let C t+1 = g + σ c ε t+1 Let v t = Vt C t and write value function as v t = 1 β + βe t (v 1 α t+1 ( Ct+1 Since consumption is iid v is constant. C t ) ) 1 ρ 1 α 1 α 1 1 ρ

28 SDF with iid consumption With v t = v S t,t+1 = β Take logs: = β ( Ct+1 C t ( Ct+1 C t ) ρ ( ) ρ α Vt+1 R t (V t+1 ) ) α 1 ( ( ) ) 1 α E Ct+1 t C t (1 θ) log S t,t+1 = log β α c t+1 + (1 θ) log E t exp((1 α) c t+1 ) = log β α c t+1 + (α ρ) g + (1 θ) (1 α) 2 σ c 2

29 Risk free rate with iid consumption Risk free rate: r f = log E t S t,t+1 = ( ) E t log S t,t+1 + σ2 s 2 = log β + ρg [ (1 θ) (1 α) + α 2] σ 2 c 2 If ρ = α this is the standard expression r f = log β + ρg ρ 2 σ2 c 2

30 Price-Dividend ratio Log dividend price ratio: Conjecture a constant q where q = E t S t,t+1 C t+1 C t (1 + q) log S t,t+1 + c t+1 = log β + (1 α) c t+1 + (1 θ) log E t exp((1 α) c t+1 ) = log β + (1 α) c t+1 + (α ρ) g + (1 θ) (1 α) 2 σ c 2

31 Gordon-Growth Formula So the price-dividend ratio satsifies: log q 1 + q = log β + (1 ρ)g (1 α) 2 θ σ2 c 2 ( ) = r f + g + σ2 c ασc 2 2 where the term in brackets is expected consumption growth log(e t C t+1 /C t ). Hence this is a risk-adjusted Gordon growth formula.

32 Risk Premium The risk premium on a consumption claim is then q + 1 C t+1 log E t R t+1 = log E t q C t so that r m + σ m 2 r f = ασ 2 c

33 Consumption-Wealth Ratio: Start with the identity: W t+1 = R m,t+1 (W t C t ) to obtain the log-linear equation: w t+1 = r m,t+1 + k + where ρ = 1 exp(c w). Rearrange to obtain ( 1 1 ) (c t w t ) ρ (1 ρ) (c t w t ) = ρr m,t+1 ρ w t+1 + ρk Present value relationship: = ρr m,t+1 + ρ [ (c t+1 w t+1 ) c t+1 ] + ρk c t w t = ρ (r m,t+1 c t+1 ) + ρ (c t+1 w t+1 ) + ρk = ρ s [r m,t+s c t+s ] + ρ 1 ρ k s=1

34 Present value expression Now combine the risk free and market rate Euler equations to obtain: r m,t+s c t+s = (1 ψ) r m,t+s µ m where µ m is a constant that depends on conditional covariances etc.. c t w t = (1 ψ) E t s=1 ρ s r m,t+s + ρ (κ µ m) 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.

35 Unexpected changes in consumption Now use c t+1 E t c t+1 = W t+1 E t W t+1 + (1 ψ) (E t+1 E t ) = r m,t+1 E t r m,t+1 + (1 ψ) (E t+1 E t ) ρ s r m,t+s+1 s=1 ρ s r m,t+s+1 s=1 Unexpected returns increase consumption growth. Unexpected future returns increase current consumption growth if the IES < 1.

36 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.

37 Asset pricing implications: We can now compute cov t (r i,t+1, c t+1 ) = σ ic = σ im + (1 ψ) σ ih where σ ih is the covariance of r i,t+1 with the surprise in future market returns: σ ih = cov(r i,t+1, (E t+1 E t ) ρ s r m,t+s+1 ) s=1

38 Epstein-Zin preferences and Risk Premiums: Using EZ preferences, the risk premium is: E t r i,t+1 r f,t+1 + σ2 i 2 = θ σ ic ψ + (1 θ) σ im The risk premium for asset i depends on its covariance between current returns and its covariance with news about future market returns: E t r i,t+1 r f,t+1 + σ2 i 2 = ασ im + (α 1) σ ih 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.

39 EZ preferences and the Equity premium puzzle: For EZ preferences we can now write the risk premium on the market return as:. E t r m,t+1 r f,t+1 + σ2 m 2 = ασ2 m + (α 1) σ mh If returns are unforecastable, σ ih = 0. Given σ im = 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 on 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 σ mh < 0 we would need a higher α. Since mean-reversion is difficult to determine, the estimate could be substantially higher.

40 Predictable consumption growth: Consumption growth: c t+1 = g + x t + u t x t = φx t 1 + v t Again use Campbell-Shiller decomposition: r t+1 = ρq t+1 q t + c t+1 where q t = P t /C t the price of a consumption claim and Solving forward ρ = q 1 + q < 1 q t = E t ρ s [ c t+s+1 r t+s+1 ] s=1

41 Price-dividend ratio Euler equation where ψ is the IES so that c t+s = ψr t+s [ c t+s+1 r t+s+1 ] = The price-dividend ratio satisifies ( ρφ q t = 1 ρφ [ 1 1 ] c t+s+1 ψ ) [ 1 1 ψ ] x t

42 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 opposite 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.

43 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).

44 Time-varying volatility: Now add time-varying volatility: x t = φx t 1 + σ t u t σ t = (1 γ)σ + γσ t 1 + v t We then have a solution of the form ( ) [ ρφ q t = 1 1 ] x t + aσt 2 1 ρφ ψ 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.

45 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.