Empirical Methods in Finance - Section 5 - Consumption-Based Asset Pricing Models

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1 Empirical Methods in Finance - Section 5 - Consumption-Based Asset Pricing Models René Garcia Professor of finance EDHEC Nice, June 2010

2 Introduction The central question: what is the nature of macroeconomic risk that drives risk premia in asset markets? The central fact is that expected returns vary over time. They are correlated with business cycles: high in bad times, low in good times. Average returns are high in bad times because the marginal value of wealth is high. Assets that pay off poorly in bad times have low prices or high returns.

3 Introduction The central idea of modern finance is that prices are generated by expected discounted payoffs p i t = E t (m t+1 x i t+1 ) Using the definition of covariance and the real risk-free rate R f = 1 E(m), we can write the price as: pt i = E t (xt+1 i ) Rt f + cov t (m t+1,xt+1 i ). The discount factor m t + 1 is equal to the growth in the marginal value of wealth VW (t + 1) m t+1 =. VW (t)

4 Introduction The traditional theories of finance, CAPM, ICAPM, and APT, measure the marginal utility of wealth by the behavior of large portfolios of assets. CAPM: return on the market portfolio. Multifactor models: returns on multiple portfolios. To make the link between the real economy and financial markets, we measure the growth in marginal utility of wealth by the growth in consumption. Idea that consumption is the payoff on the market portfolio.

5 Consumption Capital Asset Pricing Model (CCAPM) Max (Ct,w i,t+1 ) E t β i U(C t+i ),i = 1,...,n i=0 under the constraint: n n C t + p it w i,t+1 (p it + D it )w i,t + y t i=1 i=1 One consumption good, infinite horizon, additive and time separable utility p it = price of asset i at time t, D it = dividend paid on asset i at t, beginning of period, w it = units of asset i held at beginning of period t, y t = labor income exogenous at time t. FOC: U (C t ) = βe t [ Ri,t+1 U (C t+1 ) ] Then: M t+1 = β U (C t+1 ) U. (C t ) with isoelastic utility function: U(C) = C1 γ 1 γ ( ) ] Ct+1 γ E t [R i,t+1 β = 1 (1) C t This condition derived for the first time by Grossman-Shiller (AER, 1981).

6 Empirical Strategies 1 Objective of empirical work: estimate preference parameters β and γ and test restrictions imposed by (1). Assumption about distributions: Joint log-normality of consumption and returns. No assumption about distributions: estimation of Euler equations by GMM 2 Are the intertemporal asset pricing models compatible with SDF extracted from data? 3 Are the models able to reproduce empirical facts such as moments of returns, predictability of returns, lack of predictability of dividend growth? Assume a stochastic process for consumption: calibration of moments and of some statistics.

7 1- Estimation of the CCAPM

8 Estimation of Model with distributional assumptions Joint log-normality of consumption and returns and conditional homoskedasticity. Definition and properties X d conditionally lognormal log E t X = E t logx Var t logx Var t logx E t [ (logx Et logx) 2] Conditional homoskedasticity Var t logx = E [(logx E t logx) 2] = Var(logX E t logx)

9 Joint log-normality of consumption and returns Let us take log of E t [R i,t+1 β( Ct+1 C t γ )] = 1 0 = E t r i,t+1 + logβ γe t c t [σ 2 i + γ 2 σ 2 c 2γσ ic ] Writing this equation for the riskless asset: r ft+1 = logβ γ 2 σ2 c 2 + γe t c t+1 Then, implication in cross-section: for the risk premium E t [ ri,t+1 r f,t+1 ] = σ 2 i 2 + γσ ic

10 Cross-sectional Regressions

11 Equity Premium Puzzle Equity premium and risk free rate puzzle. These figures imply a γ of about 19 to rationalize a 6.0% equity premium given that log consumption is smooth (σ( lnc) = 3.3%. Is it too high? Some (most) argue it is. Invoke simple introspection. Others (Kandel and Stambaugh, JME, 1991) argue that introspection can lead to very different estimates of risk aversion depending on the size of the gamble considered.

12 Risk-free Rate Puzzle But if γ is not too high, then there is a second puzzle Mean r ft = 1,8%, mean Er ft = logβ γ 2 σ2 ] c 2 + γe t [ Ct+1 [ Ct+1 ] = 1.8% γ = 19 β = 1.12, a negative rate of time preference. This is the risk-free rate puzzle, Weil (1989).

13 Intuitive Explanation The risk-free puzzle illustrates the fact that with power utility, investors who are extremely averse to risk are also very reluctant to substitute consumption intertemporally. For u = C1 γ 1 γ Coefficient of relative risk aversion u u C = γ Elasticity of intertemporal substitution ) Ct+1 u ( ( ) C t+1 C t u ε = ( ) u (C t ) 1 = (Ct+1 ) C t+1 γ u (C t ) C t Given positive average consumption growth, a low interest rate and a positive rate of time preference, such investors would have a strong desire to borrow from the future. A low riskless interest rate is possible in equilibrium only if investors have a negative rate of time preference that reduces their desire to borrow.

14 Estimation by GMM Another way to assess the model is to estimate the preference parameters without making any assumption on the distribution of consumption. Isoelastic utility and GMM Hansen and Singleton (Ecta, 1982) Estimate β and γ Moment Condition : ( ) γ Ct+1 u (X t+1,θ) = β Z C i,t+1 1 t θ = (β γ);α γ 1 f t+1 (θ) L t u (X t+1,θ) L t = instruments (1 NL) N u assets N u NL elements in f t+1 (θ)

15 Empirical Results (Hansen and Singleton, 1982)

16 Legend: EWR: Equally-weighted returns (NYSE) VWR: Value-weighted returns (NYSE) NDS: Non-Durables Plus Services ND: Non-Durables Both NDS and ND: per capita, real consumption Discussion of results NLAG: Number of lags of X t+1 = P t+1+d t+1 P t, C t+1 C t DF = NLAG (constant) - 2 (nb of parameters) Results: Estimation of α, β are plausible. Prob: Prob (XDF 2 < T Statistic computed) less evidence against the model with VWR than with EWR less evidence against the model when more lags

17 Empirical Results (Hansen and Singleton, 1982)

18 Discussion of results DF = NLAG 3 instruments 2 Moment conditions (EWR and VWR) + 2 (constant) - 2 parameters Results less favorable to model when both VWR and EWR are used together Results less favorable also when industry-average returns are used (Chemical products, Transportation and Equipment, Retail Trade).

19 2- Assessment of Models with SDF Frontier

20 SDF without models From the relations, E t [ Mt+1 R i,t+1 ] = 1 or E [ M t R i,t ] = 1 (1) What can we say about the behavior of the stochastic discount factor just observing a set of returns? Hansen and Jagannathan (1991, JFE) have shown how to derive a lower volatility bound for the SDF given a set of returns. For a given mean-value m of the SDF, we construct: M t (m) = m + (R t E (R t )) β m Var ( M t (m)) = (ι m E(R t )) 1 (ι m E(R t )) This is the volatility lower bound for any SDF with mean m. Computation of β m

21 Derivation of Hansen-Jagannathan SDF Volatility Bound Mt (m) = m + (R t E(R t )) β m (*) Mt (m) is a linear combination of returns and by (1): E ( R t Mt (m)) = ι Financial theory gives us β M. ι = E ( Mt (m)) E(R t ) + Cov ( R t,mt (m)) by covariance formula = me(r t ) + E [ (R t E(R t )) ( Mt (m) m)] = me(r t ) + E [ (R t E(R t ))(R t E(R t )) ] β m by (*) ι = me(r t ) + β m where is the unconditional variance-covariance matrix of asset returns So: β m = i me(r t ) β m = 1 (i me(r t ))

22 Derivation of Hansen-Jagannathan SDF Volatility Bound Var ( [ (M Mt (m)) = E t (m) m ) ] 2 = β me [ (R t E(R t )) (R t E(R t )) ] β m = (i me(r t )) 1 1 (i me(r t )) by (*) VarM t (m) = [i me(r t)] 1 [i me(r t )]

23 Derivation of Hansen-Jagannathan SDF Volatility Bound Is it the minimal bound? Take another M t (m), i.e. another SDF with same mean which also obeys (1) So : E [ [ R t Mt (m) Mt (m)]] = 0 Since M t (m) and Mt (m) have the same mean, the covariance is also zero: Cov ( R t,m t (m) Mt (m)) = 0 But since Mt (m) is a linear combination of asset returns: Cov ( Mt (m),m t (m) Mt (m)) = 0 So: Var (M t (m)) = Var ( M t (m) + M t (m) M t (m)) = Var ( M t (m)) + Var ( M t (m) M t (m)) Var ( M t (m)) Since Var ( M t (m) M t (m)) 0. Why is it useful to have a lower bound for volatility? Graphically, asset pricing models can be judged with reference to the mean-standard deviation frontier. But both the volatility bound for the SDF and the points computed for various models of asset pricing are estimated with error.

24 Hansen and Jagannathan (1991) Volatility Bound.

25 Duality with the canonical mean-variance portfolio problem

26 Generalization of HJ Bound Almeida and Garcia (2008) (AG) generalize the HJ approach by taking into account higher-order moments of returns This is important when considering nonlinear asset pricing models Bansal and Viswanathan (JF, 1993), Harvey and Siddique (JF, 2000), Dittmar (JF, 2002), Vanden (RFS, 2006), Timmermann and Guidolin (RFS, 2008), among others....or evaluating the performance of portfolios with options and other nonlinear assets (hedge funds)

27 Generalization of HJ Bounds But, how to account for higher moments? - Snow (J. Finance, 1991) Stutzer (J. Econometrics, 1995), Bansal and Lehmann (Macroeconomic Dynamics, 1997), Chabi-Yo (RFS, 2008). SDF information bounds are based on discrepancy functions that directly generalize the HJ variance bound. Duality arguments link the solution of these SDF information bounds to the solutions of HARA portfolio problems. AG completely characterize the SDFs implied by their bounds and show that they are useful in a variety of empirical applications.

28 Minimum Discrepancy SDF Bounds Let R denote the vector of primitive securities returns whose realizations {R i } i=1,...,t are given in a K-dimensional space. For a fixed mean a, a SDF m will be associated with a probability measure π by m i = a T π i,i = 1,...,T. The moment conditions for the MD estimator will be the Euler equations that any admissible SDF should satisfy to price the primitive securities: 1 T T i=1 ( m i R i 1 ) = 0 a K (2) The Minimum Discrepancy SDF bound is defined by: 1 ˆm MD = arg min T T i=1 φ(m i), {m 1,...m T } subject to 1 T T i=1 m i ( ) R i a 1 1 = 0, T T i=1 m i = a,m i > 0 i. (3)

29 Choosing the discrepancy: The Cressie Read (CR) Family The CR discrepancy functions are given by φ(π) = (π)γ+1 1 γ(γ+1). They are interesting because we can interpret their dual problems as HARA utility maximizing problems: 1 ˆλCR = arg sup λ Λ CR T T i=1 1 ( (1 + γλ R γ + 1 i 1 ) ) ( γ+1 γ ) a The implied SDF can be recovered via the first order conditions of the problem: ˆm MD i = T a (1+γˆλ CR(R i 1 a )) 1 γ T j=1 (1+γˆλ CR(R j 1 a )) 1 γ The SDF-related frontier is found by solving (4) for a grid of meaningful values for the SDF mean A = {a 1,a 2,...,a J }. The SDF-related frontier is given by: I CR (a l,γ) = 1 T T ˆm i MD (a l) γ+1 1,l = 1,2,...,J (5) γ(γ + 1) i=1 (4)

30 Skewness and Kurtosis Weights given by Cressie Read Estimators The goal is to analyze how the coefficient of risk aversion γ affects the weights given to skewness and kurtosis in the specific solutions of our CR HARA-utility problems. E[u(v)] u(v0) u 2(v0)λ 2 opt E(R E(R)) u 3(v0)λ 3 opt E(R E(R)) u 4(v0)λ 4 opt E(R E(R))4 where v0 = λ opt E(R 1 a ) is a scaled expected excess return.

31 3- Calibration of CCAPM with Exogenous Consumption Process

32 Exogenous Process for the endowment and Calibration Suppose now an exogenous process for consumption as in Mehra and Prescott (JME 1985). The growth rate of consumption follows a Markov process Pr [ x t+1 = λ j x t = λ i ] = φij where x t+1 = C t+1 C t. Suppose 2 states for the Markov chain. The growth rate can take 2 values: λ 1 = 1 + µ + δ λ 2 = 1 + µ δ µ controls the average growth level in the economy

33 Calibration of the model - Mehra and Prescott, 1985 In the Lucas model, production=consumption=dividends δ controls the variability of the growth rate φ 11 = φ 22 = φ. The φ parameter controls for the degree of serial correlation in growth rates The parameter values are obtained by calibration. µ, δ, and φ set so that the theoretical mean, variance and autocorrelation of the process correspond to the empirical moments of observed aggregate consumption. Derivation of the parameters by calibration [ ] Ct+1 E = 1 C t 2 (1 + µ + δ) + 1 (1 + µ δ) = 1 + µ 2 µ = Similarly: [ ] Ct+1 Var = 1 C t 2 δ δ2 = δ 2 δ = 0.036

34 Calibration of the Model ρ 1 = E [ (X t E(X t ))(X t 1 E(X t 1 ) ] /σ(x t )σ(x t 1 ) = 1 δ 2 E [ X t X t 1 E(X t )E(X t 1 ) ] E ( X t X t 1 ) = Pr(X t = λ 1,X t 1 =λ 1 )λ 2 1 +Pr(X t =λ 1 =X t 1 =λ 2 +Pr ( X t =λ 2 X t 1 =λ 1 ) λ1 λ 2 +Pr ( X 1 =λ 2 X t 1 =λ 2 ) λ 2 2 Pr [ X t =λ 1 X t 1 =λ 2 ] =Pr [ Xt =λ 1 Xt 1 =λ 2 ] Pr [ Xt 1 =λ 2 ] Pr [ X t 1 = λ 2 ] Π 2 1 φ 11 2 φ 11 φ 12 = = unconditional probability 1 φ 2(1 φ) = 1 2 = Π 1

35 Calibration of the Model Then: E ( X t X t 1 ) = = 1 2 φ(1 + µ + δ)2 + 1 (1 φ)(1 + µ + δ)(1 + µ δ) (1 φ)(1 + µ + δ)(1 + µ δ) + 1 φ(1 + µ δ)2 2 1 [ 2 φ (1 µ) 2 + δ 2] + 1 [ 2 φ (1 + µ) 2 + δ 2] +(1 φ) [(1 + µ) 2 δ 2] = 2φδ 2 δ 2 + (1 + µ) 2 = (2φ 1)δ 2 + (1 + µ) 2 So: ρ 1 = 1 [ δ 2 (2φ 1)δ 2] = 2φ 1 = 0.14 φ = 0.43.

36 Solution of the Model If we rewrite the first-order condition with this process for consumption: p e 2 (c,i) = β φ ij (λ j c) α [ p e ] (λ j c,j) + cλ j c α j=1 p e (c,i) is homogeneous of degree 1 in c p e (c,i) = w i c w i = 2 β φ ij λ (1 α) j (w j + 1) j=1 r e ij = pe (λ j c,j) + λ j c p e (c,i) p e (c,i) = λ j(w j + 1) w i 1

37 Solution of the Model [ w 1 = β w 2 = β w = β {[ ] 2 (w 2 + 1) ] φλ (1 α) 1 (w 1 + 1) + (1 φ)λ (1 α) [ (1 φ)λ 1 α 1 (w 1 + 1) + φλ (1 α) 2 (w 2 + 1) φλ (1 α) 1 (1 φ)λ (1 α) 2 (1 φ)λ (1 α) 1 φλ (1 α) 2 (I βa)w = βaι w = (I βa) 1 βaι ] (w + ι) }

38 Solution of the Model ( ) ( )] R e λ1 (w = Π 1 [φ λ2 (w 1 + (1 φ) 2 + 1) 1 w 1 w 1 ( ) ( )] λ1 (w +Π 2 [(1 φ) 1 + 1) λ2 (w 1 + φ w 2 w 2 R f = Π 1 R 1f + Π 2 R 2 f R 1f = 1 p 1f 1, R 2f = 1 p 2f 1 p 1f = β [ φλ α 1 + (1 φ)λ α 2 ], p2f = β [ (1 φ)λ α 1 + φλ α 2 Using the calibrated parameter values and a grid of values for the preference parameters, Mehra and Prescott (1985) could not reproduce the observed equity premium. ]

39 Review of Empirical Evidence about Basic CCAPM We have used three methodologies to assess the empirical validity of the CCAPM. All three reject the model. Reasonable values for the parameters are not able to explain the levels of the risk premium and the risk-free rate (joint log-normality of consumption growth and returns and calibration of consumption growth by a Markov-chain process). Overidentification restrictions of the intertemporal asset pricing model are usually rejected when tested with data on consumption growth and asset returns, especially when additive time-separable utility with constant relative risk aversion is assumed for the representative agent

40 Potential Misspecifications Gallant and Tauchen (Econometrica, 1989) use seminonparametric techniques to determine whether (and to what extent) misspecification errors can explain the rejections of the model. Seminonparametric methods - The stochastic processes for returns and consumption and - the utility function are represented in an augmented parameter space. Gaussian VARs are a particular case of this generalized parameterization In a VAR, variables are linearly projected on lagged variables. In a seminonparametric approach, variables are projected on lagged variables non linearly (can accommodate conditional heteroskedasticity or nonlinear conditional functional forms of lagged variables).

41 Seminonparametric Approach Example: h ( ) y T y L h = α 0 + α 1 y L + K β K g ( y ) L, where g can be logistic functions k=1 exp(1), polynomial expansions,etc. 1+exp(1) Similarly, the utility function can be characterized by a general functional form whereby utility depends on past consumption and includes as a special case additively separable utility with constant relative risk aversion. Example: U (C t J,,C t ) = (1 γ) 1 { C (1 γ) t [ 1 + poly( Ct J C t,, C t 1 C t )]} where poly (.) is a polynomial of degree K in J arguments.

42 Results SNP models for consumption growth and real returns show that there exists a strong conditional heterogeneity. But the misspecification of the stochastic process is not the reason of the rejection obtained by Hansen and Singleton (1983) with a Gaussian model. The rejection comes more from the mispecification of the utility function. This result is implicit in the rejection of the model when estimating by GMM and specifying an additively separable utility function with constant relative risk aversion. Once the intertemporal non-separabilities are accounted for, the model is not rejected. How to characterize these non-separabilities.

43 Generalizing Preferences in the Consumption-Based Asset Pricing Model

44 Recursive Utility Models Recursive utility (Epstein and Zin, Ecta 1989) In a deterministic context, the recursive utility framework is defined as follows. V (C 0,C 1,C 2, ) = W (C 0,V (C 1,C 2, )) Current utility is determined by current consumption and future utility through an aggregation function W. In a stochastic framework, future utility is random and one needs to form a certainty equivalent µ of random future utility, so: V (C 0,m) = W (C 0,µ[V (m)]) where m represents vector of future consumption.

45 Certainty Equivalents The aggregating function W is a constant elasticity of substitution function W (x,y) = ( x ρ + βy ρ) 1 ρ so: U t = W (C t,µ t ) U t = [ C ρ t + βµ ρ ] 1 ρ t The models will differ by the certainty equivalent µ. A special case: expected utility µ(p) = ( x ρ dp(x)) 1 ρ = ( Ep x ρ) 1 ρ The ρ parameter which weights the realizations of x is the same as the weighting parameter in the aggregation function. [ ] 1 V (C 0,m) = Co ρ ρ + Em β t C ρ t 1 This is the standard power utility model.

46 Kreps-Porteus Model Other class: Kreps-Porteus The utility function is given by: µ(p) = (E p x γ ) 1 γ [ ] V (C o,m) = C ρ 0 + βe mv γ (.) ρ 1 ρ γ With these preferences, the investor can have various attitudes towards resolution of uncertainty over time. ρ is interpreted as a parameter for intertemporal substitution σ = 1 1 ρ γ is interpreted as risk aversion parameter γ < ρ the agent prefers early resolution of uncertainty γ > ρ the agent prefers late resolution of uncertainty

47 Resolution of Uncertainty

48 Kreps-Porteus Model - SDF The utility function can be parameterized as follows: { U t = C 1 γ θ t with: θ (1 γ)/(1 1/σ). ( + β E t U 1 γ t+1 ) 1 θ } θ 1 γ The intertemporal budget constraint for a representative agent can be written as follows: W t+1 = (1 + R m,t+1 )(W t C t ) where (1 + R m,t+1 ) is the return on the market portfolio of all invested wealth. We then obtain the following Euler condition: { ( ) 1 } θ { } 1 = E t Ct+1 σ 1 1 θ β (1 + R C t (1 + R m,t+1 ) i,t+1)

49 Under Lognormality of Returns Assuming that returns and consumption are homoskedastic and jointly lognormal: Riskless real interest rate: r f,t+1 = logβ + θ 1 2 V mm θ 2σ 2 V cc + 1 σ E t c t+1 Risk premium on market portfolio. E t [ rm,t+1 r f,t+1 ] = (1/2 θ)vmm + θ σ V cc Risk premium on an asset other than the market portfolio: E t r i,t+1 r f,t+1 = V ii 2 + θv ic σ + (1 θ)v im

50 Interpretation Interpretation of: E t r i,t+1 r f,t+1 = V ii 2 + θv ic σ + (1 θ)v im The risk premium on an asset i is a weighted combination of asset i covariance with consumption growth (divided by the elasticity of intertemporal substitution σ ) and asset i covariance with the market return. θ = 1 CCAPM θ = 0 CAPM Martin and Shapiro (1986) found that betas with the market have greater explanatory power for the cross-sectional pattern of returns than do betas with consumption.

51 Tests of KP model by GMM Test of this class of preferences by GMM (Epstein-Zin, JPE 1991). Moments conditions (change of notation). γ = α ρ α parameter of risk aversion E t β γ [ Ct+1 C t ] γ(ρ 1) (1 ) γ 1 ( ) + Rm,t Ri,t+1 = 1,i = 1,...,N [ E t ) ] ρ 1 Ct+1 ( ) β( C 1 + Rm,t+1 t γ 1 = 0 Test of γ = 1 is a test of expected utility. t - test in finite sample not invariant to nonlinear transformations of the assumption. So we have two estimations with and without imposing γ = 1. The difference between the values of the objective function is a test statistic equivalent to a likelihood ratio test, X 2 (1).

52 Estimation and Tests of KP model by GMM

53 Estimation and Tests of KP model by GMM

54 Discussion of Results σ is small, always less than 1.; Risk preferences: γ close to zero, logarithmic preferences. δ often negative but estimated with little precision. The results are sensitive to the measure of consumption and to the instruments. More evidence against the model for Non Durables and Services. For the two first sets of instruments, expected utility model is rejected. Results are not robust.

55 Disappointment Aversion Other class of recursive utility preferences called disappointment aversion µ(.) is defined implicitly as ( ) y φ DA dp(y) = 0 µ(p) with: { v(x) v(1) x 1 φ DA (x) = A(v(x) v(1) x < 1 { x α 1 v(x) = α α 0 log(x) α = 0 x = 1 when realization is equal to certainty equivalent.

56 Loss Aversion This is similar in spirit to loss aversion (Kahmeman and Tversky, 1979, prospect theory) in a static context. Preferences are defined over gains caused losses relative to a benchmark outcome. { x 1 γ 1 1 γ v(x) = 1 1 of x 0 λ x 1 γ 2 1 γ 2 1 of x < 0

57 Allais Paradox Disappointment aversion can explain Allais paradox. An agent prefers p 1, a degenerate lottery which gives m for sure, to a lottery p 2, which provides m (>> m) with a 0.9 probability and 0 with a 0.1 probability. However, he prefers p 4 which provides m with a 0.45 probability and 0 with a 0.55 probability, to p 3 which provides m with a 0.5 probability and 0 with a 0.5 probability. The paradox: p 1 p 2 u(m) > 0.9 u(m ) u(0) p 4 p u(m) u(0) < 0.45 u(m ) u(0) Dividing by 0.5 the second inequality: u(m) + u(0) < 0.9 u(m ) u(0) u(m) < 0.9 u(m ) u(0), a reversal of preferences Such an agent cannot maximize expected utility.

58 Estimation and Tests of EU model by GMM

59 Estimation and Tests of KP model by GMM

60 Estimation and Tests of DA model by GMM

61 Long-run Risks Model What are the risks affecting consumption? Bansal and Yaron (2004) (BY): Long-run risks (LRR) model a small long-run predictable component driving consumption fluctuating economic uncertainty measured by consumption volatility

62 Long-run Risks Model The LRR model driving fundamentals in the economy: c t+1 = x t + σ t ε c,t+1 d t+1 = (1 φ d )µ x + φ d x t + ν d σ t ε d,t+1 x t+1 = (1 φ x )µ x + φ x x t + ν x σ t ε x,t+1 σ 2 t+1 = (1 φ σ)µ σ + φ σ σ 2 t + ν σ ε σ,t+1 c t logarithm of real consumption, d t logarithm of real dividends. Correlation between innovations in consumption growth and dividend growth; independence between innovations in consumption growth and volatility.

63 Potential empirical issues with the BY LRR model The existence of a long-run risk component in expected consumption growth is a source of debate: it is hard to detect statistically by univariate methods - consumption is close to a random walk; the effect on asset prices depends on investors detecting it; it makes (counterfactually) consumption growth predictable by the price-dividend ratio. A more recent calibration Bansal, Kiku and Yaron (2007) (BKY) shifts the weight towards the second source of long-run risk - persistent volatility - reducing the predictability of consumption growth. In their model, the two sources interact, but the volatility risk is not priced when expected consumption growth is not persistent (BKY 2009).

64 Disappointment Aversion and Long-Run Risks BGMT (forthcoming RFS) propose consumption-based asset pricing model: with Long-run risk in consumption growth volatility only and Generalized Disappointment Aversion preferences Solution of the model Solve analytically the model. Assessment of the model Reproduction of stylized facts: analytical formulas for stylized moments of returns and asset valuation ratios, predictive regressions Advantage: test robustness of model with respect to changes in parameters of of fundamentals and preferences Point to deficiencies of model otherwise non-apparent

65 Benchmark Random Walk Heteroscedastic Model c t+1 = µ c + σ t ε c,t+1 d t+1 = µ c + ν d σ t ε d,t+1 σ 2 t+1 = (1 φ σ)µ σ + φ σ σ 2 t + ν σ ε σ,t+1

66 GDA Preferences A generalization of Gul s (1991) disappointment aversion preferences introduced by Routledge and Zin (2009). It overweights outcomes below a threshold - κ times the certainty equivalent. The kink makes it specially sensitive to volatility risks. Interaction of long-run volatility risks and GDA preferences generates interesting asset pricing dynamics.

67 Calibration of the parameters φ x, φ σ parameters of special interest. BKY assume: φ x = and φ σ = long-run risks φ d = 2.5 and ν d = 6.5 ν x = Assume φ σ = (half life of 11.5 years) as in Lettau, Ludvigson and Wachter (2008), instead of the (half life of 58 years) in BKY. The other parameters are as in BKY.

68 Generalized Disappointment Aversion Introduced by Routledge and Zin (2009), generalizing Gul (1991): where + (1 α 1) (,κr ) R 1 γ 1 γ (, ) = ( V 1 γ 1 γ V 1 γ df (V ) 1 γ ) (κr )1 γ 1 γ α 1 measures the intensity of disappointment aversion disappointment aversion: α < 1 Kreps-Porteus: α = 1. df (V ) κ 1 (6) κ measures the place of the kink in terms of percentage of the certainty equivalent R.

69 GDA SDF ( M t,t+1 = z 1 γ ( ) t+1 R m ( α 1 1 ) ) I (z t+1 < κ) t κ ( 1 γ α 1 1 ). (7) E t I (z t+1 < κ) where z t+1 = ( δ ( Ct+1 C t ) 1 ψ R m t+1 ) ψ R m is the return on the portfolio that generates the flow of aggregate consumption.

70 Intuition about Disappointment Aversion Particular case:γ = 0 and ψ =. ( ) ( ) M t,t+1 = δ 1 + α 1 1 I Rt+1 m < κ δ 1 + κ ( α 1 1 ) ( ) E t I Rt+1 m < κ δ For each state in t the sdf has only two possible values: one for non-disappointing outcomes and another α 1 times greater for disappointing outcomes - it generates variability in the sdf - necessary to produce sizeable risk premia. The probability of disappointing outcomes may differ for different states. - it generates state-dependent risk premium.

71 A Large Equity Premium: How it works Since E t [Rt+1 e R f procyclical: ] = Cov t(m t,t+1,rt+1) e, and stock market returns are E t [M t,t+1 ] in states where disappointing outcomes have sizeable probabilities: - when return on the market portfolio is low, R) e is low and M is high (disappointment). Thus, Cov t (M t,t+1,rt+1 e << 0, generating sizeable equity premia. in states where the probability of disappointing outcomes is very small: ) - M is almost a constant and Cov t (M t,t+1,rt+1 e is small.

72 LRR and GDA Expected utility: LRRs do not matter ( ) 1 Ct+1 ψ M t,t+1 = δ C t Kreps-Porteus: LRRs and R t (V t+1 ) - depends on γ > 1 γ : ( ) 1 ( ) 1 Ct+1 ψ V ψ γ M t,t+1 = δ t+1 C t R t (V t+1 ) GDA: additional channel: LRRs and the kink - does not depend on ψ. ( ) 1 ( Ct+1 ψ V M t,t+1 = δ t+1 C t R t (V t+1 ) ( ) 1 + α 1 1 ) 1 ψ γ I ( Vt+1 R t (V t+1 ) < κ ) 1 + κ 1 γ ( α 1 1 ) E t I ( Vt+1 R t (V t+1 ) < κ )

73 Solving the Model Bansal and Yaron (2004) use an approximate solution based on Campbell and Shiller (2004) log linearization. Hansen, Heaton and Li (2005) use an approximation around a unitary value for the elasticity of intertemporal substitution ψ. Since the GDA utility is non-differentiable at the kink where disappointment sets in, one cannot rely on the same approximation techniques to obtain analytical solutions of the model.

74 Solving the Model To sidestep this problem, use the following procedure: Approximate the LRR process for consumption and dividends using a Markov Switching process. BGMT derive analytical formulas for: the population moments of asset returns coefficients and R 2 of predictability regressions.

75 Approximating Endowment Process Let s t be the Markov state at time t. For BKY process we combine two states in mean and in volatility to obtain four states, s t {µ L σ L,µ L σ H,µ H σ L,µ H σ H }. c t+1 = µ c (s t ) + (ω c (s t )) 1/2 ε c,t+1 (8) d t+1 = µ d (s t ) + (ω d (s t )) 1/2 ε d,t+1, (9) where ε c,t+1 and ε d,t+1 follow a bivariate normal process with mean zero and correlation ρ. The states evolve according to the 4 by 4 transition probability matrix P. For random walk in mean the process above is reduced to two states in volatility.

76 Matching Procedure 1 The expected means of the consumption and dividend growth rates are a linear function of the same autoregressive process of order one denoted x t ; We assume that the expected means of the consumption and dividend growth rates are a linear function of the same Markov chain with two states given that a two-state Markov chain is an AR(1) process. 2 The conditional variances of the consumption and dividend growth rates are a linear function of the same autoregressive process of order one denoted h t ; The conditional variances of the consumption and dividend growth rates are a linear function of the same two-state Markov chain. 3 The variables x t+1 and h t+1 are independent conditionally to their past; The two Markov chains should be independent; hence four states. 4 The innovations of the consumption and dividend growth rates are correlated given the state variables.

77 Matching Parameters of the LRR Model of BY Panel A σ L σ H µ c ( µ d) /2 ω c ( ) 1/ ω d ρ P σ L σ H Π Panel B µ L σ L µ L σ H µ H σ L µ H σ H µ c ( µ d) /2 ω c ( ) 1/ ω d ρ P µ L σ L µ L σ H µ H σ L µ H σ H Π

78 Benchmark Preference Parameters ψ = 1.5 as in BY and Lettau, Ludvigson and Watcher (2008). γ = 2.5 and α = 0.3 are consistent with estimation of Epstein and Zin (2001). κ = as in Routledge and Zin (2009).

79 Analytical Formulas for Statistics Reproducing Stylized Facts Moment Matching: We derive analytical formulas for E [R t+1:t+h J t ] and Var [ ] [R t+1:t+h ] [ (equity ] returns, risk-free rate, excess returns) as well as E PD and Var PD. Predictability Regressions: Typically, when one runs the linear regression of a variable, say y t+1:t+h, onto by another one, say x t, and a constant, one gets y t+1:t+h = a(h) + b (h)x t + η y,1,t+h (h) where b (h) = Cov (y t+1:t+h,x t ) Var [x t ] while the corresponding population coefficient of determination denoted R 2 (h) is given by: R 2 (h) = (Cov (y t+1:t+h,x t )) 2 Var [y t+1:t+h ]Var [x t ].

80 Asset Pricing Implications: GDA with Benchmark Model Data GDA 50% PV δ γ 2.5 ψ 1.5 α 0.3 κ Panel A. Asset Pricing Implications E [R R f ] σ[r] E [R f ] σ[r f ] E [P/D] σ[d/p] Panel B. Predictability of Excess Returns R 2 (1) [b (1)] R 2 (3) [b (3)] R 2 (5) [b (5)]

81 Asset Pricing Implications: GDA with Benchmark Model Data GDA 50% PV δ γ 2.5 ψ 1.5 α 0.3 κ Panel C. Predictability of Consumption Growth R 2 (1) [b (1)] R 2 (3) [b (3)] R 2 (5) [b (5)] Panel D. Predictability of Dividend Growth R 2 (1) [b (1)] R 2 (3) [b (3)] R 2 (5) [b (5)]

82 Asset Pricing Implications: Robustness to Preference Parameters

83 Asset Pricing Implications: Robustness to Preference Parameters

84 Sensitivity to Persistence in Volatility - Asset Pricing E[R R f ] σ[r R f ] E[R f ] σ[r f ] E[P/D] φ σ σ[d/p] GDA KP DA0 GDA φ σ

85 Sensitivity to Persistence in Volatility - Predictability 1Y 3Y 5Y R GDA KP DA0 GDA SLOPE φ σ φ σ φ σ

86 Comparison of Models in the BY Environment We compare all preferences used above in BY environment: GDA preferences do at least as well as above. KP has good performance for moments, as we know from BY. KP has difficulties in generating predictability for excess returns and tends to generate excess predictability for consumption growth, although it is not always rejected in small sample tests.

87 Comparison of Models in the BY Environment

88 Sensitivity to Persistence in Expected Consumption Growth - Asset Pricing E[R R f ] σ[r R f ] GDA KP DA0 GDA E[R f ] 1 0 σ[r f ] E[P/D] 20 σ[d/p] φ x φ x

89 Sensitivity to Persistence in Expected Consumption Growth - Asset Pricing 40 R R f, 1Y 40 R R f, 3Y 40 R R f, 5Y R SLOPE GDA KP DA0 GDA φ x φ x φ x

90 Other generalizations of preferences

91 State-Dependent Preferences With disappointement preferences, the risk aversion is changing above and below the certainty equivalent µ( ). More generally, one can make the risk aversion parameter dependent on an observable or inobservable state variable.

92 State-Dependent Preferences Melino and Yang (2003) generalize the model of Epstein and Zin (1989) by allowing the representative agent to display state dependent preferences. The parameters β, α, ρ are all state-dependent and then denoted as β(u t ),α(u t ) and ρ(u t ), where U t represents a state variable. Melino and Yang (2003) show that the SDF is: m t+1 = β(u t )( Ct+1 C t ) ρ(ut ) ρ(u t ) ρ(u t+1 ) γ(u t ) R α(u t ) ρ(u t+1 ) 1 mt+1 P α(u t ) ρ(u t+1 ) α(u t ) ρ(u t ) t (10) where γ(u t ) = α(u t ) ρ(u t ) and P t is the time t price of the market portfolio.

93 State-Dependent Preferences When β(u t ), α(u t ), ρ(u t ) = ρ(u t+1 ) are constants, this pricing kernel reduces to the Epstein and Zin SDF. For simplicity, they work within the simple environment introduced by Mehra and Prescott (1985). They investigate various combinations of state dependent CRRA with state dependent EIS. In the case of constant elasticity of intertemporal substitution and time-varying CRRA, the results look very similar to those generated without state dependence. However, combining a state-dependent EIS with a state-dependent CRRA allows them to match perfectly the US historical data.

94 Habit formation and consumption externalities Time non-separabilities can be caused by habit formation (Constantinides (1990), Sundaresan(1989). Today s consumption has a positive effect on tomorrow s marginal utility of consumption. So utility is U (C t,x t ), where X t is a time-varying habit or subsistence level.

95 Modeling issues First issue: functional form of U. Ratio models: U(.) power function of the ratio C t /X t (Abel 1990, 1996). Difference models: U(.) power function of the difference C t X t (Constantinides, 1990; Sundaresan (1989), Campbell and Cochrane (1999)). Second issue: effect of an agent s own decisions on future levels of habit. Internal habit: habit depends on an agent s own consumption (Constantinides (1990), Sundaresan (1989)). External habit: habit depends on aggregate consumption which is unaffected by any one agent s decisions. (Abel, 1990, 1996: Catching up with the Joneses ; Campbell and Cochrane (1999)).

96 Ratio models U t = j=0 using one lag of consumption: X t = C κ t 1 for internal habit δ j ( Ct+j /X t+j ) 1 γ 1 1 γ X t = C t 1 κ for external habit, where C t 1 represents aggregate past consumption. The parameter κ governs the degree of time non-separability. Note: Since it is a representative agent economy, in equilibrium the agent s consumption must equal aggregate consumption, but the two formulations yield different Euler equations.

97 Ratio Model with External Habit Euler equation with external habit Then : U = C κ(γ 1) C t 1 t C γ t ( ) κ(γ 1) ( ) ] γ Ct Ct+1 1 = δe t [(1 + R it+1 ) C t 1 C t

98 Ratio Model with External Habit Assuming homoskedasticity and joint lognormality of asset returns and consumption growth: Risk-free rate r f, t+1 = logδ γ 2 σ2 c 2 + γe t [ c t+1 ] κ(γ 1) c t E t [ ri,t+1 r f,t+1 ] = σ 2 i 2 + γδ ic The riskless real interest rate is equal to its value under power utility plus the extra term κ(γ 1) c t. The risk premium is exactly the same as the risk premium with power utility.

99 Ratio Model with External Habit Abel (1990,1996) argues that Catching up with the Joneses helps to explain the equity premium puzzle. (1) The average level of the riskless real interest rate: r f = logδ γ 2 σ2 c 2 + (γ κ(γ 1))g, g is the average consumption growth rate. For γ large and κ > 0, r is lower. So one can increase γ to solve the equity premium puzzle without encountering the risk free rate puzzle. (2) A positive κ is likely to increase the riskless rate volatility because of κ(γ 1) c t

100 Ratio Model with External Habit (3) The last point is a disadvantage since the real interest rate does not vary greatly in the short run. σ(r) = 5.5% σ( c t ) 3%. Large values of κ and γ = too much volatility in the expected real interest rate. This is a fundamental problem for habit formation models. With time-nonseparable preferences, consumers derive utility from consumption relative to its recent history rather than from the absolute level of consumption. So expected marginal utility is subject to large swings at successive dates, which implies large movements in the real interest rate.

101 Difference models U t = E t [ J=0 X t is considered external. ( ) 1 γ ] δ j Ct+j X t+j 1 1 γ Two important differences with previous model (ratio). 1 In the difference model, the agent s risk aversion varies with the level of consumption relative to habit (in the ratio model, risk aversion is constant). 2 In the difference model, consumption must always be above habit for utility to be well defined.

102 First point Define consumption surplus S t : Difference models S t C t X t C t CUcc Uc = γ S t Risk aversion rises as S t declines (when consumption declines toward habit). Second point For exogenous consumption processes, this poses a problem. To handle this problem: Campbell-Cochrane (1999) specify a nonlinear process by which habit adjusts to consumption, remaining below consumption at all times.

103 Campbell-Cochrane (1999) Log surplus consumption Ratio s t log(s t ) Log consumption is random walk plus drift g. c t+1 = g + v t+1 Dynamics of log surplus consumption ratio s t+1 = (1 φ) s + φs t + λ(s t )v t+1 where s is the steady-state surplus consumption ratio. φ persistence of s t+1 λ(s t ) sensitivity of s t+1 (and therefore log X t+1 ) to innovations in consumption growth v t+1

104 Campbell-Cochrane (1999) Since habit is external SDF: M t+1 δ u (C t+1 ) u (C t ) Power utility S t = 1. u (C t ) = (C t X t ) γ = S γ t C γ t ( ) = δ St+1 C γ t+1 S t C t

105 Campbell-Cochrane (1999) In the habit-formation model, a volatile SDF can be obtained from a volatile consumption ratio S t. Riskless real interest rate: In logs: 1 + R f t+1 = 1 E t (M t+1 ) r f t+1 = log(δ) + γg γ(1 φ)(s t s) γ2 σ 2 v 2 [λ(s t) + 1] 2

106 Campbell-Cochrane (1999) First two terms of left-hand side, same as in power utility. Third term: linear function of s t s, intertemporal substitution. If s t is low, marginal utility of consumption is high But s t is expect to revert to its mean, so marginal utility is expected to fall in the future The consumer would like to borrow, driving up the equilibrium risk-free interest rate. Fourth term: linear function of σ 2 v in [λ(s t ) + 1] 2, precautionary savings. As σ 2 v increases, consumers would like to save, which drives down the equilibrium interest rate.

107 Campbell-Cochrane (1999) To generate stable interest rates like those observed in the data, serial correlation φ must be near one; λ(s t ) should decline with s t so uncertainty is high when s t is low. The precautionary saving term offsets the intertemporal substitution In Campbell-Cochrane, they choose λ(s t ) so that the two effects cancel each other, implying a constant riskless interest rate Even with a constant rate, the model produces a large equity premium, volatile stock prices, and predictable excess stock returns. The basic mechanism is time-variation in risk aversion. Campbell-Cochrane calibrate their model to US data on consumption and dividends, solving for equilibrium stock prices in the tradition of Mehra and Prescott (1985).

108 Campbell-Cochrane (1999)

109 Campbell-Cochrane (1999)

110 Campbell-Cochrane (1999)

111 Campbell-Cochrane (1999)