The Long Run Risks Model

Transcription

1 1 / 83 The Long Run Risks Model René Garcia EDHEC Business School Lectures Center for Applied Economics and Policy Research, Indiana University September 2012

2 2 / 83 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 3 / 83 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 4 / 83 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 5 / 83 Consumption Capital Asset Pricing Model (CCAPM) under the constraint: Max (Ct,w i,t+1 )E t β i U(C t+i ),i = 1,...,n i=0 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 ( ) ] γ Ct+1 E t [R i,t+1 β = 1 (1) C t

6 6 / 83 Empirical Strategies 1 Objective of empirical work: estimate preference parameters β and γ and test restrictions imposed by (1). Hansen and Singleton (1982, 1983) Assumption about distributions: Joint log-normality of consumption and returns. No assumption about distributions: estimation of Euler equations by GMM 2 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. Mehra and Prescott (1985)

7 7 / 83 Review of Empirical Evidence about Basic CCAPM Most studies that assessed the empirical validity of the CCAPM 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, and additive time-separable utility with constant relative risk aversion is assumed for the representative agent

8 8 / 83 The Recursive Utility Kreps-Porteus Model The utility function can be parameterized as follows: U t = { (1 β)ct 1 γ θ ( + β E t U 1 γ t+1 ) 1 θ } θ 1 γ with: θ (1 γ)/(1 1/ψ). The intertemporal budget constraint for a representative agent can be written as follows: W t+1 = R c,t+1 (W t C t ) where (R c,t+1 ) is the return on the aggregate portfolio of all invested wealth. The logarithm of the intertemporal marginal rate of subsitution (IMRS) m t+1 is: m t+1 = θlogβ θ ψ g t+1 + (θ 1)r c,t+1

9 9 / 83 Asset Pricing Restrictions and Fundamentals The asset pricing restriction on any asset continuous return r i,t+1 is: E t [exp (θlogβ θψ )] g t+1 + (θ 1)r c,t+1 + r i,t+1 = 1 where g t+1 equals log(c t+1 /C t and r c,t+1 is the log of the return on an asset that delivers aggregate consumption at each time period (it is unobservable). The return on an asset that delivers aggregate dividends is the observed return of a market portfolio, r m,t+1 Aggregate consumption includes aggregate dividends but is much larger, the difference between the two being labor income.

10 10 / 83 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

11 11 / 83 Long-Run Growth and Economic Uncertainty Risks The LRR model driving fundamentals in the economy: g t+1 = µ + x t + σ t η t+1 x t+1 = ρx t + ϕ e σ t e t+1 σ 2 t+1 = σ2 + ν 1 (σ 2 t σ2 ) + σ w w t+1 g d,t+1 = µ d + φx t + ϕ d σ t u t+1 c t logarithm of real consumption, d t logarithm of real dividends. All innovations are ℵ, i.i.d.(0, 1).

12 12 / 83 Solution of the Model To derive solutions, BY (2004) use the standard approximations of the return formula used in Campbell and Shiller (1988): r c,t+1 = κ 0 + κ 1 z t+1 z t + g t+1 κ 0 and κ 1 are approximating constants that depend only on the average level of z. The relevant state variables in solving for the equilibrium price-consumption ratio are x t and σ 2 t. The approximate solution for z t is conjectured to be: z t = A 0 + A 1 x t + A 2 σ 2 t

13 13 / 83 Solution of the Model Since g and g d are exogenous, a solution for z t characterizes completely the return r c,t+1. Using the Euler condition and this expression for z t one obtains the following solutions: A 1 = A 2 = 1 1 ψ 1 κ 1 ρ 0.5[ ( θ θ ψ ) 2 + (θa1 κ 1 ϕ e ) 2 ] θ(1 κ 1 ν 1 )

14 14 / 83 Interpretation of the Solution A 1 is positive if the IES ψ is greater than 1, that is if the intertemporal substitution effect dominates the wealth effect. Higher expected growth means higher expected returns so agents buy more assets and the wealth-consumption ratio rises. In the standard power utility, risk aversion greater than 1 means ψ is lower than 1, so prices fall. If the IES and risk aversion are larger than 1, then θ is negative and a rise in volatility lowers the price-consumption ratio. An increase in the persistence of volatility shocks (increase in ν 1 ) magnifies the effect of volatility shocks on valuation ratios. Effects are similar and stronger for the price-dividend ratio.

15 15 / 83 The solution for the price-dividend ratio Similar expressions can be derived for the coefficients A 1,m and A 2,m of z m,t. The approximate solution for z m,t is conjectured to be: z m,t = A 0,m + A 1,m x t + A 2,m σ 2 t Using the Euler condition for the return on the market portfolio (that delivers aggregate dividends) and this expression for z m,t one obtains the following solutions: A 1,m = φ 1 ψ 1 κ 1 ρ A 2,m = (1 θ)a 2(1 κ 1 ν 1 ) + 0.5H m (1 κ 1,m ν 1 ) with: H m [λ 2 m,ν + ( λ m,e + β m,e ) 2 + ψ 2 d ] and β m,e κ 1,m A 1,m ψ e

16 16 / 83 The Pricing Kernel Innovations in the pricing kernel m t+1 E t [m t+1 ] = θ ψ (g t+1 E t [g t+1 ]) + (θ 1)(r c,t+1 E t [r c,t+1 ]) Innovations in the consumption growth: g t+1 E t [g t+1 ] = σ t η t+1 Innovations in the wealth portfolio return: Finally: r c,t+1 E t [r c,t+1 ] = σ t η t+1 + κ 1 A 1 ψ e σ t e t+1 + A 2 κ 1 σ w w t+1 m t+1 E t [m t+1 ] = ( θ ψ +θ 1) σ t η t+1 +(θ 1)κ 1 A 1 ψ e σ t e t+1 +(θ 1)A 2 κ 1 σ w w t+1

17 17 / 83 Market Prices of Risk Therefore, the prices of risk that correspond to the three sources of risk η t+1, e t+1 and w t+1 are: λ m,η ( θ ψ + θ 1) Then: λ m,e (1 θ)κ 1 A 1 ψ e λ m,w (1 θ)a 2 κ 1 m t+1 E t [m t+1 ] = λ m,η σ t η t+1 + λ m,e σ t e t+1 + λ m,w σ w w t+1

18 18 / 83 Risk Premia As asset returns and the pricing kernel are conditionally normal, the risk premium on any asset i is E t [r i,t+1 r f,t ] = cov t (m t+1,r i,t+1 ) 0.5σ 2 r i,t. Risk Premium for the wealth portfolio E t [r c,t+1 r f,t ] = λ m,η σ 2 t + λ m,eκ 1 A 1 ϕ e σ 2 t + λ m,w κ 1 A 2 σ 2 w 0.5var t (r c,t+1 ) with var t (r c,t+1 ) = (1 + (κ 1 A 1 ϕ e ) 2 )σ 2 t + (κ 1A 2 ) 2 σ 2 w. Equity Premium (Market Portfolio) E t [r m,t+1 r f,t ] = κ 1,m A 1,m ϕ e λ m,e σ 2 t + λ m,w κ 1,m A 2,m σ 2 w 0.5var t (r m,t+1 ) with var t (r m,t+1 ) = (ϕ 2 d + (κ 1,mA 1,m ϕ e ) 2 )σ 2 t + (κ 1,mA 2,m ) 2 σ 2 w.

19 19 / 83 Calibration Parameters for Fundamentals: The decision interval of the agent is monthly and the targeted data to match are annual. Parameters are intended to match mean, variance and autocorrelation of consumption and dividend processes. µ = µ d = ρ = σ = φ = 3 ϕ e = ϕ d = 4.5 ν 1 = σ w = Parameters for Preferences: γ = 7.5 or 15 ψ = 1.5

20 Asset Pricing Implications 20 / 83

21 21 / 83 Comments on Asset Pricing Implications The long-run growth risk is critical for explaining the equity risk premium. It accounts for a significant portion of the risk premium and magnifies the contribution of the volatility risk. If the variance of x t is zero (no long-run growth risk), the annualized equity premium is less than 1%. The population value of the volatility of the price-dividend ratio in this case is about If the long-run growth risk is present but the volatility channel is shut down, the annualized equity premium is 3.95% but the variance of the price-dividend ratio drops to Thus, the long-run growth risk is important for the level of the equity risk-premium, while the volatility channel is important for the variability of asset prices.

22 22 / 83 Predictability of Returns, Growth Rates and Price-Dividend Ratios

23 23 / 83 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). A value greater than one for ψ is a source of debate.

24 24 / 83 New calibration Bansal, Kiku and Yaron (2009) New equation for dividend growth: New values for the Model Parameters g d,t+1 = µ d + φx t + πσ t η t+1 + ϕσ t u d,t+1

25 25 / 83 Dynamics of Growth Rates and Prices, Bansal, Kiku and Yaron (2009)

26 26 / 83 Predictability of Consumption Growth by PD-ratio, Bansal, Kiku and Yaron (2009)

27 27 / 83 Predictability of Excess Return by PD-ratio, Bansal, Kiku and Yaron (2009)

28 28 / 83 Conclusive Comments The model of BY needs the presence of a small predictable component in expected consumption growth. The presence of this component makes consumption growth too predictable by the price-dividend ratio. Even though volatility is made very persistent in the calibration of BKY (2009) the volatility of the price-dividend ratio is not high enough for the price-dividend ratio to predict excess returns as in the data. All results rely on a limited set of parameters. Hard to evaluate the sensitivity of results to changes in parameters.

29 29 / 83 Challenges Is it possible to explain stylized facts about asset prices and predictability in a model without predictability in consumption growth (random walk consumption)? Is it possible to obtain these results without relying on a value of ψ greater than one. Can we find an efficient way to test the robustness of model results with respect to changes in parameters of fundamentals and preferences?

30 30 / 83 Disappointment Aversion and Long-Run Risks BGMT (RFS, 2011) propose a 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 fundamentals and preferences Point to deficiencies of model otherwise non-apparent

31 31 / 83 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

32 32 / 83 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.

33 33 / 83 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.

34 34 / 83 Generalized Disappointment Aversion Introduced by Routledge and Zin (2009), generalizing Gul (1991): + (1 α 1) (,κr ) R 1 γ 1 γ (, ) = V 1 γ df (V ) 1 γ ) ( V 1 γ (κr )1 γ 1 γ 1 γ df (V ) κ 1 (1) where α 1 measures the intensity of disappointment aversion disappointment aversion: α < 1 Kreps-Porteus: α = 1. κ measures the place of the kink in terms of percentage of the certainty equivalent R.

35 35 / 83 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 ). (2) 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.

36 36 / 83 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.

37 37 / 83 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.

38 38 / 83 Expected utility: LRRs do not matter LRR and GDA ( Ct+1 M t,t+1 = δ C t ) 1 ψ Kreps-Porteus: LRRs and R t (V t+1 ) - depends on γ > 1 γ : ( ) 1 ( ) 1 Ct+1 ψ Vt+1 ψ γ M t,t+1 = δ C t R t (V t+1 ) GDA: additional channel: LRRs and the kink - does not depend on ψ. ( ) 1 ( ) 1 Ct+1 ψ Vt+1 ψ γ M t,t+1 = δ C t R t (V t+1 ) 1 + ( α 1 1 ) ( ) V I t+1 R < κ t(v t+1) 1 + κ ( 1 γ α 1 1 ) ( ) V E t I t+1 R < κ t(v t+1)

39 39 / 83 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.

40 40 / 83 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.

41 41 / 83 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 (3) d t+1 = µ d (s t ) + (ω d (s t )) 1/2 ε d,t+1, (4) 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.

42 42 / 83 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.

43 43 / 83 The Matching Procedure We would like to match an AR(1) process, say z t, like x t or σ 2 t by a two-state Markov chain, say zt. We assume that zt = a + by t where y t is a two-state Markov chain that takes the values 0 (first state) and 1 (second state), and where the transition matrix P y of y t is given by: ( ) Py py,11 1 p = y,11. 1 p y,22 p y,22 The (unconditional) stationary distribution of y t is π y,1 = Prob (y t = 0) = 1 p y,22 1 p y,11, π y,2 = Prob (y = 1) =. 2 p y,11 p y,22 2 p y,11 p y,22

44 44 / 83 The Matching Procedure The vector of parameters of the two-state Markov chain that matches the AR(1) process z t is given by: p y,11 = 1 + φ z 2 p y,11 = 1 + φ z 2 b = 1 φ z φ z 2 k z 1 σ z πy,1 π y,2, a = µ z bπ y,2 k z + 3, p y,22 = 1 + φ z + 1 φ z 2 2 k z 1 k z + 3, p y,22 = 1 + φ z 1 φ z 2 2 k z 1 k z + 3 if s z 0, k z 1 k z + 3 if s z > 0, where k z = k y = π2 y,1 π y,2 + π2 y,2 π y,1 s z = s y = π y,1 π y,2 πy,1 π y,2

45 45 / 83 Asset Pricing Solutions The Markov property of the model is crucial for deriving analytical formulas for all the stylized facts we have put forward. We solve for the valuation ratios entering the SDF (λ 1z and λ 1v ), as well as the valuation ratios of the assets (wealth portfolio, market portfolio and risk free rate, λ 1c, λ 1d and λ 1f respectively. R t (V t+ ) C t = λ 1z ζ t, V t C t = λ 1v ζ t, P d,t D t = λ 1d ζ t, P d,t C t = λ 1c ζ t and P f,t = λ 1f ζ t. (5) Solving for these ratios amounts to characterize the vectors λ 1z, λ 1v, λ 1d, λ 1c and λ 1f as functions of the parameters of the consumption and dividends dynamics and of the recursive utility function defined above.

46 46 / 83 Asset Pricing Solutions These vectors are computed in two steps. In the first step, we characterize the ratio of the certainty equivalent of future lifetime utility to current consumption and the ratio of lifetime utility to consumption. R t ( Vt+1 ) C t = λ 1z ζ t and V t C t = λ 1v ζ t, where the components of the vectors λ 1z and λ 1v are given by: ( λ 1z,i = exp µ c,i + 1 γ ) ( ) 1 N 2 ω c,i pij 1 γ λ1 γ 1v,j j=1 λ 1v,i = { [ ] while the matrix P = pij 1 i,j N p ij = p ij (1 δ) + δλ 1 1 ψ 1z,i is defined by: ( ) 1 + α 1 1 Φ } 1 1 1ψ if ψ 1 and λ 1v,i = λ δ 1z,i if ψ = 1, ( ln κ λ ) 1z,i µ λ c,i 1v,j ω 1/2 c,i 1 + ( α 1 1 ) κ 1 γ N p ij Φ j=1 ( ln κ λ 1z,i (1 γ)ω 1/2 c,i. ) µ λ c,i 1v,j ω 1/2 c,i

47 47 / 83 Asset Pricing Solutions In the second step, we characterize the price-consumption ratio, the equity price-dividend ratio, and the single-period risk-free rate. These characterizations are done by solving the Euler equation for different assets. P d,t D t = λ 1d ζ t, P c,t C t = λ 1c ζ t and 1 R f,t+1 = λ 1f ζ t, where the components of the vectors λ 1d, λ 1c, and λ 1f are given by: ( 1 λ 1d,i = δ λ 1z,i ( 1 λ 1c,i = δ λ 1z,i ( λ 1f,i = δexp ) 1 ψ γ ( exp ) 1 ψ γ ( exp µ cd,i + ω cd,i 2 γµ c,i + γ2 2 ω c,i ) N ) ( 1 λ ψ γ 1v µ cc,i + ω ) ( 1 cc,i ψ λ γ 2 1v ( p ij λ1v,j λ j=1 1z,i ) ( ( P Id δa µ cd + ω )) 1 cd ei, 2 ) ( ( P Id δa µ cc + ω )) cc 1 ei, 2 ) 1 ψ γ.

48 48 / 83 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

49 49 / 83 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).

50 50 / 83 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 [ ] Var PD. [ PD ] and 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 where y t+1:t+h = a(h) + b (h)x t + η y,1,t+h (h) 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 ].

51 51 / 83 Asset Pricing and Return Predictability 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)]

52 52 / 83 Predictability of Consumption and Dividend Growth: GDA with Data GDA 50% PV δ γ 2.5 ψ 1.5 α 0.3 κ Benchmark Model 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)]

53 53 / 83 Asset Pricing Implications: Robustness to Preference Parameters

54 54 / 83 Predictability of Returns and Consumption Growth: Robustness to Preference Parameters

55 55 / 83 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

56 56 / 83 Sensitivity to Persistence in Volatility - Predictability 1Y 3Y 5Y R GDA KP DA0 GDA SLOPE φ σ φ σ φ σ

57 57 / 83 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.

58 Comparison of Models in the BY Environment 58 / 83

59 59 / 83 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

60 60 / 83 Sensitivity to Persistence in Expected Consumption Growth - Predictability 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

61 61 / 83 New LRR-based research on the Risk-Return Trade-Off Finding a strong empirical support for a systematic trade-off between conditional volatility and expected returns has been hard. Some recent studies have provided reasons for the weak and unstable relationship and have suggested ways to detect it more readily. The dependence is statistically mild at short horizons but strong in the long run (Bandi and Perron, 2008). Leverage effect and asymmetric dependence between volatility and returns is captured at daily horizons (Bollerslev, Sizova and Tauchen, 2012). The variance premium (VIX 2 E t (RV )) has forecasting power in the short run (one to three months), Bollerslev, Tauchen and Zhou (2009), Drechler and Yaron (2011). Equilibrium models are proposed to rationalize some of these facts. Our paper proposes a long-run volatility risk model that aims at matching all these stylized facts plus asset returns moments and predictability regressions.

62 62 / 83 Related Papers with Equilibrium Models Bollerslev, Sizova and Tauchen (2012): Set the model to reproduce daily asymmetries and dynamic dependencies (fractionally integrated process for volatility). Bollerslev, Tauchen and Zhou (2009): LRR model with no predictability in consumption growth. Set a model with time-varying volatility of volatility in consumption growth. Focus on the short-run risk-return trade-off (predictability of returns by variance premium). Drechler and Yaron (2011): More ambitious. Extend the LRR model with rich transient dynamics (large spikes in the level of consumption growth volatility and infrequent jumps in the small, persistent component of consumption and dividend growth). Focus on sets of asset pricing moments, long-run predictability (consumption growth and returns) and short-run risk-return trade-off. Bonomo, Garcia, Meddahi and Tédongap (2011): Consumption and dividends are heteroskedasctic random walks and GDA preferences; explain sets of aset pricing moments, long-run predictability of returns with analytical formulas In this paper, we keep same calibration of preferences and add a transient component in consumption growth volatility. Aim at explaining additional long and short-run risk-return trade-offs with analytical formulas.

63 63 / 83 Moments: Fundamentals, Asset Returns; Predictability Regressions Moments E [g c ] 1.84 σ[g c ] 2.20 AC1(g c ) 0.48 E [g d ] 1.05 σ[g d ] AC1(g d ) 0.11 Corr (g c,g d ) 0.52 E [pd] 3.33 σ[pd] 0.45 AC1(pd) 0.85 AC2(pd) 0.75 E [r f ] 0.65 σ[r f ] 3.79 E [r] 5.35 σ[r] β(1y ) R 2 (1Y ) 3.53 β(3y ) R 2 (3Y ) β(5y ) R 2 (5Y ) 23.75

64 64 / 83 Long-Horizon Risk-Return Trade-Off Regression: r t,t+h h = α h + β h σ 2 t h,t h + ε t,t+h January December 2010 h β h se R

65 65 / 83 Variance Premium Moments: January September 2010 Moments Based on Daily Monthly L Monthly H E [VRP] σ[vrp] [ E VIX 2] [ σ VIX 2] E [RV ] σ[rv ]

66 66 / 83 Autocorrelations - VIX, Realized Volatility and Variance Premium VIX 2 VRP 0.8 autocorrelations lag (days)

67 67 / 83 Cross-correlations - VIX, RV and Variance Premium cross-correlations cross-correlations VIX 2 VRP leads and lags of returns leads and lags of returns

68 68 / 83 Short-Run Risk-Return Trade-Off: January September 2010 Regression: r t,t+l l = α 0l + β 1,0l vp t + β 2,0l z d,t 1,t + ε (0) t,t+l h h RV Based on Daily Returns (RVL) β 1,0l se β 2,0l se Rl

69 69 / 83 The Model - Preferences We assume that the investor s decision interval corresponds to the frequency so that dynamics of preferences, endowments and other exogenous state variables are given at the frequency. The investor derives utility from consumption, recursively as follows: { } V t = (1 δ)c 1 ψ 1 1 t + δ[r t (V t+ )] 1 ψ 1 1 ψ 1 if ψ 1 = C 1 δ t [R t (V t+ )] δ if ψ = 1. With Generalized Disappointment Aversion preferences (Routledge and Zin, 2010) the risk-adjustment function R ( ) is implicity defined by: R 1 γ 1 1 γ = ( V 1 γ 1 κr df (V ) (α 1 1) 1 γ ) (κr ) 1 γ 1 V 1 γ 1 df (V ), 1 γ 1 γ

70 70 / 83 The Model - Stochastic Discount Factor When α = 1, we obtain the stochastic discount factor with Kreps-Porteus preferences: ( ) 1 ( ) 1 Mt,t+ = δ Ct+ ψ Vt+ ψ γ = δ C t R t (V t+ ) ( Ct+ C t ) 1 ψ Z 1 ψ γ t+, where ( Z t+ = V ( t+ R t (V t+ ) = Ct+ δ C t ) 1 ψ Rc,t+ ) ψ With GDA preferences (α < 1): M t,t+ = M t,t+ ( 1 + (α 1 ) 1)I (Z t+ < κ) 1 + (α 1 1)κ 1 γ, E t [I (Z t+ < κ)] where I ( ) is an indicator function (value 1 if condition is met, 0 otherwise).

71 71 / 83 The Model - Consumption and Dividend Growth Dynamics We assume that: g c,t+ = µ x + σ t ε c,t+ g d,t+ = µ x + ν d σ t ε d,t+ (6) where lnσ 2 t has the mean µ σ, the volatility σ σ and the persistence φ σ, and where ( ) (( εc,t+1 0 J ε t N ID d,t+1 0 ) ( 1 ρt, ρ t 1 )),

72 72 / 83 Matching the volatility process We assume that lnσ 2 t = a z + b 1z z 1,t + b 2z z 2,t where z 1,t and z 2,t are two independent two-state Markov chain that can take values 0 and 1. The chain z 1,t has a persistence φ 1z and a unitary kurtosis (which for a two-state Markov is also equivalent to a zero skewness) so that its conditional volatility is constant. The chain z 2,t has a persistence φ 2z and a kurtosis k 2z, with a positive skewness. Their transition probability matrices P 1z and P 2z are given by ( ) ( ) P1z = p1z,11 1 p 1z,11 and P 1 p 1z,22 p 2z = p2z,11 1 p 2z,11 1z,22 1 p 2z,22 p 2z,22 where p 1z,11 = 1 + φ 1z 2 p 2z,11 = 1 + φ 2z 2 and p 1z,22 = 1 + φ 1z 2 k 2z 1 k 2z φ 2z 2 and p 2z,22 = 1 + φ 2z 2 1 φ 2z 2 k 2z 1 k 2z + 3. Given φ 1z, φ 2z and k 2z, we solve for a z, b 1z and b 2z to match the mean, the persistence and the volatility of lnσ 2 t.

73 73 / 83 Matching the volatility process We find that σ σ φ b 1z = 2z φ σ σ σ φ σ φ and b 2z = 1z π1z,1 π 1z,2 φ 2z φ 1z π2z,1 π 2z,2 φ 2z φ 1z a z = µ σ b 1z π 1z,2 b 2z π 2z,2 π 1z,1 = P ( z 1,t = 0 ) = π 2z,1 = P ( z 2,t = 0 ) = 1 p 1z,22 2 p 1z,11 p 1z,22 and π 1z,2 = P ( z 1,t = 1 ) = 1 π 1z,1 1 p 2z,22 2 p 2z,11 p 2z,22 and π 2z,2 = P ( z 2,t = 1 ) = 1 π 2z,1 In our calibration analysis, we assume that the first component is very persistent (with φ 1/ 1z close to one) and that the second component is not persistent (with φ 1/ 2z less than 0.50) and has a very high kurtosis (large k 2z ). In this case, there is an unfrequent jump component of size b 2z in log volatility.

74 74 / 83 Markov Switching Dynamics for Consumption and Dividends Consumption and dividend growth rates evolve according to a Markov variable s t which takes N values, s t {1,2,...,N}, when N states of nature are assumed for the economy. The sequence s t evolves according to a transition probability matrix P defined as: P = [ p ij ]1 i,j N and p ij = Prob (s t+ = j s t = i). (7) Let ζ t = e st, where e j is the N 1 vector with all components equal to zero but the jth component is equal to one. Therefore, the dynamics of consumption and dividends growths are given by: ( ) Ct+ g c,t+ = ln = µ c ζ t + ω c ζ t ε c,t+ C t ( ) Dt+ g d,t+ = ln = µ d D ζ t + ω d ζ t ε d,t+ t where ( ) εc,t+ ε ε c,j,ε d,j,j t (( ) ( 0 1 ρ d,t+ ;ζ k,k Z N, ζ t 0 ρ ζ t 1 )).

75 75 / 83 Asset Pricing Solutions The Markov property of the model is crucial for deriving analytical formulas for all the stylized facts we have put forward. We solve for the valuation ratios entering the SDF (λ 1z and λ 1v ), as well as the valuation ratios of the assets (wealth portfolio, market portfolio and risk free rate, λ 1c, λ 1d and λ 1f respectively. R t (V t+ ) C t = λ 1z ζ t, V t C t = λ 1v ζ t, P d,t D t = λ 1d ζ t, P d,t C t = λ 1c ζ t and P f,t = λ 1f ζ t. (8) Solving for these ratios amounts to characterize the vectors λ 1z, λ 1v, λ 1d, λ 1c and λ 1f as functions of the parameters of the consumption and dividends dynamics and of the recursive utility function defined above.

76 76 / 83 Calibrated Model Parameters Preferences 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). Fundamentals µ M c = µ M d = , νm d = µ M σ = , φ M σ = 0.95, σ M σ = φ M 1z = 0.999, κ 1z = 1, φ M 2z = 0.297, κ 2z =

77 77 / 83 Moments: Fundamentals, Asset Returns; Predictability Regressions Moments Model E [g c ] σ[g c ] AC1(g c ) E [g d ] σ[g d ] AC1(g d ) Corr (g c,g d ) E [pd] σ[pd] AC1(pd) AC2(pd) E [r f ] σ[r f ] E [r] σ[r] β(1y ) R 2 (1Y ) β(3y ) R 2 (3Y ) β(5y ) R 2 (5Y )

78 78 / 83 Long-Horizon Risk-Return Trade-Off Regression: r t,t+h h = α h + β h σ 2 t h,t h + ε t,t+h January December 2010 h β h se R Model h β h R

79 79 / 83 Variance Premium Moments: January September 2010 Moments Based on Daily Monthly L Monthly H Model E [VRP] σ[vrp] [ E VIX 2] [ σ VIX 2] E [RV ] σ[rv ]

80 80 / 83 Autocorrelations - VIX, Realized Volatility and Variance Premium VIX 2 VRP autocorrelations lag (days)

81 81 / 83 Cross-correlations - VIX, RV and Variance Premium 0.15 VIX VRP cross-correlations 0.05 cross-correlations leads and lags of returns leads and lags of returns

82 82 / 83 Short-Run Risk-Return Trade-Off: Data and Model Results Regression: r t,t+l l = α 0l + β 1,0l vp t + β 2,0l z d,t 1,t + ε (0) t,t+l h h RV Based on Daily Returns (RVL) β 1,0l se β 2,0l se Rl Model β 1,0l β 2,0l Rl

83 83 / 83 Conclusion An equilibrium model with GDA preferences and economic uncertainty with one persistent component and a transient jump-like process is able to reproduce short- and long-run stylized facts relating volatility and returns. The model delivers analytical formulas for the population statistics characterizing these stylized facts. A major advantage to this analytical solving is that it allows to conduct readily thorough sensitivity analyses withe respect to the model parameters (both preferences and fundamentals dynamics). This is essential when calibration is used. Since we have population moments, it is also important to simulate the model and compute the quantities of interest for the finite sample size that we use to compute the stylized facts. We can then place the data statistics in the distribution of these simulated statistics and obtain p-values.