Endogenous Counter-Cyclical Risk Aversion and the Cross Section

Transcription

1 Endogenous Counter-Cyclical Risk Aversion and the Cross Section Anastasiya Ostrovnaya Carnegie Mellon University Bryan R. Routledge Carnegie Mellon University Stanley E. Zin Carnegie Mellon University and NBER December 25 Abstract Asset-pricing models that allow risk aversion to increase in recessions and decrease in expansions are capable of generating equilibrium asset returns that resolve the equity premium puzzle. Routledge and Zin (24) demonstrate that a tightly parametrized stationary recursive utility model with risk preferences that exhibit an aversion to outcomes that are sufficiently disappointing, has precisely this countercyclical risk aversion property in equilibrium. In this paper, we extend their Generalized Disappointment Aversion (GDA) model to a broader cross-section of assets. We show that the GDA model leads to a new cross-sectional risk factor in addition to the more traditional consumption-growth and market-return betas. By assuming that asset returns and consumption growth are jointly lognormally distributed, we provide a closed-form expression for this new factor, and show that it is a highly nonlinear function of the other two factors. In particular, this new factor introduces a new and independent role for the consumption-growth factor beyond the traditional consumption-capm. We estimate the parameters of the model and test its over-identifying restrictions with GMM applied to the model s Euler-equation forecast errors. The GDA model provides a significant improvement over the standard CAPM for both size and book-to-market sorted portfolios, as well as for industry portfolios. Correspondence: Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, <

2 1 Introduction Preference models in which risk aversion increases in recessions and decreases in expansions are able to reconcile dynamic general equilibrium asset-pricing theories with the properties of observed asset prices that are puzzling from the perspective of time-additive expected utility models (see, for example, Campbell and Cochrane (1999), Gordon and St-Amour (2), Barberis, Huang, and Santos (21), and Melino and Yang (23)). Routledge and Zin (24) show that a generalized version of disappointment aversion preferences (GDA), when imbedded in a stationary recursive utility framework, exhibits endogenous counter-cyclical risk aversion and, hence, a quantitative equilibrium asset-pricing model that matches the conditional and unconditional moments of equity and bond returns. This structural approach provides a tight link between the basic axioms of choice and the parameters of their quantitative model and, hence, establishes clear theoretical foundations for the counter-cyclical risk aversion that is apparent in the data. In this paper we consider the implications of the counter-cyclical risk aversion property of the GDA model for a broader cross-section of asset returns. It is well documented that firm characteristics like market capitalization and the accounting ratio of book to market equity provide a better characterization of the cross-sectional pattern of equity returns than the theoretically motivated consumption-growth and market-return betas (for example, Fama and French (1992)). Here, we will explore how much of this apparent pattern can be attributed to a risk premium that is an artifact of an aversion to particular types of risk, namely, disappointment aversion. In particular we document how a down market beta factor arises quite naturally in equilibrium and, as in Ang, Chen, and Xing (25), we show that such a factor significantly improves the empirical performance of standard CAPM regressions. Our structural model places additional restrictions on the functional form of this factor and provides a natural framework for testing this extension against alternative CAPM-like specifications. Our analysis is strictly preference-based and we do not explore potential interactions of preferences with more general cash-flow processes. As with an expected utility model, there is certainly scope for endogenous firm-level cash-flow decisions to produce additional risk and time-variation in risk factors (as in Gourio (25), 1

3 Tuzel (25)). Likewise, we do not explore the consequences of durable consumption decisions as in Pakos (25) and Yogo (25). Rather our approach is closer in spirit to Petkova and Zhang (25) who document the effect of time-variation in risk on the cross-section patterns and Santos and Veronesi (25) who investigate time-variation in the equity premium from a habits-based model. We proceed in this structural and preference-based fashion for a number of very practical, rather than purely theoretical, reasons. It serves to eliminate the potential for postulating behavior that would be generally viewed as irrational. That is, we avoid violating the basic properties of transitivity, monotonicity, and continuity as we consider departures from expected utility. In addition, starting with basic axioms governing choice over risky outcomes leads to a better understanding of the behavioral roles played by the parameters in our model. This is helpful for assessing both the reasonableness of parameter values and the likelihood of parameter stability across different economic and government policy environments We build on Routledge and Zin (24) that provides an axiomatic generalization of the disappointment aversion preferences of Gul (1991) which allow for a more flexible definition of a disappointing outcome. This one-parameter extension of Gul s utility function allows outcomes to be treated as disappointing not when they lie below the certainty equivalent, but only when they lie sufficiently far below the certainty equivalent. This focus on more extreme tail behavior echos real-world approaches such as Value at Risk calculations in finance (see Basak and Shapiro (21)), and with much of the behavioral/experimental evidence, such as Kahneman and Tversky (1979). In addition, by imposing standard regularity conditions (i.e., transitivity, monotonicity (first-order stochastic dominance), risk aversion (second-order stochastic dominance), and time stationarity, GDA preferences do not admit irrational behavior. Moreover, since they maintain many of the properties of expected utility models, GDA produces a tractable quantitative framework for studying departures from expected utility. Our paper is organized as follows. After reviewing the details of Generalized Disappointment preferences in an asset-pricing model in Section 2, we present a lognormal version of the model in Section 3 and discuss the empirical properties of the new downmarket factor implied by the model. Section 4 presents the results from estimating the model s parameters by applying GMM to Euler equation forecast 2

4 errors, and for tests of the model against the standard CAPM. Section 5 summarizes and concludes the paper. 2 Generalized Disappointment Aversion Expected utility preferences are useful in many applied settings such as asset pricing since they are tractable and involve a small number of parameters. However, in experiments choices rarely conform to expected utility. In particular, the independence axiom is typically violated. For example, the Allais (1979) ratio paradox and the Kahneman and Tversky (1979) common consequences effect both reveal preferences that violate independence (see also Machina (1987)) and Conlisk (1989)). 1 In this paper, we will focus on Generalized Disappointment Aversion preference introduced in Routledge and Zin (24). GDA preferences define the certainty equivalent, µ, implicitly, as µ α = i p i x α i θ i p i (δ α µ α x α i ) I(x i δµ(x)) (1) where x i is an outcome with probability p i and I( ) is an indicator function. Preference parameters are α, θ, and δ. The preferences specified here are linearly homogeneous (like CRRA preferences in expected utility). The preferences are similar to expected utility with constant relative risk aversion parameter α, except they impose a penalty, proportional to θ, on outcomes that lie below the disappointment threshold of δµ(p). This preference specification nests expected utility since with θ =, the preferences are expected utility. The specification also nests the disappointment aversion preferences of Gul (1991) when δ = 1. Preferences with δ < 1 capture non-central disappointment aversion by moving the disappointment cutoff. Outcomes are disappointing only if they lie sufficiently far below the certainty 1 The Allais ratio paradox is that people express a preference for a certain $2 over a.8 chance at $3 and they also express a preference for a.4 chance at $3 over a.5 chance at $2. These choices violate the independence axiom. To see why, let p 1 be the certain $2 lottery and p 2 be the.8 chance at $3. A.5 chance at $2 is the lottery.5p 1+.5() while a.4 chance at $3 is the lottery.5p 1 +.5(). The preference p 1 p 2 and.5p 1 +.5().5p 1 +.5() is inconsistent with independence. 3

5 equivalent. Equivalently, preferences in equation (1) can be expressed as N µ α = ˆp i x α i (2) i=1 where ˆp i are decision weights that overweight items, relative to expected utility, that are disappointing. [ ] 1 + θi(x i δµ) ˆp = p i 1 + θδ α (3) E[I(x δµ)] GDA preferences belong to the Chew-Dekel class of preferences (Chew (1983), (1989) and Dekel (1986)). This class of preferences generalize expected utility but maintain much of the parsimony and tractability of expected utility by satisfying the standard axioms of completeness, transitivity, and continuity. However, in place of the empirically troubling independence axiom, these preferences satisfy a weaker betweenness axiom. An important feature of these preferences that we rely on in our empirical analysis is that the preferences, hence, first-order and Euler conditions, are are linear in probabilities. Weighted Utility (Chew (1983)), Disappointment Aversion (Gul (1991)), and the Generalized Disappointment Aversion (Routledge and Zin (24)) that we focus on here are all examples of this class of preferences. See Backus, Routledge, and Zin (25) for an overview of these preferences. The focus or overweighting of lower-tail outcomes implied by GDA preferences is consistent with risk-management approaches such as value at risk calculations (see Basak and Shapiro (21)), and with much of the behavioral/experimental evidence, such as Allais (1979) and Kahneman and Tversky (1979). However, GDA preferences have an axiomatic foundations similar to expected utility. This eliminates the potential for postulating behavior that would be generally viewed as irrational. That is, GDA preferences maintain basic properties of transitivity, continuity, monotonicity (first-order stochastic dominance), and risk aversion (second-order stochastic dominance). GDA preferences are also based solely on the outcomes and probabilities. They do not include any external reference points or rely on any past action or outcome such as a habit. To see how these preferences differ from the more familiar expected utility preferences, Figure 1 plot indifference curves in a two-outcome setting where probabilities of the two states are fixed at p(x 1 ) = p(x 2 ) =.5. Figure 1(a) shows the familiar 4

6 smooth trade-off between state 1 and state 2 payoffs implied by expected utility. Figure 1(b) shows the indifference curve for disappointment aversion (Gul (1991)). Since disappointment is defined relative to the certainty equivalent, the indifference curve has a kink at certainty. This is the first-order-risk-aversion feature of disappointment aversion. Below the forty-five degree line, it is the state two outcome that is disappointing. Above the forty-five degree line, the state one outcome is disappointing. The parameter θ controls how much extra weight the disappointing outcome receives and, hence, the severity of the kink. Figure 1(c) shows an indifference curve for GDA in state space. Here, disappointment occurs for outcomes less than 8% of the certainty equivalence (δ =.8). For gambles close to certainty (shown as the center cone in Figure 1(c)), preferences are identical to expected utility since neither outcome is disappointing. This is analogous to the lower right hand portion of Figure 1(c). Lotteries where the state-two payoff is sufficiently low (below the cone) are disappointing (similarly for low state 1 payoffs). The kink in the indifference curves is not at certainty. The GDA preferences specified in equation (1) are also homothetic. For different utility levels, all kinks lie along the rays that define the cone in Figure 1(c). Figure 1(d) shows the indifference curve for GDA with δ = 1.25 > 1. The preferences are similar to Figure 1(c). In the center cone in Figure 1(d) both outcomes are disappointing, whereas in Figure 1(c) neither outcome is disappointing in this region. However, both cases preferences in the center cone are expected-utility-like. Figure 1 shows the effect of preference parameters θ and δ on effective risk aversion. In Disappointment Aversion, Figure 1(b), an increase in θ increases the severity of the kink and makes the preferences exhibit more risk aversion. This is also true in GDA in Figure 1(c) and (d). However, as δ moves away from one, the disappointment threshold is moved further away from certainty, the kink and effective risk aversion decreases. Monotonicity of preferences limits the kink since preferences can, at most, be a vertical (horizontal) line. 5

7 2.1 GDA and Asset Pricing Embedding GDA preferences into a standard representative agent asset-pricing economy is straightforward. 2 A representative agent consumes a single perishable consumption good in each period. In period t, current consumption, c t, is known with certainty, but future consumption levels are generally uncertain. Following Epstein and Zin (1989) and (199), the intertemporal utility functional is recursive. The Bellman equation defined by optimal consumption and portfolio choice has the form [( ) ρ U(a t,ω t ) = max c γ c t t,ω t 1 + ρ + 1 ] 1/γ 1 + ρ µγ t, γ 1, ρ >, (4) subject to a wealth constraint: a t+1 = (a t c t )R ω t+1 where R ω t+1 = N ωtr i t+1 i i=1 is the portfolio return and N i=1 ωi t = 1. (See Backus, Routledge, and Zin (25) for an overview.) GDA preferences are incorporated in µ t = µ(u (a t+1,ω t+1 ) Ω t ) which is defined by equation (1). It is the certainty equivalent of random future utility using the period-t conditional probability distribution. The preference parameters in equation µ t in equation (1) are α, a measure of risk aversion, θ, a degree of disappointment aversion, and δ, the threshold for disappointment. The other preference parameters are standard: γ determines the elasticity of intertemporal substitution and ρ is the marginal rate of time preference. This problem divides into separate portfolio and consumption decisions. Given a portfolio that maximizes µ t with stochastic return Rt+1 ω, the consumption decision is given by the Euler condition µ t (z t+1 ) = 1 (5) 2 Note that since GDA preferences are homothetic and depend solely on consumption, they share all the same aggregation properties of CRRA expected utility. Namely, if consumers have identical GDA preferences the cross sectional properties of wealth do not matter. 6

8 where [ 1 z t+1 = 1 + ρ ( ct+1 c t ) γ 1 R ω t+1] 1/γ (6) The Euler condition in (5) is a generalization of a familiar tangency condition for an optimum. Note that if the portfolio return is certain, the optimum involves setting the marginal rate of substitution, governed by the parameter γ, equal to the price ratio (the portfolio return). The optimum involves setting certainty equivalent of the tangency condition to one. Of course in an economy with risk, the ex-post tangency will not hold since given the realized return savings will be too high or too low. With a GDA certainty equivalent, disappointment only relates to this ex-post savings error. Disappointment effects are not, for example, defined exogenously on a the market nor any single asset return. The pricing kernel can be derived from the first-order-condition to portfolio optimization portion of the Bellman equation in equation (4). An important consequence of the fact that GDA preferences belong to the Chew-Dekel class of preferences is that the portfolio first-order conditions are linear in probabilities. This linearity in preferences allows us to apply standard method-of-moment estimation techniques in [ Section 4. The equilibrium pricing kernel, M t+1, that satisfies E t Mt+1 Rt+1] i = 1 for all i for the GDA model is M t+1 = zα t+1 (Rm t+1 ) 1 [1 + θi(z t+1 < δ)] 1 + θδ α E t [I(z t+1 < δ)], (7) where x t+1 = c t+1 c t is the growth rate in aggregate consumption and the optimal portfolio return is that of the return on aggregate wealth, Rt+1 m. 3 Log-Normality and the disappointment factor To explore the cross-sectional implications of GDA, it is helpful to consider a lognormal example. This will illustrate how a disappointment factor effects the properties of cross section of returns. To begin, we start with the implications for the unconditional properties of returns. (The basic structure developed here is maintained in the dynamic version that we consider in the next section). 7

9 The pricing kernel in the Generalized Disappointment Aversion model in equation (7) implies ] ] E [z α Ri R m + θe [z α Ri RmI(z < δ) = 1 + θδ α E[I(z < δ)] (8) where equation (6) defines z. Note, we are suppressing the time subscripts for the moment since we initially focus on the unconditional implications of or model. Assume that consumption growth, x, and all asset returns are jointly log-normally distributed: 3 ln x x σx 2 σ xm σ x1 σ xn ln R m r m σ mx σm 2 σ m1 σ mn ln R 1 N r 1, σ 1x σ 1m σ1 2 σ 1N. (9) ln R N r N σ Nx σ Nm σ N1 σn 2 Hence, using standard properties (see Appendix), the expectations in equation (8) can solved using the result in equations (A3) and (A4): r i + σi 2 /2 = λ + (1 α γ ) r m + α (1 γ) x γ ( + λ m β im + λ x β ix + log 1 + θδ α Φ( ) 1 + θφ( m β im + x β ix ) ), (1) 3 This assumption is, of course, an approximation since individual assets and portfolios cannot both be log-normally distributed. 8

10 where Φ is the standard normal density and β im β ix = σ im /σ 2 m = σ ix /σ 2 x λ = α γ log(1 + ρ) [α γ (1 γ)]2 σ 2 x/2 (1 α γ )2 σ 2 m/2 α γ (1 γ)(1 α γ )σ xm λ m = (1 α γ )σ2 m λ x = α γ (1 γ)σ2 x = [γ log(δ) + log(1 + ρ) + (1 γ) x r m ]/σ z m = + [σ 2 m (1 γ)σ xm]/σ z α γ σ z = σ 2 m/σ z x = (1 γ)σ 2 x /σ z σ z = [(1 γ) 2 σ 2 x + σ2 m 2(1 γ)σ xm] 1/2. Given a riskless log return, r f, expected excess log returns have the form: ( r i + σ 2 i /2) r f = λ m β im + λ x β ix ( + log 1 + θφ( ) 1 + θφ( m β im + x β ix ) ). (11) 3.1 The CAPM and a down-market factor As a starting point, consider a CAPM version of the model by choosing preference parameters α = (which implies log risk aversion in an expected utility version of the model) and γ = 1 (perfect substitutability of intertemporal consumption). In this case (11) is ( ) ( r i + σi 2 /2) r f = σm 2 β 1 + θφ ( ) im + log 1 + θφ ( σ m β im )) where = (log δ +log(1+ρ) r m +σ 2 m )/σ m). Returns in this parameterization of the model depend two factors. The first is the familiar linear CAPM market beta. The second term is down-market factor that depends on disappointment preference parameters δ and θ. While both factors depend only on the market beta, the second factor is not linear in market beta. 9 (12)

11 With θ = (or δ = ), this is the well-known expected utility model where all returns are proportional to the single factor, the market return. This, of course, is not a good model of the equity premium since it is not large enough and does not permit time variation in the equity premium. 4 However, our goal here is to investigate the cross-section relationship. Setting θ = and evaluating equation (11) for the Fama-French s size-and-book-to-market portfolios and industry portfolios reproduces the well-known empirical problem of the CAPM: the Market beta does not explain the cross section of returns. Figure 2 plots the market beta against excess return for the six size and book-to-market portfolios. Figure 3 shows the same information for ten industry portfolios. Even adjusting the equity premium to its historical level, the expected utility model cannot explain the cross section. With GDA preferences, θ >, the downmarket factor is the ratio of 1+θΦ ( ) to 1 + θφ ( σ m β im )). For β im >, this ratio is positive (recall Φ is cumulative Normal distribution and is increasing). Thus GDA parameters can increase the excess returns (and the equity premium) in this model. However, this version of the model will not help explain the cross section of returns. The cross-sectional variation in the down-market depends only on β im. However, the down-market factor is monotonically increasing in β im. Since the market betas do not align with the expected returns, this model does not explain the cross-section variation in the Fama-French portfolios. 3.2 The CAPM, CCAPM, and a down-market factor A natural extension to the simple CAPM is to include consumption risk and the consumption-growth beta as in Giovannini and Weil (1989). 5 In an expected utility context (θ = ), this is captured in equation (11) with α. Recall from equation (5), optimality sets the certainty equivalent of the ex post saving error, z t+1, to one. The savings error, z t+1, is the ratio of the slope of the budget constraint, R m t+1, to the slope of the indifference curve, (1 + ρ)(c t/c t+1 ) γ 1, all raised to the power 1/γ. Hence, what is considered a bad (risky) outcome depends on the sign 4 The parameterization α =, γ = 1, and θ =, from (5), implies that exp(e t[log(r t+1)]) = 1+ρ. Hence, expected returns are constant and there can be no time variation in the equity premium. 5 More recently, Campbell and Vuolteenaho (23), Campbell, Polk, and Vuolteenaho (25), Santos and Veronesi (25) have looked at multiple beta approaches. 1

12 of γ. For this reason, an expected-utility model (with θ = ) can have market and consumption risk premia that are positive or negative; i.e, the signs of λ m and λ x depend on the preference parameters α and γ. Despite this flexibility, however, the CAPM and CCAPM have little success in explaining the cross-section of asset returns. Since consumption and market betas are highly correlated, explaining the cross-section pattern shown in Figures 2 and 3 requires λ m and λ x to be of opposite signs. While one can choose parameters to achieve this, the already small equity premium of the CAPM model becomes even smaller. Moreover, as we will explain in more detail below, such a model cannot capture the salient dynamic properties of equity premiums. Consumption-growth risk in the GDA model (θ > ) is more interesting. To explore this feature, we maintain the logarithmic risk aversion, α =, assumption but allow for inter-temporal consumption smoothing γ < 1. With these parameters, excess returns from equation (11) are given by ( ) ( r i + σ i /2) r f = σm 2 β 1 + θφ ( ) im + log 1 + θφ ( m β im + x β ix )) Consumption-growth risk enters only in the downmarket factor. That is, λ x = but x >. Interestingly, in the downmarket factor, consumption-growth risk and market risk affect the equity premium differently. The downmarket factor is increasing in market beta but decreasing in consumption-growth beta: (13) ( r i r f ) φ( m β im + x β ix ) = θ x <, (14) β ix (1 + θφ( m β im + x β ix )) where φ is the standard normal density. To see why the consumption and market betas enter with different signs, look at the source of the downmarket factor in equation (8). With α =, the downmarket factor comes from the term E[(R m ) 1 R i z < δ]. The market beta, β im, arises from the usual interaction or R i with R m and from the relationship between R i and z through R m. However, the consumption-growth beta, β ix, enters only due to the fact the expectation is conditional on z which is a function of consumption-growth, x. To understand the downmarket factor and aid with our empirical calibration, Figure 4 plots the Normal cumulative density function. The downmarket factor depends on the distance between Φ ( ) and Φ ( σ m β im + x β ix )). If lies 11

13 in the tail of the distribution, from say a small value of the disappointment threshold δ, then for any values of β im and β ix, Φ ( ) Φ ( σ m β im + x β ix )) and the downmarket factor is zero. Therefore, for the disappointment factor to matter and for consumption-growth beta to play a role, the model must be calibrated so that lies close to the center of the distribution. To get a feel for how the downmarket factor works with the data, Figure 5 plots the downmarket factor for the Fama-French size and book-to-market portfolios. The consumption-growth and market betas are shown in Table 1. 6 For concreteness, set the disappointment aversion parameter at θ = 1 and plot the downmarket factor for a range of disappointment thresholds, δ. As noted above, for small levels of δ, log δ and hence lies in the left tail of the normal distribution and the disappointment factor is zero. The horizontal lines plotted are the average pricing errors from the expected utility CAPM (θ = ) (also in Table 1). From the figure, you can see which level of the disappointment threshold, δ, will exactly fit each of the six portfolios. For example, δ =.819 will exactly fit the S/L portfolio (small firms with low bookto-market). However, to fit the S/H portfolio, a threshold of δ =.91 is needed. Figure 6 plots the same information for ten industry portfolios. Here the range of δ that exactly fit each portfolio is smaller. To fit the Manufacturing portfolio a δ =.838 is needed and to fit the Energy portfolio a δ =.882 is needed. Is a range of threshold parameters from.8 to.9 reasonable? Is the range large? First the range of parameters is slightly smaller than Routledge and Zin (24) use to fit the dynamics of the equity premium. In their calibration a threshold of.97 was needed. However, of more concern is the wide range of δ implied by the individual portfolios. To fit the equity premium, and generate time-variation in the equity premium in Routledge and Zin (24), δ had to be carefully chosen. Perhaps another way to indicate that this range for δ is troubling is to choose a single value for the preference parameter δ and calculate the consumption-growth beta, β ix, that is needed to reconcile the realized returns data with the GDA model. Figures 7 and 8 show the results for the Fama-French and industry portfolios. The plot shows the β ix needed to fit each of the portfolios for the value of δ shown (again, other parameters are held fixed in this exercise). At any fixed level of δ, the consumption-growth betas for most of the portfolios are counter-factually large 6 The data sources are standard. See Appendix B for a description. 12

14 (or small) by an order of magnitude. Perhaps, one could argue that this reflects some more complicated dynamics in consumption growth as in Bansal and Yaron (25) and Kiku (25) that makes measured unconditional consumption-growth betas misleading. Alternatively, the unconditional normality assumption may be misleading. To explore this possibility, we now turn to a dynamic version of the model. 3.3 Time varying risk aversion From Backus and Zin (1994), Campbell and Cochrane (1999), Melino and Yang (23), Routledge and Zin (24), we know that understanding the time-series property of equity returns requires a model that incorporates time-variation in the equity premium. Presumably, the time variation in equity premium has significant implications for the conditional cross sectional behavior of asset returns. The basic structure of the unconditional log-normal model maintains in a dynamic setting. Assume that the lognormality assumption in equation (9) is a statement about conditional probabilities. Assume that the conditional variancecovariance matrix is constant. That is, interpret the constant σ parameters as conditional variances and covariances, and attach a time subscript to the conditional means to denote their dependence on the state of the economy. The dynamic analog to equation (1) r i,t + σi 2 /2 = λ + (1 α γ ) r m,t + α γ (1 γ) x t + [ + λ m β im + λ x β ix + log 1 + θδ α Φ( t ) 1 + θφ(,t m β im + x β ix ) The analogue to equation (11) for excess returns is given by: ]. (15) ( r i,t + σ 2 i /2) rf t = λ m β im + λ x β ix ( ) 1 + θφ(,t ) + log 1 + θφ(,t m β im + x β ix ) (16) Note that in this model, the the dynamics enter only through the parameter t (and t, ). Therefore, the only source of dynamics is the GDA downmarket factor. This is, perhaps, not surprising since it is difficult for an expected utility preference 13

15 specification to yield time variation in the equity premium (see Routledge and Zin (24)). However, even an expected utility-like model with time-variation in the equity premium is not sufficient to explain the cross-section of asset returns. 7 In addition to maximum likelihood estimation, imposing the cross-equation restrictions of equation (16), we can estimate and test this model using restrictions on Euler equation forecast errors and GMM. 4 GMM Estimation The pricing kernel in equation (7), for any assets i and j, implies ( )] E t [z t+1 α (Rm t+1 ) 1 [1 + θi(z t+1 < δ)] Rt+1 i Rj t+1 =, (17) where, recall from equation (6), z t+1 = [ ] 1 1/γ 1 + ρ (x t+1) γ 1 Rt+1 m (18) Note that this equation is linear in probabilities (a feature of the fact GDA belongs to the Chew-Dekel class of preferences). Therefore we can use a method of moments estimator. Define the ex-post Euler equation error for portfolio i as ( ) ǫ i t = zα t+1 (Rm t+1 ) 1 [1 + θi(z t+1 < δ)] Rt+1 i Rf t+1 (19) We obtain initial consistent estimates of the GDA parameters θ and δ by minimizing the equally weighted sample analogs of the population moment E[ǫ i t] =. Since we are primarily interested in the GDA parameters, here we hold fixed the other preference parameters ρ =.4/12 and α = and γ =.5. We discuss this issue further below. In our estimation on a sample of excess returns, δ is only estimated up to an interval over which the number of disappointing outcomes in the sample increments by one. We arbitrarily use the mid-point for this interval as the 7 For example, through exogenous time variation in the risk-aversion parameter as in Melino and Yang (23) or Gordon and St-Amour (2). 14

16 point estimate for δ. 8 Given these consistent parameter estimates, we then estimate the efficient weighting matrix and obtain efficient parameter estimates, as well as asymptotic standard errors and test statistics, using standard methods. 4.1 Estimates To begin, Table 2 shows GMM estimates for the notoriously hard-to-price Fama- French size and book-to-market portfolios. In general, the GDA model has both lower average Euler equation errors than the CAPM and the errors are also more volatile. Both attributes contribute to making the GDA model harder to reject. The estimate for disappointment aversion parameter θ is which is significantly different from the expected utility level of zero. More interesting is the parameter for δ of.76. This choice of disappointment threshold implies that, in this sample, there are 4 ex-post disappointing events. To see the effect of disappointment on the pricing kernel and the ex post Euler errors, Figures 9 and 1 plot the ex-post Euler errors in both the CAPM model and the GDA model. For most of the sample, the ex-post errors in the CAPM and GDA model are identical since for most of the period, z t > δ and events are not disappointing. The top panel plots the realized z t and the threshold δ. The four disappointing outcomes for z t are denoted. On the four disappointing months, the ex-post error in the GDA model is significantly larger (multiplied by 1 + θ). Effectively, risk aversion is higher in these months. On the plot the CAPM (θ = ) Euler errors are indicated by a o. The GDA Euler error is indicated with a. The reason the GDA model is able to improve the fit, over the CAPM, is that the four disappointing months happen to correspond to bad outcomes of the portfolio return. It is important to note that disappointment is not judged stock-by-stock (or portfolio-by-portfolio). Disappointment effect enters the pricing decision through the aggregate consumption-savings/portfolio decision. The high average returns seen in some of the Fama-French size/book-to-market portfolios is because, in the GDA model, they are risky; their returns are low when the pricing kernel is amplified with the disappointment effect. The disappointment effect is present (or not) for all stocks simultaneously. As emphasized in the previous section, correlation is still 8 Since the population moments are continuous in δ, we can still estimate a standard error for this parameter using standard methods. 15

17 the source of risk. However, when the implied risk aversion in the pricing kernel is state-dependent, the correlation is amplified during disappointing periods. Note that for the GDA model, the endogenous time variation in the risk aversion is important. To fit the data, disappointment effects appear infrequently. It is crucial to our model that the risk aversion be not constant. For example, the Gul (1991) version of DA with δ = 1, will not work. Setting δ = 1 implies that roughly half of the dates have disappointing z t outcomes. This means that these dates will have the 1 + θ amplification. This will amplify the ex post pricing errors to a much greater degree. However, recall the correlation between z t and the size-book-tomarket portfolios is weak (e.g., the CAPM and CCAPM models do not work very well). Therefore, the GDA model with δ = 1 will amplify both the good and bad returns. The extra volatility in the pricing kernel reflects the higher effective risk aversion but cannot explain the cross section. The key to the GDA model is the disappointment effect that comes from the risky consumption/savings decision as seen in the Euler equation in (5). Recall that the z t represents the ex-post savings error. This is the degree to which the consumptionsavings decision missed setting the marginal rate of intertemporal substitution equal to the budget constraint slope. It is determined by consumption growth and the market return (and the the preference parameter of patience, ρ, and intertemporal substitution, γ). Low outcomes of z t < δ are disappointing and trigger the magnification by 1 + θ. The GMM estimate of δ for the size/book-to-market portfolios implies that there are four disappointing outcomes in the realized z t series. These are the four months represent the four lowest realizations of z t in the sample. Since consumption growth in the data is so smooth, the key disappointment dates, not surprisingly, are driven by low market returns. They are November, 1973, March, 198, October 1987, and August Turning to the Industry portfolios, we repeat the GMM exercise. Table 3 reports the results. The estimate of θ is similar to the FF portfolios at However, in this sample the estimate disappointment threshold δ, is higher at With this value of δ there are 7 disappointing events in the realized series z t. Figures 11, 12, and 13 show the realized z t and the ex-post Euler errors for the industry portfolios. 9 This indicates that the results are sensitive to the measurement of the data. In quarterly data, for example, the October 1987 event is far less prominent. 16

18 Since other preference parameters of ρ and γ are constant across these estimations, the series z t is identical to that used in the size/book-to-market portfolios. We have more to say on this below. Here the seven disappointing outcomes include the same four dates mentioned above: November 1973, March 198, October 1987, and August However, now fitting the industry data is improved by including April 197, September 1974, and October Lastly, Table 4 estimates the preference parameters on sample of six size/book to market and ten industry portfolios jointly. The results are broadly similar to Tables 2 and 3. The estimate for disappointment aversion, θ, is The estimate of disappointment threshold, δ, is.7927 indicating there are six disappointing realizations for the consumption/savings error, z t. 4.2 Discussion Estimating the disappointment threshold parameter δ in our model is somewhat non-standard. In our sample of portfolio returns for the six size/book-to-market portfolios, for example, we estimate that there are four key dates that are disappointing. A value for δ anywhere in the range.753 to.7758 will yield the same four disappointing outcomes in our sample of realized ex-post saving errors, z t. In any finite sample, excess returns can only estimate a range for δ. More precise point estimates could be obtained by using level for returns rather than excess returns we focus on here. That is, since the denominator in equation (7) is continuous in δ, estimating E t [M t+1 Rt+1 i ] = 1 can uniquely identify δ. However, even in our excess-return exercise, we seem to generate fairly precise information about δ since both the estimated range is small, and since the model s performance would worsen dramatically for the size/book-to-market portfolios if we were to lower δ to imply only 3 (or fewer) disappointing dates, or raise δ to imply 5 (or more) disappointing dates. Our GMM strategy here, is to estimate the key GDA parameters δ and θ. To do this, we fixed ρ =.4/12, α =, and γ =.5 across all the estimations. Given these parameters, we can calculate the z t ex post savings errors. Holding ρ and γ constant across our estimation implies that z t are the same in each of our estimations. Fixing the parameter ρ is of little importance (it is mostly a scaling parameter). The effects 17

19 of altering the expected utility risk aversion parameter α are also straightforward. However, setting γ > is important. This implies that intertemporal consumption at different dates (along certain paths) are substitutes. Besides the usual importance this has in an expected utility model, the sign of γ determines what type of savings errors are disappointing. Recall, in the consumption-savings problem that underlies our asset pricing model, the definition of disappointment depends on the intertemporal substitution parameter γ. Recall from equation (5), optimality sets the certainty equivalent of the ex post saving error, z t+1, to one. z t+1 is the the ratio of the slope of the constraint, Rt+1 ω, to the slope of the indifference curve, (1 + ρ)(c t /c t+1 ) γ 1, raised to the power 1/γ. So with γ >, z t+1 < δ implies that the realization that period-t savings was larger than would have been optimal ex post, is the disappointing outcome. Conversely, if intertemporal consumption is complementary, γ <, the realization that period-t savings was smaller than would have been optimal ex post, is the disappointing outcome. Therefore, the nature of disappointment depends on the sign of the substitution parameter γ. The fact that the cross-sectional evidence suggests γ > is interesting and is worth considering in future research. 5 Conclusion [TO BE ADDED] 18

20 References Allais, M. (1979): The Foundations of a Positive Theory of Choice Involving Risk and a Criticism of the Postulates and Axioms of the American Schhol, in Expected Utility Hypothesis and the Allais Paradox, ed. by M. Allais, and O. Hagen. D. Reidel Publishing Co., Dordrecht, Holland. Ang, A., J. Chen, and Y. Xing (25): Downside Risk, NBER Working Papers Backus, D. K., B. R. Routledge, and S. E. Zin (25): Exotic Preferences for Macroeconomists, in NBER Macroeconomics Annual 24, ed. by M. Gertler, and K. Rogoff, vol. 19, pp MIT Press, Cambridge, MA. Backus, D. K., and S. E. Zin (1994): Reverse Engineering the Yield Curve, NBER Working Paper # W4676. Bansal, R., and A. Yaron (25): Risks For the Long Run: A Potential Resolution of Asset Pricing Puzzles, Journal of Finance (forthcoming), Wahrton Working Paper. Barberis, N., M. Huang, and T. Santos (21): Prospect Theory and Asset Prices, The Quarterly Journal of Economics, CXVI(1). Basak, S., and A. Shapiro (21): Value-at-Risk Based Risk Management: Optimal Policies and Asset Prices, Review of Financial Studies, 14, (2), Campbell, J. Y., and J. H. Cochrane (1999): By Force of Habit: A Consumption- Based Explanation of Aggregate Stock Market Behavior, Journal of Political Economy, 17, Campbell, J. Y., C. Polk, and T. Vuolteenaho (25): Growth or Glamour? Fundamentals and Systematic Risk in Stock Returns, NBER Working Paper No Campbell, J. Y., and T. Vuolteenaho (23): Bad Beta, Good Beta, NBER Working Paper No Chew, S. H. (1983): A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox, Econometrica,, 51(4), (1989): Axiomatic Utility Theories with the Betweeness Property, Annals of Operations Research, 19, Conlisk, J. (1989): Three Variants on the Allais Paradox, The American Economic Review, 79(3), Dekel, E. (1986): An Axiomatic Characterization of Preferences under Uncertainty, Journal of Economic Theory, 4, Epstein, L. G., and S. E. Zin (1989): Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework, Econometrica, 57(4), (199): First-Order Risk Aversion and the Equity Premium Puzzle, Journal of Monetary Economics, 26(3), Fama, E. F., and K. R. French (1992): The Cross-Section of Expected Stock Returns, Journal of Finance, 47,

21 Giovannini, A., and P. Weil (1989): Risk Aversion and Intertemporal Substitution in the Capital Asset Pricing Model, NBER Working Paper No Gordon, S., and P. St-Amour (2): A Preference Regiem Model of Null and Bear Markets, American Economic Review, 9(4), Gourio, F. (25): Operating Leverage, Stock Market Cyclicality, and the Cross-Section of Returns, Working Paper, Boston University. Gul, F. (1991): A Theory of Disappointment Aversion, Econometrica, 59(3), Kahneman, D., and A. Tversky (1979): Prospect Theory: An Analysis of Decision under Risk, Econometrica, 47(2), Kiku, D. (25): Is the Value Premium a Puzzle?, Duke University Working Paper. Machina, M. J. (1987): Choince Under Uncertainty: Problems Solved and Unsolved, Journal of Economic Perspectives, 1(1), Melino, A., and A. X. Yang (23): State Dependent Preferences Can Explain the Equity Premium Puzzle, Review of Economic Dynamics, 6(2), Pakos, M. (25): Asset Pricing with Durable Goods and Non-Homothetic Preferences, Working Paper, Carnegie Mellon University. Petkova, R., and L. Zhang (25): Is Value Riskier than Growth?, Journal of Financial Economics, pp Routledge, B. R., and S. E. Zin (24): Generalized Disappointment Aversion and Asset Prices, NBER Working Paper No. w117. Santos, T., and P. Veronesi (25): Cash-Flow Risk, Discount Risk, and the Value Premium, NBER Working Papers Number Tuzel, S. (25): Corporate Real Estate Holdings and the Cross Section of Stock Returns, University of Souther California Working Paper. Yogo, M. (25): A Consumption-Based Explanation of Expected Stock Returns, Journal of Finance, 61 forthcoming(2), April 26. 2

22 Appendix A Consider the expectation where» y1 y 2 E[exp {αy 1 + y 2} y 1 < a],»ȳ1 N, ȳ 2» σ 2 1 σ 12 σ 21 σ 2 2 (A1) «, (A2) and α and a are arbitrary real numbers. Using standard results, this expectation can be written as «a ȳ1 σ 12 E[exp {αy 1 + y 2}]Φ ασ 1, (A3) σ 1 where and Φ( ) is the standard normal cdf. E[exp {αy 1 + y 2}] = exp {αȳ 1 + ȳ 2 + α 2 σ 2 1/2 + σ 2 2/2 + ασ 12}, (A4) The assumption of log-normality of consumption growth and returns in equation (9), implies that ln z and ln R i lnr m are jointly normally distributed. Thus we can apply (A3) and (A4) to (8) to get equation (1). Appendix B - Data The data we use is standard. We use monthly data for the period February 1956 to April 25 (556 months). Consumption growth is calculated as per-capita monthly growth in real consumer non-durable and service chain-weighted expenditures (Source: Datastream). For asset returns, we use the data provided from Ken French on the nominal monthly returns for the one-month Treasury bill rate, the value-weighted return on all NYSE, AMEX, and NASDAQ stocks. In addition, we use the nominal monthly returns on the ten industry portfolios as well as six portfolios constructed by sorting on size (market capitalization) and book-to-market ratio (the ratio shareholders equity to the market capitalization). Additional details are available at: < library.html> These nominal returns are converted to real returns using the the price level change calculated from the ratio of nominal to real consumer expenditures on non-durable and services chain-weighted expenditures (Source: Datastream). 21

23 Table 1: Data Mean St. Dev. Log Consumption Growth Log Equity Returns Log Risk-Free Rate Portfolio Return Moments Market Consumption CAPM Mean St. Dev. Beta Beta Error Ê[r i r f ] + ˆσ i 2/2 ˆσ i ˆβ im ˆβix Ê[ǫ i ] Market S/L S/M S/H B/L B/M B/H NoDur Durbl Manuf Enrgy HiTec Telcm Shops Hlth Utils Other The data consists of 256 monthly real rates of return for the period February 1959 to April 25. The portfolios are the Fama-French six size and book-to-market portfolios and ten industry portfolios. Consumption Growth and price-level data are from Datastream. ˆβim is the market beta and ˆβ ix is the consumption-growth beta calculated from the data. Ê[r i r f ] + ˆσ 2 i /2 is the estimated average excess return on the portfolios. Ê[ǫ i] is the average CAPM pricing error relative to the unconditional expected utility CAPM in equation (12). That is ǫ i,t = (r i,t r f,t + σ 2 i /2) σ 2 mβ im 22

24 Table 2: GMM Size and Book-to-Market Portfolios - Unconditional CAPM GDA Parameter Fixed Estimate st. error θ δ n.a.7644 n.a. freq. of z t < δ n.a 4 Euler Equation Errors mean st. dev. mean st. dev. S/L S/M S/H B/L B/M B/H J-statistic d.f 6 4 P-value GMM estimation of Euler equation restriction on cross section of returns. The CAPM column fixes θ = and δ is arbitrary. The GDA model estimates δ and θ. The standard error for the θ estimate is calculated. The standard error for δ is not provided. The frequency of z t < δ is implied by the choice of δ and is the number of occurrences in the 256 month sample. The data is monthly real excess returns in six Fama-French size and book-to-market portfolios for the period February 1959 to April 25. Other model parameters are fixed at α =, γ =.5, and ρ =.4/12. 23

25 Table 3: GMM Industry Portfolios - Unconditional CAPM GDA Parameter Fixed Estimate st. error θ δ n.a.7929 n.a. freq. of z t < δ n.a 7 Euler Equation Errors mean st. dev. mean st. dev. NoDur Durbl Manuf Enrgy HiTec Telcm Shops Hlth Utils Other J-statistic d.f 1 8 P-value GMM estimation of Euler equation restriction on cross section of returns. The CAPM column fixes θ = and δ is arbitrary. The GDA model estimates δ and θ. The standard error for the θ estimate is calculated. The standard error for δ is not provided. The frequency of z t < δ is implied by the choice of δ and is the number of occurrences in the 256 month sample. The data is monthly real excess returns in ten industry portfolios for the period February 1959 to April 25. Other model parameters are fixed at α =, γ =.5, and ρ =.4/12. 24

26 Table 4: GMM FF Size and Book-to-Market and Industry Portfolios - Unconditional CAPM GDA Parameter Fixed Estimate st. error θ δ n.a.7927 n.a. freq. of z t < δ n.a 6 Euler Equation Errors mean st. dev. mean st. dev. S/L S/M S/H B/L B/M B/H NoDur Durbl Manuf Enrgy HiTec Telcm Shops Hlth Utils Other J-statistic d.f 1 14 P-value GMM estimation of Euler equation restriction on cross section of returns. The CAPM table fixes θ = and δ is arbitrary. The GDA model estimates δ and θ. The standard error for the θ estimate is calculated. The standard error for δ is not provided. The frequency of z t < δ is implied by the choice of δ and is the number of occurrences in the 256 month sample. The data is monthly real excess returns on six size/book-to-market portfolios and ten industry portfolios for the period February 1959 to April 25. Other model parameters are fixed at α =, γ =.5, and ρ =.4/12. 25

27 Figure 1: Indifference Curves 3 (a): θ= (Expected Utility) 3 (b): θ=2. δ=1 (Gul) x x x x (c): θ=2. δ=.8, (GDA) x < δ µ(p) (d): θ=2. δ=1.25, (GDA) x < δ µ(p) 1 x x 1, x 2 > δ µ(p) x x 1, x 2 < δ µ(p) 1 x 2 < δ µ(p) 1 x 2 < δ µ(p) x x 1 Indifference Curve over two outcomes x 1 and x 2 with prob(x 1) =.5. Shown are: (a) Expected Utility (θ = ), (b) Gul Disappointment Aversion θ = 2. (δ = 1), (c) Generalized Disappointment Aversion θ = 2. and δ =.8 (d) Generalized Disappointment Aversion θ = 2. and δ = 1.2. The dashed lines are expected utility with distorted odds. The indifference curve is the upper envelope (solid line). The rays from the origin indicate the threshold where x 1 and/or x 2 are disappointing. 26