Survey Expectations and the Equilibrium Risk-Return Trade

ISSN 2279-9362

Survey Expectations and the Equilibrium Risk Return Trade Off

Roberto Marfè

No. 408 May 2015

www.carloalberto.org/research/working-papers

Trade O∗

Roberto Marfè†

Collegio Carlo Alberto

ABSTRACT

Intuition and leading equilibrium models are at odds with the empirical evidence that expected returns are weakly related to volatility at the market level. This paper proposes a closedform general equilibrium model, which connects the investors’ expectations of fundamentals with those of market returns, as documented by survey data. Forecasts suggest that investors feature procyclical optimism and, then, overestimate the persistence of aggregate risk. The forwardlooking component of stock volatility oset the transient risk and leads to a weak riskreturn relation, in line with survey data about market returns. The model mechanism is robust to many features of ﬁnancial markets. Keywords: riskreturn trade o survey expectations general equilibrium optimism asset pricing puzzles heterogeneous preferences closedform expression JEL Classiﬁcation: D51 D53 D83 G11 G12

∗ An earlier version of this paper was circulated under the title “Asset Prices and the Joint Formation of Habit and Beliefs.” I would like to thank my supervisor Michael Rockinger and the members of my Ph.D. committee Philippe Bacchetta, Pierre CollinDufresne, Bernard Dumas and Lukas Schmid for stimulating conversations and many insightful comments and advices. I would also like to acknowledge comments from Hengjie Ai, Michela Altieri, Daniel Andrei, Ravi Bansal, Giovanni BaroneAdesi, Cristian Bartolucci, MarieClaude Beaulieu, Harjoat Bhamra, Tim Bollerslev, Andrea Buraschi, Stefano Colonnello, Giuliano Curatola, Marco Della Seta, Jérôme Detemple (Gerzensee discussant), Theodoros Diasakos, Darrel Due, Andras Fulop, José Miguel Gaspar, Dino Gerardi, Paolo Ghirardato, Christian Gollier, Edoardo Grillo, Campbell Harvey, Toomas Hinnosaar, Julien Hugonnier, Leonid Kogan, Jia Li, Elisa Luciano, Hanno Lustig, Loriano Mancini, Jocelyn Martel, Ian Martin, Ignacio Monzón, Giovanna Nicodano, Piotr Orlowski, Andrew Patton, Fulvio Pegoraro, Patrice Poncet, Andrei Savochkin, Yuki Sato, Daniel Schmidt, Hersh Shefrin, Aleksey Tetenov, Viktor Todorov, Raman Uppal, Adrien Verdelhan and from the seminar participants at the Duke Finance and Econometrics Workshop (2012), at the Financial Management Association Conference (2012), at the 11th Swiss Doctoral Workshop in Finance (2012) and at the Collegio Carlo Alberto (2012 and 2013). All errors remain my only responsibility. Part of this research was written when the author was a Ph.D. student at the Swiss Finance Institute and at the University of Lausanne. Part of this research was conducted when the author was a visiting scholar at Duke University. Financial support by the NCCR FINRISK of the Swiss National Science Foundation and by the Associazione per la Facoltà di Economia dell’ Università di Torino is gratefully acknowledged. Usual disclaimer applies. † Postal address: Collegio Carlo Alberto, Via Real Collegio 30. 10024 Moncalieri (TO), Italy. Email: [email protected]. Webpage: http://robertomarfe.altervista.org/. I. Introduction

Standard equilibrium consumer models fail to explain the main properties of asset prices and their connection with the macroeconomy. Most of these models rely on the assumption that market participants rationally maximize their expected utility under the true probabilities of uncertain economic states. However, economic fundamentals are not directly observable: a mounting evidence suggests the presence in the ﬁnancial markets of traders with biased beliefs and a growing literature studies if such a presence has a price impact or can be neglected. Instead, one aspect that has attracted little investigation, so far, concerns the role of the investors’ survey expectations in the context of equilibrium asset pricing. Namely, the literature has usually disregarded how survey expectations –that is, the actual expectations computed by the investors over time– relate with the economic conditions and which eects they produce on the equilibrium dynamics of asset prices.1

The aim of this paper is to question whether a stylized representation of the investors’ survey expectations about fundamentals can help to explain in equilibrium a stylized representation of the investors’ survey expectations of market returns, as well as some standard asset pricing puzzles. In particular, I focus on the key relation between market risk and its expected remuneration, that is the riskreturn trade o and its dynamics.

Survey data from professional forecasters (SPF) suggest that investors’ beliefs about economic growth (i.e. quarterly

GDP growth rates) are barely unbiased in average but feature procyclical optimism. The latter implies that investors are too optimistic in good times and too pessimistic in bad times and, in turn, they overestimate the persistence of the economic conditions (Baker and Wurgler (2006, 2007)). I use such survey data to parsimoniously characterize the investors’ beliefs about fundamentals.2 Namely, I consider a closedform and parsimonious continuoustime general equilibrium model with investors who have unbiased beliefs about economic growth in the longrun, but are optimistic and pessimistic respectively in good and bad times, consistently with the empirical evidence. The analysis bases on a single parameter capturing the strength of the bias in beliefs and shows that the model can capture many features of the asset returns. In particular, the equilibrium leads to a weak or negative riskreturn trade o of the stock markets, in line with the empirical data but at odds with standard ﬁnance theory, such as the dynamic CAPM of Merton (1980) and more recent leading asset pricing models.

Lettau and Ludvigson (2010) show that the unconditional correlation between expected excess returns and return volatility is negative, whereas general equilibrium models able to produce a timevarying Sharpe ratio –such as the

1Recently, Bacchetta, Mertens, and van Wincoop (2009) and Greenwood and Shleifer (2014) have empirically documented the importance, the robustness and the incompatibility with rational expectation models of actual expectations as proxied by survey data. 2Recently, Patton and Timmermann (2010) study the evolution of survey expectations about economic growth over the business cycle.

1 models by Campbell and Cochrane (1999), Barberis, Huang, and Santos (2001) and Bansal and Yaron (2004)– predict a strongly positive relationship. The left panel of Figure 1 shows the riskreturn trade o through a scatter plot when conditional moments are computed using a small set of instruments as conditioning information.

Insert Figure 1 about here.

In particular, the analysis of Lettau and Ludvigson (2010) documents that the smaller the information set, the more negative the riskreturn trade o: intuitively, the negative relationship arises as long as investors overweight simple measures of economic conditions, e.g. the cay, when computing expectations.3 Such a result is conﬁrmed by survey data. Duke/CFO survey is, to the best of my knowledge, the only data source providing information about the investors’ expectations of both the ﬁrst and the second moment of market returns. These survey data suggest that investors’ expectations about market returns lead to a nonlinear riskreturn trade o, as documented in Graham and Harvey (2001,

2012) and as shown in the right panel of Figure 1.

I document that this nonlinear, namely humpshaped, dynamics is a result of the procyclical variation of the trade o: market risk and its expected remuneration are positively and negatively correlated respectively in good and bad times. Such a dynamics generates a weak or even negative unconditional trade o and cannot obtain in leading equilibrium models.4 Such an empirical evidence suggests that the riskreturn relation is closely connected with the formation of investors’ beliefs about fundamentals. This paper proposes a general equilibrium model which generates such a connection. Beliefs deviate from the Bayesian assessment generating an alternation of waves of optimism and pessimism. The stronger such a deviation, the more procyclical the relation between market risk and its expected remuneration and, hence, the lower and even negative their unconditional correlation.

To my knowledge, this is the ﬁrst work providing a rationale for the riskreturn trade o and its dynamics in terms of agents behavior and aggregation result. In particular the model oers a theoretical link among the survey expectations of fundamentals and those of market returns. That is the most authentic empirical guidance about respectively the input and the output of an equilibrium asset pricing model. Surprisingly, beyond a huge empirical literature, there is a lack of general equilibrium literature concerning the relationship between the ﬁrst two moments of market returns.5

A notable exception is Whitelaw (2000). His model focuses on the physical dynamics of consumption growth in a

3A correlation still negative but close to zero can obtain extending the conditioning information set to a broad battery of financial indicators. At the opposite, if the information set reduces to the only cay as predictive variable (in addition to the lags), the correlation becomes strongly negative, about 0.70. See Lettau and Ludvigson (2010) and references therein. 4Survey data seem consistent with most recent works using realized returns: Rossi and Timmermann (2015) find strong evidence of nonmonotonic humpshaped risk return relation, Ghosh and Linton (2012) account for nonlinearities by verifying structural breaks and Lustig and Verdelhan (2012) document that variations of the first moment of returns are more closely related to business cycle than those of return volatilities. 5Backus and Gregory (1993) show that the riskreturn relation is potentially largely unrestricted from a theoretical point of view.

2 representative agent model under full information and captures a rich riskreturn trade o through multiple regimes with statedependent transition probabilities. Therefore, the approach that I propose in this paper is both alternative and complementary to the model of Whitelaw (2000). However, my model –which preserves timeseparable utility and a simple, namely homoscedastic, speciﬁcation of fundamentals– also captures many other empirical patterns of asset returns, such as unconditional and conditional moments and longhorizon predictability, which instead fail to obtain in

Whitelaw (2000).6

There are three main ingredients which produce the economic implications of the model. First, to model procyclical optimism in the formation of beliefs, I need that agents recognize the state of the economy through a simple state variable, akin a business cycle indicator. A well known approach in the ﬁnance literature is external (linear) habit. I assume that investors feature “catching up with the Joneses” preferences (CuJ, hereafter), in spirit of Abel (1990, 1999): agents have constant relative risk aversion but interpret good and bad states of the economy through the ratio of aggregate consumption over its smoothed past evolution.

Second, to study the riskreturn trade o, I need the model generates timevariation in the price of risk. This can obtain exogenously by nonlinear habit as in Campbell and Cochrane (1999) or endogenously by preference heterogeneity as in Chan and Kogan (2002). I follow the latter approach since it provides an economically pertinent microfoundation as well as additional testable implications, which impose discipline to the calibration of the model. Namely, under preference heterogeneity the crosssectional distributions of consumption and wealth vary over time and, in turn, lead endogenously to countercyclical aggregate risk aversion and price of risk, in line with the real data.

The third main ingredient of the model concerns the formation of investors’ beliefs and represents the main novelty of the paper. I extend the framework by Chan and Kogan (2002) along two dimensions. On the one hand, expected growth is stochastic and mean reverting. On the other hand, information is incomplete: investors do not observe the aggregate consumption drift but can infer information from the consumption realizations and a noisy signal. The main model mechanism is the following. Investors’ beliefs deviate from the Bayesian assessment by a simple term driven by the habit statevariable. Namely, waves of optimism and pessimism alternate over time.7 In turn, beliefs aect investors’ perception about the future evolution of the habit statevariable. Good (bad) states are excessively persistent when investors are optimistic (pessimistic). Such a mutual feedback eect between the habit statevariable and expected growth makes these variables more volatile, more persistent and more positively correlated than in the Bayesian case.

6As suggested by Whitelaw (2000) himself, a more sophisticated modelling of beliefs and preferences could improve many results of his framework, which enhances the complementarity of the two approaches. 7Alternation of optimism and pessimism as a driving source of the business cycle and the ﬁnancial markets has been studied since the seminal works of Pigou (1920) and Mitchell (1927).

3 This leads to equilibrium asset prices which help to explain a number of empirical patterns of the stock markets. The riskreturn trade o is as follows. Under Bayesian beliefs, return volatility is always countercyclical. This obtains because transient risk –which inherits the countercyclical dynamics of the price of risk– dominates the procyclical variation of the forwardlooking component of volatility. The latter is given by the elasticity of prices with respect to the current state. Under procyclical optimism, the habit statevariable is excessively persistent and, hence, the forwardlooking component of volatility becomes more important. Such an eect is stronger in bad times because risk aversion makes prices more sensitive to the state. Hence, the premium and the return volatility are both countercyclical in good times, but the premium is more countercyclical than volatility in bad times. As a consequence, the riskreturn relation switches from positive to negative in bad times, leading to a procyclical dynamics and to a low or negative unconditional level.

Therefore, the model endogenizes the dynamic riskreturn tradeo from survey data.

Although simple and parsimonious the model mechanism is robust to many other aspects of ﬁnancial markets.

Indeed, in addition to i) the weak or negative riskreturn relation and its procyclical dynamics, the model helps to explain ii) the equity premium and riskfree rate puzzles, iii) the smooth dynamics of the riskfree rate and the excess volatility of risky returns, iv) the stronger persistence of the dividend yield than that of the riskfree rate, v) the long horizon predictability of excess returns by dividend yields without predictable dividend growth, vi) the low and upward sloping term structure of interest rates with low and downward sloping volatilities, vii) the countercyclical behavior of the price of risk and of the equity premium, viii) the countercyclical dynamics of trading volume. Furthermore, the model ix) matches the little dispersion of the crosssectional distribution of consumption shares and x) features procyclical optimism about economic growth. These ten empirical patterns, on the one hand, provide robustness to the model and, on the other hand, suggest that a stylized representation of survey data about fundamentals is a key ingredient of an equilibrium asset pricing model.

Finally, it is worth emphasizing that, to my knowledge, this is the ﬁrst paper in which closed form expressions are available for the pricing kernel, the consumption and portfolio strategies, the asset prices and their moments in a multipleagents economy under incomplete information with a realistic pattern of preference heterogeneity. On the one hand, an indeep analysis of equilibrium outcomes is feasible. On the other hand, since the equilibrium is stationary, an easy calibration of cashﬂows and asset pricing quantities is possible.

The paper is organized as follows. Section II provides further empirical and theoretical motivation to the model and relates the paper to the literature. Section III describes the economy and characterizes the formation of beliefs. Sections

IV and V respectively solve for the equilibrium and derive asset prices. Section VI calibrates the model and investigates the asset pricing implications. Section VII highlights potential extensions. Section VIII concludes.

4 II. Empirical Evidence, Model Intuition and Related Literature

A. Survey Expectations

A.1. Expectations about fundamentals

A large literature studies survey data of investors’ forecast: Pesaran and Weale (2006) provide a recent review. La Porta

(1996) and, more recently, Bergman and Roychowdhury (2008) study the forecast errors and their ﬁndings are consistent with the idea that agents are excessively optimistic and pessimistic respectively in good and bad times and, there fore, they underestimate longrun mean reversion. Moreover, individual beliefs are heterogeneous: Veldkamp (2005),

Van Nieuwerburgh and Veldkamp (2006) and Patton and Timmermann (2010) document that forecasts dispersion moves with the business cycle. Investors’ priors, their heterogeneity and their cyclicality jointly determine the “average belief” about economic growth. In particular, aggregation of individual agents seems to lead to a suboptimal or irrational evolution of the “average belief,” even if individual investors were rational. Therefore, procyclical optimism in aggregate beliefs is not necessarily at odds with individual rational behavior.

The main assumption of the model concerns the procyclical optimism of investors’ beliefs about economic growth.

In order to provide empirical support to such an assumption, I consider the following simple model –which is similar to

Patton and Timmermann (2010). Let the aggregate output be a random walk: Yt+1 = Yt +µt +σyεy,t+1, with unobservable drift µt 1 = (1 κ)µt + σµεµ t 1. Moreover, deﬁne a simple business cycle indicator given by: + − , +

ωt 1 = (1 λ)ωt + ∆Yt 1. (1) + − +

KF Then, denote the “optimal forecast” µˆt t+1 as the Kalman ﬁlter estimate of expected growth from the observations Yt . | { } KF Prior I assume that the average belief obtains by combining the optimal forecast µˆt t+1 with a prior µt t+1: | |

α Prior α KF µˆt t+1 = t µt t+1 + (1 t )µˆt t+1. (2) | | − |

In line with Patton and Timmermann (2010), I set both the prior and the weight to be potentially statedependent:

Prior ω ω µt t+1 =a0 + a1( t ¯ ), (3) | − 2 2 αt =E[e ]/(qt + E[e ]), with log(qt )= b0 + b1 ωt ω¯ . (4) t t | − |

5 ω E ω Survey Survey where ¯ = [ t ] and et is the model error: µˆt t+1 µˆt t+1. I denote with µˆt t+1 the average forecast from the Survey | − | | of Professional Forecasters. I estimate this model by maximum likelihood using quarterly data on the sample 19682005.

Table I reports the estimates for several values of the business cycle parameter λ.8

Insert Table I about here.

Prior Both a1 and b1 are statistically signiﬁcant and respectively positive and negative. This means that the prior, µt t+1, | is procyclical and that investors put more weight, αt , on the prior when ωt is far from its average, i.e., when agents face a scenario they are not experienced with. The resulting forecasts µˆt t+1 reduce the historical mean square error { | } produced by µˆKF for various levels of the parameter λ. These estimates support the idea that investors feature { t } procyclical optimism.

KF Notice that asset pricing implications can be considerable even if µˆt t+1 diers little from µˆt t+1 . For a1 = { | } { | } ω 0,b1 = 0, beliefs driven by Eq. (2) lead to a first feedback eect from t to µˆt ,t+1. However, also a second feedback | eect takes place in the mind of the investors. Namely, the investors belief µˆt ,t+1 aects the perceived evolution of the | economy ∆ωt 2. This can be observed by writing Eq. (1) as ∆ωt 2 = λ(ωˆ t 1 ωt 1)+ σyεy t 2. The meanreversion + + + − + , + ω λ threshold ˆ t+1 = µˆt t+1/ is increasing with the investors belief µˆt t+1. Therefore, for a1 > 0,b1 < 0, the persistence | | of the business cycle indicator ωt is overestimated because the meanreversion threshold ωˆ t is too high and too low respectively in good a bad times. In summary, a stylized representation of survey data (i.e. a1 > 0,b1 < 0) leads to a ω mutual feedback between t and µˆt t+1 and makes these variables more persistent, more volatile and more correlated. | Asset prices will reflect investors beliefs about both expected growth (the socalled cashflows channel) as well as business cycle (the socalled discount rate channel). I will show that this mechanism is at the heart of the riskreturn trade o, once embedded in a general equilibrium model. The next scheme highlights the formation of beliefs implied by the model.

a1,b1 perceived evolution economic = 0 formation of meanreversion of economic conditions (ωt ) beliefs (µˆt t+1) threshold (ωˆ t+1) | conditions (∆ωt+2)

fundamentals (Yt )

8 Prior The term µt t+1 is not formally a prior since the left hand side of Eq. (2) corresponds to an aggregate quantity. However, it is intended to capture in reduced| form the cyclicality due to the aggregation of individual but heterogeneous beliefs (see Patton and Timmermann (2010)).

6 A.2. Expectations about market returns

The riskreturn trade o of the market is captured by the unconditional correlation between the expected excess returns and the return volatility, that is the unconditional correlation among the two components of the Sharpe ratio. There is some disagreement among studies that seek to determine the empirical relation between these two components.9 Recently, Lettau and Ludvigson (2010) document that such a disagreement is likely associated to the amount of conditioning information used in empirical studies. They verify that when using a small information set the correlation between the conditional mean and conditional volatility is strongly negative. When the information set is widely extended to an array of ﬁnancial indicators, the correlation is still negative but not necessarily statistically dierent from zero. Harvey (2001) and Graham and Harvey (2001) also document a negative correlation.

These works focus on the riskreturn trade o, as it is computed by the “econometrician”, and provide very puzzling evidence. Indeed, such a weak or negative relation obtains under various speciﬁcations of conditioning information and, then, appears as a robust result. However, the true riskreturn trade o –that is the relation based on the actual expectations of the agents, in contrast to those computed expost by the “econometrician”– has been yet neither investigated in the empirical literature nor explained by equilibrium models.

Investors’ expectations about market returns from Duke/CFO survey data allow to document a number of stylized facts. Table II reports estimates of the regression of survey expectations on a business cycle indicator.

Insert Table II about here.

P P Namely, expected premia (µ rt ) and volatility (σ ) are regressed on a state variable, ωt , computed as in Eq. (1): t − t

P P µ rt =α0 + β0ωt + εt and σ = α0 + β0ωt + εt t − t

The ﬁrst column of Panels A and B show that ωt strongly negatively explains variation in the expected mean and volatility of market returns (β0 < 0). Therefore, survey expectations are countercyclical. The second column of Panels A and B includes in the regressions a nonlinear function of ωt :

P µ rt =α0 + α11ω ω¯ + β0ωt + β1ωt 1ω ω¯ + εt t − t < t < σP α α β ω β ω ε t = 0 + 11ωt <ω¯ + 0 t + 1 t 1ωt <ω¯ + t

9Among others Bollerslev, Engle, and Wooldridge (1988), Harvey (1989), Ghysels, SantaClara, and Valkanov (2005) and Guo and Whitelaw (2006) ﬁnd a positive relation, while Campbell (1987), Breen, Glosten, and Jagannathan (1989), Pagan and Hong (1991), Glosten, Jagannathan, and Runkle (1993), Whitelaw (1994), Brandt and Kang (2004) and Conrad, Dittmar, and Ghysels (2013) ﬁnd a negative relation.

7 This allows to verify the dynamics of survey expectations over the business cycle. We observe that the negative relation between expected return and the statevariable ωt does not vary with the level of ωt (β0 < 0, β1 0). Instead, the ≈ correlation between expected volatility and ωt switches from strongly negative (β0 < 0) to close to zero when economic conditions deteriorates (β1 β0 > 0). ≈− Finally, panel C reports the estimates from the regression of expected return on expected volatility:

P P µ rt =α0 + β0σ + εt t − t P P P µ rt =α0 + α11ω ω¯ + β0σ + β1σ 1ω ω¯ + εt t − t < t t t <

In the ﬁrst column we observe that the unconditional riskreturn relation is not statistically dierent from zero (β0 0). ≈ The second column shows that, conditioning on the level of ωt , we can recover a strongly positive trade o in good times (β0 > 0) and a strongly negative trade o in bad times (β1 < β0 < 0). Such a procyclical dynamics explains the − low unconditional riskreturn relation and is due to the lack of countercyclical dynamics of expected volatility in bad times.

While the data sample is probably too small to assess an exact value for the unconditional trade o, it suggests that survey expectations are at odds with the strongly positive riskreturn relation implied by equilibrium models. Instead, the procyclical dynamics of the trade o seems a robust feature of the data: indeed, the sample covers the various phases of the business cycle, including the recent ﬁnancial crisis.

The three scatter plots of Figure 2 report the ﬁt from the regressions of Table II (both ﬁrst and second column).

Insert Figure 2 about here.

In particular, the right panel of Figure 2 clearly shows how the riskreturn trade o features a pronounced positive or negative sign, depending on the level of the statevariable.

I show that, in general equilibrium, a parsimonious characterization of investors’ beliefs about fundamentals, in line with the model of Eq. (2)(3)(4), leads to an endogenous osetting mechanism among the components of the return volatility, which exactly captures a weak or negative riskreturn trade o and its procyclical dynamics, as documented by survey data of market returns.

B. Other Related Literature

Behavioral ﬁnance oers empirical and theoretical arguments that cast doubts on the hypothesis that security markets are fully justiﬁed by economically pertinent news, despite market imperfections. Among others, Saunders (1993), Elster (1998),

8 Hirshleifer and Shumway (2003) and Baker and Wurgler (2007) document that markets are systematically inﬂuenced by investors’ psychology and mood. Charness and Levin (2005) stress that the processing of new information involves some form of reinforcement or extrapolation which often produces deviations from the Bayesian updating and makes choices more conform with recent economic conditions. Similar results are conﬁrmed by neuroeconomics research as in Kuhnen and Knutson (2011). In particular, Shefrin (2008) discusses several studies concerning the perception that risk and return are negatively related.

In the ﬁnance literature, Cecchetti, Lam, and Mark (2000) and Abel (2002) show that models in which consumers who exhibit pessimism and doubt could help to explain the average stock return and riskfree rate. More recently,

Fuster, Laibson, and Mendel (2010) and Fuster, Hebert, and Laibson (2011) introduce the socalled natural expectations: agents form expectations based on a simpler and more intuitive model than the true one, then they adjust towards ratio nal expectations. The appeal of simple models, the cost of processing information and cognitive biases are used to justify an incomplete adjustment. Extrapolative expectations lead to similar results and are investigated by Hirshleifer and Yu

(2011) and Barberis, Greenwood, Jin, and Shleifer (2015) in general equilibrium. Dumas, Kurshev, and Uppal (2009) ex tend Scheinkman and Xiong (2003) assuming that, when estimating the expected growth, some agents overreact to a noisy signal and, in turn, generate a sentiment risk. In particular, Xiong and Yan (2010) and Jouini and Napp

(2010) show that investors with irrational beliefs survive and have price impact as long as they are rational on av erage, in the sense that incorrect beliefs are symmetric and there is not a longrun bias at the aggregate level.

BaroneAdesi, Mancini, and Shefrin (2012) empirically verify that excessive optimism and overconﬁdence lead to the negative riskreturn relation, in line with survey data (e.g. Duke/CFO survey). Baker and Wurgler (2006) and Yu and Yuan

(2011) empirically document how investors’ sentiment aects respectively the crosssection of equity return and the riskreturn trade o. Duarte, Kogan, and Livdan (2012) recover a negative riskreturn relation by assuming preference shocks with timevarying volatility correlated with output. This paper complements the above literature about the role of biased beliefs in ﬁnancial markets. As a peculiarity, this paper uses survey data about fundamentals as an empirical input of an equilibrium model, whose output (i.e. asset prices) can explain some empirical patterns of survey data about market retuns.

This paper also relates with the literature of preference heterogeneity.10 In particular, Chan and Kogan (2002) show that preference heterogeneity can lead to a number of interesting asset pricing implications, such as countercyclical price of risk and equity premium, and therefore can endogenize some of the results of Campbell and Cochrane (1999).

10Dumas (1989) and Wang (1996) study twoagent economies respectively with production and in the pure exchange case. Bhamra and Uppal (2014) and Cvitanic,´ Jouini, Malamud, and Napp (2011) study the interaction of heterogeneous beliefs and preferences with power utility investors. Weinbaum (2010) and Curatola and Marfè (2011) provide closed formulas with inﬁnitely many agents and a particular pattern of heterogeneity.

9 Nonetheless, Xiouros and Zapatero (2010) ﬁnd that just a marginal eect on asset price dynamics does obtain when the model is calibrated with a realistic amount of heterogeneity in risk attitudes. This paper shows that, even under a realistic pattern of heterogeneity, the results of Chan and Kogan (2002) continue to hold because of the peculiar formation of beliefs. A nonnegligible redistribution of wealth among agents takes place because, under their subjective probability, investors overestimate risk and persistence of economic conditions. This leads to endogenous ﬂuctuations in asset prices and also to a weak or negative riskreturn trade o.

III. The Model

In a continuoustime inﬁnitehorizon pure exchange economy of Lucas (1978) type, agents feature “catching up with the

Joneses” (CuJ) preferences in spirit of Abel (1990). I study both the case of homogeneous agents and the case of agents, who dier from each other with respect to their relative risk aversion as in Chan and Kogan (2002).

A. Preferences and Heterogeneity

In the economy, inﬁnitely many investors maximize expected utility of the form

∞ ˆ ρt γ γ 1 1 γ γ E0 e− U(Ct ,Xt ; )dt , where U(Ct ,Xt ; )= Ct − Xt , (5) Z0 1 γ −

Eˆ is the expectation operator under the probability measure of the investors, Ct is consumption and Xt is an exogenous process interpreted by all agents as a standard of living. The marginal utility from consumption is given by

γ γ γ UC(Ct ,Xt ; )= Ct− Xt , (6)

γ γ γ γ 1 which implies UCX (Ct ,Xt ; )= Ct− Xt − > 0. Thus, for all the agents, the standard of living Xt and the current consumption Ct act as complementary goods. All agents in the economy have the same time discount rate ρ but they dier with respect to their preference parameter γ.

Since in general closed form solutions are not feasible under heterogeneous risk aversion, I assume a speciﬁc pattern of heterogeneity to obtain explicit solutions for the optimal consumption and portfolios, the stateprice density, the risky asset prices and their moments. I assume that agents have relative risk aversion in the interval between zero and an arbitrary upper bound γ¯. Each agent type i is characterized by its constant relative risk aversion and social weight:

10 γ¯ ν i γi γ j γi = i N, with γ¯ N, = p − , p > 0, i, j N. (7) i ∀ ∈ ∈ ν j ∀ ∈

Therefore, the distribution of types is deﬁned on a realistic range of values, (0,γ¯).11 Moreover, the average value and the variation of the representative agent’s risk aversion will be characterized in closed form. Section VI.B shows that for reasonable parameters the equilibrium distribution of consumptionshares closely approximates individual consumption data. The homogeneousagents economy obtains by assuming that all social weights but one are equal to zero.

B. Primitives

The aggregate dividend, Yt , serves as a numeraire and follows the stochastic process:

dYt =µtYt dt + σyYt dBy,t , (8)

dµt =κ(µ¯ µt )dt + σµdBµ t . (9) − ,

Expected growth, µt , is stochastic and follows an OrnsteinUhlenbeck (OU) process with long run mean µ¯ and speed of mean reversion κ. The volatilities σy and σµ are constant and the standard Brownian motions By,t and Bµ,t are mutually independent for the sake of exposition. Eq. (8)(9) represent the most common setting in the literature on asset pricing under incomplete information (see, e.g., Kim and Omberg (1996), Brennan and Xia (2001), Pastor and Veronesi (2009)).

As usual in habit models, the standard of living process is a weighted geometric average of past realizations of the aggregate dividend: t λt λ(t u) xt = x0e− + λ e− − yudu, such that dxt = λ(yt xt )dt, (10) Z0 − where yt = logYt , xt = log Xt and x0,λ > 0. The habit state variable, ωt = yt xt , has the interpretation of a business − cycle indicator. Indeed, it characterizes good and bad times in the economy in the mind of agents with CuJ preferences.12

An application of Ito’sˆ lemma shows that

dωt = λωt dt + dyt = λ(ω¯ t ωt )dt + σydBy t , (11) − − ,

2 where ω¯ t = (µt σ /2)/λ. Thus, ωt is a mean reverting process and is conditionally normally distributed. − y 11 ω¯ I set p = e with ω¯ = Eˆ 0[logYt /Xt ] to center the sensitivity of the price of risk at the steadystate. See Section V for the details. 12Section IV shows that individual marginal utilities as well as the representative agent’s marginal utility evaluated at optimal consumption are functions of ωt only.

11 I assume incomplete information in the sense that investors cannot observe µt . However, they have to infer it from observations of the dividend process Yt and a noisy signal, st , about expected growth:

dst = µt dt + σsdBs,t (12)

where σs is constant and the standard Brownian motion Bs,t is independent of By,t and Bµ,t . Dynamics in Eq. (8) to (12) deﬁne the true model.

C. Investors’ Beliefs

Investors make their consumption and investment decisions without observing µt . The investors’ estimate is denoted by µˆt = Eˆ t [µt ], where the expectation is calculated under the investors’ subjective probability measure. Economies with incomplete information have been studied in Detemple (1986), Dothan and Feldman (1986), and Gennotte (1986), among others. The analysis consists of the ﬁltering and of the consumption and investment decision. I consider at ﬁrst the benchmark case of Bayesian investors and then I introduce procyclical optimism.

C.1. Optimal forecast

Investors know the structure and the parameters of the true model and update their beliefs in a Bayesian way. Priors are normal with mean µˆ0 and variance σµ,0. Standard ﬁltering theory, as in Liptser and Shiryaev (2001), leads to the following dynamics of the aggregate dividend, the estimated drift and the habit state variable:13

dYt = µˆtYt dt + σyYt dBˆy,t , (13)

dµˆt = κ(µ¯ µˆt )dt + σµ ydBˆy t + σµ µdBˆµ t + σµ sdBˆs t , (14) − , , , , , ,

dωt = λ(ωˆ t ωt )dt + σydBˆy t , (15) − ,

2 where ωˆ t = (µˆt σ /2)/λ and Bˆy t ,Bˆµ t and Bˆs t are independent standard Brownian motions under the investors’ − y , , , subjective measure. With a perfect signal, i.e. a signal with zero noise, the drift is observable, therefore µˆt = µt . It follows that σµ,µ = σµ, σµ,y = σµ,s = 0, Bˆy,t = By,t and Bˆµ,t = Bµ,t . With an imperfect signal, i.e. a signal with nonzero σ σ 1 σ σ 1 σ noise, µ,y = v∞ y− , µ,s = v∞ s− and µ,µ = 0 with

13For the sake of exposition and analytic tractability, I assume a long enough horizon of past observations, so that the variance has already converged towards its steady state. This is an usual assumption, as in Dumas, Kurshev, and Uppal (2009) for instance.

12 2 2 2 2 2 2 1 v∞ = κ + σ σy− + σs− κ σ− + σ− − . (16) µ − y s Since investors do not observe the true drift, Bˆµ,t vanishes from Eq. (14). This will be the case considered in the following of the paper. Finally, the case of a signal with inﬁnite noise obtains by taking the limit σs ∞: the signal becomes → useless and both the Bˆµ,t and Bˆs,t vanish from Eq. (14).

C.2. Procyclical optimism

I consider now the case of investors who deviate from the Bayesian formation of beliefs, consistently with the empirical

ﬁndings of Section II.A. The aim is to model procyclical optimism through a simple and intuitive mechanism, governed by a single parameter χ, which resembles Eq. (2)(3)(4). Such a parameter will capture the impact of current economic conditions on the formation of beliefs (i.e. ωt dµˆt ) and, in turn, the impact of the perceived growth on the perceived → evolution of economic conditions (i.e. µˆt dωt ). Namely, I assume a simple deviation term, Ψ(ωt ), as follows: →

dµˆt = κ(µ¯ µˆt )dt + Ψ(ωt )dt + σµ ydBˆy t + σµ µdBˆµ t + σµ sdBˆs t . (17) − , , , , , ,

Bayesian Then, the bias process ςt = µˆt µˆ satisﬁes − t

t κt κ(t u) ςt = ς0e− + e− − Ψ(ωu)du, such that dςt = κςt dt + Ψ(ωt )dt. (18) Z0 −

The deviation term Ψ(ωt ) has a simple functional form since it is just a centered version of the habit state variable scaled by the deviation parameter χ:

Ψ(ωt )= χ(ωt ω¯ ), (19) −

2 where ω¯ = (µ¯ σ /2)/λ. For χ > 0, in good times (i.e. high ωt ) the investors tend to overestimate expected growth, − y while in bad times they tend to underestimate it. A positive parameter χ parsimoniously captures the eects of both Bayesian positive a1 and negative b1 in Eq. (3) and (4). Indeed, dµˆt procyclically deviates from dµˆt (i.e. a1 > 0) and such a deviation is larger in magnitude when ωt is far from its average (i.e. b1 < 0). For χ > 0, waves of optimism and pessimism obtain and alternate over time.14

14The same dynamics for the expected growth in Eq. (17) can obtain by assuming that the agents are still Bayesian but conditionally to the misleading interpretation of the signal. They observe a signal as in Eq. (12), but they interpret it erroneously as

dst =(µt + Ψ′(ωt ))dt + σsdBs,t,

13 Finally, I set Ψ(ωt ) to have zero expectation to rule out from equilibrium prices the mechanical eects of the

15 preponderance of optimism over pessimism or viceversa. The key implication of the term Ψ(ωt ) concerns the perceived dynamics of ωt in the mind of the investors. Under their subjective probability measure, the reversion threshold, ωˆ t , is positively related to µˆt :

dωt = λ(ωˆ t ωt )dt + σydBˆy t , (20) − , 2 ωˆ t = (µˆt σ /2)/λ. (21) − y

Consequently, for χ > 0, in good times (i.e. high ωt ) expected growth is overestimated and the threshold ωˆ t is higher than in the Bayesian case. It follows that the persistence of good states is overestimated too. Similarly, in bad times

(i.e. low ωt ) expected growth is underestimated and the threshold ωˆ t is lower than for a Bayesian investor. Therefore, bad times appear to the investors more persistent than in the Bayesian case. Such a joint formation of habit (dωt ) and beliefs (dµˆt ) leads to a perceived evolution of the state of the economy which is riskier than under the true probability measure. Then, the proposed approach provides a rationale for the excess variability of the equilibrium pricing kernel and is consistent with the large ﬂuctuations of stock prices beyond the smooth dynamics of fundamentals.

The left and middle panels of Figure 3 show respectively the cumulative distribution function and the autocorrelation function of ωt as perceived by the investors for both the Bayesian case (χ = 0%) and the procyclical optimism

16 (χ = 7.5%). The right panel reports the scatter plot of ωt and µˆt as they evolve under the investors probability measure for both the two levels of the deviation parameter: the larger χ, the more correlated ωt and µˆt .

Insert Figure 3 about here.

The Bayesian updating of beliefs obtains when χ goes to zero. Instead, for χ < 0, waves of optimism and pessimism occur respectively in bad and good times: in such a case –in contrast with the empirical evidence– the perceived persistence of ωt is weaker than under the true probability measure.

IV. The Equilibrium

including a deviation term Ψ′(ωt ) ∝ Ψ(ωt ). See the Appendix A for the details. Such an interpretation is similar to Dumas, Kurshev, and Uppal (2009), where some agents are overconﬁdent about the noisy signal. 15 The asymmetry in the waves of optimism and pessimism can be easily achieved setting Ψ(ωt ) = χ(ωt ω¯ + ε), where ε could be eventually calibrated to capture the longrun bias in the forecast errors. See Section VII for further details. − 16Results are obtained through simulations: parameters are the same used in the empirical analysis. See Section VI.A for the details.

14 This section considers the representative agent problem to derive the optimal consumption policy and the state price density. Under the investors’ subjective measure, uncertainty is described by the Brownian motions Bˆy,t and Bˆs,t . I assume there exist assets such that markets are complete. Namely, the instantaneously riskfree asset and two longlived securities: the market asset, which pays the aggregate dividend, and a perpetual bond.17

Market completeness implies the existence of a representative agent maximizing a linear combination with positive coecients of the agents’ utility functions. He acts as a social planner in such a way the resulting allocation is Pareto optimal. Given the social weights, νi, the representative agent solves:

∞ ρ sup Eˆ e t ∑ν U(C ,X ;γ )dt subject to ∑C Y t 0. (22) 0 Z − i i,t t i i,t t Ci,t 0 i i ≤ ∀ ≥ { }

Since there is no intertemporal transfer of resources, this optimization reduces to a static problem:

sup ∑νiU(Ci,t ,Xt ;γi), (23) Ci t i { , } subject to the resource constraint. Lemma 1 characterizes the optimal consumption policy.

Lemma 1 The optimal consumption sharing rule is given by

ω ω i e t ¯ − (ωt ω¯ ) Ci,t = ci,tYt , with ci,t = ω ω e− − . (24) 1 + e t ¯ −

Individual consumption, as a fraction of the aggregate endowment, is a stationary function of ωt . Thus, consumption (and wealth) of all agents grows at the same average rate and no single agent dominates the economy in the longrun.18

Let ξt,u denote the state price density in an ArrowDebreu economy which supports in equilibrium the Pareto optimal allocation of Lemma 1 (see Due and Huang (1985)). The price at time t of an arbitrary payo stream Fu,u (t,∞) is { ∈ } Eˆ ∞ ξ given by t [ t t,uFudu]. Corollary 1 characterizes the state price density and the aggregate relative risk aversion. R

17The market can be completed considering at least two longlived securities. However, since the price of these securities would be determined endogenously, one would have to verify endogenous completeness. This can be done by using the techniques of Hugonnier, Malamud, and Trubowitz (2012) or by verifying that the prices’ diusion matrix is invertible for almost all states and times. Oth erwise, one can just assume a derivative asset, written on Bˆs,t in zero net supply, with endogenous drift but exogenous volatility such that the market is complete. 18This result is due to the CuJ preferences. As in Chan and Kogan (2002), external habit has an equalizing eect on marginal utilities of the heterogeneous agents such that individual consumption and wealthshares remain stationary over time. At the opposite, under standard timeseparable CRRA preferences, the least risk averse agent will eventually dominate the economy, as in Wang (1996).

15 Corollary 1 The equilibrium state price density is given by

RA ω ω¯ γ¯ ρ U (Y ,X ) ρ γ ω ω 1 + e u ξ (u t) Y u u (u t) ¯( u t) − t,u = e− − = e− − − − ω ω , (25) U RA(Y ,X ) 1 + e t ¯ Y t t −

RA ν γ where U (Yt ,Xt )= ∑i iU(Ci,t ,Xt ; i). The aggregate relative risk aversion is given by

RA γ UYY (Yt ,Xt ) ¯ γ˜(ωt )= Yt = . (26) RA ωt ω¯ − UY (Yt ,Xt ) 1 + e −

Corollary 1 states that the aggregate relative risk aversion is stationary and decreasing in the habit state variable.

Furthermore, it is bounded above (below) by the coecient of relative risk aversion of the most (least) risk averse agent. γ 1 Intuitively, the risk tolerance of the representative agent is a weighted average of the individual risk tolerances i− with weights given by consumption shares.19

As a result of heterogeneity, the logarithm of the state price density is a nonlinear function of ωt . Therefore, its conditional variance depends on the habit statevariable and leads to an endogenous timevariation of the price of risk, even if ωt is an homoscedastic process. Instead, the equilibrium state price density of an homogeneousagents economy obtains when the social weights of all agents are equal to zero but one. If such an agent has arbitrary risk aversion

γ > 0, then the state price density is

ρ(u t) γ(ωu ωt) ξt,u = e− − − − . (27)

Such a state price density is exponential ane in ωt and therefore inherits its homoscedasticity. To compare asset pricing implications in the two economies, I set the relative risk aversion coecient of the homogeneousagents economy equal to the steady state aggregate relative risk aversion in the heterogeneous economy: γ = γ˜(ω¯ )= γ¯/2. Although a mounting evidence suggests that investors are heterogeneous in both beliefs and risk preferences, modelling simultaneously both these forms of heterogeneity is quite dicult. Online Appendix OA.A oers an equilibrium microfoundation of the procyclically biased but homogeneous beliefs assumed in Eq. (17). Namely, under heterogeneous risk attitudes, equilibrium aggregation of heterogeneous (e.g. symmetric) beliefs can lead to procyclical optimism for the representative agent. Then, the deviation term Ψ(ωt ) in Eq. (19) can be interpreted as a reduced form approach: it is useful to build a direct link with the survey data (Section II.A) and to provide analytic tractability.

19 Indeed, in the limit good state (ωt +∞) the least risk averse agent dominates the economy and his consumption share tends to one. → Viceversa, in the limit bad state (ωt ∞), the most risk averse agent dominates. → −

16 V. Equilibrium Asset Prices and Portfolio Strategies

This section studies the behavior of security prices. The assets of interest are the riskfree asset, the inﬁnitely lived market security, Pt , that pays the aggregate dividend, and the perpetual bond, Qt :

∞ ∞ P = Eˆ ξ Y du and Q = Eˆ ξ du . (28) t t Z t,u u t t Z t,u t t

I derive analytically the instantaneous rate of interest and the price of risk, the asset prices and their moments, the individual wealth and the portfolio strategies. The analysis characterizes at ﬁrst the homogeneousagents economy and then the heterogeneous one: dierences arise endogenously and can be understood as a result of the ﬂuctuations in the crosssectional distribution of wealth.

A. Homogeneous Agents Economy

Agents have equal relative risk aversion and, hence, the pricing kernel in Eq. (27) allows to discount cashﬂows.

Corollary 2 Under homogeneous risk attitudes, the instantaneous rate of interest is given by

1 2 2 rt = ρ + γλ(ωˆ t ωt ) γ σ , (29) − − 2 y

2 where ωˆ t = (µˆt σ /2)/λ, and the price of risk is equal to − y

θt = γσy. (30)

The price of risk is constant and thus stateindependent. Instead, the interest rate decreases with the habit state variable,

ωt . Moreover, such a variation increases with both reversion, λ, and risk aversion, γ. The estimated drift, µˆt , aects the riskfree rate through the reversion threshold ωˆ t . As commented in Section III.C, procyclical optimism (χ > 0) increases the correlation of ωt and µˆt . Then, it reduces the volatility of ωˆ t ωt and, in turn, that of the riskfree rate. −

Proposition 1 Under homogeneous risk attitudes, the equilibrium market price-dividend ratio equals

P ∞ t γωt γωt A(u t,c¯)+B(u t,c¯)ωt+C(u t,c¯)µˆt = e F(ωt ,µˆt )= e e − − − du, (31) Yt Zt

17 where c¯ = (0,1, γ,0), F( , ) is implicitly deﬁned and A(τ,φ), B(τ,φ) and C(τ,φ) are deterministic functions of time − derived in Appendix C. The equilibrium volatility of the market asset is given by

σP σP 2 σP 2 t = y + s , (32) where ω ω ω σP = σ + θ + σ Fω( t,µˆt ) + σ Fµˆ ( t ,µˆt ) and σP = σ Fµˆ ( t,µˆt ) . (33) y y t y F(ωt,µˆt ) µ,y F(ωt,µˆt ) s µ,s F(ωt ,µˆt )

Fω and Fµˆ denote the partial derivatives of F(ωt ,µˆt ). The equity premium is given by

P P µ rt = θt σ . (34) t − y

Similarly to the rate of interest, also the pricedividend ratio, the market volatility and equity premium are stationary functions of ωt and µˆt . Since under homogeneous preferences the price of risk is constant, the equity premium and the σP priced component of the stock volatility y are perfectly correlated. σP σP σP The volatility of the market asset has two components: y and s . Notice that y is the priced component of volatility and is given by the sum of four terms. The ﬁrst is the volatility of fundamentals such that the other three components lead to excess volatility. The second and the third terms are due to the CuJ preferences: the former is exactly the price of risk, the latter accounts for the persistence of the state variables. The fourth term arises when the σP expected dividend growth is stochastic and captures the eects of the incomplete information. Similarly, s captures the eects of the incomplete information due to the noisy signal, but it is not priced in equilibrium.

Proposition 2 Under homogeneous risk attitudes, the equilibrium perpetual bond price equals

∞ γωt ε γωt A(u t,c¯)+B(u t,c¯)ωt+C(u t,c¯)µˆt Qt = e F (ωt ,µˆt )= e e − − − du, (35) Zt where c¯ = (0,0, γ,0), Fε( , ) is implicitly deﬁned and A(τ,φ), B(τ,φ) and C(τ,φ) are deterministic functions of time − derived in Appendix C. The equilibrium volatility of the bond is given by

σQ σQ 2 σQ 2 t = y + s , (36) where

18 ε ε ω ε ω Q Fω(ωt ,µˆt ) Fµˆ ( t ,µˆt ) Q Fµˆ ( t,µˆt ) σ = θt + σy ε + σµ y ε and σ = σµ s ε . (37) y F (ωt ,µˆt ) , F (ωt ,µˆt ) s , F (ωt,µˆt )

ε ε ε ω Fω and Fµˆ denote the partial derivatives of F ( t ,µˆt ). The bond premium is given by

Q Q µ rt = θt σ . (38) t − y

The equilibrium yield of a zero-coupon bond with maturity τ is given by

1 1 ε(t,τ)= logEˆ t [ξt t τ]= (γωt + A(τ,c¯)+ B(τ,c¯)ωt +C(τ,c¯)µˆt ). (39) − τ , + − τ

The perpetual bond price is a stationary function of ωt and µˆt . Its volatility and premium have a decomposition similar to that of the market asset. Real yields are ane in ωt and µˆt . In turn, yields volatility is stateindependent.

B. Heterogeneous Agents Economy: Asset Prices

This section assumes agents have the heterogeneous relative risk aversion as in Eq. (7) and, hence, the pricing kernel in

Eq. (25) allows to discount cashﬂows.

Corollary 3 Under heterogeneous risk attitudes, the instantaneous rate of interest is given by

γσ¯ 2 ωt ω¯ ωˆ t ωt y γ¯+e − ρ γλ¯ ω ω rt = + −t ¯ 2 ωt ω¯ 2 , (40) 1+e − − (1+e − )

2 where ωˆ t = (µˆt σ /2)/λ, and the price of risk is equal to − y

γσ θ ¯ y t = ωt ω¯ . (41) 1+e −

The equilibrium price of risk can be written as θt = γ˜(ωt )σy: its variation is due heterogeneity. In particular, the price

ω ω¯ ω ω¯ 2 of risk inherits the countercyclical behavior of γ˜(ωt ): ∂ωθt = γσ¯ ye t (1 + e t ) < 0, with maximal sensitivity − − − − at ωt = ω¯ . The price of risk reduces to that of an homogeneousagents economy with risk aversion equal to γ¯ or zero when the habit state variable goes respectively to minus or plus inﬁnity. Indeed, in these cases, respectively the most or the least risk averse agent dominates the economy.

19 Using the aggregate relative risk aversion, the interest rate can be written as

σ2 2 y 2 ωt ω¯ rt = ρ + (µˆt σ /2)γ˜(ωt ) λωt γ˜(ωt ) γ˜(ωt ) (1 + e − /γ¯). (42) − y − − 2

Investors’ patience is inversely related to the common discount rate ρ which positively aects the level of r(ωt ,µˆt ).

The interest rate increases with µˆt . The higher the rate of consumption growth, the lower the expected marginal utility (relative to the present). The second term on the right hand side of Eq. (42) shows two additional eects. On the one hand, expected growth is scaled by aggregate relative risk aversion. In good (bad) states, marginal utility is low

(high) relative to the steady state. Consequently, a low (high) γ˜(ωt ) reduces (increases) the impact of expected growth on the equilibrium interest rate. On the other hand, incomplete information aects µˆt . In particular, for χ > 0, ωt and µˆt are more correlated than in the Bayesian case. Hence, procyclical optimism mitigates the volatility of the sum of the second and the third term in Eq. (42). Habit persistence measures the incentive to postpone consumption and therefore to save more when consumption today is high with respect to habit: λωt γ˜(ωt ) captures such an eect. − Heterogeneity adds a countercyclical adjustment proportional to the aggregate relative risk aversion. The volatility of consumption growth increases the expected marginal utility and thus induces a precautionary savings component in the equilibrium interest rate. The last term on the right hand side of Eq. (42) describes this eect. While it is constant in the homogeneousagents economy, under heterogeneity it also measures the incentive to save less when the price of risk in the economy is low. Therefore, habit persistence and precautionary savings terms are inversely proportional to each other. Consequently, heterogeneity generates an endogenous osetting mechanism which reduces the level of the interest rates and helps to capture the countercyclical variation of risk premia.

Proposition 3 Under heterogeneous risk attitudes, the equilibrium market price-dividend ratio equals

γ¯ γ¯ ∞ P ωt ω¯ ωt ω¯ γ¯ γ t e − e − ¯ A(u t,c¯)+B(u t,c¯)ωt+C(u t,c¯)µˆt ω ω F ω µˆ ω ω ∑ e du = 1+e t ¯ ( t , t )= 1+e t ¯ j=0 j − − − , (43) Yt − − Zt where c¯ = ( ω¯ ,1, j γ¯,0), F( , ) is implicitly deﬁned and A(τ,φ), B(τ,φ) and C(τ,φ) are deterministic functions of − − time derived in Appendix C. The volatility of the market asset is given by

σP σP 2 σP 2 t = y + s , (44) where

20 ω ω P Fω(ωt,µˆt ) Fµˆ ( t ,µˆt ) P Fµˆ ( t,µˆt ) σ = σy + θt + σy + σµ y and σ = σµ s . (45) y F(ωt,µˆt ) , F(ωt,µˆt ) s , F(ωt ,µˆt )

Fω and Fµˆ denote the partial derivatives of F(ωt ,µˆt ). The equity premium is given by

P P µ rt = θt σ . (46) t − y

Similarly to the homogeneous case, the pricedividend ratio, the market volatility and equity premium are stationary functions of the ωt and µˆt . Such a stationarity property does not obtain under heterogeneity in absence of CuJ preferences. While the above expressions only hold for a speciﬁc pattern of heterogeneity, it is worth emphasizing that, to my knowledge, they are the only closed form expressions available for a stationary equilibrium in a heterogeneous multipleagents economy under incomplete information.

The equilibrium price dividend ratio is given by the product of two terms: the ﬁrst comes from the current level of the state price density, while the second one, F(ωt ,µˆt ), captures the expected evolution of the states ωt and µˆt . Under heterogeneous risk attitudes, this term is given by a weighted sum of integrals of exponential ane functions of the states. This weighted sum obtains from the nonlinearity of the log state price density and, therefore, leads to additional endogenous heteroscedasticity in returns. σP σP σP The volatility of the market asset has two components: y and s . The term y is the priced volatility and is given σP by the sum of four terms. The ﬁrst term is the volatility of fundamentals. The second term of y is the price of risk and, under preference heterogeneity, is decreasing with ωt . Therefore, the whole volatility can be eventually counter cyclical, whereas it is nondecreasing with the habit state variable in the homogeneousagents economy. Similarly to the σP σP homogeneous case, s and the last two terms of y are forwardlooking terms given by the semielasticity of prices σP with respect to the states. In particular, s is the unpriced volatility due to the noisy signal. Since the price of risk is not constant in the heterogeneousagents economy, the equity premium is not a scaled version of the priced volatility. Consequently a rich riskreturn trade o –that is the unconditional correlation between

µP r and σP– can obtain, as it will be shown in Section VI. −

Proposition 4 Under heterogeneous risk attitudes, the equilibrium perpetual bond price equals

γ¯ γ¯ ∞ ωt ω¯ ε ωt ω¯ γ¯ γ ω e − ω e − ¯ A(u t,c¯)+B(u t,c¯) t+C(u t,c¯)µˆt Qt = 1+eωt ω¯ F ( t ,µˆt )= 1+eωt ω¯ ∑ j 0 j e − − − du, (47) − − = Zt

21 where c¯ = ( ω¯ ,0, j γ¯,0), Fε( , ) is implicitly deﬁned and A(τ,φ), B(τ,φ) and C(τ,φ) are deterministic functions of − − time derived in Appendix C. The volatility of the bond is given by

σQ σQ 2 σQ 2 t = y + s , (48) where ε ε ω ε ω Q Fω(ωt ,µˆt ) Fµˆ ( t ,µˆt ) Q Fµˆ ( t ,µˆt ) σ = θ σ + σ ε + σ ε and σ = σ ε . (49) y t y y F (ωt ,µˆt ) µ,y F (ωt ,µˆt ) s µ,s F (ωt ,µˆt )

ε ε ε ω Fω and Fµˆ denote the partial derivatives of F ( t ,µˆt ). The bond premium is given by

Q Q µ rt = θt σ . (50) t − y

The equilibrium yield of a zero-coupon bond with maturity τ is given by

γ¯ 1 1 eωt ω¯ γ¯ γ¯ A(τ,c¯)+B(τ,c¯)ω +C(τ,c¯)µˆ ε τ Eˆ ξ τ − t t (t, )= τ log t [ t,t+ ]= τ log ωt ω¯ ∑ j e . (51) − − 1+e − j=0

Under heterogeneous risk preferences, the bond price and its moments are still stationary functions of ωt and µˆt . The endogenously heteroscedastic state price density leads to real yields which are nonane in the state variables. In turn, yields volatility is statedependent. A similar term structure of real interest rates obtains in Buraschi and Jiltsov (2007).

C. Heterogeneous Agents Economy: Wealth and Portfolio Strategies

This section characterizes the equilibrium individual wealth and portfolio holdings.

Proposition 5 Under heterogeneous risk attitudes, the equilibrium wealth of agent i equals

γ¯ γ¯ ∞ ωt ω¯ ωt ω¯ U ω e − i ω e − ϒ A(u t,c¯)+B(u t,c¯) t+C(u t,c¯)µˆt Wi,t = Yt 1+eωt ω¯ F ( t ,µˆt )= Yt 1+eωt ω¯ ∑ j 0 (i, j) e − − − du (52) − − = Zt where c¯ = ( ω¯ ,1,ℓ(i, j),0), Fi( , ) is implicitly deﬁned. U, ϒ(i, j), ℓ(i, j) and the deterministic functions of time − A(τ,φ), B(τ,φ) and C(τ,φ) are derived in Appendix C. πP πQ The proportions of wealth invested in the market asset, i , and in the perpetual bond, i , satisfy

σWi σP σQ πP y y y i   =   , (53) σWi σP σQ πQ s s s i           22 σP σP σQ σQ where y , s and y , s are from Eq. (45) and (49),

i ω Fi(ω ,µˆ ) Fi(ω ,µˆ ) σWi σ θ σ Fω( t ,µˆt ) σ µˆ t t σWi σ µˆ t t = y + t + y i + µ y i and = µ s i . (54) y F (ωt ,µˆt ) , F (ωt ,µˆt ) s , F (ωt,µˆt )

Individual wealth is proportional to the aggregate dividend, consequently both wealth shares, Wi,t /Pt , and wealth consumption ratios, Wi,t /Ci,t , are stationary functions of the states ωt and µˆt only. This result comes out from the equalizing eect of the standard of living Xt , which makes marginal utilities and, in turn, the state price density stationary as well. Wealth distribution ﬂuctuates over time around a steadystate. Procyclical optimism (χ > 0) reinforces such an endogenous timevariation –which is otherwise quite limited under a realistic degree of preference heterogeneity.

Indeed, for χ > 0, investors overestimate the persistence of good and bad times and, hence, the unconditional volatility of aggregate risk. πP πQ The proportions of wealth invested in the market asset, i , and in the perpetual bond, i , can be recognized since under completeness the diusion matrix of the risky assets is invertible. Here, endogenous completeness can be veriﬁed

P Q P Q as long as σ σs σ σy = 0. y − s

VI. Asset Pricing Results

This section investigates whether the model can capture the main empirical stylized facts of asset returns together with the weak or negative riskreturn trade o and its dynamics. The analysis relies on the single parameter χ which captures procyclical optimism. Such an approach leads to a number of predictions in line with empirical evidence and improves the Bayesian benchmark (χ = 0) over many dimensions. At ﬁrst, I describe the model calibration and the crosssectional implications of heterogeneity. Then, the focus turns on the equilibrium riskreturn trade o and its dynamics. Finally, for robustness, I analyse the unconditional and conditional moments of returns, their predictability, the term structure of the interest rates and the dynamics of portfolios and trading volume.

A. Model Calibration

I use annual data from 1933 to 2006 from the S&P Composite Index, the Consumer Price Index and the threemonth

Treasury bill from the Federal Reserve Bank of St. Louis. Statistics for the pricedividend ratio, the riskfree rate and the market return are computed from the data. I also report other data to account for the recent ﬁnancial crisis.20

20For the purpose of comparison, the empirical analysis considers the same data about stock markets as in Xiouros and Zapatero (2010), Marfè (2011), and Curatola and Marfè (2011) which study CuJ preferences under heterogeneity but full information.

23 The values of the unconditional mean µ¯ (1.8%) and the instantaneous volatility σy (3.6%) of the aggregate dividend growth are calibrated to match the implied moments in Mehra and Prescott (1985). The speed of reversion κ (.35) and

21 the instantaneous volatility σµ (1%) of expected growth are set in the range of values considered in the literature. Preference parameters are the subjective timediscount rate ρ, the maximum individual relative risk aversion γ¯ and the habit persistence λ. The maximum individual relative risk aversion is set to γ¯ = 10, which is a conservative upper bound in line with Mehra and Prescott (1985). In turn, the steady state aggregate relative risk aversion is γ˜(ω¯ )= 5, in line with the estimate by Xiouros and Zapatero (2010) from individual consumption data. In the homogeneousagents economy, I set γ = γ˜(ω¯ )= 5 for the sake of comparison. Habit persistence is somewhat not observable and there is some disagreement in the literature about its value. Since agents have timeseparable utility, I set λ = .35 to capture enough persistence in the pricedividend ratio but without increasing too much the riskfree rate volatility.22 The timediscount rate is set to ρ = 3.5%, which is in the range of values considered in the literature, in order to match the unconditional riskfree rate, given the other parameters.

23 The formation of beliefs involves two parameters. I set the volatility of the noisy signal σs = 5%. Procyclical optimism depends on χ: I consider several speciﬁcations and verify how equilibrium outcomes evolve as a function of such a parameter. In particular, I focus on the Bayesian case (χ = 0%) and a target value χ = 7.5%, which captures many empirical patterns of asset returns. Table III summarizes the calibration of the model.

Insert Table III about here.

B. Preference Heterogeneity and Equilibrium Distributions

This section investigates the equilibrium distributions when preferences are heterogeneous. The speciﬁcation of agents types in Eq. (7) allows for analytical solutions. It is important to notice that such a speciﬁc pattern of agents’ types is economically irrelevant. Instead, the endogenous consumption and wealth distributions are economically important and meaningful. Indeed, these are equilibrium outcomes and can be compared to their counterpart in the real data.

Xiouros and Zapatero (2010) study individual consumption data to infer the crosssectional distribution of consumption shares as a function of risk tolerance and their results conﬁrm the empirical ﬁndings of Kimball, Sahm, and Shapiro

(2008). Namely, relative risk aversion has an average of about 5 and most of the probability mass is in the range (0, 10).

21These parameters are in line with most of longrun risk literature: see Bansal, Kiku, Shaliastovich, and Yaron (2014) and references therein. 22Accordingly, Lynch and Randall (2011) show that crosssectional returns are consistent with a low persistence of the habit statevariable when expected growth is meanreverting. 23 Although the choice of σs is arbitrary, it seems safe to assume some monotonicity in the outcomes of the model between the cases of inﬁnite noise (useless signal) and of a perfect signal (σs 0). See also Branger, Schlag, and Wu (2011). Notice that the analysis focuses instead on the marginal eect on the asset prices of the parameter→ χ.

24 Under the pattern of heterogeneity in Eq. (7), the equilibrium distributions of consumption and wealthshares are given by:

i j ωt ω¯ ω ω j W C ω e − ( t ¯ ) W ω i,t Fγ 1 ( j, t )= ∑ ωt ω¯ e− − and Fγ 1 ( j, t ,µˆt )= ∑ j = 1,2,... (55) − i=1 1+e − − i=1 Pt ∀

The upper left panel of Figure 4 shows the consumptionshare distribution at the steady state when γ¯ = 7.5,10, 15 { } together with the estimated distribution of Xiouros and Zapatero (2010). For γ¯ = 10, the equilibrium distribution closely resembles the empirical one.24 The upper right panel displays the wealthshare distribution, which is quite similar to the consumptionshare distribution but it is more sensitive to the state.

Insert Figure 4 about here.

The lower left panel of Figure 4 shows the density function of aggregate relative risk aversion γ˜(ωt ) under the investors probability measure. Under procyclical optimism (χ = 7.5%), agents overestimate the persistence of economic conditions and then a larger variance and a higher probability of extreme events than in the Bayesian case (χ = 0%) obtain. The lower right panel of Figure 4 exhibits the undiscounted log pricing kernel as a function of the habit state variable:

ωu ω¯ ωt ω¯ ρ(u t)+ logξt u = γ¯(ωt ωu)+ γ¯[log(1 + e − ) log(1 + e − )], (56) − , − −

and its linear counterpart γ(ω¯ )(ωt ωu) in the homogeneousagents economy. The degree of convexity that arises − under heterogeneity is an endogenous result of the redistribution of wealth among the agents. Consequently, the log state price density features timevarying volatility even if the state ω is homoscedastic. In turn, the price of risk and the aggregate relative risk aversion are endogenously timevarying and countercyclical as in Chan and Kogan (2002).

Nonetheless, Xiouros and Zapatero (2010) have shown that a realistic degree of heterogeneity is so small that asset prices barely dier from those of a homogeneousagents economy. However, under procyclical optimism (χ > 0), even a small and realistic amount of heterogeneity is sucient to produce interesting equilibrium outcomes, since the agents overestimate aggregate risk.

In summary, the assumed pattern of heterogeneity i) captures at equilibrium the empirical crosssectional distribution of consumption; ii) is parsimonious (it involves only one parameter) and allows for an upper bound to individual relative

24The model distribution diverges from that of Xiouros and Zapatero (2010) only for very low values of risk tolerance, that is for unrealistically high levels of risk aversion. The latter are ruled out in the model by the upper bound γ¯, whereas they are taken into account by the parametric distribution of Gamma type in Xiouros and Zapatero (2010).

25 risk aversion, γ¯ ; iii) leads to analytical solutions not only for the pricing kernel but also for prices, returns and portfolios

(under any speciﬁcation of exogenous uncertainty in the jumpdiusion ane class).

C. RiskReturn Trade O

The market riskreturn trade o is captured by the unconditional correlation between the conditional expected excess return and the conditional return volatility, that is the two components of the Sharpe ratio. The trade o, as it is computed by the “econometrician,” provides very puzzling evidence: a weak or negative relation obtains under broad settings of available information (see Section II.A). Moreover, the actual expectations of the agents (i.e. Duke/CFO survey data), document a weak unconditional trade o and a procyclical dynamics. When economic conditions deteriorate, the relation switches from strongly positive to strongly negative.

The model proposes the following explanation of the riskreturn trade o, its dynamics and its potentially negative sign. As long as agents feature procyclical optimism (i.e. χ > 0 in the main model or, similarly, a1 > 0,b1 < 0 in Section II.A), the two components of the Sharpe ratio do not move together in some states of the world. Namely, this happens in bad times, leading to a weak unconditional riskreturn relation. σP In order to see such a mechanism, consider the priced component of volatility y in Eq. (45). It has the following decomposition:

σP σ θ σ σ y = y + t + ( yFω + µ,yFµˆ )/F priced volatility fundamentals transient risk forwardlooking risk. (57)

∂ω=0, ∂µˆ =0 ∂ω<0, ∂µˆ =0 ∂ω>0, ∂µˆ >0 Namely, the transient risk is exactly the price of risk in the economy and the forwardlooking term is the semielasticity of discounted cashﬂows with respect to the states. While transient risk moves negatively with ωt , forwardlooking risk moves positively with both ωt and µˆt . Therefore, the cyclicality of priced volatility depends on which of these two terms dominates. Under reasonable parameters and Bayesian beliefs, priced volatility inherits the countercyclical behavior of transient risk, whereas forwardlooking risk is residual. Instead, for χ > 0, the excessive persistence perceived by the agents enhances the forwardlooking term in comparison with the Bayesian case. In particular, this eect is large in bad times because agents are risk averse and prices are more sensitive to the state. As a consequence, forwardlooking risk oset transient risk in bad times. Then, procyclical optimism breaks the cyclicality of volatility in bad times. This eect is consistent with the empirical evidence from survey data (see Table II and Figure 2).

26 Consider now the equity premium: it is simply the product of the price of risk and the priced volatility:

P θ σP µ r = t y − × (58) premium transient risk priced volatility.

∂ω<0, ∂µˆ =0 The above osetting mechanism, due to excessive persistence, is less pronounced for the equity premium than for the priced volatility. Indeed, such an eect is mitigated by the transient risk: the latter is not directly aected by µˆt and moves negatively with ωt . Consistently with survey data, the equity premium always moves negatively with ωt . Consider now the riskreturn relation. In good times, both volatility and premium are countercyclical and move together. This leads to a positive conditional trade o. However, in bad times, the premium is still countercyclical, whereas the cyclicality of volatility depends on the strength of procyclical optimism. For χ > 0 large enough, volatility can be weakly or even positively correlated with ωt . Consequently, procyclical optimism can lead to a negative conditional tradeo in bad times.25 In turn, the riskreturn relation has a procyclical dynamics and its unconditional level can be weak or even negative. Such a dynamics is consistent with survey data.

In summary, the endogenous osetting mechanism among the components of the return volatility leads to the following dynamics for the two components of the Sharpe ratio and for the equilibrium riskreturn trade o:

corr(ω, µP r low ω) corr(ω, µP r) corr(ω, µP r high ω), − | ≈ − ≈ − | corr(ω, σP low ω) > corr(ω, σP) > corr(ω, σP high ω), (59) y | y y | corr(µP r, σP low ω) < corr(µP r, σP) < corr(µP r, σP high ω). − y | − y − y |

These inequalities supply, as an equilibrium outcome, a stylized representation of the three scatter plots in Figure 2.

Table VI provides further support. We can observe that, for both the survey data and the model, the trade o is positive in good times (ωt > ω¯ ) and becomes strongly negative when economic conditions deteriorate (ωt < ω¯ ). This obtains because return volatility is not countercyclical in bad times. Therefore, the equilibrium trade o behaves in a similar fashion with what we observe from survey data.

Insert Table VI about here.

Under heterogeneous risk attitudes, procyclical optimism can generate a large range of values for the unconditional correlation between the mean and the volatility of returns. In the Bayesian case (χ = 0) such a correlation is positive

25 σP Unpriced volatility s can further reduce the riskreturn trade o, however at the market level its role it is supposed to be quite limited.

27 and almost perfect. As long as the agents feature procyclical optimism (χ > 0), the correlation at ﬁrst decreases below zero and then increases again up to one. The decrease is due to the osetting mechanism among the components of the return volatility, as commented above. For the target range of χ (.07,.08), which allows to capture many features ∈ of asset returns, the model leads to an unconditional correlation from close to zero to strongly negative, which is in line with the empirical evidence (Lettau and Ludvigson (2010)), but at odds with most of equilibrium asset pricing models.26

Figure 5 reports the unconditional correlation between the expected excess returns and the conditional return volatility as a function of χ in both the homogeneous and the heterogeneousagents economy.

Insert Figure 5 about here.

Providing a theoretical link among survey expectations of fundamentals and those of returns, the model oers a potential general equilibrium solution to the puzzling evidence about a key quantity in asset pricing, that is the relation among perceived risk and its expected remuneration.

D. Unconditional Moments of Returns

Statistics of the model, as a function of the parameter χ, are compared with the data for the cases of the homogeneous and heterogeneousagents economy respectively in Table IV and V.

Insert Table IV and V about here.

In all the considered cases, the average riskfree rate is quite low (about 1.8%) in line with the data (1.7%). Heterogeneity in risk aversion reduces the unconditional level of the riskfree rate because of the statedependent behavior of the precautionary savings term. However, such an eect is quantitatively quite limited for a realistic degree of heterogeneity.

Instead, the average riskfree rate is not sensitive to the value of the parameter χ, since Ψ(ωt ) does not aect the perceived unconditional mean of both µˆ and ω.

In the Bayesian case (χ = 0), the volatility of the riskfree rate (about 6.4%) is signiﬁcantly lower than that of the stock returns, but it is somewhat higher than in real data (4.4%). Heterogeneity does not produce substantial dierences with respect to the homogeneousagents economy. Instead, procyclical optimism (χ > 0) helps to explain the smooth dynamics of the riskfree rate: the larger χ, the lower the riskfree rate volatility. Such an eect can be understood from

Eq. (29) or (40). The larger the deviation parameter, the stronger the correlation between the states µˆt and ωt and, in

26For unrealistically high values of the deviation parameter χ, forwardlooking risk becomes almost insensitive to the states. Therefore, the whole variation in volatility is due to the price of risk. In turn, the volatility and the price of risk move together and the equity premium moves with the same sign. As a result, the unconditional tradeo increases up to one for very large χ values.

28 turn, the less volatile the dierence ωˆ t ωt (recall ωˆ t is increasing with µˆt ). Consequently the riskfree rate volatility − reduces too. For χ = 7.5%, the volatility of the riskfree rate (about 4.6%) matches the historical data. The unconditional equity premium is unaected by the choice of homogeneous or heterogeneous risk attitudes. In the Bayesian case (χ = 0), the average expected excess return is quite low (about 3.8%): the historical equity premium is almost twice this value (about 7.4%). Instead, procyclical optimism (χ > 0) increases the unconditional equity premium. Waves of optimism and pessimism alternate over time and induce a riskier perception about the economy than in the

Bayesian case. In turn, they generate a higher equity premium, all other things being equal. Notice that such an increase in the premium is not the mechanical result of a longrun bias in the estimation of the expected growth. For χ = 7.5%, a substantial increase in the equity premium (about 4.8%) can be observed, but it remains still below the historical value. A further increase in the parameter χ allows to ﬁt the real data but at the cost of too much stock return volatility.

Notice that including the recent ﬁnancial crisis the historical equity premium and price of risk reduce respectively to

5.6% and 28.3%, which are signiﬁcantly closer to the model predictions for χ = 7.5%.27 Because of the timeseparable utility and the realistic aggregate relative risk aversion (5 in average), the unconditional price of risk (about 18%) is lower than the historical Sharpe ratio (41% or 28.3%). Furthermore, neither preference heterogeneity nor procyclical optimism (χ > 0) alter the average price of risk.28 A relatively low price of risk implies that a high unconditional equity premium should be accompanied by a high unconditional return volatility. This is forthright in the homogeneousagents economy since the price of risk is constant and the equity premium is just a scaled version of the priced volatility. Under preference heterogeneity, the price of risk decreases with ωt as in Eq. (41): while the conditional moments of returns are signiﬁcantly aected (see Section VI.E), the unconditional premium and volatility are close to their counterparts in the homogeneousagents economy. Therefore, the model can achieve a high equity premium only at a cost of a high stock return volatility. However, for χ = 7.5%, a substantial equity premium is accompanied by a return volatility (about 26%) which is somewhat higher than in real data (18%) but quite realistic.

The equilibrium pricedividend ratio shows some interesting implications. First, procyclical optimism (χ > 0) produces additional volatility with respect the Bayesian case (χ = 0), almost in line with the real data. Second, pro cyclical optimism allows to disentangle the persistence of the pricedividend ratio from that of the riskfree rate, a feature of the data which is dicult to obtain in most of asset pricing models. The ﬁrst order autocorrelation of the pricedividend ratio and the riskfree rate are equal in the Bayesian case (about 71%). Instead, the larger the parameter

27A better quantitative description of the equity premium can obtain by relaxing some of the simple assumptions of the model (made for the purpose of exposition). Section VII highlights some model extensions. 28 θ σ γ x ω¯ 1 The unconditional price of risk under preference heterogeneity is given by E( ) = y¯ R2 (1 + e − )− dGω,µˆ (x,z), where Gω,µˆ denotes the joint distribution of ω and µˆ. As long as Ψ(ωt ) does not alter the unconditional mean or theR symmetry of the distribution of ω, the unconditional x ω¯ 1 mean of θ is only barely aected. Indeed, (1 + e − )− is almost linear in the range of values where Gω,µˆ puts most of the probability mass.

29 χ, the stronger the autocorrelation of the pricedividend ratio and the weaker that of the riskfree rate. For χ = 7.5%, a signiﬁcant divergence obtains (about 74% and 54%) and goes in the direction of the real data (91% and 33%). The autocorrelation of the pricedividend ratio is somewhat below its historical counterpart since λ is set to capture also the unconditional stock volatility and its cyclical behavior (see Section VI.E).

The bond premium and volatility decrease with χ since the bond oers an hedge opportunity beyond the stock and its cashﬂows do not depend directly on expected growth, µˆt .

Insert Figure 6 about here.

Even in a simple and parsimonious form, procyclical optimism (χ > 0) helps to explain some of the stylized features of the markets and improves over the benchmark (χ = 0). Figure 6 shows a summary of the analysis: asset pricing quantities are plotted as a function of the parameter χ under heterogeneous risk attitudes (similar results obtain for the homogeneous case).

E. Conditional Moments of Returns

The conditional pricedividend ratio, stock and bond volatility and premia as functions of ωt and µˆt are plotted in Figure 7 for the case of heterogeneous preferences. I consider both the Bayesian case (χ = 0) and procyclical optimism for the target value of χ = 7.5%.

Insert Figure 7 about here.

The ﬁrst line of panels shows that the log pricedividend ratio increases with ωt , capturing the procyclical behavior of prices. Two additional eects obtain. First, heterogeneity in risk attitudes makes the pricedividend ratio a little steeper

(ﬂatter) in bad (good) times than in the homogeneous case. Second, for χ > 0, the pricedividend increases more in good times and decreases more in bad times, as a result of the waves of optimism and pessimism.

The second line of panels shows the conditional stock return volatility. The model captures the excess of volatility over fundamentals. The speciﬁcation of risk attitudes signiﬁcantly aects the dynamics of return volatility. In the homogeneousagents economy, volatility is procyclical, i.e. increasing with ωt . Indeed, both the terms in Eq. (33) ω σP increase with t . Under preference heterogeneity, this is not necessarily true: indeed the second term of y in Eq. (45) ω λ κ σP is the price of risk which is decreasing with t . Consequently, if and are large enough, the residual terms of y σP and s –which are forwardlooking terms– do not oset the countercyclical behavior of the price of risk. Then, under heterogeneous preferences return volatility increases in bad states and decreases in good times.

30 Also procyclical optimism aects the dynamics of return volatility. The eect is quite limited in the homogeneous agents economy, but an important result obtains under preference heterogeneity (see also Section VI.C). Indeed, for χ > 0, the states ωt and µˆt feature a larger unconditional variance and correlation. Therefore, the forwardlooking components σP σP of volatility are larger in magnitude than in the Bayesian case. In particular, y becomes less countercyclical and s becomes more markedly procyclical. Furthermore, such an eect is asymmetric: an increase in χ produces a larger σP σP ω χ eect on volatility in bad times than in good ones. Figure 8 shows y and s as a function of t for = 7.5%.

Insert Figure 8 about here.

The third line of panels in Figure 7 reports the conditional equity premium. Since in the homogeneousagents economy the price of risk is constant, the equity premium inherits the procyclical behavior of priced volatility. This is not the case under heterogeneous risk attitudes. The equity premium is strongly countercyclical, in line with the empirical evidence. χ σP For > 0, the equity premium inherits the above osetting mechanism between the components of y , but such an eect is mitigated by the countercyclical price of risk. As a result, for χ > 0, the equity premium is more countercyclical than conditional volatility, in line with the real data (see Lettau and Ludvigson (2010)). Such a result is at the heart of the market riskreturn trade o.

The last two lines of panels in Figure 7 show the conditional volatility and premium of the perpetual bond. Similarly to the stock, both the ﬁrst two moments of bond returns are decreasing with ωt under preference heterogeneity. However, the eect of χ is weaker than that on the stock. Indeed, bond returns reﬂect the excess of persistence in ωt but do not directly account for the perceived dynamics of µˆt . In turn, the bond premium and volatility are positively correlated.

F. Predictability and Other Model Predictions

This section investigates the predictability of the excess returns. For each model, 1000 simulations are run, with each simulation accounting for 100 observations. For each simulation, I run regressions of cumulative excess market returns from one to seven years on the current pricedividend ratio and on the real 10 years term spread. Tables VII and VIII show the results respectively for the homogeneous and the heterogeneousagents economies.

Insert Table VII and VIII about here.

In the Bayesian case (χ = 0), none of the model coecients is statistically signiﬁcant. These results are in line with Xiouros and Zapatero (2010): predictability is quite modest since a realistic degree of preference heterogeneity is too small to produce enough variation in the equity premium. Procyclical optimism (χ > 0) leads to some interesting results: the negative coecients increase in size with the horizon as well as their tstatistics and the explanatory power.

31 For χ = 7.5%, the regression coecients are highly signiﬁcant at all the horizons, in line with the empirical evidence. Intuitively, investors overestimate the persistence of aggregate risk. In turn, the pricedividend ratio is more persistent and captures the additional variation in the excess return. Figure 9 shows the tstatistics and the R2 from the regressions as functions of χ under heterogeneous risk aversion.

Insert Figure 9 and 10 about here.

Furthermore, the predictability of stock returns does not come with the counterfactual predictability of dividend growth by valuation ratios. The latter obtains for instance in longrun risk models, as documented by Beeler and Campbell

(2012). Figure 10 reports a little explanatory power for any value of the deviation parameter χ and regression coecients that are not statistically signiﬁcant. Such a result is due to the incomplete information and procyclical optimism.

Although persistent, true dividend growth is not predictable by equilibrium prices.

The term premium –i.e. the spread between the 10 year interest rate and the riskfree rate– predicts future cumulative excess returns. For χ = 7.5%, the regression coecients are highly signiﬁcant at all the horizons but the ﬁrst one. The explanatory power obtains because the term premium captures the lack of meanreversion in ωt perceived by the agents.

The real yield is an ane in ωt and µˆt in the homogeneousagents economy, while it is a nonlinear function of the states under heterogeneous risk preferences. A similar result obtains in Buraschi and Jiltsov (2007) because of the nonlinear speciﬁcation of their habit state variable. Here, nonlinearity is due to the endogenous redistribution of wealth among the agents.

Insert Figure 11 about here.

The left panel of Figure 11 shows that the real term structure decreases with ωt in the heterogeneousagents economy.

This happens because investors’ marginal utility is high (low) when ωt is low (high) implying a higher (lower) demand for consumption. Moreover, the yield curve is upward sloping across dierent maturities in good states, while it is downward sloping in bad states. Intuitively, in bad states ωt is expected to move toward its longrun mean leading to lower risk for the investors at longer maturities, while the opposite holds in good times. Similar results arise in the homogeneousagents economy. The middle and right panels of Figure 11 report the unconditional mean and volatility of the real yield as a function of maturity. Their term structures are respectively increasing and decreasing. Therefore, the model can lead to a positive slope of the nominal curve (as in actual data), which is not due to an inﬂation risk premium only. When χ > 0, the curve is slightly lower than in the Bayesian case since the perceived aggregate risk is more volatile and, then, the bond is a better hedge against consumption risk. Nonetheless, for large maturities the curves converge since the agents have unbiased beliefs in the longrun.

32 Although timeseparable utility, the model accounts for a number of features of both interest rates and stock returns.

Therefore, this paper complements the literature which considers recursive utility, such as Bansal and Shaliastovich (2013).

Under preference heterogeneity, the model has predictions about crosssectional portfolios and the resulting trading volume dynamics. Figure 12 reports the wealth proportions invested in the stock and in the bond by the agents types with relative risk aversion between γ¯ = 10 and one. For reasonable parameters these agents own almost all wealth in the economy. Allocations are plotted as a function of ωt . Investments in stock and bond are respectively countercyclical and procyclical for the low risk averse agent. The opposite holds for the high risk averse agent. In bad times (low ω), wealth shifts towards the high risk averse agents, who are long in the bond and allow the low risk averse agents to fund their stock investment beyond their wealth. In good times (high ω), all agents tend to invest their entire wealth in the stock since the price of risk and, in turn, prices are little sensitive to the state.

Insert Figure 12 about here.

Under procyclical optimism (χ > 0), portfolio allocations become more extreme, that is they are more dispersed in bad times and more aligned in good times. Indeed, allocations reflect the excessive persistence of the habit state variable perceived by the investors. Furthermore, allocations endogenously lead to countercyclical trading volume in line with the empirical evidence. Notice that the model is defined in continuous time: then, continuous trading and Brownian innovations generate infinite trading volume in any finite time interval. However, since the economy has a stationary equilibrium, the state dependent dynamics of trading can be studied by assuming discrete trading at a fixed time interval. ρP ω πP Namely, portfolio holdings i ( t ,µˆt )= i,tWi,t /Pt are functions of the states only and turnover is given by

n ω ω 1 ρP ω ρP ω T( ′,µˆ′ , ′′,µˆ′′ )= 2 ∑ i ( ′′,µˆ′′) i ( ′,µˆ′) , { } { } i=1 | − | for each pair of states: ω ,µˆ and ω ,µˆ . The sum is truncated to the ﬁrst n agents such that the remainder owns { ′ ′} { ′′ ′′} a negligible share of total wealth.

VII. Extensions

Relaxing some assumptions –made for the purpose of exposition– could help to improve the quantitative implications of the model. Two approaches are: a more general source of uncertainty and a richer construction of beliefs.

33 Exogenous Uncertainty: Consumption dynamics can be generalized by accounting for stochastic volatility in spirit of

Campbell and Hentschel (1992), whereas the dividend process, representative of the stock market, can be a distinct (in spirit of Campbell (1996)) but cointegrated process with consumption:

dYt = µtYt dt + √νtYt dBy,t ,

dµt = κµ(µ¯ µt )dt + φµ√νt dBµ t , − ,

dνt = κν(ν¯ νt )dt + φν√νt dBν t , − ,

and log Dt = logYt zt with dzt = κz(z¯ zt )dt + φz√zt dBz and z¯ > 0. This setting of uncertainty is similar to the − − t longrun risk literature and can be further generalized to account for disasters (Wachter (2013)). As long as exogenous uncertainty belongs to the jumpdiusion ane class, all closed forms are preserved. A more realistic exogenous uncertainty can help to match the equity premium, to increase the variation of the price of risk and to capture the downwardsloping termstructure of dividend volatility.

Beliefs formation: The paper analyses the case of symmetric waves of optimism and pessimism. A simple way to include a longrun bias, which preserves closed form solutions, is a modiﬁed deviation term: Ψ(ωt )= χ(ωt ω¯ + ε), − 29 where ε captures the bias in the longrun forecast errors. Such a bias aects the reversion threshold ωˆ t . For ε > 0, good states are perceived as more persistent than bad states. Instead, for ε < 0, waves of pessimism are more persistent than waves of optimism. Figure 13 shows the unconditional equity premium as a function of the deviation parameter

χ and the bias parameter ε. For ε > 0, we observe a negative longrun bias in forecast errors about growth and an increase in the equity premium.

Insert Figure 13 about here.

An interesting extension concerns expected growth volatility. Assuming that the volatility of µˆt , is a function of the habit state variable can be interpreted as timevarying investors’ attention and/or timevarying quality of available information.

Predictability of forecast errors dispersion could provide empirical support (see Patton and Timmermann (2010)). On a technical side, a characterization, which preserves analytical solutions, can be dicult and is left to future research.

29The model solution is unchanged: an additional term equal to χεC appears on the right hand side of the partial dierential equation (C4) in the Appendix C and aects the solution of the deterministic coecient A, while B and C remain unchanged.

34 VIII. Conclusion

This paper proposes a closed form general equilibrium model under incomplete information. Investors feature heteroge neous risk preferences and procyclical optimism, as suggested by the survey data about fundamentals. In turn, investors overestimate the persistence of the economic environment. This provides a rationale for the excess variability of the equilibrium pricing kernel and, hence, for the disconnect between the smooth dynamics of fundamentals and the large

ﬂuctuations of asset prices.

The analysis, based on a single parameter governing procyclical optimism, shows that the model explains:

i the weak or negative empirical relation between market risk and its remuneration;

At the same time, many other asset pricing quantities move in line with the data providing further robustness:

ii the equity premium increases, whereas the riskfree rate stays low;

iii the riskfree rate (and bond) volatility decreases, whereas stock volatility increases;

iv the persistence of the pricedividend ratio and of the riskfree rate respectively increases and decreases;

v the predictability of excess returns by dividend yields increases, predictability of dividend growth does not obtain;

vi the term structure of real yields is upward sloping and yields volatilities decrease with the maturity;

vii the price of risk and the expected returns are more countercyclical than return volatility in bad times;

viii portfolio strategies are more sensitive and dispersed in bad times, leading to countercyclical trading volume.

Finally, the assumptions of the model are consistent with the data:

ix preference heterogeneity provides a microfoundation to the heteroscedasticity of the kernel and imposes discipline

on the model calibration by matching the empirical crosssectional distribution of consumption;

x procyclical optimism about economic growth provides a stylized representation of survey data from professional

forecasters.

These ten empirical patterns support the main model mechanism which generates in equilibrium a connection between survey data about fundamentals and those about market returns.

The analysis of this paper can be extended in a number of directions. The model can easily account for more general speciﬁcations of both exogenous uncertainty and beliefs formation schemes, without loosing closed form solutions. An interesting extension of the model –that I leave to future research– is to make the formation of beliefs to depend on both exogenous and endogenous quantities in order to study in general equilibrium, as an additional feature, the informational and selffulﬁlling role of prices, in spirit of Bacchetta, Tille, and van Wincoop (2012).

35 Appendix A – Filtering and state variables

In this appendix I consider the ﬁltering problem of the incomplete information setting of Section III. The agents observe the dividend and the signal, whose dynamics is deﬁned in Eq. (8) and (12), and estimate the unobservable drift in Eq. (9). The dynamics for the estimated drift in the Bayesian case follows from an application of the KalmanBucy theorem:

σ2 1 σ2 1 y 0 − dY /Y y 0 − 1 1 t t κ κ 1 1 1 dµˆt = vt [ ] σ2 ds + µdt¯ + vt [ ] σ2 1 µˆt dt (A1) 0 s t − − 0 s µt dt + σydBy t µt dt + σsdBs t = κ(µ¯ µˆ )dt + v , + , µˆ (σ 2 + σ 2)dt , (A2) t t σ2 σ2 t y− s− − y s − where vt satisﬁes the following Riccati equation:

2 2 2 2 v˙t = 2κvt v (σ− + σ− )+ σ , (A3) − − t y s µ v0 = σµ,0. (A4)

At the steady state vt is as in Eq. (16) and therefore the estimated drift has dynamics

v∞ v∞ dµˆt = κ(µ¯ µˆt )dt + σ dBˆy,t + σ dBˆs,t . (A5) − y s which leads to Eq. (14). If the agents interpret the signal as dst = (µt + Ψ′(ωt ))dt + σsdBs,t (see Section III.C), therefore Eq. (A5) becomes κ v∞ ˆ v∞ ˆ v∞ Ψ ω dµˆt = (µ¯ µˆt)dt + σ dBy,t + σ dBs,t + σ2 ′( t )dt. (A6) − y s s Ψ ω χ ω ω χ v∞ χ where v∞ is unchanged, ′( t )= ′( t ¯ ) and the dynamics in Eq. (17) is obtained for = σ2 ′. − s Both µˆt and ωt are processes in the ane class under the investors subjective probability measure in both the Bayesian and nonBayesian cases of Section III.C. Therefore, they have conditional joint Laplace transform of exponential ane form (a special case of Eq. (C2)): aωu+bµˆu A(u t)+B(u t)ωt+C(u t)µˆt Mt,u(a,b)= Eˆ t [e ]= e − − − (A7) for some deterministic functions of time A, B and C. Conditional central (co)moments can be computed dierentiating at zero the logarithm of the Laplace transform. In both the Bayesian and nonBayesian cases, it can be veriﬁed that µˆt and ωt have unconditional expectation equal respectively to µ¯ and (µ¯ σ2/2)/λ. In the nonBayesian case, covariance stationarity requires the − y additional condition: λ + κ > (λ κ)2 + 4χ. − Appendix B – Equilibrium Solution: Proofs

Proof of Lemma 1: Pareto optimality requires that optimal consumption solves the problem (22). Taking the ﬁrst order condition γ γ ν ρt i i ζξ ζ we get ie− Ci−,t Xt = 0,t , where is the Lagrange multplier of the resource constraint. Hence optimal consumption is equal to

1 γ ρ γ γ 1 γ γ γ γ ν / i t/ i ζ 1/ i ξ / i ν ν 1/ i 1/ i Ci,t = i e− − 0−,t Xt = ( i/ 1) (C1,t /Xt ) Xt . (B1)

Market clearing implies that γ γ γ 1 ν ν 1/ i 1/ i Yt /Xt = Xt− ∑i Ci,t = ∑i( i/ 1) (C1,t /Xt ) . (B2) Under the assumption about preference heterogeneity of section III.A, we get

(1 i)ω¯ i i ω¯ ω¯ Yt /Xt = ∑ e − C X − = e C1,t /(e Xt C1,t ). (B3) i 1,t t −

36 ωt ωt ω¯ Then C1,t = e Xt /(1 + e − ) and the optimal consumption Ci,t in Eq. (24) automatically follows. q.e.d.

Proof of Corollary 1: Armed of optimal consumption of agent i as in Eq. (24), the representative agent’s utility evaluated at the optimal consumption is

νi γ γ γ γ γ RA ν γ ν ν 1/ i 1/ i 1 i i U (Yt ,Xt )= ∑i iU(Ci,t ,Xt ; i)= ∑i (( i/ 1) (C1,t /Xt ) Xt ) − Xt (B4) 1 γi − Then dierentiating with respect to the aggregate consumption we get

γ γ γ γ γ γ γ γ ∂C /X U RA(Y ,X )= ∑ ν ((ν /ν )1/ i (C /X ) 1/ i X ) i X i (ν /ν )1/ i X (γ /γ )(C /X ) 1/ i 1 1,t t (B5) Y t t i i i 1 1,t t t − t i 1 t 1 i 1,t t − ∂Yt

Under the assumption about preference heterogeneity of section IIIA, marginal utility reduces to

2ω¯ 2 RA ν e Xt γ¯ ω¯ i 1 ν γ¯ U (Yt ,Xt )= 1 ω¯ 2 (C1,t /Xt )− ∑ i(e− C1,t /Xt ) − = 1(C1,t /Xt )− . (B6) Y (e Xt +Yt ) i

ξ RA The equilibrium state price density t,u in Eq.(25) automatically follows. Dierentiating again with respect to Yt leads to UYY (Yt ,Xt )= γ¯ 1 2ω¯ ω¯ 2 γν¯ 1(C1,t /Xt ) e Xt (e Xt +Yt ) . Therefore, the ArrowPratt coecient, γ˜(ωt ), is given by − − − − RA γ UYY (Yt ,Xt ) ¯ γ˜(ωt )= Yt = (B7) RA ωt ω¯ − UY (Yt ,Xt ) 1 + e −

γ ω ∑ γ 1 1 1 as in Eq. (26). It can be easily veriﬁed that the aggregate relative risk aversion is also equal to ˜( t )= i i− Ci,tYt− − . For further details about the optimal consumption and the preferences of the representative agent see Curatola and Marfè (2011). In the case of homogeneousagents, since consuming the aggregate dividend must be optimal for the representative agent, the marginal rate of substitution identiﬁes the equilibrium state price density which must consequently be of the form in Eq. (27). q.e.d.

Appendix C – Equilibrium Prices: Proofs

Consider the following discounted conditional Laplace transform:

ρ(u t) c1 logYu+c2(ωu+c0)+c3µˆu Mt,u(c¯)= e− − Eˆ t [e ] (C1) where c¯ = (c0,c1,c2,c3) is a coecient vector such that the expectation exists, and guess an exponential ane solution of the kind: c1 logYt +A(u t,c¯)+B(u t,c¯)ωt +C(u t,c¯)µˆt Mt,u(c¯)= e − − − . (C2) FeynmanKac gives that M has to meet the following partial dierential equation

1 2 2 1 2 2 Mt + MY µˆtYt + MYY σ Y + Mωλ(ωˆ t ωt )+ Mω,ω σ + MYωσ Yt + Mµˆ (κ(µ¯ µˆt )+ Ψ(ωt)) 2 y t − 2 y y − 1 σ2 σ2 σ2 σ σ σ σ ρ +Mµˆµˆ 2 ( µ,y + µ,µ + µ,s)+ MYµˆ y µ,yYt + Mωµˆ y µ,y = M (C3) where the arguments have been omitted for ease of notation. Plugging the resulting partial derivatives from the guess solution into the pde and simplifying gives

1 2 2 1 2 2 At + Bt ωt +Ct µˆt + c1µˆt + c1(c1 1)σ + B(µˆt λωt)+ B(c1 1/2)σ + σ B 2 − y − − y 2 y 1 2 1 2 2 2 2 +(κ(µ¯ µˆt)+ χ(ωt µ¯/λ)+ c1σyσµ,y + χσ )C + (σ + σ + σ )C + σyσµ,yBC = ρ (C4) − − 2λ y 2 µ,y µ,µ µ,s

37 This equation has to hold for all ωt and µˆt . We thus get three ordinary dierential equations for A, B and C:

1 2 1 2 2 1 2 At = ρ c1(c1 1)σ σ B C(κµ¯ χµ¯/λ + c1σyσµ,y χσ ) − 2 − y − 2 y − − − 2λ y 1 2 2 2 2 2 (σ + σ + σ )C B(c1 1/2)σ σyσµ,yBC (C5) − 2 µ,y µ,µ µ,s − − y − Bt = λB χC (C6) − Ct = κC B c1 (C7) − − with initial conditions A(0,c¯)= c2c0, B(0,c¯)= c2 and C(0,c¯)= c3. The solution is

(κ+λ+z)τ/2 τ κ λ τ τ 1 τ e− χ z ( + +z) /2 z κ λ κλ χ B( ,c¯)= 2(κλ χ)z c1z(1 + e 2e ) + (1 e )(c1( + ) 2c3( )) − − − − − − 1 1 zτ zτ + c2(κλ χ)((λ κ)(1 e )+ z(1 + e )) , (C8) 1 − − −

(κ+λ+z)τ/2 τ τ τ 1 τ e− z κ λ λ χ λ z z /2 C( ,c¯)= 2(κλ χ)z c1 (e 1)(( ) 2 ) z(1 + e 2e ) − − − − − − 1 1 zτ zτ zτ + (κλ χ)(2c2(e 1) c3((λ κ)(1 e ) z(1 + e ))) , (C9) 1 − − − − − − where z = (λ κ)2 + 4χ and A(τ,c¯) can be computed in closed form but it is too long to be reported. In the limit case χ 0 − τ τ → (i.e. under Bayesian beliefs), B( ,c¯) and C( ,c¯) reduce to: λτ B(τ,c¯)= c2e− , (C10) λτ κτ κτ c2κe + c1(κ λ)(1 e )+ κ(c3(κ λ) c2)e C(τ,c¯)= − − − − − − − . (C11) κ(κ λ) −

Proof of Corollary 2: Applying Itô’s lemma to the state price density ξ0,t as in Eq. (27) and using the dynamics in Eq. (15) and dξ0,t = ξ0,t rt dt ξ0,t θt dBˆy,t gives the expressions for rt and θt : − − ∂ξ ∂ ∂ξ ∂ω σ2 ∂2ξ ∂ω2 ∂ξ ∂ω 0,t / t λ ω ω 0,t / t y 0,t / t θ σ 0,t / t rt = ξ ( ˆ t t ) ξ ξ and t = y ξ (C12) − 0,t − − 0,t − 2 0,t − 0,t which lead to the Eq. (29) and (30). q.e.d.

Proof of Proposition 1: To compute the pricedividend ratio, recall that

∞ ∞ γωt ρ(u t) logYu γωu Pt = Eˆ t ξt,uYudu = e e− − Eˆ t e − du, (C13) Z0 Z0 where the latter expectation is a conditional Laplace transform of the type in Eq. (C1). It has solution as in Eq. (C2) where the system (C5)(C6)(C7) is solved for c¯ = (0,1, γ,0). Therefore, the pricedividend ratio, Pt /Yt , is a function of the habit state variable − and the perceived expected growth only as in Eq. (31). The dynamics of the stock price is obtained by applying Itô’s Lemma:

∂ ∂ ∂ dP = [.]dt + Pt σ Y dBˆ + Pt σ dBˆ + Pt σ dBˆ + σ dBˆ + σ dBˆ (C14) t ∂Yt y t y,t ∂ωt y y,t ∂µˆt µ,y y,t µ,µ µ,t µ,s s,t and therefore the stock return volatility is given by

2 2 2 P 1 ∂Pt ∂Pt ∂Pt ∂Pt ∂Pt σ = P− σ Y + σ + σ + σ + σ (C15) t t ∂Yt y t ∂ωt y ∂µˆt µ,y ∂µˆt µ,µ ∂µˆt µ,s

38 which leads to Eq. (32), where σµ,µ is equal to zero since the expected growth is not observable. The premium is given by P 1 dP dξ µ rt = , . q.e.d. t − − dt P ξ t Proof of Proposition 2: To compute the real yield, recall that

ρτ+γωt γωt+τ Eˆ t [ξt,t+τ]= e− Eˆ t e− , (C16) where the latter expectation is a conditional Laplace transform of the type in Eq. (C1). It has solution as in Eq. (C2) where the system (C5)(C6)(C7) is solved for c¯ = (0,0, γ,0). Therefore, the real yield, ε(t,τ), is a function of the maturity, the habit state − ∞ Eˆ ξ variable and the perceived expected growth only as in Eq. (39). The perpetual bond price is given by Qt = t t [ t,u]du. The dynamics of the perpetual bond price is obtained by applying Itô’s Lemma: R

∂ ∂ dQ = [.]dt + Qt σ dBˆ + Qt σ dBˆ + σ dBˆ + σ dBˆ (C17) t ∂ωt y y,t ∂µˆt µ,y y,t µ,µ µ,t µ,s s,t and therefore the bond return volatility is given by

∂ ∂ 2 ∂ 2 ∂ 2 σQ = Q 1 Qt σ + Qt σ + Qt σ + Qt σ (C18) t t− ∂ωt y ∂µˆt µ,y ∂µˆt µ,µ ∂µˆt µ,s which leads to Eq. (36), where σµ,µ is equal to zero since the expected growth is not observable. The premium is given by Q 1 dQ dξ µ rt = , . q.e.d. t − − dt Q ξ t Proof of Corollary 3: Applying Itô’s lemma to the state price density ξ0,t as in Eq. (25) and using the dynamics in Eq. (15) and dξ0,t = ξ0,t rt dt ξ0,t θt dBˆy,t gives the expressions for rt and θt : − − ∂ξ ∂ ∂ξ ∂ω σ2 ∂2ξ ∂ω2 ∂ξ ∂ω 0,t / t λ ω ω 0,t / t y 0,t / t θ σ 0,t / t rt = ξ ( ˆ t t ) ξ ξ and t = y ξ (C19) − 0,t − − 0,t − 2 0,t − 0,t which lead to the Eq. (40) and (41). q.e.d.

Proof of Proposition 3: To compute the pricedividend ratio, recall that

∞ γ¯ ∞ γ¯ ωt ω¯ ρ ωu ω¯ Eˆ ξ e − (u t)Eˆ 1+e − Pt = t t,uYudu = 1 eωt ω¯ e− − t eωu ω¯ Yu du. (C20) Z0 + − Z0 − ωu ω¯ γ¯ In order to evaluate the latter expectation, since γ¯ is a positive integer, it is possible to apply the Binomial formula (1 + e − ) = γ¯ γ ω ω ∑ ¯ j( u ¯ ) j=0 j e − and therefore to obtain:

γ¯ ρ ωu ω¯ ρ γ¯ γ γ ω ω (u t)Eˆ 1+e − (u t) ¯ Eˆ logYu+( j ¯)( u ¯ ) e− − t ωu ω¯ Yu = e− − ∑ t e − − . (C21) e − j=0 j The latter expectation is a conditional Laplace transform of the type in Eq. (C1). It has solution as in Eq. (C2) where the system (C5)(C6)(C7) is solved for c¯ = ( ω¯ ,1, j γ¯,0) j = 0,...,γ¯. Therefore, the pricedividend ratio, Pt /Yt , is a function of the habit − − ∀ state variable and the perceived expected growth only as in Eq. (43). The dynamics of the stock price is obtained by applying Itô’s Lemma: ∂ ∂ ∂ dP = [.]dt + Pt σ Y dBˆ + Pt σ dBˆ + Pt σ dBˆ + σ dBˆ + σ dBˆ (C22) t ∂Yt y t y,t ∂ωt y y,t ∂µˆt µ,y y,t µ,µ µ,t µ,s s,t and therefore the stock return volatility is given by

2 2 2 P 1 ∂Pt ∂Pt ∂Pt ∂Pt ∂Pt σ = P− σ Y + σ + σ + σ + σ (C23) t t ∂Yt y t ∂ωt y ∂µˆt µ,y ∂µˆt µ,µ ∂µˆt µ,s

39 which leads to Eq. (44), where σµ,µ is equal to zero since the expected growth is not observable. The premium is given by P 1 dP dξ µ rt = , . q.e.d. t − − dt P ξ t Proof of Proposition 4: To compute the real yield, recall that

γ¯ γ¯ ρτ ωt ω¯ ωt+τ ω¯ Eˆ ξ τ e − Eˆ 1+e − t [ t,t+ ]= e− ωt ω¯ t ωt τ ω¯ . (C24) 1+e − e + − ωt+τ ω¯ γ¯ In order to evaluate the latter expectation, since γ¯ is a positive integer, it is possible to apply the Binomial formula (1+e − ) = γ¯ γ ω ω ∑ ¯ j( t+τ ¯ ) j=0 j e − and therefore to obtain:

γ¯ ρ ωt τ ω¯ ρ γ¯ γ γ ω ω (u t)Eˆ 1+e + − (u t) ¯ Eˆ ( j ¯)( t+τ ¯ ) e− − t ωt τ ω¯ = e− − ∑ t e − − . (C25) e + − j=0 j The latter expectation is a conditional Laplace transform of the type in Eq. (C1). It has solution as in Eq. (C2) where the system (C5)(C6)(C7) is solved for c¯ = ( ω¯ ,0, j γ¯,0) j = 0,...,γ¯. Therefore, the real yield, ε(t,τ), is a function of the maturity, the − − ∀ habit state variable and the perceived expected aggregate dividend growth only as in Eq. (51). The perpetual bond price is given by ∞ Eˆ ξ Qt = t t [ t,u]du. The dynamics of the perpetual bond price is obtained by applying Itô’s Lemma: R ∂ ∂ dQ = [.]dt + Qt σ dBˆ + Qt σ dBˆ + σ dBˆ + σ dBˆ (C26) t ∂ωt y y,t ∂µˆt µ,y y,t µ,µ µ,t µ,s s,t and therefore the bond return volatility is given by

∂ ∂ 2 ∂ 2 ∂ 2 σQ = Q 1 Qt σ + Qt σ + Qt σ + Qt σ (C27) t t− ∂ωt y ∂µˆt µ,y ∂µˆt µ,µ ∂µˆt µ,s which leads to Eq. (48), where σµ,µ is equal to zero since the expected growth is not observable. The premium is given by Q 1 dQ dξ µ rt = , . q.e.d. t − − dt Q ξ t Proof of Proposition 5: To compute the individual wealth, recall that i N ∀ ∈ ∞ γ¯ ∞ γ¯ ωt ω¯ ρ ωu ω¯ i ω ω Eˆ ξ e − (u t)Eˆ 1+e − − ( u ¯ ) Wi,t = t t,uCi,udu = 1 eωt ω¯ e− − t eωu ω¯ e− − Yu du. (C28) Z0 + − Z0 − ωu ω¯ γ¯ i In order to evaluate the latter expectation, since γ¯ is a positive integer, it is possible to apply the Binomial formula (1+e − ) − = ω ω ∑U ϒ j( u ¯ ) j=0 (i, j)e − where

γ¯ i γ¯ i γi 1 − γi 1 U = − ≥ and ϒ(i, j)= j ≥ (C29) ℓ(i, j) j ∞ γi < 1 ( 1) γi < 1 j − with ℓ(i, j)= j (γ¯ i + 1) and therefore to obtain: − − γ¯ ρ ωu ω¯ i ω ω ρ U ω ω (u t)Eˆ 1+e − − ( u ¯ ) (u t) ϒ Eˆ logYu+ℓ(i, j)( u ¯ ) e− − t ωu ω¯ e− − Yu = e− − ∑ (i, j) t e − . (C30) e − j=0 The latter expectation is a conditional Laplace transform of the type in Eq. (C1). It has solution as in Eq. (C2) where the system (C5)(C6)(C7) is solved for c¯ = ( ω¯ ,1,ℓ(i, j),0) j = 0,...,γ¯. Therefore, the ratio, Wi,t /Yt , is a function of the habit state variable − ∀ and the perceived expected growth only as in Eq. (52). The dynamics of individual wealth is obtained by applying Itô’s Lemma:

P P Q Q Wi Wi dWi,t = Wi,t (r + π (µ r)+ π (µ r) Ci,t/Wi,t )dt + σ dBˆy,t + σ dBˆs,t (C31) i − i − − y s

40 where

∂W ∂W ∂W σWi πPσP πQσQ 1 i,t σ i,t σ i,t σ y = i y + i y = Wi−,t ∂Y y + ∂ω y + ∂µˆ µ,y , (C32) ∂W σWi πPσP πQσQ 1 i,t σ s = i s + i s = Wi−,t ∂µˆ µ,s , (C33) πP πQ as in Eq. (54). Wealth proportions invested in the risky assets, i and i , satisfy Eq. (53) as long as the market asset and the P Q P Q perpetual bond lead to endogenous completeness, that is their diusion matrix is invertible (i.e., σ σs = σ σy for almost all states y s and times). q.e.d.

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44 Figures & Tables

Figure 1. Empirical riskreturn trade o

Expost expectations from realized returns Survey expectations HDukeCFOL Conditional mean Conditional mean 0.08 0.08

0.06 0.06 0.04

0.02 0.04 0.00

0.02 0.02

0.04 Conditional Conditional 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 volatility 0.04 0.05 0.06 0.07 0.08 volatility

Left panel. Scatter plot of the conditional mean and conditional volatility of excess returns on the CRSP value weighted stock index. Source: Lettau and Ludvigson (2001). The conditioning information set is given by the cay and two lags of volatility. The sample is quarterly and spans the period 1953:Q4 to 2000:Q4. Right panel. Scatter plot of the conditional mean and conditional volatility of returns from the Duke/CFO survey (see Graham and Harvey (2012)). The sample is quarterly and spans the period 2000:Q32012:Q2. The solid line denotes the nonlinear (quadratic) ﬁt (panel C, third column in Table II).

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Figure 2. Survey riskreturn trade o vs economic conditions

Conditional Conditional Conditional mean mean volatility 0.08 0.08 0.08

0.06 0.07 0.06

0.06 0.04 0.04

0.05 0.02 0.02

0.04 Conditional 0.00 0.00 0.00 0.02 0.04 0.06 0.08 0.00 0.02 0.04 0.06 0.08 0.04 0.05 0.06 0.07 0.08 volatility

The left and middle panels show the scatter plot of state variable ω (built as in Eq. (1))and respectively the conditional mean and the conditional volatility of returns from the Duke/CFO survey. The right panel shows the scatter plot the survey conditional mean and volatility of returns. The sample is quarterly and spans the period 2000:Q32012:Q2. Blue and red colors denote observations associated to values of ω respectively above and below its longrun average ω¯ . Lines denote predicted values from regressions in panels A, B and C of Table II. Black lines correspond to the case where the only predictive variable is the variable on the horizontal axis (ﬁrst column, Table II); coloured lines correspond to the case where also a dummy is considered (second column, Table II).

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45 Figure 3. Procyclical optimism

Joint distribution of Μ and Habit state variable: autocorrelation function Habit state variable: cumulative distribution Μ 1.0 1.0è

è é 0.04 é éé ééé 0.8 é é ééé é 0.8 è éé éééé é é ééééééééé éé éé é é è é é éééééééé ééééééééé éé é è éé ééééééééééééééééééééééééééé é é é é ééé ééééééééééééééééééééééééééééééééééééééé é é é ééé éééééééééééééééééééééééééééééééééééééééééééééé è 0.02 é éééééééééééééééééééééééééééééééééééééééééééééé ééé ééééééééééééééééééééééééééééééééééééééééééééééééééé ééééé é 0.6 0.6 è é éééééé éééééééééééééééééééééééééééééééééééééé ééééé ééé éééééééééééééééééééééééééééééééééééé éé é è é ééééééééé ééééééééééééééééééééé éé ééé é è é éé éééé é éééééééééééééééééééééééééééé éé è é é éééééé éééé ééééééé éé éé é é é ééééé é ééééé éééé éééééé éé é è éé é éééééééé ééé ééé é ééé è 0.00 é ééé éééééé é é ééé éééééééé é éé 0.4 0.4 è è é é éé é é éé é éé é é é é é é é ééé éé é é è é é é é é é é é 0.02 é é 0.2 0.2 è é éé è é é é è é é è é è è 0.04 0.0 0.0 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0 2 4 6 8 10 0.2 0.1 0.0 0.1 0.2

Cumulative distribution function (left panel), autocorrelation function (middle panel) of the habit state variable and scatter plot of the habit state variable and expected dividend growth (right panel). Colors blue and red denote respectively the Bayesian beliefs (χ = 0%) and procyclical optimism (χ = 7.5%). Model calibration is from Table III.

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Figure 4. Preference Heterogeneity and Equilibrium Distributions

Equilibrium distribution of consumption 1.0

0.8

0.6

0.4

0.2

0.0 1 Γ 0.0 0.2 0.4 0.6 0.8 1.0 Log pricing kernel Aggregate relative risk aversion

3.5 6

3.0 4

2.5 2 2.0 0 1.5

2 1.0

0.5 4

0.0 ΓHL 4.4 4.6 4.8 5.0 5.2 5.4 5.6 1.0 0.5 0.0 0.5 1.0

Upper left panel: Equilibrium cumulative distribution of consumptionshares at the steady state ωt = ω¯ . Colours green, red and blue denote respectively the model distribution for γ¯ = 7.5,10,15 . Black line denotes the Gamma distribution estimated by Xiouros and Zapatero (2010) { } from individual consumption data. Upper right panel: Equilibrium cumulative distribution of consumption and wealthshares and for γ¯ = 10. Black line denotes wealthshares at the steady state ωt = ω¯ : red and blue areas denote respectively the range of variation of consumption and wealthshares. Lower left panel: Density function of aggregate relative risk aversion under Bayesian beliefs (χ = 0%, blue line) and procyclical optimism (χ = 7.5%, red line). Lower right panel: Undiscounted log state price density under homogeneous (blue line) and heterogeneous risk attitudes (red line) as a function of ωt . Model calibration is from Table III.

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46 Figure 5. Equilibrium riskreturn trade o

Unconditional correlation between equity premium and return volatility 1.0

0.5

0.0

0.5

Χ 0.00 0.02 0.04 0.06 0.08 0.10

Equilibrium unconditional correlation between expected excess return and stock return volatility as a function of the parameter χ. Colours blue and red denote respectively the homogeneous and the heterogeneousagents economy. Model calibration is from Table III.

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Figure 6. Unconditional moments and procyclical optimism

Equity premium, bond premium and riskfree rate 0.10 Riskfree rate volatility 0.070

0.08 0.065

0.060 0.06

0.055

0.04 0.050

0.02 0.045

0.040 0.00 Χ Χ 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10

Equity and bond volatility Equity, bond and riskfree rate 1st order autocorrelation 0.45 0.9

0.40 0.8

0.35 0.7 0.30

0.25 0.6

0.20 0.5 0.15

0.4 0.10 Χ Χ 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10

Unconditional moments as a function of the parameter χ. First panel: expected stock (solid) and bond (dashed) returns and riskfree rate (dotted line). Second panel: riskfree rate volatility. Third panel: stock (solid) and bond (dashed line) return volatility. Fourth panel: ﬁrst order autocorrelation of pricedividend ratio (solid), bond (dashed) and riskfree rate (dotted line). Model calibration is from Table III.

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47 Figure 7. Conditional moments of asset prices in the heterogeneousagents economy

Log pricedividend ratio (ﬁrst line), stock return volatility (second line), equity premium (third line), bond return volatility (fourth line) and bond premium (ﬁfth line) are plotted as a function of ωt and µˆt . Left and right panels consider respectively Bayesian beliefs (χ = 0%) and procyclical optimism (χ = 7.5%). Model calibration is from Table III.

Back to the text. → 48 Figure 8. Stock return volatility and procyclical optimism

Priced (left) and unpriced (right) components of stock return volatility are plotted as a function of the habit state variable. Solid and marked lines denote the cases: µˆt = µ¯ 3%. Colours blue and red denote respectively Bayesian beliefs (χ = 0%) and procyclical optimism (χ = 7.5%). Model calibration is from Table ±III.

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Figure 9. Longhorizon stock return predictability and procyclical optimism

price dividend ratio: tstat price dividend ratio: R2 0 0.4

2 0.3

4 0.2

6 0.1

8 Χ 0.0 Χ 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10

term spread: tstat 2 0 term spread: R

0.15

0.10 2

0.05 3

4 Χ 0.00 Χ 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10

Regressions of cumulative excess stock returns on the log pricedividend ratio (upper panels) and on the 10 years term premium (lower panels). Colours blue, green, red, cyan and orange denote the horizon of respectively 1,2,3,5 and 7 years. tstatistics (left panels) and R2 (right panels) are plotted as a function of the parameter χ. Model calibration is from Table III.

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49 Figure 10. Longhorizon dividend growth predictability and procyclical optimism

price dividend ratio: tstat price dividend ratio: R2

0.6 0.05

0.4 0.04

0.2 0.03

0.0 0.02

0.2 0.01

0.4 Χ 0.00 Χ 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10

term spread: tstat term spread: R2 0.6 0.05

0.4 0.04

0.2 0.03

0.0 0.02

0.2 0.01

0.4 Χ 0.00 Χ 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10

Regressions of cumulative dividend growth on the log pricedividend ratio (upper panels) and on the 10 years term premium (lower panels). Colours blue, green, red, cyan and orange denote the horizon of respectively 1,2,3,5 and 7 years. tstatistics (left panels) and R2 (right panels) are plotted as a function of the parameter χ. Model calibration is from Table III.

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Figure 11. Term structure of interest rates and procyclical optimism

Real yeald Real yeald volatility 0.040 0.08

0.035 0.06

0.030

0.04

0.025

0.02 0.020

Τ 0.00 Τ 0 10 20 30 40 50 0 10 20 30 40 50

Left panel: Real yield as a function of the habit state variable and the maturity (with µˆt = µ¯) under procyclical optimism (χ = 7.5%)and heterogeneous preferences. Middle and right panels: Unconditional mean (middle) and volatility (right) of the real yield as a function of the maturity under Bayesian beliefs (χ = 0%, blue line) and procyclical optimism (χ = 7.5%, red line). Model calibration is from Table III.

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50 Figure 12. Portfolio allocations and procyclical optimism

Stock allocations HΧ 0L Stock allocations HΧ 7.5L

1.4 1.4

Γ10 1

Γ10 1 1.3 1.3

1.2 1.2

1.1 1.1

1.0 1.0

Γ 1 10 Γ 1 10 0.9 0.9 0.4 0.2 0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4

Bond allocations HΧ 0L Bond allocations HΧ 7.5L 0.1 0.1

Γ 1 10 Γ 1 10

0.0 0.0

0.1 0.1

0.2 0.2

Γ 1 0.3 10 0.3

Γ10 1

0.4 0.4 0.4 0.2 0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4

Wealth proportions invested in the market asset (upper panels) and in the perpetual bond (lower panels) as a function of ωt (with µˆt = µ¯) respectively under Bayesian beliefs (χ = 0%, left panels) and procyclical optimism (χ = 7.5%, right panels). Model calibration is from Table III.

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Figure 13. Longrun bias in forecast errors and equity premium

Unconditional equity premium as a function of the deviation parameter χ and the bias parameter ε (model extension of Section VII). Model calibration is from Table III.

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51 Table I Investors’ Beliefs and Forecast Errors

Maximum Likelihood Estimation

λquarterly = .15 λquarterly = .20 λquarterly = .25 Kalman ﬁlter

a0 .006 .006 .006 tstat 5.58 5.88 6.22

a1 .179 .251 .313 tstat 2.57 3.55 4.18

b0 9.07 9.08 9.15 tstat 34.59 35.68 34.93

b1 37.40 49.43 53.19 tstat 3.75 5.81 4.55

σe .003 .003 .003 tstat 17.15 17.15 17.15

1000 RMSE 3.01 2.99 2.97 3.32 ×

Maximum likelihood estimation of Survey σ µˆt t+1 = µˆt t+1 + et , et N(0, e), | | ∼ Survey where µˆt t+1 is the onequarter ahead average forecast from the Survey of Professional Forecasters (19682005) and µˆt t+1 is the model forecast | given by | α Prior α KF µˆt t+1 = t µt t+1 +(1 t )µˆt t+1, | | − | with Prior ω ω µt t+1 =a0 + a1( t ¯ ), | − 2 2 αt =E[e ]/(qt + E[e ]), and log(qt ) = b0 + b1 ωt ω¯ , t t | − | ω λquarterly ω ω¯ KF where t is the habit state variable computed as in Eq. (1) (with adjusted to quarterly frequency and 0 = ) and µˆt t+1 is the Kalman | ﬁlter estimator of expected log GDP growth from the observation of realized log GDP Yt . RMSE denotes the root mean square error.

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52 Table II Survey RiskReturn Trade O

Panel A – expected returns vs economic conditions: P µ rt = α0 + α1 1ω <ω¯ + β0 ωt + β1 ωt 1ω <ω¯ + εt t − t t (1) (2)

β0 .49∗∗∗ 1.56∗∗ tstat − 3.86 − 2.16 − − β1 1.05 tstat 1.38 R2 .24 .28

Panel B – expected volatility vs economic conditions: σP α α β ω β ω ε t = 0 + 1 1ωt<ω¯ + 0 t + 1 t 1ωt<ω¯ + t (1) (2)

β0 .36∗∗∗ .99∗∗∗ tstat − 5.17 − 2.74 − − β1 .91∗∗ 2.39 tstat − R2 .37 .49

Panel C – expected returns vs expected volatility: P P P P 2 µ rt = α0 + α1 1ω <ω¯ + β0 σ + β1 σ 1ω <ω¯ + β2 (σ ) + εt t − t t t t t (1) (2) (3) (4)

β0 .25 1.17∗∗ 8.28∗∗∗ 5.71∗∗ tstat 1.02 2.10 3.45 2.24 β1 1.82∗∗∗ 1.34∗∗ tstat − 2.83 − 1.97 − − β2 69.88∗∗∗ 41.68∗ tstat − 3.36 − 1.82 − − R2 .02 .26 .22 .32

P P Regressions are run with data about expected returns (µ rt ) and expected return volatility (σ ) from the Duke/CFO survey t − t 2000:Q32012:Q2. The state variable ωt is built as in Eq. (1) using GDP growth data (yearly parameter λ = .35 is adjusted to quarterly 1/4 quarterly frequency: λ = 1 (1 λ) and ω0 starts at the steady state ω¯ ). Symbols ∗, ∗∗ and ∗∗∗ denote 10%, 5% and 1% signiﬁcance levels. − −

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53 Table III Model Calibration

Symbol Value Description Aggregate Dividend Process µ¯ 1.8% Longrun growth.

σy 3.6% Instantaneous volatility. κ 0.35 Speed of reversion of longrun growth.

σµ 1.0% Instantaneous volatility of longrun growth. Investors Preferences ρ 3.5% Timediscount rate. γ¯ 10 Maximum relative risk aversion in the heterogeneousagents economy. λ 0.35 Speed of reversion of the habit state variable. Beliefs Formation

σs 5.0% Instantaneous volatility of the noisy signal. χ {0%, 5%, 7.5%, 10%} Procyclical optimism parameter.

Implied Parameters

σµ,y 0.33% Volatility of perceived longrun growth due to dividend observations.

σµ,s 0.24% Volatility of perceived longrun growth due to signal observations. ω¯ 4.96% Steadystate level of the habit state variable. γ˜(ω¯ ) 5 Steadystate aggregate relative risk aversion.

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54 Table IV Unconditional Moments – HomogeneousAgents Economy

Data Model (a) (b) χ = 0% χ = 5% χ = 7.5% χ = 10% E(r)(%) 1.70 0.60 1.93 1.92 1.90 1.90

σr(%) 4.38 3.00 6.40 5.17 4.67 4.32 E(µP r)(%) 7.38 5.60 3.86 4.26 4.78 7.37 − σP(%) 17.88 19.80 20.80 23.10 26.10 41.10 E(θ)(%) 41.27 28.28 17.90 17.90 17.90 17.90 E(logP/Y) 3.36 3.38 4.09 4.17 4.33 6.12 σ(logP/Y) .44 .45 .24 .28 .33 .58 AC(logP/Y) .91 .88 .71 .72 .74 .77 AC(r) .33 .67 .72 .60 .52 .45

Unconditional statistics of the model, in the homogeneousagents economy and for various levels of χ, are compared with the data sample estimates. In (a) I use annual data between 1933 and 2006: pricedividend ratio and stock returns refer to the S&P Composite Index and the riskfree rate is equal to rate of return on the threemonth Treasury bill. In (b) data are from 1931 to 2009 as in Constantinides and Ghosh (2011). Model calibration is from Table III.

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55 Table V Unconditional Moments – HeterogeneousAgents Economy

Data Model (a) (b) χ = 0% χ = 5% χ = 7.5% χ = 10% E(r)(%) 1.70 0.60 1.93 1.87 1.84 1.82

σr(%) 4.38 3.00 6.04 5.10 4.60 4.20 E(µP r)(%) 7.38 5.60 3.86 4.26 4.79 7.37 − σP(%) 17.88 19.80 21.46 23.10 26.10 41.10 E(θ)(%) 41.27 28.28 17.90 17.90 17.90 17.90 E(logP/Y) 3.36 3.38 4.09 4.17 4.33 6.12 σ(logP/Y) .44 .45 .24 .28 .33 .57 AC(logP/Y) .91 .88 .71 .72 .74 .77 AC(r) .33 .67 .72 .60 .51 .44

Unconditional statistics of the model, in the heterogeneousagents economy and for various levels of χ, are compared with the data sample estimates. In (a) I use annual data between 1933 and 2006: pricedividend ratio and stock returns refer to the S&P Composite Index and the riskfree rate is equal to rate of return on the threemonth Treasury bill. In (b) data are from 1931 to 2009 as in Constantinides and Ghosh (2011). Model calibration is from Table III.

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56 Table VI Equilibrium RiskReturn Trade O

Correlations Survey data Model χ = 0% χ = 5% χ = 7.5% χ = 10% ω corr(ω, µP r) .49 (.00) .99 .99 .98 .98 ∀ − ∗∗∗ P corr(ω, σ ) .61∗∗∗ (.00) .99 .95 .47 .48 corr(µP r, σP) .15 (.32) .99 .97 .32 .29 − ω > ω¯ corr(ω, µP r) .43 (.06) .99 .99 .99 .99 − ∗ P corr(ω, σ ) .76∗∗∗ (.00) .99 .97 .42 .64 corr(µP r, σP) .42 (.02) .99 .99 .54 .72 − ∗∗ ω < ω¯ corr(ω, µP r) .43 (.02) .99 .99 .95 .92 − ∗∗ corr(ω, σP) .12 (.54) .99 .83 .75 .81 corr(µP r, σP) .39 (.04) .99 .83 .52 .51 − ∗∗

Data about expected returns (µP r) and expected return volatility (σP) are from the Duke/CFO survey 2000:Q32012:Q2. The state R − R variable ω is built as in Eq. (1) using GDP growth data (λ is adjusted to quarterly frequency). Symbols ∗, ∗∗ and ∗∗∗ denote 10%, 5% and 1% signiﬁcance levels. Numbers in parentheses denote pvalues. Model calibration is from Table III. Model correlations are computed from simulations.

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57 Table VII LongHorizon Regressions – HomogeneousAgents Economy

Data χ = 0% χ = 5% χ = 7.5% χ = 10% Coe. tstat R2 Coe. tstat R2 Coe. tstat R2 Coe. tstat R2 Coe. tstat R2 Horizon Log pricedividend ratio 1 .11 2.70 .072 .09 .78 .015 .17 1.64 .032 .20 2.09 .047 .20 2.58 .067 2 .19 2.80 .112 .16 1.04 .026 .31 2.31 .061 .38 2.99 .091 .40 3.76 .131 3 .22 2.68 .121 .22 1.19 .035 .44 2.75 .084 .52 3.63 .128 .56 4.64 .187 5 .33 3.19 .164 .30 1.38 .047 .62 3.31 .119 .75 4.48 .183 .81 5.89 .271 7 .46 3.57 .223 .37 1.54 .056 .74 3.64 .141 .89 4.99 .218 .96 6.68 .325 Horizon 10 years term premium 1 .51 .79 .015 1.04 1.30 .024 1.20 1.30 .024 1.50 1.09 .020

58 2 .91 1.04 .026 1.98 1.84 .044 2.36 1.92 .046 3.19 1.74 .040 3 1.21 1.19 .035 2.75 2.19 .060 3.36 2.35 .065 4.74 2.22 .059 5 1.71 1.41 .049 3.93 2.65 .084 4.89 2.91 .093 7.22 2.87 .090 7 2.09 1.57 .058 4.69 2.87 .098 5.86 3.19 .111 8.80 3.22 .111

The iyears cumulative excess log returns are regressed on the log pricedividend ratio (upper panel) and the 10 years term premium (lower panel): e e α β ε rt+1 + ... + rt+i = + xt + t . Regression estimates are averages of ten thousand simulations with each simulation having 100 observations.

Back to the text. → Table VIII LongHorizon Regressions – HeterogeneousAgents Economy

Data χ = 0% χ = 5% χ = 7.5% χ = 10% Coe. tstat R2 Coe. tstat R2 Coe. tstat R2 Coe. tstat R2 Coe. tstat R2 Horizon Log pricedividend ratio 1 .11 2.70 .072 .09 .80 .015 .17 1.66 .033 .20 2.11 .048 .21 2.59 .067 2 .19 2.80 .112 .17 1.07 .027 .32 2.34 .062 .38 3.02 .093 .40 3.78 .133 3 .22 2.68 .121 .22 1.22 .035 .44 2.79 .086 .53 3.67 .130 .56 4.67 .189 5 .33 3.19 .164 .31 1.42 .048 .62 3.36 .121 .75 4.53 .186 .81 5.93 .273 7 .46 3.57 .223 .38 1.58 .058 .74 3.69 .144 .89 5.04 .222 .96 6.73 .327 Horizon 10 years term premium 1 .53 .81 .016 1.07 1.31 .024 1.22 1.30 .024 1.50 1.08 .020

59 2 .95 1.07 .027 2.03 1.86 .044 2.39 1.92 .046 3.19 1.72 .040 3 1.26 1.22 .036 2.81 2.21 .060 3.40 2.34 .064 4.74 2.19 .058 5 1.77 1.45 .049 4.02 2.67 .085 4.95 2.90 .093 7.22 2.83 .089 7 2.16 1.60 .059 4.79 2.89 .099 5.93 3.19 .110 8.80 3.18 .109

Back to the text. → Online Appendix of “Survey Expectations and the Equilibrium RiskReturn Trade O”

Online Appendix A – Equilibrium Homogenization of Beliefs

This section provides an explanation of the cyclical bias Ψ(ωt ) in Eq. (17) as an aggregation result of investors featuring heterogeneity in both beliefs and risk preferences. Indeed, both the two sources of heterogeneity can be embedded in a model with heterogeneity only in risk aversion and an endogenously adjusted dynamics for the aggregate dividend. Such an endogenous adjustment can be proxied, in reduced form, by the deviation term Ψ(ωt ). To gain intuition the framework of this section is simpliﬁed: the economy is populated by only two agents who do not learn about expected growth and traded assets are such that agents face complete markets. The economy consists of two agents i = A,B , who dier both in their risk preferences as well as in their beliefs. Namely, { } they maximize utility of the form

∞ ∞ γ γ Ei ρt i Ei ρt 1 1 i i 0 e− U (Ci,t )dt = 0 e− 1 γ Ci,t− Xt dt , i = A,B , Z0 Z0 i { } − i where E is the expectation operator under the probability measure of the investor i, Ci,t is individual consumption and Xt is the exogenous standard of living as in Section III.A. The agents dier in their beliefs about the growth rates of the aggregate dividend and their probability measures are characterized as follows:

Pi t t P i d 1 ηi 2 ηi t = = exp 2 ( u) du udBy,u , i = A,B , dP − Z0 − Z0 { } t ηi i ηi where the function represents the deviation of agent i’s beliefs on the growth rate µt from its true value µt : when is negative, the agent i is optimistic with respect to the physical measure and viceversa. Indeed, the agents observe the realizations of the aggregate dividend but interpret the Brownian shocks under their subjective information set:

A σ A dYt B σ B µt dt + ydBy,t = = µt dt + ydBy,t Yt where

A A ηA ηA µt µt dBy,t = dBy,t + t dt, t = − , σy B B ηB ηB µt µt dBy,t = dBy,t + t dt, t = − . σy

The two agents assign dierent but equivalent measures to the future uncertain dividend process and still can trade in ﬁnancial assets such that the market is complete. Therefore, a representative agent optimization problem can be constructed to account for heterogeneous beliefs: ∞ ∞ ρ ρ sup νAEA e uU A(C )du + νBEB e uU B(C )du. 0 Z − A,u 0 Z − B,u CA,t +CB,t =Yt 0 0

60 The optimization problem can be written under the physical measure (see Detemple and Murthy (1994) and Basak (2005)) as

∞ ∞ E νA P A ρu A νB P B ρu B sup 0 u e− U (CA,u)du + u e− U (CB,u)du . Z0 Z0 CA,t +CB,t =Yt Then, the economy is characterized by the following ﬁrstorder conditions:

ρ γ ρ γ νAP A t A ξ νBP B t B t e− (CA,t /Xt )− = 0,t = t e− (CB,t /Xt )− ,

ωt and the market clearing condition: CA,t +CB,t = Yt or, equivalently, CA,t /Xt +CB,t /Xt = e . Let denote with Θt the following process: t t Θ = exp β˘ du + δ˘ dB , t Z u Z u y,u 0 0 where ηA 2 ηB 2 ηA ηB ˘ ( t ) /2 ( t ) /2 ˘ t t βt = − and δt = − . γA γB γA γB − − γ γ Θ A P A Θ B P B It is easy to verify that t− / t = t− / t . Consider now the following simple transformation:

CA,t C˘A,t ΘtCA,t , → ≡ CB,t C˘B,t ΘtCB,t , → ≡ Yt Y˘t ΘtYt , → ≡ with ω˘ t logY˘t logXt . Under such a transformation the dynamics from the ﬁrst order conditions simpliﬁes to ≡ − ρ γ ρ γ A t A ˘ B t B ν e− (C˘A,t /Xt )− = ξ0,t = ν e− (C˘B,t /Xt )− ,

γ ξ˘ ξ Θ A P A ˘ ˘ ˘ ω where 0,t = 0,t t− / t , market clearing requires CA,t +CB,t = Yt and ˘ t fully characterizes the equilibrium stateprice density. The latter equation represents the ﬁrst order conditions of a twoagent CuJ economy whose agents dier only in their relative risk aversion whereas they have homogeneous beliefs (similarly to the problem in Eq. (22)). Heterogeneous beliefs have been homogenizated at equilibrium by a change in the (perceived) dynamics of the aggregate dividend. Such a result can be generalized to an arbitrary number of agents, to a wide class of biases in beliefs and to heterogeneous time preferences: see Tran and Zeckhauser (2012) for further details. In particular, assume the case where individual beliefs feature a cyclical bias which evolves with the habit state variable, ωt :

i µ = a0,i + a1,i(ωt ω¯ ), i = A,B . t − { } ηA ηB β˘ δ˘ ω Consequently, t , t , t and t are functions of t and the adjusted dynamics of the aggregate dividend becomes:

˘ β˘ δ˘ 2 σ δ˘ ˘ σ δ˘ ˘ dYt =(µt + t + t /2 + y t )Yt dt + ( y + t )Yt dBy,t

=µ˘(µt ,ωt )Y˘t dt + σ˘ (ωt )Y˘t dBy,t

Statedependent biases in individual beliefs translate, as an endogenous result, into an adjusted dynamics for the aggregate dividend of an economy where risk aversion is the only form of heterogeneity. Consider the simple case a0,A = a0,B and a1,A = a1,B > 0 − but γA > γB. The degree of optimism in good times (and pessimism in bad times) of agent A exactly oset the degree of pessimism in good times (and optimism in bad times) of agent B. However, the former agent is more risk averse than the latter. Under such simple conditions the adjusteddrift µ˘(µt ,ωt ) can be increasing in ωt in most of the states, whereas the aggregate relative risk aversion is decreasing in ω˘ t , as shown in Figure OA.1. This is an endogenous result of heterogeneity and does not need asymmetry in individual beliefs. Procyclical optimism about expected growth and countercyclical aggregate risk aversion are the two key ingredients of the model of this paper and drive all asset pricing implications. They can be understood as an aggregation result and

61 Figure OA.1. Equilibrium homogenization of beliefs

Γ Μî

5.4

0.0215 5.3

5.2 0.0210 5.1

0.0205 5.0

4.9 0.0200 4.8

î 0.05 0.05 0.10 0.05 0.05 0.10

Left panel: Adjusted expected growth µ˘t (red line) and longrun growth µ¯ (black line) are plotted as a function of the habit state variable. Right ˘ ˘ panel: Aggregate relative risk aversion γ˜(ω˘ t ) = ∂ω˘ ξ0 t /ξ0 t is plotted as a function of the (adjusted) habit state variable. Parameters: − t , , µ¯ = a0 A = a0 B = .02,σy = .035,a1 A = a1 B = .05,γA = 10,γB = 2. , , , − , are not the mechanical implication of either a longrun bias or asymmetry in individual beliefs. Therefore, the homogeneous belief driven by Ψ(ωt ) in Eq. (17) can be interpreted as a reduced form of the equilibrium homogenization of heterogeneous beliefs and is adopted for the sake of exposition and analytic tractability throughout the paper.