Income Insurance and the Equilibrium Term-Structure of Equity

Transcription

1 Income Insurance and the Equilibrium Term-Structure of Equity Roberto Marfè Job Market Paper EFA 2014 Comments welcome Most recent version available here ABSTRACT Leading asset pricing models are inconsistent with the recent empirical findings which document downward sloping term structures of equity risk and premia. This paper shows that a simple general equilibrium model can accommodate these stylized facts as long as dividend distributions endogenously obtain from an explicit model of labor relations, where wages do not correspond to the marginal product of labor but incorporate an income insurance from shareholders to workers. Fluctuations in the income insurance that workers exploit over the business cycle lead to short-term risk of equity returns. Therefore, the model captures simultaneously the negative slope of the term structures of both equity and dividends as well as traditional asset pricing facts. An empirical analysis documents the connection between income insurance and the term-structures of wage and dividend risk. Moreover, the model is consistent with additional stylized facts such as the predictability of dividend growth and excess stock returns, the cross-sectional value premium and the upward-sloping term-structure of interest rates. Finally, the model implies that short-run discounted cash-flows contribute more to the equity premium than long-run ones, whereas the opposite holds in leading models. Keywords. term structure of equity income insurance dividend strips distributional risk equilibrium asset pricing JEL Classification. D53 E24 E32 G12 An earlier version of this paper was circulated under the title Labor Relations, Endogenous Dividends and the Equilibrium Term Structure of Equity. I would like to thank my supervisor Michael Rockinger and the members of my Ph.D. committee Swiss Finance Institute Philippe Bacchetta, Pierre Collin-Dufresne, Bernard Dumas and Lukas Schmid for stimulating conversations and many insightful comments and advices. I would also like to acknowledge comments from Michela Altieri, Ainhoa Aparicio-Fenoll, Ravi Bansal, Cristian Bartolucci, Luca Benzoni, Harjoat Bhamra, Vincent Bogousslavsky, Andrea Buraschi, Gabriele Camera, Gian Luca Clementi EFA discussant, Stefano Colonnello, Giuliano Curatola, Marco Della Seta, Jérôme Detemple Gerzensee and Paris discussant, Theodoros Diasakos, Jack Favilukis, Christian Gollier, Michael Hasler, Campbell Harvey, Marit Hinnosaar, Toomas Hinnosaar, Leonid Kogan, Kai Li, Jeremy Lise, Elisa Luciano, Loriano Mancini, Guido Menzio, Antonio Merlo, Ignacio Monzón, Giovanna Nicodano, Andrea Prat, Birgit Rudloff, Raman Uppal and from the participants at the 41st European Finance Association EFA Annual Meeting 2014, at the 11th International Paris Finance Meeting 2013 AFFI/EUROFIDAI, at the 12th Swiss Doctoral Workshop in Finance 2013, and at seminar at the Collegio Carlo Alberto. 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 Moncalieri TO, Italy. [email protected]. Webpage:

2 I. Introduction Leading equilibrium asset pricing models can describe some important features of financial markets, such as i the high equity premium, ii the excess return volatility, and iii the low and smooth risk-free rate. These models have different rationale e.g. habit formation, time-varying expected growth, disasters, prospect theory but share an important feature: priced risk comes from variation in long-run discounted cash-flows. 1 An important implication of this is that these models are inconsistent with some recent empirical evidence, such as i the decreasing variance ratios of dividend distributions, ii the downward sloping term-structures of equity risk and premia at least at short and medium horizons, and iii the cyclicality and mean-reversion of the dividend-share of consumption. 2 This is important because the term-structures of both cash-flows and equity returns provide information about how prices are determined in equilibrium. Hence, a term-structure perspective offers additional testable implications to asset pricing frameworks and can help us to understand the macroeconomic determinants of asset prices. The main contribution of the paper is the following. I propose a simple model of labor relations which allows us to reconcile the two sets of stylized facts mentioned above in the context of a parsimonious and closed-form general equilibrium model. The rationale behind this is as follows. The paper argues that the short-term risk of dividends and the dynamics of the dividend-share are due to a distributional risk. The model assumes that workers benefit from income insurance from shareholders and this leads to counter-cyclical labor-share dynamics. 3 As a consequence, the risk of owning capital increases and, hence, dividend distributions are more volatile and the dividend-share is negatively correlated with the labor-share. Therefore, equity is a bad hedge against aggregate risk and commands a high compensation. In addition, as long as income insurance leads to stationary labor- and dividend-shares, a term-structure effect takes place: the transitory component of aggregate risk is shared asymmetrically among workers and shareholders, whereas the permanent component is faced by both. Namely, shareholders bear a high short-term risk because dividend risk shifts toward the short horizon, whereas workers bear a strong long-run risk because the riskiness of wages increases with the horizon. Finally, under standard preferences, such a term-structure effect also preserves in the financial markets and downward-sloping term-structures of equity risk and premia obtain. An empirical investigation supports the main model mechanism. Indeed, consistently with the model assumption of income insurance from shareholders to workers, in the real data the labor-share is counter-cyclical and mean-reverting and is the main determinant of the fluctuations in the dividend-share. 4 Moreover, real GDP growth rates have approximately a flat term-structure of variance, whereas wages and dividends feature variances which, respectively, increase and decrease with the horizon. In accord with the model, these term-structures and the co-integrating relationship among GDP, wages, and dividends are consistent with the idea that both workers and shareholders are subject to permanent shocks but they share transitory shocks in such a way that workers are subject more to long-run risk than short-run risk and shareholders more to 1 Examples are the seminal works by Campbell and Cochrane 1999, Barberis, Huang, and Santos 2001, Bansal and Yaron 2004 and Gabaix 2012, among others. 2 Belo, Collin-Dufresne, and Goldstein 2014 document the downward sloping term-structure of aggregate dividend volatility; van Binsbergen, Brandt, and Koijen 2012 and van Binsbergen, Hueskes, Koijen, and Vrugt 2013 provide empirical evidence about the term-structures of equity returns by use of dividend strips prices; Boldrin and Horvath 1995, Longstaff and Piazzesi 2004, Danthine, Donaldson, and Siconolfi 2006, Santos and Veronesi 2006 and Ríos-Rull and Santaeulàlia-Llopis 2010 investigate the dynamics of the labor- and dividend- shares. 3 The idea that distributional risk is at the heart of labor relations and, also, that the very role of the firm is that of insurance provider have a long tradition since Knight 1921 as well as Baily 1974 and Azariadis They suggest that, on the one hand, labor s claim on output is partially fixed in advance and, hence, shareholders bear most of aggregate uncertainty and, on the other hand, in exchange of income insurance, shareholders gain flexibility in labor supply. 4 In particular, I show that at the aggregate level, the shareholders remuneration, as a residual choice of the firm, is mainly affected by variation in the remuneration of human capital and only marginally by investment and financing decisions. 1

3 short-run risk than long-run risk. In addition, the labor-share predicts dividend growth and excess stock returns, coherently with the positive and negative intertemporal relationships implied by the model. 5 Notice that standard real business cycle models fail to explain financial markets mainly because Walrasian labor markets lead to constant labor-share. In turn, dividends include a hedge component and, hence, do not deserve a sizeable premium. 6 Instead, a stylized representation of labor relations, which captures the deviation from the Walrasian benchmark due to income insurance, leads to a source of short-run but persistent risk and, hence, to the correct specification of the term-structures of both cash-flows and equity returns. The paper provides two additional noteworthy contributions. First, the model offers an extensive analysis of asset pricing in spite of a peculiar perspective, based on the properties of the equilibrium term-structures. Consider the following question: whence the equity premium? The equity premium is the compensation of the claim on the future stream of aggregate dividends. As suggested by van Binsbergen et al. 2012, the properties of the single discounted cash-flows at each future horizon provide a lot of information about the way stock prices are formed. For the first time, this paper derives a closed-form equilibrium density which describes the relative contribution of each time horizon to the whole equity premium, as shown in Figure 1. Insert Figure 1 about here. As shown in Figure 1, the long-run risk model black dashed line as well as other leading models implies that long-run cash-flows are the main determinant of the equity premium. Instead, the model presented in this paper red solid line, which does a better job at modeling the properties of the term-structures of dividends and equity, implies a strong shift of priced risk toward the short-run. Figure 1 shows that investors have a very different interpretation about investment and saving decisions as well as about which risks are priced. Indeed, in the long-run risk model, the density peaks at about 15 years far in the future, whereas in the model of this paper it is just the immediate future that contributes more to the market compensation. Therefore, the paper shows that in order to reconcile the standard asset pricing facts with the new evidence about the term-structures in general equilibrium and given standard preferences we need a radically different explanation of the driving forces beyond stock markets. Accounting for labor relations provides a rationale: the stronger the income insurance to workers, the larger the shift of equity risk toward the short horizon. As a second additional contribution, the paper analyzes the joint effect on the equilibrium term-structures of both recursive preferences and flexible dynamics of fundamentals. Exogenous uncertainty represents the total resources of the representative firm which should be distributed to workers and shareholders. Such a process is usually stationary in real business cycle literature and integrated in asset pricing literature. This paper analyzes the role of both transitory and permanent shocks jointly and shows how they can differently affect equilibrium outcomes. Under standard preferences, the transitory and the permanent shocks contribute, respectively, negatively and positively to the slope of the term-structures of equity risk and premia. Therefore, a long-run risk model can match downward-sloping term-structures only under a different parametrization of preferences which, in turn, deteriorates all standard asset pricing implications. Instead, a model with both types of shocks can reconcile an otherwise standard model with the recent evidence about dividend strips, as long as short-run but persistent risk is large enough. A simple model of labor relations provides a rationale. The model works as follows. In spirit of Danthine and Donaldson 1992, 2002, I model labor relations through a contract rule which qualitatively captures the implications of contract arrangements such as in Boldrin and Horvath 1995 and Gomme and Greenwood 1995 that produces, as a result of a simple bargaining model, income insurance to workers as long as those 5 A similar analysis of the non-financial corporate sector data confirms the results concerning the total economy and provides further robustness. 6 For instance, production-based asset pricing models, such as those in Jermann 1998, Boldrin, Christiano, and Fisher 2001, and Kaltenbrunner and Lochstoer 2010, have a hard time to capture either the smooth dynamics of wages or the riskiness of excess returns on equity. 2

4 are more risk averse than shareholders, in line with the empirical literature see Moskowitz and Vissing-Jørgensen Such a model of labor relations makes it possible to capture the counter-cyclical dynamics of the labor-share and to enhance the riskiness of owning capital. Indeed, income insurance to workers within the firm endogenously leads to a strong desire for the shareholders to smooth consumption intertemporally. 7 For the sake of exposition and in order to make unambiguous the equilibrium implications of labor relations, production is modeled in reduced form and limited market participation is assumed such that workers do not own capital. Such simplifying assumptions prevent some forms of consumption smoothing which can hide the role of income insurance in the determination of wage payments and dividend distributions. Distributional risk due to labor relations increases the short-run riskiness of dividends even if the latter are not riskier than the firm s operational cash-flows in the long-run. Namely, income insurance loads on shareholders most of the fluctuations due to the transitory component of aggregate risk. Therefore, dividends are risky at short horizons and feature downward sloping term structures of volatility. This result is consistent with the empirical work by Guiso, Pistaferri, and Schivardi 2005 which documents economically significant income insurance from employers to employees, mostly related to transitory, instead of permanent, productivity shocks. Similar results are documented by Han, Maug, and Schneider Moreover, in line with the real data, labor relations induce counter-cyclical dynamics to the labor-share and, hence, reduce the hedge component of dividends. Equilibrium returns on equity depend on the marginal rate of substitution of shareholders, who act as a representative agent under limited market participation. In presence of both long-run and short-run persistent shocks, the term-structure of equity risk is non-monotone. The long-run shock contributes positively to the slope of the equilibrium term-structure, whereas the short-run shock contributes negatively if the elasticity of intertemporal substitution is larger than unity and vice-versa. Negative slopes at short and medium horizons obtain as long as income insurance enhances the role of the short-run shock. Moreover, the model generates long-run excess return volatility over dividends and consumption. Instead, the long-run risk model of Bansal and Yaron 2004 leads to counter-factual upward-sloping volatilities and cash-flows riskier than equity at long horizons, as pointed out by Beeler and Campbell The term-structures of equity premia depend on risk preferences. Namely, short-run persistent shocks contribute negatively to the slope if the intertemporal substitution effect dominates the wealth effect and vice-versa. On the other hand, long-run persistent shocks contribute negatively to the slope if either i the elasticity of intertemporal substitution is larger than one and shareholders have preference for the late resolution of uncertainty or ii the elasticity of intertemporal substitution is smaller than one and shareholders have preference for the early resolution of uncertainty. Both the cases have poor equilibrium implications about standard asset pricing facts in the context of a long-run risk model. Indeed, the latter requires elasticity of intertemporal substitution larger than unity and preference for the early resolution of uncertainty. Instead, the model presented in this paper reconciles the equilibrium term-structures with the standard asset pricing facts by accounting for both long-run and short-run persistent shocks, given the usual parametrization of preferences. As a peculiarity the model calibration exploits the information from the term-structures of wage and dividend risk. On the one hand, they represent additional stylized facts captured by the model, and, on the other hand, they provide information about the persistence of both transitory and permanent shocks a main issue in long-run risk models. Interestingly, the model leads to four important additional testable implications. First, the model improves on the description of the term-structure of interest rates. The real yield increases with the horizon even if the intertemporal substitution effect dominates the wealth effect. This is not the case in the long-run risk model but is important for two reasons. On the one hand, increasing real yields help to describe a positive nominal term-spread as in the real data without 7 Such an effect of income insurance essentially substitutes for exogenous approaches such as habit formation or some form of market incompleteness. 3

5 referring only to inflation risk. On the other hand, increasing real yields help to produce downward sloping premia on the equity yields, in line with the empirical findings by van Binsbergen, Hueskes, Koijen, and Vrugt Second, the model provides a rationale for the value premium and offers an equilibrium explanation to the framework of Lettau and Wachter They define growth and value firms through the longer and shorter duration of their cash-flows and then obtain a value premium by an exogenously specified pricing kernel. This paper endogenizes the pricing kernel and shows that capturing the downward-sloping term-structure of equity automatically leads to cross-sectional equity prices featuring a value premium. A counter-factual growth premium would obtain if labor relations do not enhance enough the effect of short-run risk. Third, van Binsbergen et al and Muir 2014 refine the findings of van Binsbergen et al suggesting that the negative slope of equity premia is an unconditional property of the data, whereas conditional equity premia are slightly increasing in good times and strongly decreasing in bad times. I show that such a dynamics obtains in the model without deteriorating other results once fundamentals feature countercyclical heteroscedasticity. Fourth, income insurance leads to a countercyclical dynamics of conditional premia, volatility, and Sharpe ratios of stock returns. Moreover, the labor-share predicts dividend growth and excess stock returns, respectively, with positive and negative signs, in line with the results of the empirical analysis. 8 Related literature. A number of recent papers points out the importance of labor relations for asset pricing. Namely, the high risk of owning capital is due to a large extent to the smooth dynamics of wages. The latter can obtain because of preference heterogeneity as in Danthine and Donaldson 1992, 2002, infrequent wage resettling as in Favilukis and Lin 2012, search frictions as in Kuehn, Petrosky-Nadeau, and Zhang 2012, labor mobility as in Donangelo All these models imply an explicit or implicit income insurance from shareholders to workers, which takes place within the firm and is consistent with the empirical findings of Guiso et al and Han et al This paper complements such a literature studying the term-structure implications of income insurance in general equilibrium. Differently from those works, I provide empirical and theoretical support to the idea that labor relations alter not only the cyclicality of dividends but also the timing of dividend risk. The latter is a key ingredient to reconcile long and short term patterns of equity returns. My paper differs from the above mentioned works also on another dimension. Danthine and Donaldson 1992, 2002, Favilukis and Lin 2012 and Kuehn et al propose big macro-finance models and rely on numerical solutions. Instead, I consider a simpler and more parsimonious economy and provide analytical solutions. 9 This allows for i transparent asset pricing implications and a thorough investigation of term-structures under recursive preferences; ii a direct comparison with the long-run risk model of Bansal and Yaron 2004, that is a standard asset pricing benchmark which is severely challenged by the empirical findings of van Binsbergen et al. 2012; Ai, Croce, Diercks, and Li 2012 and Belo, Collin-Dufresne, and Goldstein 2014 suggest alternative channels which can contribute to explain the term-structure of equity. 10 Similarly to the present paper, Ai et al proposes a general equilibrium model where dividends are a direct outcome of the firm decisions. However, the focus is not on labor but on production: heterogeneous investment risk due to vintage capital leads to a downward-sloping term-structure of equity premia. Ai et al does not investigate the term-structure of return volatility and requires an extremely high risk aversion to obtain a sizeable equity premium. Belo et al shows in partial equilibrium that fluctuations in financial leverage can 8 For the sake of exposition and analytic tractability, these results are presented under the special case of power utility. However, they also hold under recursive preferences as long as numerical solutions avoid the log-linearization of dividends dynamics. 9 Even if my model is silent on production and workers saving decisions, Danthine, Donaldson, and Siconolfi 2006 have shown that a similar form of distributional risk improves the asset pricing implications of a standard real business cycle model without deteriorating business cycle predictions. 10 Croce, Lettau, and Ludvigson 2014 and Marfè 2013 consider the term-structure of equity in the context of a long-run risk model. Berrada, Detemple, and Rindisbacher 2013, Curatola 2014 and Marfè 2014 focus on non-standard beliefs formation and state-dependent risk preferences. 4

6 shift the risk from EBIT to dividends toward the short horizon. However, at the aggregate level EBIT risk and dividend risk have essentially the same term-structure. This implies that the leverage channel is quantitatively negligible and that the determination of short-term risk takes place earlier than financing decisions. Moreover, a downward-sloping term-structure of equity premia obtains because of the exogenous co-movements between the leverage and the priced factors. 11 Therefore, labor relations, financial leverage, and investment decisions are three channels which can contribute to the equilibrium explanation of the timing of equity risk. However, at the aggregate level, small fluctuations in the labor-share are likely more relevant than leverage and investment adjustments: the cost of labor is more than half of the representative firm output, while the service of debt and the investment-share are significantly smaller. The empirical analysis of Section II provides support to the idea that, at the aggregate level, variations in the labor-share are a more important determinant of the dividend-share and, in turn, of the timing of dividend risk than financing and investment decisions. Namely, the gap between the flat term-structure of GDP risk and the downward-sloping term-structure of dividend risk can be almost entirely imputed to the upward-sloping term-structure of wage risk. This paper is also related to the literature about preference heterogeneity. Under complete markets, Chan and Kogan 2002 show that preference heterogeneity leads to counter-cyclical price of risk as a result of endogenous time-variation in the aggregate relative risk aversion. 12 The same endogenous counter-cyclical dynamics obtains in the model of this paper and is due to variations in dividend volatility instead of risk aversion. Such a counter-cyclical variation is still due to preference heterogeneity but obtains through the income insurance channel. The paper is organized as follows. Section II provides empirical support to the main model mechanism. Section III describes workers and shareholders problems, the representative firm and labor relations. Section IV and V derive the pricing kernel, cash-flows, asset prices and their term structures. Asset pricing results are presented in Section VI. Alternative model specifications are discussed in Section VII. Section VIII concludes. II. Empirical Support In this section, I document several properties of aggregate dividends, wages and other GDP components in the US data. These properties provide empirical support for the key features of the model and drive the main results. First, I show that i GDP, dividends and wages are co-integrated, ii wages are smoother and more persistent than dividends, and iii the labor-share moves counter-cyclically. Second, I document that the term-structure of GDP volatility is approximately flat over long horizons, whereas those associated with wages and dividends are respectively increasing and decreasing with the horizon. Third, I provide support to the idea that variations in the labor-share are the main driving force of the dividend-share and, hence, of the timing of dividend risk. Fourth, I show that the labor-share forecasts both aggregate dividend growth and market excess returns. In a nutshell, the analysis offers empirical support to the main economic mechanism of the model, that is the role of income insurance from shareholders to workers and its implications for both dividends and equity returns. In the online appendix OA.A a similar analysis, which focuses on the non-financial corporate sector only, provides further robustness. 11 Namely, Belo et al assume that leverage is correlated with all priced factors and that the levels of the latter also affect the leverage dynamics. In particular, the expositions to the levels are used as free parameters to calibrate the model on asset prices. A general equilibrium approach would produce economic restrictions on those parameters and offer additional insights into the relationship between leverage and equity term-structures. 12 Therefore, Chan and Kogan 2002 model can be interpreted as a micro-founded version of nonlinear habit models such as Campbell and Cochrane See Xiouros and Zapatero 2010 for the empirical shortcomings of such an approach. 5

7 A. Data The key variables of the analysis are the GDP shares of aggregate wages and dividends. The main data source is the National Income and Products Account NIPA, available through the Bureau of Economic Analysis BEA website. Real GDP levels are from section 1.1.6, aggregate wages Compensation of employees paid, dividends Net dividends and investments Gross private domestic investment are from section 1.10 and As a second and broader measure of shareholders compensation I also consider aggregate corporate profits after tax Profits after tax with inventory valuation and capital consumption adjustments from section Data are collected at yearly frequency since 1932 to I also consider a measure of the aggregate leverage ratio, computed as in Belo et al. 2014, from the Flow of Funds Accounts of the US Board of Governors of the Federal Reserve System table B.102. Data of excess returns on the S&P500 are from Robert Shiller s webpage. B. Shares of GDP Table I summarizes a number of statistics about GDP growth rates as well as both growth rates and GDP shares of wages, investments, corporate profits and net dividends. Insert Table I about here Some properties are noteworthy. First, as shown in panel A, the growth rates of wages are smooth like those of GDP, whereas the growth rates of net dividends as well as corporates profits and investments are substantially more volatile. Second, the labor-share is larger, smoother and more persistent than the dividend- and investment- shares. Moreover, the labor-share is counter-cyclical: its variations are negatively related with GDP growth rates. Instead, corporate profits move pro-cyclically and net dividends and investments do not feature a correlation significantly different from zero. Third, panel B and C show that the GDP shares are stationary. On the one hand, they all feature negative coefficient in the regressions of the share differences on their lagged levels. On the other hand, the levels of GDP, wages, investments, corporate profits and net dividends are cointegrated and share a unique stochastic trend: a Johansen tests indicates that there are four cointegrating equations in a vector error-correction model VECM. 14 Figure 2 shows the times series of the labor-share, the investment-share and the aggregate leverage-ratio as well as their scatter plots with the dividend-share, the profit-share and the GDP growth rates. We can observe a clear negative relation between the labor-share and the dividend-share as well as between the labor-share and the growth rates of GDP: correlations are respectively -.67 and -.26 consistently with Ríos-Rull and Santaeulàlia-Llopis Insert Figure 2 about here The above results support the model assumption about income insurance commented in the Introduction. Workers benefit of an insurance from shareholders which smoothes wages but makes dividends distributions more volatile; moreover, such an insurance protects wages in bad times as implied by the strong persistence and counter-cyclical dynamics of the labor-share. Finally, cointegration suggests that the insurance mechanism concerns the transitory component of GDP, whereas both workers and shareholders are subject to permanent shocks. If this interpretation of the data is correct, we should observe a coherent term-structure effect on the riskiness of both wages and dividends, as shown below. 13 Quarterly data from 1947 to 2012 are used to compute with higher precision the term-structures of variance ratios, but similar results obtain using annual data. 14 Reported results make use of the Johansen s trace statistics but also the maximum eigenvalue statistic and the minimization of information criteria confirm a unique common stochastic trend. 6

8 C. Variance ratios Now the focus turns on the term-structures of growth rates volatility. Consider at first the GDP: the term-structure of its variance ratios VR s is close to one. This implies that GDP variance increases linearly with the observation interval i.e. the volatility over n intervals is about n times larger than the volatility over one interval. The upper panels of Figure 3 reports the term-structures of VR s for GDP, wages, investments, corporate profits and net dividends, computed accounting for heteroskedasticity and overlapping observations. Insert Figure 3 about here The VR s of wages are larger than one and increasing with the horizon. Instead the VR s of net dividends as well as corporate profits and investments are lower than one and strongly decreasing with the horizon, consistently with Beeler and Campbell 2012 and Belo et al These ratios reach a positive long-run limit, consistently with cointegration. The interpretation is as follows. The term-structure of GDP risk is flat because the upward-sloping effect of the permanent shock e.g. time-variation in expected growth is approximately offset by the downward-sloping effect of the transitory shock. 15 Wages load less on the transitory shock such that the upward-sloping effect dominates and the VR s increase with the horizon; whereas dividends as well as profits and investments load more on the transitory shock such that the downward-sloping effect dominates and the VR s are decreasing. These term-structure effects are exactly consistent with the main model mechanism. Indeed, income insurance from shareholders to workers concerning the transitory shocks of GDP induces an asymmetry in the short-run risk of wages and dividends. Wages have to be less subject to short-run risk than GDP, such that the long-run risk dominates and the VR s are increasing. Vice-versa, dividends should load more short-run risk than GDP, such that the long-run risk is more than offset and decreasing VR s obtain. The middle panels of Figure 3 show two patterns. First, GDP minus wages feature downward-sloping VR s similar to those of dividends. This suggests that wages are the main determinant of the timing of dividend risk, whereas alternative channels, such as financial leverage, can have only a second order effect. Second, VR s of consumption are slightly increasing and lie between those of wages and GDP: this obtains because wages weigh more to consumption than to GDP. Approximating consumption with wages plus dividends as I do in the model is safe from a term-structure perspective: VR s feature a similar upward-sloping shape. As it will be shown later, in the model the excess volatility of dividends due to income insurance is increasing in the ratio of the workers remuneration over the shareholders remuneration: L/D. On the one hand, a negative transitory shock increases and decreases the fraction of resources devoted, respectively, to workers and shareholders. On the other hand, a negative transitory shock also enhances the short-run risk of dividends. The lower left panel of Figure 3 shows the time-series of the wages-to-dividends ratio. In order to test the above mechanism, I divide the 1947:2012 quarterly sample in three sub-samples of equal length. In the middle sub-sample the L/D ratio is in average substantially higher than in both the initial and final sub-samples. Consistently with the model, we should expect that in the middle sub-sample the decreasing term-structure of dividend risk would be steeper. The lower right panel shows the VR s of dividends. The red line, which denotes the middle sub-sample, is clearly steeper than the other two term-structures, documenting that dividend risk strongly shifts toward the short-horizon when the wages-to-dividends ratio is high. This result further supports the idea that income insurance is a main driver of the timing of dividend risk. 15 The alternative hypothesis that the term-structure of GDP risk is flat because GDP is indeed a random-walk with i.i.d. increments is not consistent with the upward-sloping term-structure of wage risk, given cointegration and the stationarity of the labor-share. 7

9 D. Determinants of the dividend-share To provide further support to the income insurance mechanism, the focus turns on the determinants of the dividend-share. The term-structure of GDP risk is approximately flat and, hence, the downward-sloping term-structure of dividend risk should be imputed to the dividend-share. The latter is the shareholders remuneration and can be interpreted as a residual choice of the representative firm with respect to the remuneration of human capital as well as the investment and financing decisions. Therefore, I investigate both the contemporaneous and intertemporal relationship between the dividend-share D/Y and the wage-share L/Y, investment-share I/Y and aggregate leverage ratio Lev. Table II reports the estimation results from the following regressions: D i /Y t+ j = b 0 + b 1 Lev t + b 2 I/Y t + b 3 L/Y t + ε t, j {0,1}, i {1,2}. Both net dividends D 1 and after tax corporate profits D 2 are used as measures of shareholders remuneration. Insert Table II and III about here The investment-share is never significant, whereas the leverage ratio shows some explanatory power about the share of corporate profits but is essentially uncorrelated with dividends. Instead, the labor-share negatively and significantly explains both dividends and profits in both the contemporaneous and the intertemporal regression. Moreover, this negative relationship and its statistical significance do not weaken when the three independent variables are considered simultaneously. In addition, standardized coefficients suggest that the economic significance of the correlation between labor-share and dividends is more than twice that of leverage about -.50 versus The results of Table II clearly support the negative relationship between labor- and dividend- shares implied by the main model mechanism of income insurance. To provide further robustness, I proceed as follows. At first, I regress either the shares of net dividends or corporate profits on their lags as well as the lags of the labor-share, the investment-share and the leverage ratio: D i /Y t = b 0 + j=1,2 b0 j D i /Y t j + b 1 j Lev t j + b 2 j I/Y t j + b 3 j L/Y t j + εt, i {1,2}. Then I test for Granger causality. Table III reports the estimation results. When I control for the lags of the dependent variable, the leverage-ratio is not significant, whereas the labor-share explains both the future dividend- and profit- shares the economic significance of the first lag is about The investment-share shows some explanatory power about corporate profits but its correlation with dividends is essentially zero. As a result, testing for Granger causality, we observe that the labor-share Granger-causes the dividend-share while leverage-ratio and investment-share do not. Corporate profits are Granger-caused by both the labor-share and the investment-share but not by the leverage ratio. Tables II and III support the idea that the labor-share, by affecting level of the dividend-share, also determines the properties of the term-structure of dividend risk. E. Long-horizon predictability While the previous analysis is supportive of the role of income insurance in the determination of the dividend policy and its term-structure properties, now I look at additional testable implications. As long as the labor-share affects the dividend-share, I expect that the labor-share has some explanatory power of future dividend growth. Therefore, I consider long-horizon predictability regressions of dividend growth on the leverage-ratio, the investment-share and the labor-share. Table IV reports the results for 1, 5, 10, 15 and 20 years horizons. Insert Table IV about here 8

10 In accord with Belo et al. 2014, the aggregate leverage-ratio shows a substantial explanatory power of future dividend growth. However, also the investment-share and the labor-share strongly explain future dividends over long horizons. Moreover, standardized coefficients show that the economic significance is quite comparable. When the three independent variables are jointly included in the regression equation, we observe that the labor-share is still significant. Therefore, the channels through which leverage and labor-share predict dividend growth are distinct and do not offset each other. These results support the model mechanism of income insurance which implies a positive relationship between the labor-share and the expected dividend growth. Indeed, the latter is decreasing in the transitory shock of GDP, whereas the labor-share is decreasing in the transitory shock as long as workers benefit of a hedge component due to income insurance within the firm. Differently, corporate profits are significantly less predictable than dividends, by the same independent variables. I also investigate whether the labor-share forecasts future excess stock returns. The model mechanism of income insurance has the following implication. In presence of a negative and persistent transitory shock, workers suffer from a reduction in the total resources GDP but they benefit from income insurance since the fraction of total resource devoted to their remuneration increases. Instead, shareholders suffer from both the reduction in the total resources as well as in the dividend-share. As a result, owning capital should reflect such excess of short-run risk and equity premia are expected to be negatively related with movements in the labor-share. Table V reports the results from the regression of cumulative excess market returns over 1, 5, 10, 15, and 20 years on the current level of the leverage-ratio, the investment-share and the labor-share. Insert Table V about here Consistently with the model and in accord with Santos and Veronesi 2006, labor remuneration negatively predicts excess returns at some horizons. Moreover, such a relationship survives when also the leverage-ratio and the investment-share are taken in consideration and substantial adjusted R 2 can be observed. III. The Economy A. Agents The economy consists of two classes of agents, workers w and shareholders s. Both are equipped with an utility process featuring recursive preferences as in Duffie and Epstein Given an initial consumption C, the utility at each time t is defined as J t = E t u t f C u,j u du, t 0, 1 where E t is the conditional expectation operator under full, public and rational expectations and f c,v is an aggregator function. J t is the unique solution to the SDE: dj t = f C t,j t 1 2 AJ tσ J σ J dt + σj db t. 2 Under usual technical conditions, I consider A = 0 and the normalized aggregator 16 given by f C,J = βχj C 1 1/ψ 1 γj 1/χ 1, 3 16 For any consumption process C L 2, utility U : L 2 R is a map defined by two primitive functions f,a, where f : R + R R and A : R R, and J is unique with boundary condition J T = 0 for T. The utility function exists and is monotone and risk averse for A 0. Let U and U be the utilities associated to f,a and f,a, then U is more risk averse than U if A A. 9

11 where χ = 1 γ 1 1/ψ, γ is relative risk aversion, ψ is elasticity of intertemporal substitution and β is the subjective time-discount factor. Power utility obtains for ψ γ 1. Hereafter, workers and shareholders utilities are denoted by UC w,t and V C s,t, where C w,t and C s,t stand for their consumption levels. I allow risk attitudes to differ as follows. Assumption 1 Preference heterogeneity. Workers feature larger relative risk aversion than shareholders: γ w > γ s. It will be shown that preference heterogeneity is crucial to both the endogenous riskiness of dividend distributions as well as the equilibrium pricing of risk. In order to emphasize the workers need for income smoothing in the context of their employment relationship, I make the following assumption. Assumption 2 Limited market participation. Workers do not participate in the financial markets, inelastically supply labor and consume their wage. Such an assumption provides a simplistic and extreme representation of workers and their permanent relation with the firm, but makes transparent the role of labor relations by avoiding alternative channels of income smoothing. Consequently, workers solve: sup UC w,0 = sup E 0 0 f C w,u,uc w,u du, 4 C w,t,n w,t C w,t,n w,t s.t. C w,t W t n w,t, n w,t 1, where W t is the time t wage and n w,t is the labor supply. The problem has solution: C w,t = W t and n w,t = 1. Shareholders act as rentiers: they consume dividends and trade securities in the financial markets. Under Assumption 2, shareholders own all traded securities, do not supply labor services and solve: sup V C s,0 = sup E 0 0 f C s,u,v C s,u du, 5 C s,t,n s,t C s,t,n s,t s.t. dq s,t = Q s,t rt + n s,t µ P t rtdt C s,t dt + Q s,t n s,t σ P,x tdb x,t + σ P,µ tdb µ,t + σ P,z tdb z,t, C s,t D t n s,t, n s,t 1. where Q s,t is the shareholders wealth process and n s,t is the wealth proportion invested in the claim on the flow of firm s dividends D t. The risk-free rate rt, the stock expected return µ P t and the volatilities σ P,x t, σ P,µ and σ P,z t have to be determined in equilibrium. The Brownian shocks B x,t, B µ,t and B z,t are defined later. Under standard concavity and differentiability conditions, the above problem has solution C s,t = D t and n s,t = 1. B. The Firm A representative firm behaves competitively and lives forever. The firm activity is simplified for the sake of tractability and to emphasize the role of labor relations, as the unique channel for income smoothing. A cash-flows process embeds the firm decisions and operations: Y t = A t FK t,n t I t GK t,n t,i t = X t Z t, 6 10

12 where the first equality is the usual profit identity and the second one captures the reduced form approach with d logx t = dx t =µ t σ 2 x/2dt + σ x db x,t, 7 dµ t =λ µ µ µ t dt + σ µ db µ,t, 8 d logz t = dz t =λ z z z t dt + σ z db z,t. 9 Aggregate risk is then described by an integrated process X t and a stationary process Z t. Long-run growth µ t is a stationary and mean-reverting process. All the above processes are assumed to be homoscedastic only for the sake of simplicity. The sub-cases Y t = X t and Y t = Z t are the exogenous processes usually considered respectively in finance literature, such as in Bansal and Yaron 2004, and real business cycle literature, such as in Danthine and Donaldson 1992, Here, permanent and transitory shocks are jointly taken into account to study a flexible and general class of term-structure models. Namely, the simplest case Y t = X t with µ t = µ implies a flat term-structure for both growth and volatility of logy t. Instead, persistent fluctuations in µ t and z t lead respectively to upward- and downward-sloping term-structures, whereas their joint dynamics leads to non-monotone term-structures. The cash-flows process also determines the resources to be distributed to the workers and the shareholders: Y t = W t + D t. 10 This resource constraint captures the endogenous nature of wages and dividends due to labor relations. C. Labor Relations The standard Walrasian Cobb-Douglas economy leads to constant capital and labor shares consistent with workers paid at their marginal productivity: W t = αy t, with α 0,1. The opposite extreme scenario would be the case where workers are promised perfect income smoothing: wages are constant and potentially equal to the unconditional expected marginal productivity: W t = W = E[αY t ]. 17 Here, I postulate that workers and shareholders agree in advance on an optimal risk sharing rule. Namely, a contract C is designed to effect the optimal risk sharing between workers and shareholders: wages smoother riskier than output obtain at the cost of riskier smoother dividends. A simple and intuitive approach is as follows. The two agent types arrange an agreement such that wages and dividends share the same long-run dynamics of total resources i.e. Y t, W t and D t are cointegrated but the former feature an insurance to transitory shocks, implying a levered exposition of the latter. Assumption 3 Income Insurance. The contingent wage payments and dividend distributions are governed by the following sharing rule: { } C = W t,d t : W t = X t W t φ, D t = X t D t φ, W t + D t = Y t t, 11 where W t φ = αz t 1 φ and D t φ = Z t αz t 1 φ, 12 α 0,1 and φ 0,1 denotes the optimal degree of income insurance. Hence, under Assumption 2 in equilibrium, 17 The second equality can only hold under the assumption of stationarity for Y t. C w,t = X t W t φ, and C s,t = X t D t φ

13 The optimal degree of income insurance φ represents the smoothing effect of labor relations on wages and the corresponding leverage effect on dividend distributions. Namely, wages are a concave function of Z t for φ > 0, implying a countercyclical labor share W/Y, as suggested by real data. Instead, wages reduce to the standard Walrasian case with constant labor-share for φ 0. The degree of income insurance is constrained in the interval 0,1 because φ < 0 implies a counter-factual income insurance from workers to shareholders and a pro-cyclical labor-share, whereas φ > 1 implies wages decreasing in the aggregate shock Z t, which appears implausible. The next Lemma provides a foundation of the income insurance from shareholders to workers. Lemma 1 Contract Rule and Bargaining Power. The optimal degree of income insurance from shareholders to workers satisfies the following objective: where φ = 1 + 2λ z z+logα σ 2 z φ = arg max φ 0,1 J φ = arg max Ez [ Wφ] ϑ Ez [ Dφ] 1 ϑ = max0,min1, φ, 14 φ 0,1 + 2 σ 2 z q0 + q 1 q 2, with q 0,q 1 and q 2 derived in Appendix A. The objective J φ is intended to reflect the relative bargaining power of the two parties and, jointly with the contract rule of Eq. 11, determines simultaneously both their consumption shares. Namely, Eq. 14 can be interpreted as the outcome of bargaining in labor negotiations that unions and firms accomplish in behalf of workers and shareholders. 18 The bargaining parameter ϑ is set: ϑ = αγ w αh = αγ w + 1 αγ s 1 + αh 1, 15 and is determined by both heterogeneity in risk aversion, h = γ w /γ s, and the consumption shares in the Walrasian benchmark, α i.e. the model with constant labor-share W/Y, which obtains for h = 1. Namely, ϑ is increasing in h, in spirit of the risk sharing model of Danthine and Donaldson 1992, 2002: h > 1 leads to an insurance wage component upon the labor productivity level αy. The optimal degree of income insurance is increasing in the heterogeneity in risk aversion, h, and in the Walrasian benchmark, α: h φ > 0, α φ > 0, if γ w > γ s. As a consequence, income insurance described by Assumption 3 and Lemma 1 leads to smooth wages and risky dividends distributions with the additional feature of counter-cyclical labor-share. However, the endogenous cointegration among Y, W and D implies that distributional risk due to income insurance is a short-term risk and provides a rationale for potentially downward sloping term-structures of both dividends and equity. Figure 4 shows the objective function and the optimal degree of income insurance in the baseline calibration of the model, as in Section VI.A. as Insert Figure 4 about here. 18 Assuming workers and shareholders negotiate directly with each other, a proper formulation of a Nash bargaining solution would require to define φ φt = arg maxuc w,t ϑ V C s,t 1 ϑ φ 0,1 where ϑ 0, 1 is a bargaining power parameter and heterogeneity in risk aversion induces the incentive to bargain, given an outside option standardized to zero i.e. bargaining on the concavity of the wage schedule obtains in exchange of long-term contract arrangements and full employment. Unfortunately, such a solution is neither tractable nor stationary. Instead, the objective in Eq. 14 guarantees tractability and provides a similar result. Lifetime utilities are replaced by expectations over the stationary distribution of z t of the components of workers and shareholders consumptions affected by income insurance i.e. Wφ and Dφ and the heterogeneity in preferences is embedded in the parameter ϑ which, with a slight abuse of terminology, can be interpreted as the bargaining power of workers. 12

14 This stylized model of income insurance is intended to capture the main stylized facts needed to generate the features of dividend distributions which matter from an asset pricing perspective. 19 The smoothing effect on wages due to income insurance, as in Eq. 13, can obtain also in an indirect way. For instance, recently Shimer 2005, Gertler and Trigari 2009 and Favilukis and Lin 2012 study the importance of wage rigidity in real business cycle models. Also other deviations from the standard Walrasian labor markets, which account for the dynamics of employment, can lead to similar implications for the dynamics of dividends. 20 In this paper, I explicitly model income insurance since it represents the general and intuitive economic mechanism which explains the dynamics of dividends despite the exact friction or combination of frictions in labor markets. As it will be shown, the dividend dynamics implied by Assumption 3 is sufficient to reconcile the traditional asset pricing facts with the recent evidence about the term-structures of both cash-flows and equity returns. Finally, the model is written in a parsimonious and transparent way such that the effect of income insurance on asset prices is fully captured by a unique endogenous parameter, φ. As long as such a parameter is interpreted as exogenous, all the equilibrium asset pricing implications characterized in the paper continue to hold even if alternative or complementary explanations for the dividend dynamics are taken into consideration such as financial leverage, as in Belo et al. 2014, and heterogeneous investment risk, as in Ai et al IV. The Equilibrium The equilibrium for the economy in Section III is a pair D t and W t which jointly satisfies Eq. 6, 10 and 13. Dividend distributions are optimal to shareholders, given the contract rule and the resources constraint. A. Value function and state price density In order to preserve both analytic tractability and the economic meaning of primitive parameters, consider the approximation based on the linearization of the logarithm of the dividend process: logd t logx t + log D + z logd z= z z t z = x t + d 0 + d z z t, 16 where log D = d 0 + d z z = log e z α 1 φ e 1 φ z captures the steady-state of Dφ, and d z satisfies: d z = 1 + φ α α φ e z α. 17 Notice that under the log-linearization, as usual, the dividend process is still increasing in the states x t and z t but inherits their homoscedasticity. Preferences in Eq. 2-3 and the log-linearized dividend dynamics of Eq. 16 guarantee a model solution which emphasizes the role of the state-variables µ t and z t in the formation of prices but preserves analytic tractability. A first order approximation of the shareholders consumption-wealth ratio around its endogenous steady state provides closed form solutions for prices and return moments up to such approximation. In particular, I extend Hore 2008, who focuses on the case EIS = 1, and follow Benzoni, Collin-Dufresne, and Goldstein 2011 and Marfè Even if the objective in Eq. 14 and the bargaining parameter in Eq. 15 could look somewhat arbitrary, they allow for a clear economic interpretation and transparent asset pricing results at equilibrium. Slightly different specifications of Lemma 1 could easily lead to untractable asset pricing implications or even to non-stationary equilibria without either altering the main economic mechanism of the model or providing additional insights. This would be the case of the model by Danthine and Donaldson 2002, as long as cash-flows are cointegrated a crucial feature of the data. 20 See the recent works by Kuehn, Petrosky-Nadeau, and Zhang 2012 and Donangelo

15 Proposition 1 The shareholders utility process under preferences in Eq. 3 and dynamics in Eq is given by where u 0, JX t,µ t,z t = 1 1 γ s X 1 γ s t expu γ s d 0 + u µ µ t + u z + 1 γ s d z z t, 18 u µ = 1 γ s e cq, u z = λ zγ s 1 φ α + λ µ e cq λ z α φ e φ z, α and cq = E[cq t ] are endogenous constants depending on the primitive parameters derived in Appendix B. The shareholders consumption-wealth ratio is equal to cq t = logc s,t /Q s,t = logβ χ 1 u 0 + u µ µ t + u z z t. 19 As usual, the consumption-wealth ratio reduces to β when ψ 1 and equals the market dividend-price ratio under the limited market participation of Assumption 2. Coefficients u 0 and {u µ,u z } determine respectively the unconditional level of the consumption-wealth ratio and both the prices of risk and the growth rate of wealth. From the shareholders perspective, that is the point of view of market participants, the marginal utility evaluated at optimal consumption is the valid state-price density Duffie and Epstein 1992: ξ t,u = e u t f V C s,τ,v C s,τ dτ f C C s,u,v C s,u f C C s,t,v C s,t Hence, the price of an arbitrary payoff {F u,u t, } is given by E t [ t ξ t,u F u du]. Proposition 2 The equilibrium state price density ξ 0,t satisfies:, u t. 20 e z t Y t = I C [ξ 0,t ] e d, 21 0+d z z t where I C [ξ 0,t ] = { C s,t : ξ 0,t = e t 0 f J sds f C t X t,µ t,z t } denotes the time t shareholders optimal consumption implied by ξ0,t. In absence of income insurance φ 0, we have: Y t = I C [ξ 0,t ]1 α 1. Although the state price density equals the first order condition of shareholders, it depends on the risk attitudes of both agent types. Eq. 21 represents the equilibrium state-price density by means of the resource constraint of Eq. 10. The left hand side is given by the exogenous flows of total resources produced by the firm. The right hand side is given by the product of two terms: the former is the optimality condition for market participants, that is the shareholders; the latter is a time-varying term which captures the distributional risk due to income insurance. Namely, such a term equals the inverse of the dividend share, i.e. Y t,/d t, and its variability increases with φ. Instead, in absence of income insurance φ 0, the model reduces to the Walrasian benchmark and, hence, the labor-share is constant and equal to α: only shareholders preferences matter and the second term on the right hand side of Eq. 21 reduces to the constant 1 α 1 = Y t /D t. Under CRRA preferences ψ γ 1 s, the optimality condition for the shareholders takes the usual power form: indeed, I C [ξ 0,t ] = ξ 0,t e βt 1/γ s and the term capturing the distributional risk is unchanged. Instead, for φ 0, the equilibrium state-price density reduces to ξ 0,t = e βt 1 αy t γ s, as in a Lucas economy. Notice that, even if the equilibrium state-price density is an involved function of the integrated process Y t, the economy is characterized by a stationary equilibrium: this is a necessary condition to produce realistic testable implications but usually fails to hold in models of preference heterogeneity. Whereas many real business cycle models circumvent the problem by excluding permanent shocks e.g. Y t = Z t, allowing for both permanent and transitory shocks e.g. Y t = X t Z t as in Eq. 6 14

16 and still obtaining a stationary equilibrium is of crucial importance in order to study the equilibrium implications for the term-structure of equity. Proposition 3 The equilibrium state price density has dynamics given by dξ 0,t =ξ 0,t d f C f C + ξ 0,t f J dt = rtξ 0,t dt θ x tξ 0,t db x,t θ µ tξ 0,t db µ,t θ z tξ 0,t db z,t, 22 where the instantaneous risk-free rate satisfies rt = r 0 + r µ µ t + r z z t, 23 with r 0 derived in Appendix B, r µ = 1 ψ, and the instantaneous prices of risk are given by r z = λ z φ α 1 + ψ α φ e φ z, α θ x t = X f C f C X t σ x = γ s σ x, 24 θ µ t = θ z t = µ f C f C σ µ = γ s 1/ψ z f C f C σ z = e cq σ µ, 25 + λ µ γ s λ zγ s 1/ψ φ α e cq λ z α φ e φ z σ z. 26 α The risk-free rate is an affine function of the states µ t and z t. The coefficients r µ and r z are respectively positive and negative and both decrease in magnitude with ψ, as usual under recursive preferences. Moreover, r z increases in magnitude with the degree of income insurance φ. The permanent shock commands a price of risk θ x t, which has the usual form and is a price for transient risk, that is for the contribution of x t to the instantaneous volatility of shareholders consumption. Long-run growth commands a price of risk θ µ t, which has the usual form in long-run risk models. This is a price for non-transient risk, that is a price for the contribution of µ t to the variation in the continuation utility value of shareholders. Hence, θ µ t disappears under power utility ψ γ 1 s. Such a price of risk is increasing with the degree of early resolution of uncertainty γ s 1/ψ and decreasing in magnitude with rate of reversion in long-run growth. The transitory shock leads to a price of risk θ z t, which has two components. The first, γ s d z σ z, is a positive price for transient risk, that is the contribution of z t to the instantaneous volatility of shareholders consumption. The second, θ z t γ s d z σ z, is a price for non-transient risk, that is a price for the contribution of z t to the variation in the continuation utility value of shareholders. Namely, at the opposite than θ µ, the latter is negative and increasing in the rate of reversion of z t under preferences for the early resolution of uncertainty. Also, such a term disappears under power utility. Interestingly, both the components of θ z t are proportional to the coefficient d z > 1, which is increasing in φ. Therefore, under the usual parametrization γ s > ψ > 1, income insurance leads to a positive price for its effect on the current dividend dynamics and a negative price for its effect on the evolution of shareholders utility. 15

17 B. Endogenous dividends and wages Dividends and wages evolve endogenously at equilibrium. Their dynamics is characterized by the resource constraint and the contract rule, which lead to 21 D t = X t D t φ = X t Z t αz t 1 φ and W t = X t W t φ = X t αz t 1 φ. 27 Both D t φ and W t φ are stationary functions of the state Z t, whereas Y t, D t and W t share the integrated factor X t and, hence, are cointegrated in equilibrium. Dividends and wages can be written in terms of the labor-share ωz t = W t /Y t : D t = X t Z t 1 ωz t, and W t = X t Z t ωz t, with ωz t = ααe z t φ, 28 which is the Walrasian benchmark α times an adjustment factor decreasing and convex in z t. The latter captures the effect of labor relations and reduces to one if φ 0. Since φ 0,1, it follows that dividends and wages are both increasing in z t, but the former are convex and the latter are concave in z t. The growth rates of total consumption, dividends and wages can be written as d logy t =µ Y dt + σ Y x db x,t + σ Y z db z,t = µ t σ 2 x/2 + λ z z z t dt + σ x db x,t + σ z db z,t, 29 d logd t =µ D dt + σ D x db x,t + σ D z db z,t = µ t σ 2 x/ φ α α φ e φ zt α λ z z z t φ 2 α 1+φ e φ zt σ 2 α φ e φ zt α 2 z /2 dt + σ x db x,t φ α σ α φ e φ zt α z db z,t, d logw t =µ W dt + σ W x db x,t + σ W z db z,t = µ t σ 2 x/2 + 1 φ λ z z z t dt + σ x db x,t + 1 φ σ z db z,t. 31 Lemma 2 Cash-flows persistence, smoothing and leverage. i The instantaneous expected growth rates with respect to µ t and z t satisfy: 22 µ µ W = µ µ Y = µ µ D = 1, 30 and z µ W < z µ Y = z E[dz t ]/dt < z µ D, 32 which holds with equalities for φ 0. ii The instantaneous volatilities of growth rates associated to permanent shocks are state-independent and satisfy σ W x = σ Y x = σ D x = σ x, 21 Notice that for low values of Z t, D t φ could become negative: this can be avoided by setting Z t > 1 t e.g. if z t > 0 is a square-root process. However, for reasonable parameters, the Ornstein-Uhlenbeck dynamics for z t in Eq. 9 guarantees that the probability of D t < 0 is negligible.001% and provides tractability. Hence, throughout the paper I assume D t φ > 0. Morover, such an issue does not arise in the equilibrium solution of the previous section, which makes use of the approximation in Eq. 16, provided z The second inequality in Eq. 32 holds for σ 2 z not too large. Since the labor-share ωz t is very smooth in the real data, Eq. 32 is satisfied for reasonable parameters. 16

18 and the instantaneous volatilities of growth rates associated to transitory shocks satisfy which holds with equalities for φ 0. σ W z < σ Y z = σ z < σ D z, 33 iii The instantaneous volatilities of growth rates with respect to the degree of income insurance satisfy φ σ W z < 0, φ σ D z > 0. Total consumption, dividends and wages share the same exposition to long-run expected growth since labor relations do not alter the impact of permanent shocks on either workers or shareholders consumption. Expected consumption growth also depends on z t : the rate of reversion λ z equals the sensitivity to z t of the instantaneous expected growth rate of consumption, z µ Y = λ z. Instead, the persistence of the expected growth rates of dividends and wages is altered by income insurance: the larger φ, the larger the rate of reversion of d logd t i.e. z µ D > λ z and the smaller the rate of reversion of d logw t i.e. z µ W < λ z. Therefore, the stronger the income insurance, the stronger the persistence of wages and the weaker that of dividends. Similarly to long-run expected growth, total consumption, dividends and wages share the same exposition to permanent shocks, σ x. Instead, the exposition of dividends and wages to transitory shocks is affected by the degree of income insurance. The volatility of wages associated to transitory shocks is constant but lower than that of consumption σ W z < σ z and decreasing in the degree of income insurance. Therefore, income insurance induces a smoothing effect on wages at the cost of more volatile dividends. Instead, the volatility of dividends associated to transitory shocks is state-dependent: such a heteroscedasticity is an endogenous result of income insurance. 23 σ D z > σ z : Dividends are more volatile than total consumption σ z < σ D z = 1 + φ α σ α φ e φ zt α z = 1 + φ ωz t 1 ωz t σ z. 34 Such an excess volatility or leverage is proportional to both the optimal degree of income insurance φ and the wages-todividends ratio: ωz t /1 ωz t. The leverage effect increases with workers bargaining power. The larger the labor-share, the more volatile is the instantaneous dividends dynamics i.e. ω σ D z > 0. Hence, dividends volatility is endogenously countercyclical in the transitory component of total consumption i.e. z σ D z < Figure 5 shows the volatilities of total consumption, dividends and wages due to the transitory shock respectively in black, red and blue under the baseline calibration of Section VI.A. Insert Figure 5 about here. In the first panel, volatilities are plotted as a function of z t : we can observe the smoothing effect on σ W z and the leverage effect on σ D z as well as the counter-cyclical dynamics of the latter. Instead, the second panel shows the volatilities as a function of the labor-share and, hence, the increasing dynamics of σ D z. Finally, the third panel plots the three volatilities as a function of φ at the steady-state z t = z: the larger the degree of income insurance, the stronger both the smoothing and the leverage effects respectively on wages and dividends. 23 The state-dependent dynamics of σ D z is ruled out by the log-linearization of Eq. 16. Section VII.B solves analytically for the equilibrium under power utility ψ γ 1 s : in such a case the log-linearization of Eq. 16 is not needed and the endogenous heteroscedasticity is preserved for both dividends and equity returns dynamics. 24 This mechanism is similar to that of an economy with preference heterogeneity but without limited market participation: in good times wealth shifts from high risk averse agents towards low risk averse agents i.e. from workers to shareholders, the aggregate risk aversion decreases too and so does the price of risk. The opposite holds in bad times. Here, labor relations lead to a similar countercyclical dynamics of the aggregate price of risk but this mechanism obtains through the endogenous dynamics of dividends instead of the aggregate risk aversion. 17

19 The focus now turns on the term structure of cash-flows and in particular on the slope of growth rates volatility. For the sake of tractability I make use of the log-linearized dynamics of dividends as in Eq The main quantity of interest is the moment generating function of the logarithm of dividends. We need to compute the following expectation. Corollary 1 The moment generating function of the logarithm of dividends has the following approximation: D t τ,n = E t [Dt+τ] n e nx t+b 0 n,τ+b µ n,τµ t +B z n,τz t, 35 where n and model parameters are such that the expectation exists finite, the approximation makes use of Eq. 16 and B 0,B µ and B z are deterministic functions of time derived in the Appendix B. With this result in hand, the term structures of expected dividend growth and volatility are computed as 26 g D t,τ = 1 τ log Dt τ,1 D t 0,1 The following relations automatically obtain: and σ D t,τ = 1 τ log Dt τ,2 D t τ, g D t,τ =ḡ D,τ + g D,τ,µ µ t + g D,τ,z z t 37 σ 2 Dt,τ =v D,τ,x σ 2 x + v D,τ,µ σ 2 µ + v D,τ,z σ 2 z 38 with g D,τ,µ > 0, g D,τ,z < 0, τ v D,τ,x = 0, τ v D,τ,µ > 0 and τ v D,τ,z < 0. Notice that, first, g D t,τ is respectively increasing and decreasing in µ t and z t ; second, µ t and z t lead respectively to an upward-sloping and a downward-sloping effect on the growth rates volatility. Interestingly, the stronger the degree of income insurance, the more pronounced the downward-sloping effect: φ τ v D,τ,z > 0. Therefore, income insurance enhances the strength of the transitory shock and leads to an excess of short-run risk in dividends distributions with respect to total consumption. Such an effect is the model counterpart of the shift of dividend risk toward the short horizon in comparison with GDP risk, as documented in Section II and in Figure 3. The transitory shock affects the level of the short-run limits τ 0 of g D t,τ and σ 2 D t,τ, whereas long-run growth affects the level of the long-run limits τ : however, both contribute positively to such limits. Therefore, the term-spreads at the steady state g D t, g D t,0 = 1 2 σ 2 µ d 2 λ 2 z σ 2 z µ and σ 2 Dt, σ 2 Dt,0 = σ2 µ λ 2 µ d 2 z σ 2 z, can be either positive or negative depending on the model parameters. In particular, negative spreads can obtain if the leverage coefficient d z is large enough, that is if income insurance makes dividends highly risky in the short-run. Furthermore, it is possible to compute the term-structures of correlations in closed form. Total consumption and dividend changes correlate over time satisfying: ρ Y,D t,τ = 1 τσ Y t,τσ D t,τ log YDt τ,1,1, Y t τ,1d t τ,1 25 The term structures of dividend expected growth rates and volatility can be derived in closed-form even under the exact dynamics of Eq. 27, as shown in Appendix B; however, such results are more involved and less intuitive. 26 Also, expectations W t τ,n and Y t τ,n and term-structures {g W t,τ,σ W t,τ} and {g Y t,τ,σ Y t,τ} can be computed respectively for wages and total consumption. 18

20 where YD t τ,n,m = E t [Y n t+τd m t+τ]. Such a correlation is equal to one for any τ when φ 0, whereas it is U-shaped when φ > 0 but converges to one in the long-run, given the cointegration relation. Income insurance also alters the volatility and correlations of the dividend- and labor- shares i.e. D t /Y t = 1 ωz t and W t /Y t = ωz t : σ 1 ω t,τ = σ ω t,τ = 1 τ log Et [1 ωz t+τ 2 ] 1, ρ Y,1 ω t,τ = τσ Y t,τσ 1 ω t,τ log Et [Y t+τ 1 ωz t+τ ] Y t τ,1e t [1 ωz t+τ ] 1 τ log Et [ωz t+τ 2 ] 1 E t [ωz t+τ, ρ ] 2 Y,ω t,τ = τσ Y t,τσ ω t,τ log Et [Y t+τ ωz t+τ ] Y t τ,1e t [ωz t+τ ], E t [1 ωz t+τ ] 2 In the Walrasian benchmark φ 0, both volatilities and correlations are equal to zero at any horizon. Instead, under income insurance φ > 0, we have the following results. Workers and shareholders consumption shares are not constant with 0 < σ ω t,τ < σ 1 ω t,τ, and zero long-run limits: σ ω t, = σ 1 ω t, = 0. The shares are respectively counter-cyclical and pro-cyclical: ρ Y,ω t,τ < 0 < ρ Y,1 ω t,τ, monotonically decreasing with τ in magnitude and with zero long-run limits: ρ Y,ω t, = ρ Y,1 ω t, = 0. The sign of such correlations is an endogenous result of income insurance, but also an important feature of the data, as shown in Section II which standard models usually fail to generate. V. Equilibrium Asset Prices A. Equilibrium dividend strips Proposition 4 The equilibrium price of the market dividend strip with maturity τ is given by P t,τ = E t [ξ t,t+τ D t+τ ] X t e A 0τ+A µ τµ t +A z τz t, 39, where model parameters are such that the expectation exists finite, the approximation makes use of Eq. 16 and A 0,A µ and A z are deterministic functions of time derived in the Appendix B. The price of the dividend strip is linear in the permanent shock X t and exponential affine in the states µ t and z t. Hence, the strip s price-dividend ratio is a stationary function of the states: log P t,τ D t = A 0 τ d 0 + A µ τµ t + A z τ d z z t. The functions A µ τ and A z τ are respectively the semi-elasticity of the price with respect to µ t and z t : A µ τ = µ logp t,τ = 1 e λ µτ 1 1/ψ, 40 λ µ 1 A z τ = z logp t,τ = ψ 1 e λ zτ + e λ φ zτ α 1 + α φ e φ z. 41 α Two properties are worth noting. First, A µ τ increases with ψ, whereas A z τ decreases with ψ. Second, A z τ increases with the degree of income insurance φ. Therefore, the leverage effect on dividend distributions due to income insurance also 19

21 affects prices: namely, such an effect is amplified for ψ < 1 and vice-versa. Finally, for ψ 1, A µ τ = 0 and A z τ = d z and, hence, the strip s price-dividend ratio reduces to a state-independent function of the horizon τ. 27 Proposition 5 The instantaneous volatility and premium on the dividend strip with maturity τ are given by σ P t,τ = σ 2 x + 1 e λ µτ 2 ψ 1 2 λ 2 µψ 2 σ 2 µ + e 2λzτ ψ + e λzτ 1 2 ψ 2 dz 2 σ 2 z, 42 µ P rt,τ =γ s σ 2 x + 1 e λ µτ 1 1/ψγ s 1/ψ λ µ e cq σ 2 µ + + λ µ 1 ψ 1 e λzτ + e λ zτ λ z + ψγ s e cq ψe cq dz 2 σ 2 z λ z The permanent shock as well as the states µ t and z t contribute to the return volatility and command a premium. Permanent shocks do not lead to excess volatility. Instead, the expositions to the states µ t and z t, are proportional to the fundamentals volatilities σ µ and σ z, but also depend on the horizon τ, the elasticity of intertemporal substitution and the degree of persistence of the states. Namely, the exposition to long-run growth is increasing in ψ and decreasing in the rate of reversion λ µ. The reverse holds for the exposition to the transitory shock, which is decreasing in ψ and increasing in λ z. The latter is also amplified by the coefficient d z which captures the leverage effect due to income insurance. The premium on the dividend strip is given by the sum of the compensations to the three shocks of the model. The compensation for the permanent shock is positive and has the usual form: γ s σ 2 x. Such a premium is a compensation for transient risk. Instead, the compensations associated to the states µ t and z t depend also on the horizon τ, the elasticity of intertemporal substitution and the persistence of the states. Long-run growth commands a premium which is increasing in ψ and decreasing in the rate of reversion λ µ : γ s 1/ψ λ µ e cq +λ µ A µτσ 2 µ. 44 Such a term is compensation for non-transient long-run risk, that is the contribution of µ t to the variability of the continuation utility value of shareholders. If the intertemporal substitution effect dominates the wealth effect and shareholders have preferences for the early resolution of uncertainty i.e. the usual parametrization γ s > ψ > 1 long-run growth leads to a positive premium on the dividend strip. The premium for the exposition to the transitory shock z t is given by the sum of two terms: γ s A z τd z σ 2 z, and λ zγ s 1/ψ e cq +λ z A z τd z σ 2 z. 45 The former term represents the compensation for transient risk. Such a premium is always positive and decreases with ψ. Instead, the latter term is the compensation for non-transient risk. This premium for non-transient short-run risk is decreasing in ψ and is negative positive under the shareholders preference for the early late resolution of uncertainty. Moreover, both terms depend on the horizon τ and are proportional to d z, and, hence, are increasing in magnitude in the degree of income insurance. Panel A of Table VI shows the signs of transient and non-transient components of dividend strips premia under several scenarios, which are all subcases of exogenous uncertainty in Eq. 6. Insert Table VI about here. Namely, I consider the cases of a an i.i.d. economy Y t = X t,µ t = µ, b an economy with only permanent shocks and time-varying long-run growth Y t = X t in spirit of long-run risk literature, c an economy with only transitory shocks Y t = Z t in spirit of real business cycle literature, and finally d and e the economies with both permanent and transitory shocks 27 Since the approximation of Eq. 16 leads to exponential affine prices, it also rules out state-dependence of the return moments. Therefore, the dividend strip s return moments only depend on the maturity. 20

22 Y t = X t Z t with and without time-varying long-run growth. Transient risk always requires a positive compensation, whereas non-transient long-run risk can command a positive premium under preference for both early or late resolution of uncertainty depending on ψ. Finally, non-transient short-run risk leads to a positive or negative premium under preference for respectively late or early resolution of uncertainty. Therefore, the whole premium has sign depending on parameters associated to both exogenous uncertainty as well as preferences. Moreover, the premium is not necessarily monotone in both γ s and ψ. Corollary 2 The slopes of the term-structures of dividend strips volatility and premia are given by τ σ 2 Pt,τ = 2 ψ 2 e 2λ µτ e λ µτ 1ψ 1 2 λ 1 µ σ 2 µ e 2λzτ ψ + e λzτ 1ψ 1λ z dz 2 σ 2 z, 46 γs 1/ψe λ µτ σ 2 µ τ µ P rt,τ = ψ 1 ψ 2 e cq + λ µ λ z + γ s ψe cq e λzτ λ z d 2 z σ 2 z e cq + λ z. 47 Corollary 2 provides a number of new results and sheds light on the role of preferences on the equilibrium term-structure of equity risk and premia. The slope of the term-structure of volatility depends on two terms, associated respectively to the states µ t and z t. The former is always positive despite preference parameters and, hence, contributes to an upward sloping term-structure of risk. Instead, the latter term is negative if the intertemporal substitution effect dominates the wealth effect and vice-versa. Therefore, the term-structure of volatility is monotone upward sloping if ψ < 1, whereas it is not necessarily monotone if ψ > 1. A non-monotone e.g. U-shaped term-structure of risk can obtain if the leverage effect captured by d z due to income insurance is large enough to offset, for some horizons τ, the upward sloping effect due to long-run growth. Also the slope of the term-structure of premia depends on two terms, associated to the states µ t and z t. The former is positive if the intertemporal substitution effect dominates the wealth effect and shareholders have preferences for the early resolution of uncertainty. The latter term is negative if the intertemporal substitution effect dominates the wealth effect and vice-versa. Under the usual parametrization γ s > ψ > 1, variation in long-run growth leads to an upward-sloping effect, whereas transitory shocks lead to a downward-sloping effect. Therefore, the term-structure of equity premia is not necessarily monotone as long as both permanent and transitory shocks enter the model. Such an analytical result clearly explains the failure of the standard long-run risk model to describe the recent evidence about dividend strips, that is downward sloping term-structures of equity, at least in the short-run. Indeed, the model of Bansal and Yaron 2004 rules out transitory shock z t and does a good job at matching a number of asset pricing moments as long as ψ > 1. In such a case Y t = X t, both the term-structures of volatility and premia are monotone increasing with the horizon. In the opposite scenarios, in which either only the transitory shock enters the model Y t = Z t or long-run growth is constant Y t = X t Z t with µ t = µ, monotone decreasing term-structures of risk and premia obtain for ψ > 1. However, in such a case the non-transient component of the equity premium is decreasing in ψ and the model cannot command a sizeable premium. Instead, when both µ t and z t enter the model, the sign of the slope of the term-structures depends on parameters associated to both exogenous uncertainty and preferences. As a consequence, the model can accommodate for both a high equity premium and downward sloping term-structures of equity risk and premia in the short-run. The endogenous dynamics of dividends due to income insurance plays a strong role at reconciling the above stylized facts in the context of recursive utility. Panel B of Table VI summarizes the model implications about the slope of the term-structure of equity premia under several scenarios. 21

23 Corollary 3 The instantaneous volatility and premium of the dividend strip satisfy σ 2 Pt, σ 2 Pt,0 = ψ 12 σ 2 µ ψ2 1d 2 λ 2 µψ 2 z σ 2 z, 48 ψ 2 γs 1/ψσ µ P rt, µ P rt,0 =ψ 1 2 µ ψλ µ e cq +λ µ λ z+γ s ψe cq dz 2 σ 2 z e cq +λ z. 49 ψ 2 The long-run limit of volatility depends on the price exposition to the three shocks of the model. Instead, when the maturity τ goes to zero, the volatility of the dividend strip reduces by construction to that of dividends and, hence, the term associated to µ t disappears. As a consequence, the term-spread, σ 2 P t, σ2 P t,0, is given by two terms: a term induced by variation in long-run growth, which is always positive, and a term due to transitory shocks, which is negative if ψ > 1 Therefore, a negative spread can obtain if the leverage effect in dividends due to income insurance captured by d z is large enough. Also the long-run limit of premia on the dividend strip depends on the price exposition to the three shocks of the model. Instead, when the maturity τ goes to zero, there is no compensation associated to long-run growth, since µ t does not contribute to the instantaneous volatility of dividends. Consequently, the term-spread, µ P rt, µ P rt,0, is given by two terms: a term induced by variation in long-run growth, which has sign given by the sign of ψ 1γ s 1/ψ, and a term due to transitory shocks, which has sign given by the sign of ψ 1. Therefore, similarly to volatility, a negative spread can obtain if the leverage effect of dividends due to income insurance captured by d z is large enough, provided ψ > 1. B. Equilibrium market asset Proposition 6 The equilibrium price of the market asset is given by P t = E t [ t ξ t,u D u du] X t β 1 e u 0χ 1 +d 0 +u µ χ 1 µ t + u z χ 1 +d z z t 50 where model parameters are such that the expectation exists finite, the approximation makes use of Eq. 16, and u 0,u µ and u z are endogenous constants depending on the primitive parameters derived in the Appendix B. The market price is given by the time integral of the dividend strip price over the infinite horizon: P t = 0 P t,τdτ. The market price relative to the current dividend level is a stationary function of µ t and z t : log P t D t = u 0 + u µ µ t + u z z t /χ logβ, and equals the wealth-consumption ratio of shareholders under Assumption 2. Namely, prices increase with µ t as long as the intertemporal substitution effect dominates the wealth effect and increase with z t despite preferences: µ P t 0 if ψ 1, γ s and z P t > 0, ψ,γ s. 28 Proposition 7 The instantaneous volatility and premium on the market asset are given by 1 1/ψ 2 σ P t = σ 2 x + e cq σ + λ 2 µ /ψλ 2 z µ e cq dz + λ 2 σ 2 z, 51 z µ P rt =γ s σ 2 x + 1 1/ψγ s 1/ψ e cq + λ µ 2 σ 2 µ + γ sψe cq + λ z ψe cq + λ z e cq + λ z 2 ψ 2 dz 2 σ 2 z. 52 The permanent shock x t enters the return dynamics exactly as the dividend dynamics, σ x. Instead, the price expositions to the states µ t and z t depend on the preference parameters and are respectively increasing and decreasing in ψ. The exposition to µ t always leads to an excess-volatility of market returns over the instantaneous dynamics of dividends, whereas the exposition to z t generates excess-volatility for ψ < 1 and vice-versa. However, when we look at the term-structure of dividend volatility, 28 The approximation of Eq. 16 rules out the endogenous state-dependence from return moments, given the homoscedasticity of µ t and z t. 22

24 the states µ t and z t produce respectively an upward-sloping and a downward-sloping effect. Therefore, long-run growth can lead to dividends more volatile than market returns at large horizons, 29 whereas the transitory shock can lead to long-run excess-volatility of returns over dividends even for ψ > 1. In particular the latter effect is enhanced by income insurance: the larger φ, the steeper the downward-sloping effect due to z t on the term-structure of dividend volatility. The equity premium is given by three components associated to the three shocks of the model. The permanent shock x t requires the usual positive compensation γ s σ 2 x for transient risk. Long-run growth leads to a premium, which is positive if the intertemporal substitution effect dominates the wealth effect and shareholders have preference for the early resolution of uncertainty. Moreover, such a compensation is increasing with ψ and γ s as long as, respectively, ψγ s + 1 > 2 and ψ > 1 which are satisfied for the usual parametrization γ s > ψ > 1. Instead, z t commands a premium which is always positive, decreasing with ψ and increasing with γ s. Furthermore, such a compensation term is increasing in the degree of income insurance φ, captured by d z. Finally, provided γ s > ψ > 1, the whole equity premium is increasing in γ s but non-monotone in ψ: this is due to the fact that under such a parametrization, the equilibrium pricing of µ t leads to compensations increasing with the horizon, whereas short-run but persistent uncertainty due z t requires compensations decreasing with the horizon. Once the model is calibrated with realistic parameters, such a term-structure perspective on the equity premium sheds lights about whence priced risk obtains in equilibrium. In order to answer such a question, it is useful to analyze the relative contribution of each time-horizon to the whole equity premium. Using the definition of the market asset price as the time integral of the dividend strip prices, it is possible to write the equity premium as a time integral and, hence, to derive an equilibrium density which describes the contribution of future discounted cash-flows at each time-horizon. Proposition 8 The instantaneous premium on the market asset can be written as µ P rt = 0 Πt,τdτ, 53 with density Πt,τ = βe A 0τ d 0 u 0 χ 1 +A µ τ u µ χ 1 µ t +A z τ d z u z χ 1 z t θ x tσ x + θ µ tσ µ A µ τ + θ z tσ z A z τ. The relative contribution of the horizon τ to the equity premium is defined as H t,τ = Πt,τ/µ P rt. 54 The shape of the equilibrium density H t,τ tells us whence the equity premium comes from. The more the mass of probability concentrates on either short or long horizons, the more the riskiness associated to such horizons deserves a compensation and contributes to the whole equity premium. Therefore, H t, τ is natural metric to understand the effect of the equilibrium term-structures of equity on an important asset pricing equilibrium outcome, such as the equity premium. 30 C. Equilibrium bond and equity yields Proposition 9 The equilibrium price of the zero-coupon bond with maturity τ is given by B t,τ = E t [ξ t,t+τ ] e K 0τ+K µ τµ t +K z τz t In absence of the transitory shock Y t = X t, like in the standard long-run risk model of Bansal and Yaron 2004, long-run cash-flows become counter-factually riskier than market returns, as pointed out by Beeler and Campbell Standardizing Πt,τ by µ P rt allows to compare the timing decomposition of the equity premium among models and parameter settings even if they produce different magnitudes for the premium. 23

25 where model parameters are such that the expectation exists finite, the approximation makes use of Eq. 16, and K 0,K µ and K z are deterministic functions of time derived in the Appendix B. The bond price is a stationary function of the long-run growth, the transitory shock and the maturity. The approximation of Eq. 16 leads to an exponential affine bond price: hence, the bond yield varies with µ t and z t but its volatility inherits the homoscedasticity of the states. The real yield is affine in the state-variables: εt,τ = 1 τ K 0 τ + 1 e λ µτ r µ λ 1 µ µ t + 1 e λ zτ r z λ 1 z z t. 56 The short- and long-run limits of the term-structure of real yields lead to the steady state term-spread: εt, εt,0 = r µθ µ σ µ λ µ r2 µσ 2 µ r zθ z σ z 2λ 2 µ λ z It can be either positive or negative for γ s > ψ > 1: indeed, r µ θ µ > 0 and r z θ z < r2 z σ 2 z. 57 2λ 2 z Armed with these results, the focus turns on the equity yields as introduced by van Binsbergen et al The model equity yield is defined as pt,τ = 1 τ log Pt,τ D t = 1 τ d 0 A 0 τ + 1 e λµτ 1/ψ 1 λ µ µ t + 1 e λzτ 1 1/ψd z z t 58 and, hence, is a stationary function of the states and the maturity. Moreover, it can be decomposed as: pt,τ = εt,τ g D t,τ + ρt,τ. 59 The equity yield is given by the difference among the yield on the risk-less bond, εt,τ, and the dividend expected growth, g D t,τ, plus a premium, ρt,τ. The latter is implicitly defined by Eq and is a state-independent function of the maturity: ρt,τ = K 0 τ + B 0 1,τ A 0 τ/τ, as a joint implication of the approximation of Eq. 16 and the homoscedasticity of the states. The transitory shock affects the level of the short-run limit τ 0 of ρt, τ, whereas long-run growth affects the level of the long-run limit τ : however, both contribute positively to such limits. Therefore, the term-spread ρt, ρt,0 = can be either positive or negative depending on the model parameters. γ sψλ µ +e cq λ µ +e cq λ 2 µψ σ2 µ λ z+γ s ψe cq λ z +e cq ψ d2 z σ 2 z, D. Equilibrium cross-sectional returns This section characterizes the equilibrium returns of the dividend claims of single firms. In spirit of Lynch 2003 and Menzly, Santos, and Veronesi 2004, I define a share process for the firm cash-flows, that is the fraction of aggregate dividends paid by a single firm. The share process is such that dividends paid by existing firms sum up to aggregate dividends at each point in time. As in Lettau and Wachter 2007, 2011, the share process is deterministic for the sake of simplicity. The distribution of firms is stable over time: there exists a continuum of firms defined by their residual life T 0,T max and at each point in time the firm 31 A monotone downward sloping term-structure of real yields is a well known drawback in most of long-run risk literature: in such models Y t = X t, a positive nominal term-spread can obtain as a result of inflation risk only. 24

26 with zero residual life is replaced by a new one with maximal residual life T max. The share process st is a hump-shaped function of residual life which peaks at T max /2, similarly to Lettau and Wachter 2007: st = The dividends of the firm with residual life T are simply given by D T t = st D t. sinπt /T max T max 0 sinπt /T max dt, T 0,T max. 60 Proposition 10 The equilibrium price of the dividend claim of the firm with residual life T is given by P T t [ ] = E T t 0 ξ t,t+τdt+τ T τ dτ X T t 0 eh 0τ,T +H µ τµ t +H z τz t dτ 61 where model parameters are such that the expectation exists finite, the approximation makes use of Eq. 16, and H 0,H µ and H z are deterministic functions of time derived in the Appendix B. The instantaneous return volatility and premium are given by σ T Pt = σ 2 x + µ logpt T µ T P rt =θ x tσ x + θ µ t µ logp T t 2 σ 2 µ + z logp T 2 σ 2 z, 62 t σµ + θ z t z logp T t σz. 63 The single-firm price is given by the time integral of the market dividend strip prices over the residual life weighted by dividend shares: Pt T = T 0 st τp t,τdτ. The single-firm price relative to the current dividend level is a stationary nonlinear function of the long-run growth and the transitory shock: log PT t Dt T = log T 0 eh 0τ,T +H µ τµ t +H z τz t dτ d 0 d z z t logst. Prices increase with µ t as long as the intertemporal substitution effect dominates the wealth effect and increase with z t despite preferences. Notice that the approximation of Eq. 16 does not rules out the endogenous state-dependence from both the return volatility and the premium. Similarly to Lettau and Wachter 2011, I interpret value firms as shorter-horizon equity than growth firms, because the cash-flows of the former are weighted more toward the present and those of the latter are weighted more toward the future. Therefore, the value premium is consistent with downward-sloping term-structure of equity premia: if priced risk concentrates in the short-run i.e. H t,τ is monotone decreasing with the horizon, as the red line in Figure 1, the compensation µ T P rt is decreasing with residual life T. That is, value firms deserve a higher premium than growth firms. The opposite holds if priced risk concentrates far in the future i.e. H t,τ is a hump-shaped function of the horizon, as the black line in Figure 1. Since income insurance φ > 0 shifts priced risk towards the short-run, it also helps the model to explain the value premium. Such a mechanism obtains as an endogenous result in general equilibrium. Instead, the partial equilibrium framework of Lettau and Wachter 2011 exogenously specifies a state-price density designed to captures the stylized facts. VI. Asset Pricing Results A. Model Calibration Model calibration is made setting parameters about exogenous uncertainty and labor relations by minimizing a number of moments of cash-flows growth rates. Then, preference parameters are set in the usual range of values in the literature and moments of equity returns are investigated. As a peculiarity, the model calibration exploits the information from the 25

27 term-structures of wages and dividend risk. This is important because they provide information about the persistence of both transitory and permanent shocks a main issue in most of long-run risk literature. The cash-flows dynamics involves eight parameters: θ = { µ,σ x,λ µ,σ µ,λ z,σ z,α,h}. Six parameters are from the dynamics of x t,µ t and z t with z set equal to zero for simplicity and two are from the income insurance mechanism. Therefore, I choose eight moments conditions m i θ: the yearly steady-state growth rate of total resources Y t, the yearly steady-state volatilities of Y t,w t and D t and other four moments which capture the timing of risk. Namely, I consider the variance ratios VR s at 2 and 15 years of both wages and dividends in order to capture both the short- and long-run properties of their term-structure of risk: mθ = { g Y t,1, σ Y t,1, σ W t,1, σ D t,1, V R W t,2, V R D t,2, V R W t,15, V R D t,15 }, where V R x t,τ = σ2 xt,τ. Finally, I find the parameter values by minimizing the root-mean-square-error RMSE: σ 2 xt,.25 Θ = arg min θ i=1 m i θ m empirical i 2, where the empirical moments are from the analysis of Section II. 32 Table VII and VIII report respectively the model parameters and the implied moments of cash-flows. Insert Table VII and VIII about here. The permanent shock has instantaneous volatility σ x = 4.5%, whereas long-run growth and the transitory shock have volatilities σ µ = 2.7% and σ z = 3.7% and rate of reversion λ µ = 64.4% and λ z = 27.9%. The Walrasian benchmark and the degree of preference heterogeneity are α = 85.7% and h = 1.784, which lead to an optimal degree of income insurance φ =.425. Since the model is very parsimonious, the fit is not perfect the average relative error, that is m iθ m empirical i, is m empirical i about 10%, but is accurate enough to match the main stylized facts related to cash-flows and their term-structures of risk. These stylized facts are the ingredients needed to understand the impact of the income insurance channel on asset prices. The co-integrating relationship among GDP, wages and dividends is closely related to the timing of risk. Therefore, for the sake of robustness, I look at additional moments, such as the VR s of Y t and the moments of the dividend-share D t /Y t. 33 Remaining parameters concern the shareholders preferences. I consider the values 5, 7.5 and 10 for the relative risk aversion: γ s = 5 is suggested by the empirical investigation by Kimball, Sahm, and Shapiro 2009, whereas the other two values are usually considered in the long-run risk literature. Hence, given the heterogeneity parameter h = 1.784, the implied risk aversion of workers γ w are 8.92, and The elasticity of intertemporal substitution ψ is set to values between one and two, in line with the literature. Therefore, several cases satisfying γ s > ψ > 1 are investigated. Finally, the time discount rate β is set to generate a 1% risk-free rate if possible, given all other parameters. B. The term structure of cash flows Income insurance leads to smooth wages and highly risky dividends. Under the choice of parameters commented above, on the one hand, wages feature a yearly volatility of about σ W = 5% and, hence, lower than that of total consumption σ Y = 5.7%. On the other hand, dividends are significantly riskier at yearly frequency: 18.5%. Income insurance endogenously produces the counter-cyclical dynamics of the labor-share. Although its simplicity, the model captures the stylized facts associated to the dynamics of W t /Y t, which is large in average, smooth and persistent. 32 Empirical moments about dividends are computed as the average of those of net dividends and after tax corporate profits. 33 Consistently with the model, I compute the empirical dividend-share as dividends over dividends plus wages. 26

28 Moreover, it negatively correlates with changes in logy t. Such a result fails to obtain in most of production-based models and is crucial to the modeling of the term-structure of both cash-flows and equity returns. Table VIII also shows how dividend and wage volatilities change with the degree of income insurance, φ. Anything else equal, the larger the preference heterogeneity, h, the smoother the wages and the riskier the dividends. The term structures of growth rates VR s and volatilities of Y t, W t and D t are shown in Figure 6. The distance between the term structure of Y t and those of W t and D t reflects the distributional risk due to income insurance. Insert Figure 6 about here. The left upper panel of Figure 6 shows the term-structures of the model implied VR s solid lines and their empirical counterparts dashed lines. The model matches the upward- and downward-sloping shapes of wage and dividend risk. In addition, Y t features VR s close to one, in line with the barely flat term-structure of GDP risk. The left lower panel of Figure 6 shows the model implied volatilities of Y t,w t and D t : income insurance leads to a smoothing and a leverage effect respectively on wages and dividends. The right upper and lower panels of Figure 6 show the VR s and the volatilities in the case of absence of income insurance φ = 0, which obtains by setting h = 1: wage and dividend risk collapse to that of Y t and no term-structure effects are observable. The information exploited by the term-structures of VR s leads to a co-integrating relationship between Y t,w t and D t quite in line with the data: indeed, the model captures quite well the average, standard-deviation and first order autocorrelation of the dividend-share, as documented in Table VIII. Table IX reports the values of the volatilities of cash-flows associated with several horizons and their spread over the short-run limit τ 0 in both the cases φ = φ and φ = 0. Insert Table IX about here. The slope of the term-structure of dividend risk becomes negative once income insurance is taken into account. C. The term structure of equity Under the choice of parameters in Section VI.A, shareholders have preference for the early resolution of uncertainty and from their perspective the intertemporal substitution effect dominates the wealth effect. The term structures of both premia and return volatility are decreasing with the maturity at short and medium horizons in which the downward-sloping effect due to the transitory shock dominates the upward-sloping effect due to long-run growth and slightly increasing at long horizons in which the two effects barely offset each other. Figure 7 shows the term structures of equity premia and risk as a function of the horizon. Numerical results are reported in the Panel C of Table IX. Insert Figure 7 about here. By changing preference heterogeneity, h, and, hence, the optimal degree of income insurance, φ, it is possible to alter the slope of the term structures. In particular, a larger parameter φ increases the leverage effect on dividends and, in turn, the risk associated to the transitory shock. As a consequence, if ψ > 1, a stronger downward-sloping effect obtains and we can observe term-structures of equity risk and premia which are decreasing over a longer horizon with slopes which are larger in magnitude. Dividend risk decreases with the horizon and approaches that of total consumption. At the opposite equity risk features excess volatility at any horizon and approaches a value above fundamentals risk in the long-run, as shown in Figure 8. Insert Figure 8 about here. 27

29 Such a result obtains endogenously and although homoscedastic exogenous uncertainty. In particular, the long-run excess volatility consists of the difference between the limits: σ 2 Pt, σ 2 Dt, = 1 2ψσ2 µ λ 2 µψ 2 + d2 z σ 2 z ψ 2. First, long-run excess-volatility does not depend on risk aversion. Second, µ t contributes negatively to the long-run excessvolatility for ψ > 1/2 and vice-versa. Third, z t always contributes positively to the long-run excess-volatility and, for φ large enough, such an effect can lead to a total excess-volatility even if ψ > 1/2. The lower panels of Figure 8 show the long-run excess-volatility as a function of ψ and φ. The excess-volatility is increasing in φ and decreasing in ψ. The role of ψ has the following rationale: ψ σ 2 P t, σ 2 Dt, 0 if ψ 1 d2 z σ 2 z. σ 2 µ/λ 2 µ On the left hand side ψ 1 can be interpreted as the magnitude of either the intertemporal substitution effect or the wealth effect if ψ > 1 or ψ < 1. On the right hand side, we observe a value that is larger than one if σ 2 D t,0 > σ2 D t, and vice-versa. As a consequence, the more decreasing the term-structure of dividends volatility, the larger the intertemporal substitution effect required to obtain excess-volatility of equity returns. The model generates a long-run excess volatility of equity over dividends in line with the recent empirical evidence about the decreasing variance ratios of dividend distributions, documented by Belo et al Therefore, the model improves over most of long-run risk models, which imply the opposite counter-factual relation: that is payouts to shareholders riskier than equity returns, as pointed out by Beeler and Campbell Namely, a long-run risk model Y t = X t could produce long-run excess volatility only for ψ < 1/2 a parametrization under which standard asset pricing implications would fail to obtain. D. The market asset and other asset pricing implications The model reconciles standard asset pricing facts with the evidence about the term-structure of both cash-flows and equity. Namely, on the one hand, it is possible to generate i a low and smooth risk-free rate; ii a high equity premium and Sharpe ratio and excess return volatility; iii a high volatility of dividends; on the other hand, i the term-structure of dividend volatility is decreasing with the horizon; ii the labor-share is large, persistent and counter-cyclical; iii the term-structures of equity risk and premia are downward sloping. Consider the baseline calibration γ s = 10,ψ = 1.5: it allows to match the unconditional level of the risk-free rate about 1% and a low volatility about 4% in line with the real data. Instead, long-run risk models usually produce an implausibly low volatility to match equity returns. The model leads to an equity premium of about 7% as in the real data: such a result is particularly remarkable since neither stochastic volatility and jumps nor unrealistically high and time-varying risk aversion are required. The return volatility is about 15%, which is somewhat lower than in the real data but implies a large excess-volatility over total consumption and dividends on the whole term-structure, including the long-run limit, as shown in Figure 8. As a consequence, the model produces a Sharpe ratio about 48%, which is somewhat larger than in the real data. Such a result is peculiar and due to short-run but persistent risk. The model also captures quite well the level and the volatility of the price-dividend ratio about exp3.50 and 35%. Table X summarizes the model-implied moments. Insert Table X about here Table X reports the asset pricing moments for many pairs γ s,ψ: a good fit of the risk-free rate level and of the first two return moments of the market asset obtains with γ s = 10,ψ = 1.5, γ s = 7.5,ψ = 1.25 and γ s = 5,ψ = However, decreasing the elasticity of intertemporal substitution leads to a risk-free rate and a price-dividend ratio which are respectively 28

30 too volatile and too smooth in comparison with the data. Hence, the choice γ s = 10,ψ = 1.5 seems to be preferable. The model significantly improves over the Walrasian benchmark without income insurance φ = 0. The leverage effect in dividend volatility, due to income insurance, not only is crucial to the modeling of the term-structures of cash-flows and equity returns but also helps to match the standard asset pricing moments. For none of the 12 pairs γ s,ψ a sizeable equity premium obtains in absence of income insurance. Figure 9 shows the equity premium as a function of ψ and φ upper panels. Insert Figure 9 about here The premium is decreasing in ψ and increasing in φ, all else being equal. The former relation is due to the fact that, in order to describe the term-structures in equilibrium, we need that the pricing of short-run but persistent risk dominates the pricing of long-run variations in expected growth. In order to better understand the nature of the equity premium, it is instructive to adopt a term-structure perspective. The equilibrium density H t, τ of Eq. 54 represents the relative contribution of each horizon τ to the whole premium. As commented in the Introduction and shown in Figure 1, the timing decomposition of the equity premium tells us that the model leads to a shift towards the short-run of the equilibrium density in comparison with a long-run risk model. The equilibrium density in the baseline calibration is monotone decreasing with the horizon, whereas the density associated to a long-run risk model Y t = X t is hump-shaped. Up to an horizon of about 5 years, the former stays above the latter; while after 5 years the relation reverses, implying a larger weight on the far future for the long-run risk model. The timing decomposition of the premium does not depend only on the transitory shock z t, but it also depends on how such a shock is priced in equilibrium. The lower panel of Figure 9 shows how the equilibrium density H t,τ changes for several values of the degree of income insurance φ. Indeed, income insurance leads not only to riskier dividends but also to a shift of the priced risk towards the short-run. The larger φ, the more important the contribution of the short horizons to the whole density H t,τ. The timing decomposition of the premium sheds lights on the nature of priced risk in equilibrium. Indeed, the model shows that a short-run instead of a long-run explanation of market compensations is needed in order to simultaneously describe both the standard asset pricing moments and the downward sloping term-structures of cash-flows and equity returns, given standard preferences γ s > ψ > 1. E. Bond and equity yields Now I consider the effect of income insurance on the equilibrium equity yields and their components, that is the yields on the risk-less bond, the expected growth of dividends and the premium on the equity yields. All these quantities at the steady-state are plotted in Figure 10 as a function of the horizon both in presence φ = φ, red solid lines and absence φ = 0, blue dashed lines of income insurance. Numerical results are reported in the Panel D of Table IX. Insert Figure 10 about here The left upper panel of Figure 10 shows that the equity yield is slightly decreasing with the horizon under income insurance, whereas it is strongly increasing in the Walrasian benchmark. For ψ > 1, z t induces a downward-sloping effect, which is enhanced by the degree of income insurance and more than offset the upward-sloping effect due to µ t. The right upper panel of Figure 10 shows the term-structure of the yield on the risk-less bond. In absence of income insurance, µ t is the main determinant of slope of the term-structure of interest rates. Similarly to the long-run risk literature, a counter-factual downward-sloping term-structure obtain for ψ > 1 and, hence, a positive nominal term-spread can only be imputed to inflation risk. Instead, in presence of income insurance, z t induces a stronger upward-sloping effect which leads to a positive slope for the bond yields, in accord with the real data. Income insurance makes dividends riskier at short horizons: such a 29

31 short-run risk is priced in equilibrium and bonds with short maturities are a better hedge than equity. Therefore, increasing bond yields obtain, although ψ > 1. The left lower panel of Figure 10 shows the term-structure of the expected dividend growth rates. Similarly to the equity yields, under income insurance the slope switches from slightly positive to negative. Finally, the right lower panel of Figure 10 reports the term-structure of the premia on the equity yields. Such premia increase and decrease with horizon respectively in absence and in presence of income insurance. This change in the slope is consistent with the shift of priced risk toward the short-horizon due to income insurance and the downward-sloping premia in Figure 7. In summary, the analysis of the decomposition of equity yields provides two noteworthy results. First, income insurance helps to understand the term-structures of both the real interest rates as well as the premia on the equity yields, in line with the empirical findings of van Binsbergen et al Second, income insurance helps to fix the corresponding counter-factual predictions of long-run risk models about these term-structures. F. Cross-sectional returns and the value premium Cross-sectional returns are modeled as in Section V.F and in spirit of Lettau and Wachter The life-cycle of firms is set to 100 years. The evolution of the single-firm share process and, equivalently, the stable distribution of firms is plotted in the upper left panel of Figure 11. Insert Figure 11 about here Firms with short residual life are interpreted as value firms, whereas firms with long residual life are interpreted as growth firms. Equilibrium price-dividend ratios are increasing with residual life in both the Walrasian benchmark φ = 0, blue dashed lines and in case of income insurance φ = φ, red solid lines. As shown in the upper right panel of Figure 11, price-dividend ratios are larger for growth firms than for value firms, as in the real data, but these levels are barely affected by income insurance. Instead, income insurance is important in the determination of equilibrium volatility and premia. Steady state return volatility and instantaneous premia are plotted as a function of the firm residual life in the lower panels of Figure 11. For γ s > ψ > 1 and in absence on income insurance φ = 0, blue dashed lines, long-run growth is the main determinant of priced risk and, hence, long-run cash-flows require a substantial compensation. As a consequence, premia slightly increase with residual life. Therefore, the model produces the counter-factual implication of a growth premium. Vice-versa, in presence of income insurance φ = φ, red solid lines, priced risk shifts towards the short-run and, hence, short-run cash-flows deserve a substantial compensation. Then, equilibrium premia and volatilities decrease with the firm residual life: in line with the real data, value firms require larger premia than growth firms. Hence, income insurance helps to jointly explain in general equilibrium the value premium, the term-structure of cash-flows and equity as well as standard asset pricing facts. 34 The rationale for the value premium differs from Lettau and Wachter In their paper the state-price density is exogenously specified, featuring a time-varying price of risk uncorrelated with fundamentals which is an assumption difficult to obtain in general equilibrium. Here, instead, the value premium, as well as positive and negative slopes for the term-structures of bond yields and equity premia, obtain although a constant price of risk and an economically pertinent foundation of the state-price density in general equilibrium. 34 The range of variation of premia over the firm life-cycle is relatively small. This can be due to the fact that the share-process is assumed to be deterministic and the state-variables are homoscedastic. However, such assumptions are made for the sake of simplicity and exposition and can be relaxed without altering the qualitative implications of the model. 30

32 VII. Alternative Specifications A. Countercyclical heteroscedasticity and time-varying slope of equity premia While the downward-sloping term-structure of dividend risk is a very robust feature of the data, the empirical evidence about the term structures of equity is under debate. 35 On the one hand, equity risk inherits the negative slope of dividend risk; on the other hand, the slope of equity premia depends on how and how much the sources of long-run and short-run risks are priced in equilibrium. Downward-sloping equity premia suggested by van Binsbergen et al appear as a stylized fact concerning the unconditional distribution of equity returns. Indeed, van Binsbergen et al and Muir 2014 provide some evidence supporting the idea that the term-structure of equity premia is time-varying and switches from flat or slightly increasing in good times to strongly negative in bad times. While I am agnostic about the exact properties we can infer from available data, this section shows that a minimal modification of the model allows for the above dynamics of the term-structure of equity premia. I slightly modify the dynamics of the transitory shock to include countercyclical heteroscedasticity: Z t = e z t = e z ẑ t, with dẑ t = λ z z ẑ t dt + ˆσ z ẑt db z,t, with ˆσ z = σ z / z and z > 0 such that z t is still a zero-mean reverting process but its volatility is decreasing in z t. Given the affine specification of the transitory shock, the model can be solved with the same methodology of the previous sections and all results are derived in the online appendix OA.B. Countercyclical heteroscedasticity in fundamentals leads to two results. First, under income insurance the labor-share is decreasing in z t and, hence, dividends load more on the transitory shock when the latter is high volatile. Second, the price of risk associated to z t is no more a constant but is decreasing with z t and, hence, higher compensations obtain in bad times. These two facts lead to a slope of the term-structure of equity premia which is time-varying. Indeed, long-run growth leads to an upwards-sloping effect and the transitory shock leads to a downward-sloping effect but the latter depends on the level of the volatility of z t. Therefore, the downward-sloping effect is countercyclical. As a result, the term-structure of equity premia can be flat or slightly upward-sloping in good times and strongly downward-sloping in bad times. To provide an illustration of the equilibrium results, I set all parameters as in the baseline calibration of Table VII apart z =.1. The model generates steady state moments in line with the data: 0.6% risk-free rate with 3% volatility, an equity premium of 7.6% with return volatility of 15.2% and Sharpe ratio of 50% and log price-dividend ratio of 3.48 with 34% volatility. Figure 12 shows a number of interesting results. Insert Figure 12 about here The left upper panel of Figure 12 shows the term-structures of the model implied VR s solid lines and their empirical counterparts dashed lines. Similarly to Figure 6, the model matches the upward- and downward-sloping shapes of wage and dividend risk and the barely flat term-structure of GDP risk. The right upper panel of Figure 12 shows the model implied volatility of dividends at the steady-state solid line and the 5-95% probability interval dot-dashed and dashed lines. Dividend volatility is strongly decreasing with z t at short horizons, whereas the effect of the transitory shock disappears in the long-run. The lower panels of Figure 12 shows the model implied volatility left and premia right of the dividend strip returns at the steady-state solid line and the 5-95% probability interval dot-dashed and dashed lines. Under standard preferences γ s > ψ > 1, equity risk inherits the negative slope of dividend risk. The transitory shock affects the level of the 35 See Boguth, Carlson, Fisher, and Simutin 2012 and van Binsbergen and Koijen 2012b. 31

33 term-structure, which is countercyclical, but does not alter the sign of the slope. Instead, the transitory shock affects both the level and the slope of the term-structure of equity premia. Equity compensations are low and barely flat or slightly increasing in normal and good times. However, compensations increase in size and become markedly downward-sloping in bad times. Countercyclical heteroscedasticity helps the model to match the conditional dynamics of the term-structure of equity premia and, hence, to provide further support to the main model mechanism of income insurance. B. Power utility and endogenously time-varying risk premia This section characterizes the asset pricing results in the special case of power utility ψ γ 1 s. Analytical solutions obtain even for the true dynamics of dividends in Eq. 30. All results are derived in the online appendix OA.C. Both the price of risk as well as return moments are endogenously time-varying. The state-price density is ξ 0,t = e βt Y γ s t 1 ωz t γ s. The usual power function of total consumption is multiplied by a power function of the dividend-share 1 ωz t. The latter depends on the transitory shock and embeds the effect of income insurance. Since, the dividend-share is not an exponential affine functional of z t, as an endogenous result, the logarithm of the state-price density is convex in z t and, hence, the price of risk is state-dependent. Under power utility only the transient risk of shareholders consumption is priced in equilibrium. Therefore, µ t does not deserve a premium and z t is priced only for its contribution to the instantaneous volatility of dividends. Such a contribution is affected by income insurance: θ z t = γ s σ z 1 + φ ωz t /1 ωz t. On the one hand, the stronger the degree of income insurance the larger the price of risk in magnitude; on the other hand, a negative positive transitory shock leads to a decrease increase of the fraction of total consumption devoted to shareholders remuneration and, hence, to an increase decrease in the price of risk. Such a counter-cyclical dynamics of the price of risk is a robust feature of the real data and obtains in the model as an endogenous result of income insurance, whereas it disappears for φ 0. The dividend strip price relative to the dividend is a stationary function of the states µ t and z t. However, the price is not exponential affine in the states as long as φ > 0: income insurance leads to non-linearities in asset prices and, in turn, to endogenous time-variation in the return moments. Instead, for φ 0, the price reduces to the exponential affine case and return moments are constant. Interestingly, even if the effect of income insurance on dividends only concerns the transitory shock, both µ t and z t enter the return dynamics. Therefore, the shape of the term structure of equity risk and premia is state-dependent and moves with both µ t and z t. The return volatility is state-dependent and also change with the maturity τ. The premium on the dividend strip is a compensation for transient risk only: µ P rt,τ = γ s σ 2 x + γ s P z τ,µ t,z t 1 + φ ωz t 1 ωz t σ 2 z. The first term has the usual form γ s σ 2 x. Instead, the second term has the following rationale. The volatility of the transitory shock σ z is multiplied by the price of risk θ z t and by the semi-elasticity P z of the price with respect to z t. Notice that such a term is the only one involving the maturity τ and, hence, determines the sign of the term-structure of equity premia. To gain intuition consider again the log-linearization of Eq. 16. Specializing Corollary 2 to the power utility case ψ γ 1 s, we obtain the following approximation results: τ σ 2 Pt,τ =2 e 2λ µτ e λ µτ 1γ s 1 2 λ 1 µ σ 2 µ e 2τλ z γ s 11 + e τλ z 1γ s λ z dz 2 σ 2 z, τ µ P rt,τ =e τλ z γ s 1λ z d 2 z σ 2 z. 32

34 The term-structure of equity risk is monotone increasing for γ s > 1 and non-monotone for γ s < 1, whereas the term-structure of equity premia is always monotone but the slope has the same sign of γ s 1. Such a restriction is due to power utility: when ψ γ 1 s the term-structure of equity premia is not necessarily monotone. The equilibrium price of the market asset is given by the time integral of the dividend strip price. Therefore, the logarithm of the price dividend ratio is a stationary but nonlinear function of the states µ t and z t. Consequently, the equity premium and the market volatility are endogenously time-varying as long as φ > 0. Since shareholders risk aversion has to be lower than unit in order to generate negative slopes for equity premia, the model cannot simultaneously generate a sizeable equity premium under power utility. Nevertheless, it is instructive to consider a calibration of the model and to verify the endogenous dynamics of conditional moments. I preserve all parameter values from the calibration of Section VI.A see Table VII, but the shareholders risk aversion is set to γ s = The optimal degree of income insurance is unchanged and, in turn, cash-flows unconditional moments are also unchanged. Instead, the unconditional equity premium reduces to about 3% but γ s < 1 guarantees that the term-structure of premia on the dividend strip is downward sloping. Figure 13 highlights many interesting properties. Insert Figure 13 about here The upper left panel of Figure 13 shows that the volatility of dividends decreases with the maturity τ but is also decreasing in z t. Consequently, a steeper downward sloping term structure obtains when z t is low or equivalently when the labor share ωz t or the wage-to-dividends ratio ωz t /1 ωz t are large. This result is consistent with the empirical findings of Figure 3, panels e and f. The other two panels in the first row of Figure 13 show that both the premium and the return volatility of the dividend strip are decreasing with the maturity. Moreover, the model generates an excess of volatility over both dividends and consumption, although only transient risk is priced under power utility. The second row of Figure 13 shows that both the premium and the return volatility of the dividend strip as well as the Sharpe ratio are decreasing in the transitory shock z t and, hence, income insurance endogenously leads to such countercyclical dynamics which is also a feature of the real data. The third row of Figure 13 shows the expected dividend growth and the equity yield as a function of both the maturity τ and the transitory shock z t. The former is decreasing in z t, whereas the latter is increasing. These relationships are consistent with the empirical results of Section II.E: in the data, the labor-share which is a decreasing function of z t in the model predicts dividend growth and equity returns respectively with positive and negative sign. The same holds in the model. VIII. Conclusion This paper shows that modeling of labor relations in an otherwise standard and parsimonious general equilibrium model allows to accommodate both the traditional as well as short-term patterns of dividends and equity returns, whereas the latter are at odds with leading asset pricing models. Unlike standard Walrasian models, wages do not correspond to the marginal product of labor but incorporate income insurance that workers exploit within the firm. In turn, the labor-share has counter-cyclical dynamics and, hence, the riskiness of owning capital enhances. Distributional risk due to the income insurance also implies a term-structure effect: equilibrium dividend distributions and discount rates lead to short-term risk in equity returns. Under standard preferences, downward sloping term structures of both premia and risk of equity returns obtain. Furthermore, the model simultaneously accounts for i a good fit of the first two moments of both the market asset return and the risk-free rate; ii endogenously counter-cyclical price of risk and equity premium as well as variation in return volatility; iii cross-sectional value premium of equities; iv upward sloping real yields on bonds and downward sloping premia on equity yields. 33

35 An empirical analysis supports the main model mechanism documenting a connection between income insurance and both the term-structures of wage and dividend risk as well as the dynamics of wages and dividends shares of total output. Though at the aggregate level fluctuations in the labor-share are an important channel for the equilibrium determination of both dividends and equity returns, a joint analysis of labor relations with heterogeneous investment risk, as in Ai et al. 2012, as well as financial leverage, as in Belo et al. 2014, could provide a deeper understanding about the equilibrium timing of equity risk. The paper also sheds lights on the implications of the equilibrium term-structures on the traditional asset pricing facts, such as the equity premium. A timing decomposition of the premium shows that investors have a radically different perception of priced risk than in leading models. In the latter the equity premium comes out from variation in long-run discounted cash-flows, whereas the correct modeling of the term-structure of dividends and equity implies that priced risk significantly shifts toward the short-run. References Ai, Hengjie, Mariano Massimiliano Croce, Anthony M. Diercks, and Kai Li, 2012, Production-Based Term Structure of Equity Returns, Unpublished manuscript. [4, 13, 34] Azariadis, Costas, 1978, Escalator clauses and the allocation of cyclical risks, Journal of Economic Theory 18, [1] Baily, Martin N., 1974, Wages and Employment under Uncertain Demand, The Review of Economic Studies 41, pp [1] Bansal, Ravi, and Amir Yaron, 2004, Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles, Journal of Finance 59, [1, 3, 4, 11, 21, 23, 43] Barberis, Nicholas, Ming Huang, and Tano Santos, 2001, Prospect Theory and Asset Prices, The Quarterly Journal of Economics 116, [1] Beeler, Jason, and John Y. Campbell, 2012, The Long-Run Risks Model and Aggregate Asset Prices: An Empirical Assessment, Critical Finance Review 1, [3, 7, 23, 28] Belo, Frederico, Pierre Collin-Dufresne, and Robert S. Goldstein, 2014, Endogenous Dividend Dynamics and the Term Structure of Dividend Strips, Journal of Finance forthcoming. [1, 4, 5, 6, 7, 9, 13, 28, 34] Benzoni, Luca, Pierre Collin-Dufresne, and Robert S. Goldstein, 2011, Explaining asset pricing puzzles associated with the 1987 market crash, Journal of Financial Economics 101, [13] Berrada, Tony, Jerome Detemple, and Marcel Rindisbacher, 2013, Asset Pricing with Regime-Dependent Preferences and Learning, Unpublished manuscript. [4] Boguth, Oliver, Murray Carlson, Adlai Fisher, and Mikhail Simutin, 2012, Dividend Strips and the Term Structure of Equity Risk Premia: A Case Study of the Limits of Arbitrage, Unpublished manuscript. [31] Boldrin, Michele, Lawrence J. Christiano, and Jonas D. M. Fisher, 2001, Habit Persistence, Asset Returns, and the Business Cycle, American Economic Review 91, [2] Boldrin, Michele, and Michael Horvath, 1995, Labor Contracts and Business Cycles, Journal of Political Economy 103, [1, 2] Campbell, John Y., and John H. Cochrane, 1999, By Force of Habit: A Consumption Based Explanation of Aggregate Stock Market Behavior, Journal of Political Economy 107, pp [1, 5] Chan, Yeung L., and Leonid Kogan, 2002, Catching Up with the Joneses: Heterogeneous Preferences and the Dynamics of Asset Prices, Journal of Political Economy 110, [5] Constantinides, George M., and Anisha Ghosh, 2011, Asset Pricing Tests with Long-run Risks in Consumption Growth, Review of Asset Pricing Studies 1, [65] Croce, Mariano M., Martin Lettau, and Sydney C. Ludvigson, 2014, Investor Information, Long-Run Risk, and the Term Structure of Equity, Review of Financial Studies forthcoming. [4] Curatola, Giuliano, 2014, Loss Aversion, Habit Formation and the Term Structures of Equity and Interest Rates, Unpublished manuscript. [4] Danthine, Jean-Pierre, and John B. Donaldson, 1992, Risk sharing in the business cycle, European Economic Review 36, ] [2, 4, 11, Danthine, Jean-Pierre, and John B. Donaldson, 2002, Labour Relations and Asset Returns, Review of Economic Studies 69, , 13] [2, 4, 11, Danthine, Jean-Pierre, John B Donaldson, and Paolo Siconolfi, 2006, Distribution Risk and Equity Returns, in Mehra Rajnish, eds.: Handbook of Equity Risk Premium Elsevier,. [1, 4] 34

36 Donangelo, Andres, 2014, Labor Mobility: Implications for Asset Pricing, Journal of Finance 69, [4, 13] Duffie, Darrell, and Larry G. Epstein, 1992, Stochastic Differential Utility, Econometrica 60, [9, 14] Favilukis, Jack, and Xiaoji Lin, 2012, Wage Rigidity: A Solution to Several Asset Pricing Puzzles, Working paper,. [4, 13] Gabaix, Xavier, 2012, Variable Rare Disasters: An Exactly Solved Framework for Ten Puzzles in Macro-Finance, The Quarterly Journal of Economics 127, [1] Gertler, Mark, and Antonella Trigari, 2009, Unemployment Fluctuations with Staggered Nash Wage Bargaining, Journal of Political Economy 117, [13] Gomme, Paul, and Jeremy Greenwood, 1995, On the cyclical allocation of risk, Journal of Economic Dynamics and Control 19, [2] Guiso, Luigi, Luigi Pistaferri, and Fabiano Schivardi, 2005, Insurance within the Firm, Journal of Political Economy 113, [3, 4] Han, Kim, Ernst Maug, and Christoph Schneider, 2013, Labor Representation in Governance as an Insurance Mechanism, Unpublished manuscript. [3, 4] Hore, Satadru, 2008, Equilibrium Predictability and Other Return Characteristics, Unpublished manuscript. [13] Jermann, Urban J., 1998, Asset pricing in production economies, Journal of Monetary Economics 41, [2] Kaltenbrunner, Georg, and Lars A. Lochstoer, 2010, Long-Run Risk through Consumption Smoothing, Review of Financial Studies 23, [2] Kimball, Miles S., Claudia R. Sahm, and Matthew D. Shapiro, 2009, Risk Preferences in the PSID: Individual Imputations and Family Covariation, American Economic Review 99, [26] Knight, Frank H., 1921, Risk, Uncertainty and Profit. Houghton Mifflin. [1] Kuehn, Lars-Alexander, Nicolas Petrosky-Nadeau, and Lu Zhang, 2012, An Equilibrium Asset Pricing Model with Labor Market Search, Working paper,. [4, 13] Lettau, Martin, and Jessica A. Wachter, 2007, Why Is Long-Horizon Equity Less Risky? A Duration-Based Explanation of the Value Premium, Journal of Finance 62, [24, 25] Lettau, Martin, and Jessica A. Wachter, 2011, The term structures of equity and interest rates, Journal of Financial Economics 101, [4, 24, 25, 30] Longstaff, Francis A., and Monika Piazzesi, 2004, Corporate earnings and the equity premium, Journal of Financial Economics 74, [1] Lynch, Anthony W., 2003, Portfolio Choice with Many Risky Assets, Market Clearing and Cash Flow Predictability,. [24] Marfè, Roberto, 2013, Corporate Fraction and the Equilibrium Term Structure of Equity Risk, Unpublished manuscript. [4, 13] Marfè, Roberto, 2014, Demand Shocks, Timing Preferences and the Equilibrium Term Structures, Unpublished manuscript. [4] Menzly, Lior, Tano Santos, and Pietro Veronesi, 2004, Understanding Predictability, Journal of Political Economy 112, [24] Moskowitz, Tobias J., and Annette Vissing-Jørgensen, 2002, The Returns to Entrepreneurial Investment: A Private Equity Premium Puzzle?, American Economic Review 92, [3] Muir, Tyler, 2014, Financial crises, risk premia, and the term structure of risky assets, Unpublished manuscript. [4, 31] Ríos-Rull, José-Víctor, and Raül Santaeulàlia-Llopis, 2010, Redistributive shocks and productivity shocks, Journal of Monetary Economics 57, [1, 6] Santos, Tano, and Pietro Veronesi, 2006, Labor Income and Predictable Stock Returns, Review of Financial Studies 19, [1, 9] Shimer, Robert, 2005, The Cyclical Behavior of Equilibrium Unemployment and Vacancies, American Economic Review 95, [13] van Binsbergen, Jules, Michael Brandt, and Ralph Koijen, 2012, On the Timing and Pricing of Dividends, American Economic Review 102, [1, 2, 4, 31] van Binsbergen, Jules, and Ralph Koijen, 2012b, A note on Dividend Strips and the Term Structure of Equity Risk Premia: A Case Study of the Limits of Arbitrage, Unpublished manuscript. [31] van Binsbergen, Jules H., Wouter H. Hueskes, Ralph Koijen, and Evert B. Vrugt, 2013, Equity Yields, Journal of Financial Economics 110, [1, 4, 24, 30, 31] Xiouros, Costas, and Fernando Zapatero, 2010, The Representative Agent of an Economy with External Habit Formation and Heterogeneous Risk Aversion, Review of Financial Studies 23, [5] 35

37 Appendix A. Income Insurance Proof of Lemma 1: Under Assumption 3, the transitory components of wages and dividends are given by Eq. 12. Therefore, the optimal degree of income insurance maximizes the geometric average of their unconditional expectations: Ez [ Wφ] ϑ Ez [ Dφ] 1 ϑ with bargaining parameter ϑ given in Eq. 15. We have Taking the derivative with respect to φ, 0 = 1 α φ α 1 φ e 2λ z we get three solutions: E z [ Wφ] =α 1 φ e z1 φ+σ2 z 1 φ 2 /4λ z, E z [ Dφ] =e z+σ2 z /4λ z α 1 φ e z1 φ+σ2 z 1 φ 2 /4λ z. 4λz z+σ 2 z φ 1φ 1 ϑ 4λz e z+ σ2 z 4λz α 1 φ e 2λ z z σ 2 z φ 1 + 2λ z logα, φ =1 + 2λ z z + logα, σ 2 z φ =1 + 2λ z σ 2 z φ =1 + 2λ z z + logα + σ 2 z 4λz z+σ 2 z φ 1φ 1 ϑ 4λz z+σ 2 z φ 1φ 1 4λz αe 4λz α φ e z+ σ2 z 4λz ϑ z + logα 4λ z z 2σ 2 z 4λ z logα λ z σ 2 z logϑ/α σ 2 z /2, 4λ z z 2σ 2 z 4λ z logα λ z σ 2 z logϑ/α σ 2 z /2. Since we are looking for a potential solution φ 0,1, we can exclude the first two solutions for φ given the choice z = 0, which is used in the model calibration and which guarantees D t > 0 t under the log-linearized dynamics of Eq. 16 under reasonable parameters the second term common to all solutions for φ is by far lower than minus one. Therefore, we consider the third solution and set φ = max0,min1, φ as in Eq. 14. q.e.d. Appendix B. Equilibrium, Cash-Flows and Asset Prices Proof of Proposition 1, 2 and 3: Under the infinite horizon, the utility process J satisfies the following Bellman equation: where D denotes the differential operator. Eq. 1 can be written as DJX,µ,z + f C s,j = 0, 1 0 =J X µx J X,Xσ 2 xx 2 + J µ λ µ µ µ J µ,µσ 2 µ + J z λ z z z J z,zσ 2 z + f C s,j. Guess a solution of the form JX,µ,z = 1 1 γ s X 1 γ s gµ,z. The Bellman equation reduces to 0 =µ 1 2 γ sσ 2 x + g µ λ µ µ µ g 1 γ s + 1 g µ,µ 2 g σ 2 µ 1 γ s + g z λ z z z g 1 γ s g z,z g σ 2 z 1 γ s + Under Assumption 2 and stochastic differential utility, the pricing kernel has dynamics given by β 1 1/ψ g 1/χ e 1 1/ψd 0+d z z 1. 2 dξ 0,t = ξ 0,t d f C fc + ξ 0,t f J dt = rtξ 0,t θ x tξ 0,t db x,t θ µ tξ 0,t db µ,t θ z tξ 0,t db z,t, 3 36

38 where, by use of Itô s Lemma and Eq. 2, we get rt = X f C f C µx 1 2 θ x t = X f C f C σ x X, θ µ t = µ f C f C σ µ, θ z t = z f C f C σ z, X,X f C f C σ 2 xx 2 µ f C f C λ µ µ µ 2 1 µ,µ f C f C σ 2 µ z f C f C λ z z z 2 1 z,z f C f C σ 2 z f J, An exact solution for gµ,z satisfying Eq. 2 does not exist for ψ 1. Therefore, I look for a solution of gµ,z around the unconditional mean of the consumption-wealth ratio. Aggregate wealth is given by Q s,t = E t [ t ] ξ t,u C s,u du, and, applying Fubini s Theorem and taking standard limits, the consumption-wealth ratio satisfies [ ] C s,t Q s,t = rt dt 1 E dq t Q dt 1 E t [ dξ ξ ] dq Q. 4 Guess Q s,t = C s,t β 1 gµ t,z t e γ s 1d 0 +d z z t 1/χ and apply Itô s Lemma to get dq Q. Then, plug in the wealth dynamics, the risk-free rate and the pricing kernel into Eq. 4: after tedious calculus you can recognize that the guess solution is correct. Notice that the consumption-wealth ratio approaches to β when ψ 1 as usual. Denote cq = E[logC s,t logq s,t ], hence, a first-order approximation of the consumption-wealth ratio around cq produces C s,t Q s,t = βgµ t,z t 1/χ e 1 1/ψd 0+d z z t e cq 1 cq + logβ 1 χ loggµt,z t + γ s 1d 0 + d z z t. Using such approximation in the Bellman equation 2 leads to 0 =µ 1 2 γ sσ 2 x + g µ /ψ g λ µ µ µ 1 γ s g µ,µ g σ 2 µ 1 γ s + g z λ z z z g 1 γ s + 1 g z,z 2 g σ 2 z 1 γ s e cq 1 cq + logβ 1 χ loggµ,z + 1 1/ψd 0 + d z z which has exponentially affine solution gµ,z = e u 0+1 γ s d 0 +u µ µ+u z +1 γ s d z z where β, u 0 = e cq γ s 1 2cq 1e 5cq ψ 2e cq ψlogβe cq + λ z 2 e cq + λ µ 2 2ψ 1e cq + λ z e cq + λ µ µe cq + λ z λ µ +e cq zd z λ z e cq + λ µ + e 4cq 2ψβ + 2cq 1λ z + λ µ + ψ 1 γ s σ 2 x + dz 2 γ s 1σ 2 z +2e 3cq cq 1ψλ 2 z + λ z 2βψ + 4cq 1ψλµ + ψ 1γ s σ 2 x + λµ 2βψ + cq 1ψλµ γ s σ 2 x + ψγ s σ 2 x +ψ 1d 2 z γ s 1σ 2 z + λ 2 z λ 2 µ 2βψ + ψ 1γs σ 2 x + ψ 1γs 1σ 2 µ + 2e cq λ z λ µ ψcq 1λ z λ µ + 2βλ z + λ µ +ψ 1γ s λ z + λ µ σ 2 x + ψ 1γs 1σ 2 µ + e 2cq 4λ z λ µ 2βψ + cq 1ψλµ + ψ 1γ s σ 2 x + λ 2 z 2βψ +4cq 1ψλ µ + ψ 1γ s σ 2 x + λ 2 µ 2βψ γs σ 2 x + ψγ s σ 2 x + ψ 1dz 2 γ s 1σ 2 z + ψ 1γs 1σ 2 µ 1/ 2ψ 1e cq + λ z 2 e cq + λ µ 2, u µ = 1 γ s e cq + λ µ, u z = d zλ z γ s 1 e cq + λ z, 37

39 and the endogenous constant cq satisfies cq = logβ χ 1 u 0 + u µ µ + u z z. The risk-free rate and the prices of risk take the form: r 0 = 1 2 2βψ 2 γ s 1 2 +e cq 1 ψu 0 +ψcq 1 logβγ s 1ψγ s 1 ψ1 ψγ s zu z ψu z d z γ s 1γ s λ z ψ 1+γ s 2 µu µψγ s 1λ µ ψγ s 1 r µ = ψγ s 1γ s +u µ ψγ s 1e cq +λ µ ψγ s 1, r z = ψd zγ s 1γ s λ z +u z ψγ s 1e cq +λ z ψγ s 1, θ x t =γ s σ x, γ s ψ 1 θ µ t = u µ 1 γ s σ µ, θ z t = d z γ s + u z1 ψγ s ψγ s 1 γ s 1 + γ s σ 2 x u z ψu z d z γ s 1γ s 2 σ 2 z u2 µψγ s 1 2 σ 2 µ, ψ 2 γ s 1 2 ψ ψγ s 2 σ z, and the results of Proposition 2 and 3 easily follow. q.e.d. Proof of Lemma 2: All results simply obtain from the dynamics in Eq and standard calculus. q.e.d. Proposition A The following conditional expectation has exponential affine solution: M t,τ c = E t [e c 0+c 1 logξ 0,t+τ +c 2 x t+τ +c 3 µ t+τ +c 4 z t+τ ] = ξ c 1 0,t X c 2 t e l 0τ, c+l µ τ, cµ t +l z τ, cz t, 5 where c = c 0,c 1,c 2,c 3,c 4, model parameters are such that the expectation exists finite and l 0,l µ and l z are deterministic functions of time derived in the Appendix B. Proof of Proposition A: Consider the following conditional expectation: M t,τ c = E t [e c 0+c 1 logξ 0,t+τ +c 2 logx t+τ +c 3 µ t+τ +c 4 z t+τ ] 6 where c = c 0,c 1,c 2,c 3,c 4 is a coefficient vector such that the expectation exists. Guess an exponential affine solution of the kind: M t,τ c = e c 1 logξ 0,t +c 2 logx t +l 0 τ, c+l µ τ, cµ t +l z τ, cz t, 7 where l 0 τ, c,l µ τ, c and l z τ, c are deterministic functions of time. Feynman-Kac gives that M has to meet the following partial differential equation 0 =M t M ξ r 0 + r µ µ + r z z M ξ,ξθ x t 2 + θ µ t 2 + θ z t 2 +M X µx M X,Xσ 2 xx 2 +M µ λ µ µ µ M µ,µσ 2 µ +M z λ z z z M z,zσ 2 z M ξ,x θ x tσ x X M ξ,µ θ µ tσ µ M ξ,z θ z tσ z, 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 a linear function of the states µ and z: 0 =l 0τ, c + l µτ, cµ + l zτ, cz c 1 r 0 + r µ µ + r z z c 1c 1 1θ x t 2 + θ µ t 2 + θ z t 2 + c 2 µ c 2c 2 1σ 2 x + l µ τ, cλ µ µ µ l µτ, c 2 σ 2 µ + l z τ, cλ z z z l zτ, c 2 σ 2 z c 1 c 2 θ x tσ x c 1 l µ τ, cθ µ tσ µ c 1 l z τ, cθ z tσ z. 38

40 Hence, we get three ordinary differential equations for l 0 τ, c,l µ τ, c and l z τ, c: 0 =l 0τ, c c 1 r c 1c 1 1θ x t 2 + θ µ t 2 + θ z t c 2c 2 1σ 2 x + l µ τ, cλ µ µ l µτ, c 2 σ 2 µ + l z τ, cλ z z + 2 1l zτ, c 2 σ 2 z c 1 c 2 θ x tσ x c 1 l µ τ, cθ µ tσ µ c 1 l z τ, cθ z tσ z 0 =l µτ, c c 1 r µ + c 2 l µ τ, cλ µ 0 =l zτ, c c 1 r z l z τ, cλ z with initial conditions: l 0 0, c = c 0,l µ 0, c = c 3 and l z 0, c = c 4. The explicit solutions are given by e λ2 l 0 τ, c = 1 4c 0 + 4τ µc µc 3 4τc 1 r 0 4τ µc 1 r µ 2τc 1 θ 2 x + 2τc 2 4 1θ 2 x 2τc 1 θ 2 z + 2τc 2 1θ 2 z 2τc 1 θ 2 µ + 2τc 2 1θ 2 µ + 4 z 1 + e τλ z c 4 λ z c 1 r z 1 + e τλ z + τλ z 4 µc µc 1r µ 4τc 1 c 2 θ x σ x 2τc 2 σ 2 x + 2τc 2 λ z λ µ λ 2σ 2 x µ 4c2 1 r zθ z σ z λ 2 z + 4e τλ z c 2 1 r2 z σ 2 z λ 3 z + 4c 1c 2 θ µ σ µ λ 2 µ + 4e τλ z c 2 1 r zθ z σ z λ 2 z 2c 1c 4 r z σ 2 z λ 2 z 4c2 1 r µθ µ σ µ λ 2 µ 4c 1c 4 θ z σ z λ z 2e 2τλ z c 1 c 4 r z σ 2 z λ 2 z + 4e τλz c 1 c 4 θ z σ z λ z + 4e τλ z c 1 c 4 r z σ 2 z λ 2 z + 4τc2 1 r zθ z σ z λ z + 2τc2 1 r2 z σ 2 z λ 2 z 3c2 1 r2 z σ 2 z λ 3 z + c2 4 σ2 z λ z e λ2 e 2τλ z c 2 1 r2 z σ 2 z λ 3 z e 2τλz c 2 4 σ2 z λ z 4τc 1c 2 θ µ σ µ 4c 1c 3 θ µ σ µ + 4τc2 1 r µθ µ σ µ 3c2 2 σ2 µ λ µ λ µ λ µ λ 3 + 6c 1c 2 r µ σ 2 µ µ λ 3 3c2 1 r2 µσ 2 µ µ λ 3 µ + 2τc2 2 σ2 µ λ 2 + 2c 2c 3 σ 2 µ µ λ 2 4τc 1c 2 r µ σ 2 µ µ λ 2 2c 1c 3 r µ σ 2 µ µ λ 2 + 2τc2 1 r2 µσ 2 µ µ λ 2 + c2 3 σ2 µ e 2τλµ c 2 + c 1 r µ + c 3 λ µ 2 σ 2 µ µ λ µ λ 3 µ + 4e τλ µ c 2 c 1 r µ c 3 λ µ µλ 2 µ + σ µ c 2 σ µ c 1 θ µ λ µ + r µ σ µ, l µ τ, c = c 2 c 1 r µ + e τλ µ c 2 + c 1 r µ + c 3 λ µ, λ µ 1 + e τλ z c1 r z + e τλ z c 4 λ z l z τ, c =. λ z λ 3 µ q.e.d. Proof of Corollary 1: The conditional moment generating function D t τ,n of Eq. 35 obtains as a special case of M t,τ c with c = nd 0,0,n,0,nd z. Therefore, it is given by D t τ,n = X n t e B 0τ,n+B µ τ,nµ t +B z τ,nz t with B 0 n,τ =l 0 τ, c = 1 4 4d n e τλ z zd z + 4 µ τ + 1+e τλµ λ µ nτσ 2 x + e 2τλz+λµ n e 2τλµ 1+e 2τλz dz 2 λ 3 µσ 2 z +e 2τλz λ z 1+4e τλµ +e 2τλµ 3+2τλ µ σ 2 µ λ z λ 3 µ, B µ n,τ =l µ τ, c = n e τλµ n λ µ, B z n,τ =l z τ, c = e τλ z nd z, 39

41 and B 0 0,n = nd 0,B µ 0,n = 0 and B z 0,n = nd z. Therefore, the coefficients in Eq are given by g D,τ,µ = 1 λ µ τ 1 e λ µτ, g D,τ,z = d z τ e λ zτ 1, and v D,τ,x = 1, v D,τ,µ = 4e λµτ e 2λµτ +2λ µ τ 3, v 2λ 3 D,τ,z = e λzτ sinhλ z τdz 2 µτ λ z τ. Finally, the results about the limits automatically follow. Using the true dynamics of dividends of Eq. 30, we have the following results: E t [D t+τ ] =E t [ X t+τ e z t+τ α 1 φ e 1 φ z t+τ ] =E t [X t+τ e z t+τ ] E t [ X t+τ α 1 φ e 1 φ z t+τ ] =M t,τ c M t,τ c, where c = 0,0,1,0,1 and c = 1 φ logα,0,1,0,1 φ, and [ ] [ E t D 2 t+τ =Et Xt+τ 2 ] e z t+τ α 1 φ e 1 φ z t+τ 2 [ =E t X 2 t+τ e 2z ] [ ] [ ] t+τ 2E t Xt+τα 2 1 φ e 2 φ z t+τ + E t Xt+τα 2 21 φ e 21 φ z t+τ =M t,τ c 2M t,τ c +M t,τ c, where c = 0,0,2,0,2, c = 1 φ logα,0,2,0,2 φ and c = 21 φ logα,0,2,0,21 φ. With this results in hand, it is possible to compute the term-structures of the expected growth rates and of growth rates volatility. q.e.d. Proof of Proposition 4 and 5: The equilibrium price of the market dividend strip with maturity τ of Eq. 43 obtains as a special case of M t,τ c with c = d 0,1,1,0,d z. Therefore, it is given by P t,τ = ξ 1 0,t M t,τ c = X t e A 0τ+A µ τµ t +A z τz t with A 0 τ =l 0 τ, c = 4 1 4e τλµ µ 1+r µ 1+e τλµ 1+τλ µ λ µ + e 2τλz+λµ λ 3 z λ 3 µ e 2τλ µ r z + d z λ z 2 λ 3 µσ 2 z + 4e τλ z+2λ µ rz + d z λ z λ 3 µ zλ 2 z + σ z θ z λ z + r z σ z e 2τλ z 1 + r µ 2 λ 3 z σ 2 µ + 4e τ2λ z+λ µ 1 + rµ λ 3 z σ µ θ µ λ µ r µ σ µ +e 2τλ z+λ µ λ 3 µ 4λ 2 z zr z + d z τr z λ z + λ z d 0 τr 0 + θ x σ x 4θ z λ z r z + d z τr z λ z σ z + 2d z r z λ z + d 2 z λ 2 z + r 2 z 3 + 2τλ z σ 2 z rµ θ µ λ 3 z λ µ 1 + τλ µ σ µ r µ 2 λ 3 z 3 + 2τλ µ σ 2 µ, A µ τ =l µ τ, c = 1+e τλµ 1+r µ r µ λ µ, A z τ =l z τ, c = r z+e τλz r z +d z λ z λ z, and A 0 0 = d 0,A µ 0 = 0 and A z 0 = d z. The dynamics of the market dividend strip price obtains by applying Itô s Lemma to P t,τ : Therefore the return volatility is given by dp t,τ = [ ]dt + x P t,τ σ x db x,t + µ P t,τ σ µ db µ,t + z P t,τ σ z db z,t. σ P t,τ = Pt,τ 1 x P t,τ σ x 2 + µ P t,τ σ µ 2 + z P t,τ σ z 2 = σ 2 x + A µ τσ µ 2 + A z τσ z 2, 40

42 and the premium is given by µ P rt,τ = 1 dt dξ0,t ξ 0,t, dp t,τ P t,τ = θ x tσ x + θ µ ta µ τσ µ + θ z ta z τσ z. q.e.d. Proof of Corollary 2 and 3: Given the results of Proposition 4, the limits and the slopes of the return volatility and premium for the market dividend strip obtain by standard calculus: and lim τ 0 σ2 Pt,τ = lim σ 2 x + A µ τσ µ 2 + A z τσ z 2, τ 0 lim τ σ2 Pt,τ = lim σ 2 τ x + A µ τσ µ 2 + A z τσ z 2, τ σ 2 Pt,τ = τ σ 2 x + A µ τσ µ 2 + A z τσ z 2, lim µ P rt,τ = lim θ x tσ x + θ µ ta µ τσ µ + θ x ta z τσ z, τ 0 τ 0 lim µ P rt,τ = lim θ x tσ x + θ µ ta µ τσ µ + θ x ta z τσ z, τ τ τ µ P rt,τ = τ θ x tσ x + θ µ ta µ τσ µ + θ x ta z τσ z. q.e.d. Proof of Proposition 6 and 7: Under Assumption 2, the shareholders act as a representative agent on the financial markets and, hence, the equilibrium price of the market asset is equal to the shareholders wealth. Therefore, using the results of Proposition 1, the market asset price can be written as P t = Q s,t = C s,t e cq t = X t e logβ+u 0χ 1 +d 0 +u µ χ 1 µ t +u z χ 1 +d z z t. The dynamics of the market asset price obtains by applying Itô s Lemma to P t : Therefore the return volatility is given by and the premium is given by dp t = [ ]dt + x P t σ x db x,t + µ P t σ µ db µ,t + z P t σ z db z,t. σ P t = Pt 1 x P t σ x 2 + µ P t σ µ 2 + z P t σ z 2 = σ 2 x + u µ χ 1 σ µ 2 + u z χ 1 + d z σ z 2, µ P rt = 1 dt dξ0,t ξ 0,t, dp t P t = θ x tσ x + θ µ tu µ χ 1 σ µ + θ z t u z χ 1 + d z σz. q.e.d. Proof of Proposition 8: Using the definition of the dividend strip price P t,τ = E t [ξ t,t+τ D t+τ ] and market asset price P t = 0 P t,τdτ, we can write the instantaneous premium of the market return as follows: µ P rt =θ x t xp t σ x + θ µ t µp t σ µ + θ z t zp t =θ x t = = 0 0 P t 0 x P t,τ dτ P t P t σ x + θ µ t P t σ z 0 µ P t,τ dτ P t σ µ + θ z t 0 z P t,τ dτ P t σ z Pt 1 P t,τ θ x tσ x + θ µ ta µ τσ µ + θ z ta z τσ z dτ Πt, τdτ, 41

43 and Πt,τ easily follows using the results of Proposition 4 and 6. q.e.d. Proof of Proposition 9: The equilibrium price of the zero-coupon bond with maturity τ of Eq. 55 obtains as a special case of M t,τ c with c = 0,1,0,0,0. Therefore, it is given by with K 0 τ =l 0 τ, c = e 2τλz+λµ 4λ 3 z λ 3 µ K µ τ =l µ τ, c = e τλµ 1 r µ λ µ, K z τ =l z τ, c = e τλz 1r z λ z, B t,τ = ξ 1 0,t M t,τ c = e K 0τ+K µ τµ t +K z τz t e 2τλ µ r 2 z λ 3 µσ 2 z + 4e τλ z+2λ µ rz λ 3 µ zλ 2 z + σ z θ z λ z + r z σ z e 2τλ z r 2 µλ 3 z σ 2 µ e 2τλ z+λ µ λ 2 µ 4 µrµ λ 3 z 1 + τλ µ + λ µ 4τr0 λ 3 z + 4 zr z λ 2 z 1 + τλ z +r z σ z 4θ z λ z 1 + τλ z + r z 3 2τλ z σ z 4r µ θ µ λ 3 z λ µ 1 + τλ µ σ µ + rµλ 2 3 z 3 2τλ µ σ 2 µ +4e τ2λ z+λ µ rµ λ 3 z µλ 2 µ + σ µ θ µ λ µ + r µ σ µ, and K 0 0 = 0,K µ 0 = 0 and K z 0 = 0. And the real yield in Eq. 56 easily follows. Given the definitions of pt,τ and ρt,τ in Eq. 58 and 59, it is easy to verify that the premium on the equity yield, ρt,τ = 1 τ K0 τ + B 0 1,τ A 0 τ + µ pt,τ εt,τ + gd t,τ µ t + z pt,τ εt,τ + gd t,τ z t = 1 τ K0 τ + B 0 1,τ A 0 τ, is state-independent. q.e.d. Proof of Proposition 10: Using the definition of the share process in Eq. 60 and of the price of a dividend claim on a firm with residual life T in Eq. 61, we get T Pt T = st τe t [ξ t,t+τ D t+τ ]dτ, 0 since D T τ t+τ = st τd t+τ and st is deterministic. Therefore, we can use the results of Proposition 4 to obtain: T T Pt T = st τp t,τ dτ X t st τe A 0τ+A µ τµ t +A z τz t dτ. 0 0 By setting H 0 τ,t = logst τ + A 0 τ, H µ τ = A µ τ and H z τ = A z τ, the final result automatically follows. The dynamics of the single firm price obtains by applying Itô s Lemma to P T t : dp T t Therefore the return volatility is given by and the premium is given by q.e.d. = [ ]dt + x P T t σ x db x,t + µ P T t σ µ db µ,t + z P T t σ z db z,t. σ T Pt = Pt T 1 x Pt T σ x 2 + µ Pt T σ µ 2 + z Pt T σ z 2, µ T P rt = 1 dt dξ0,t, dpt t ξ 0,t Pt T = θ x tσ x + θ µ t µp T t P T t σ µ + θ z t zp t T Pt T σ z. 42

44 Figure 1. The timing decomposition of the equity premium t,τ Τ The equilibrium density H t,τ as a function of the horizon τ represents the relative contribution of any horizon τ in years to the whole equity premium µ P rt. The red solid line denotes the baseline model calibration of Table VII. The black dashed line denotes the sub-case Y t = X t with µ = 2%,σ x = 1%,λ µ =.15,σ µ = 2%,β = 2.5%,γ s = 10,ψ = 2, which reproduces the main features of the standard long-run risk model of Bansal and Yaron Back to the text. 43

45 Figure 2. GDP components L Y I Y Lev D 1 Y 0.06 D 1 Y 0.06 D 1 Y L Y I Y Lev D 2 Y 0.08 D 2 Y 0.08 D 2 Y L Y I Y Lev y 0.20 y 0.20 y L Y I Y Lev The first line report the time-series of the GDP labor-share L/Y, the investment-share I/Y and the aggregate leverage-ratio Lev. The second the third and the foruth lines report the scatter plots of the above variables respectively with the dividend-share D 1 /Y, the profit-share D 2 /Y and the GDP growth rates y. Data are yearly on Back to the text. 44

46 Figure 3. The term-structure of variance ratios of dividends and other GDP components a Variance ratios of dividends b Variance ratios of GDP components c Variance ratios of GDP minus wages d Variance ratios of consumption e Wages to dividends ratios 1.0 f Variance ratios of dividends Upper panels a and b: The black, blue, yellow, red and dashed red lines denote polynomial fits of the variance ratios of the growth rates of respectively GDP, wages, investments, net dividends and after tax corporate profits as a function of the horizon in years. Middle panels c and d: The blue, green, red and dashed red, brown and dashed brown lines denote polynomial fits of the variance ratios of the growth rates of respectively wages, GDP minus wages, net dividends, after tax corporate profits, consumption and wages plus net dividends. Lower panels e and f: On the left, the time-series of aggregate wages over aggregate net dividends on the sample The middle sub-sample is denoted in red. On the right, the polynomial fits of the variance ratios of the growth rates of net dividends as a function of the horizon in years for the sub-samples , and The variance ratios associated to the middle sub-sample are denoted in red. Data are quarterly on from the NIPA tables. The variance ratio procedure uses the theoretical exposition of Campbell, Lo and MacKinlay 1997, pp , which accounts for heteroscedasticity and overlapping observations. Back to the text. 45

47 Figure 4. Risk sharing, bargaining and income insurance 0.75 Φ Objective function Φ Φ Optimal degree of income insurance Φ Optimal degree of income insurance Α h The upper left panel shows the objective function J φ of the bargaining model as a function of the degree of income insurance φ. The optimal degree of income insurance φ is plotted as a function of the Walrasian labor-share benchmark α and preference heterogeneity h in the upper right panel, and as a function of only α and h respectively in the bottom left and right panels. All unreported parameters are from Table VII. Back to the text. 46

48 Figure 5. Instantaneous volatilities of cash flows Σ z Y, Σ z D, Σ z W z t Σ Y z, Σ D W z, Σ z Ω z t Σ z Y, Σ z D, Σ z W Φ Instantaneous volatilities of cash-flows as a function of the transitory shock z t, the labor-share ωz t and the degree of income insurance are shown respectively in the upper, middle and lower panel. Black dot-dashed, red solid and blue dashed lines denote respectively total consumption, dividends and wages volatilities. All unreported parameters are from Table VII. Back to the text. 47

49 Figure 6. Term structures of cash flows Cash flows variance ratios Φ Φ Cash flows variance ratios Φ Τ Τ Cash flows volatilities Φ Φ Cash flows volatilities Φ Τ Τ The upper panels plot the model variance ratios of dividends red, wages blue and total resources black as a function of the horizon in the cases of φ = φ left and φ = 0 right at the steady state µ t = µ,z t = z. Dashed lines denote empirical data as in Figure 3 dividends moments are computed as the average of those of net dividends and after tax corporate profits. The lower panels plot the growth rates volatilities of dividends red, wages blue and total resources black as a function of the horizon in the cases of φ = φ left and φ = 0 right at the steady state µ t = µ,z t = z. All unreported parameters are from Table VII. Back to the text. 48

50 Figure 7. Term structures of premia and volatility on dividend strips Σ P t,τ Equity risk Τ Μ P r t,τ 0.10 Equity premia Τ The upper and lower panels plot the term structures of respectively the volatility and the premia of the dividend strips returns at the steady state µ t = µ,z t = z. The cases φ = φ and φ = 0 are denoted respectively by red solid and blue dashed lines. All unreported parameters are from Table VII. Back to the text. 49

51 Figure 8. Term structures of cash flows and equity risk and excess volatility Equity and cash flows volatilities Τ Σ p 2 Σ D Σ p 2 Σ D Ψ Φ The upper panel plots the term structures of volatility of cash flows and dividend strips returns at the steady state µ t = µ,z t = z. The volatility of dividend strips returns, dividends and total resources are denoted with red solid, blue dot-dashed and black dotted lines. The red dashed line denotes the market asset volatility. The lower panels plot the long-run excess volatility of dividend strips returns over dividends, σ 2 P t, σ2 D t,, as a function of ψ left and φ right. All parameters are from Table VII. Back to the text. 50

52 Figure 9. Equity premium, preferences and income insurance Μ P r Μ P r Ψ Φ t,τ Τ The upper panels plot the premium on the market asset, µ P rt, as a function of ψ left and φ right. The lower panel plots the equilibrium density H t,τ as a function of the horizon τ i.e. the relative contribution of any horizon τ to the whole equity premium µ P rt. The red line denotes the baseline calibration of Table VII with φ = φ. The black, blue and green lines denote respectively the cases φ = 0, φ =.15 and φ =.60. All unreported parameters are from Table VII. Back to the text. 51

53 Figure 10. Term structures of bond and equity yields p t,τ Equity yield ε t,τ 0.05 Bond yield Τ Τ g D t,τ Expected dividend growth ς t,τ 0.09 Equity yield premium Τ Τ The term structures of equity yields upper left panel, real bond yields upper right panel, dividend growth rates lower left panel and premia on equity yields lower right panel are plotted as a function of the horizon at the steady state µ t = µ,z t = z. The cases φ = φ and φ = 0 are denoted with red solid and blue dashed lines. All unreported parameters are from Table VII. Back to the text. 52

54 Figure 11. Cross-sectional returns, value premium and income insurance s T Cross sectional dividend share distribution log P t T D t T Cross sectional log price dividend ratio Value Growth100 T Value Growth100 T 0.20 Σ P T t Cross sectional return volatility Μ T P r t 0.10 Cross sectional equity premium Value Growth100 T Value Growth100 T Share distribution upper left panel, log price-dividend ratios upper right panel, return volatility lower left panel and instantaneous premium lower right panel of single firm claims are plotted as a function of the firm residual life T at the steady state µ t = µ,z t = z. The cases φ = φ and φ = 0 are denoted with red solid and blue dashed lines. All unreported parameters are from Table VII. Back to the text. 53

55 Figure 12. Cash-flows and equity term-structures with countercyclical heteroscedasticity Cash flows variance ratios 1.5 Σ D t,τ 0.30 Dividend risk Τ Τ Σp t,τ Equity risk Μp r t,τ Equity premia Τ Τ The upper left panel shows the model variance ratios of dividends red, wages blue and total resources black as a function of the horizon τ at the steady state µ t = µ,z t = 0. Dashed lines denote empirical data as in Figure 3 dividends moments are computed as the average of those of net dividends and after tax corporate profits. The upper right panel shows the volatility of dividends as a function of the horizon τ for z at the steady-state solid line and its 5-95% probability interval dot-dashed and dashed lines. The lower left panel shows the volatility of the dividend strip return as a function of the horizon τ for z at the steady-state solid line and its 5-95% probability interval dot-dashed and dashed lines. The lower right panel shows the premium of the dividend strip return as a function of the horizon τ for z at the steady-state solid line and its 5-95% probability interval dot-dashed and dashed lines. All parameters are from Table VII apart z = 0.1. Back to the text. 54

56 Figure 13. Conditional return moments under power utility Σ D t,τ Τ Μp r t,τ Τ Σp t,τ,σ D t,τ,σ Y t,τ Τ Σ P t,τ Μ P r t,τ SR t,τ z z z Upper panels: the left plot shows the dividend growth rates volatility as a function of the horizon τ for z at the steady-state solid line and its 5-95% probability interval dot-dashed and dashed lines; the central plot shows the premium of the dividend strip as a function of the horizon τ for z at the steady-state solid line and its 5-95% probability interval dot-dashed and dashed lines; the right plot shows the volatility of the dividend strip, the dividend growth rates volatility and the total consumtpion growth rates volatility respectively with red solid, blue dashed and black dot-dashed lines. Middle panels: the left, central and right plots show respectively the volatility, the premium and the Sharpe ratio on the dividend strip as a function of z, given a maturity of τ = 1 year. Lower panels: the left plot and the right plot show respectively the dividend expected growth rate and the equity yield as a function of the maturity τ and of the transitory shock z. All parameters are from Table VII apart γ s = Back to the text. 55

57 Table I GDP Shares and Growth Rates Panel A reports yearly mean, standard deviation, first order autocorrelation and cross-correlations significance levels in parenthesis about the labor-share L/Y, the investment-share I/Y, the dividend-share D 1 /Y and the profit-share D 2 /Y as well as their log differences. Panel B reports the Johansen s trace statistics and the associated 5% and 1% critical values for the number of cointegrating equations in a VECM model for cointegration among the logarithm of GDP, labor, investment, dividends and profits. Panel C reports the estimates with Newey-West corrected t-statistics from univariate regressions of first differences in labor-share, investment-share, dividend-share and profit-share on their lagged levels. Data are yearly on the sample 1932:2012 from NIPA tables. Panel A Summary Statistics Shares Growth Rates L/Y I/Y D 1 /Y D 2 /Y l i d 1 d 2 y mean std ac correlations I/Y i D 1 /Y d D 2 /Y d y y Panel B Cointegration Johansen Test for Cointegration for l, i, d 1, d 2 and y Rank Trace % c.v % c.v Panel C Stationarity of GDP Shares L/Y I/Y D 1 /Y D 2 /Y const const const const t 2.04 t 2.22 t 2.67 t 4.10 L/Yt I/Yt D 1 /Yt D 2 /Yt t t t t rmse rmse rmse rmse R R R R Back to the text. 56

58 Table II Determinants of the Dividend-Share The table reports the estimates of the model D i /Yt+ j = b0 + b1levt + b2i/yt + b3l/yt + εt, i = {1,2}, j = {0,1}, where the dependent variable is either the dividend-share D 1 /Y or the profit-share D 2 /Y at time t or t + 1; the independent variables are the aggregate leverage ratio Lev, the investment-share I/Y and the labor-share L/Y. The t-statistics are reported in parenthesis. The symbols *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels. Data are yearly on the sample 1932:2012 from NIPA tables and 1945:2012 from Flow of Funds. Panel A Net Dividends Panel B Corporate Profits D 1 /Yt D 1 /Yt+1 D 2 /Yt D 2 /Yt+1 coeff coeff coeff coeff t t t t Levt * Levt *** *** R R I/Yt I/Yt R R L/Yt *** *** L/Yt *** *** R R Levt ** * Levt *** *** I/Yt 0.074** 0.063* I/Yt 0.078* L/Yt *** *** L/Yt *** *** R R Back to the text. 57

59 Table III Determinants of the Dividend-Share: Granger Causality The table reports the estimates of the model D i /Yt = b0 + b01d i /Yt 1 + b01d i /Yt 2 + b11levt 1 + b12levt 2 + b21i/yt 1 + b22i/yt 2 + b31l/yt 1 + b32l/yt 2 + εt, i = {1,2}, where the dependent variable is either the dividend-share D 1 /Y or the profit-share D 2 /Y at time t; the independent variables are the first and second lag of the aggregate leverage ratio Lev, the investment-share I/Y and the labor-share L/Y. The t-statistics are reported in parenthesis. The symbols *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels. A set of Granger causality tests reports the F statistics and p-values of the hypothesis that all coefficients on the lags of each independent variable are jointly zero. The number of lags 2 is selected according to the final prediction error FPE, Akaike s information criterion AIC and the Hannan and Quinn information criterion HQIC. Data are yearly on the sample 1932:2012 from NIPA tables and 1945:2012 from Flow of Funds. Panel A Panel B D 1 /Yt Granger test D 2 /Yt Granger test coeff F prob coeff F prob t t D 1 /Y t *** D 2 /Y t *** t ** t Lev t Lev t t t I/Y t I/Y t *** t t L/Y t ** L/Y t ** t * t ** N 66 N 66 R All: R All: adj-r adj-r Back to the text. 58

60 Table IV Long-Horizon Predictability of Dividends The table reports the estimates of the model g i Dτ = 1 τ d i t+τ d i t = b 0 + b1levt + b2i/yt + b3l/yt + εt+τ, i = {1,2}, τ = {1,5,10,15,20}, where the dependent variable is the yearly growth rate of either net dividends d 1 = logd 1 or profits d 2 = logd 2 computed over the future horizon of τ = {1,5,10,15,20} years; the independent variables are the current level of the aggregate leverage ratio Lev, the investment-share I/Y and the labor-share L/Y. The Newey-West corrected t-statistics are reported in parenthesis. The symbols *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels. Data are yearly on the sample 1932:2012 from NIPA tables and 1945:2012 from Flow of Funds. Panel A - Net Dividends Panel B - Corporate Profits Horizon Horizon Lev Coeff *** 0.167*** 0.160*** 0.120*** ** t R adj-r I/Y Coeff * 0.281*** 0.284*** 0.251** * t R adj-r L/Y Coeff * 0.600** 0.578*** 0.574*** ** ** t R adj-r Lev Coeff *** 0.175*** 0.163*** 0.113*** ** 0.108*** t I/Y Coeff ** ** * *** t L/Y Coeff *** 0.407*** t R adj-r Back to the text. 59

61 Table V Long-Horizon Predictability of Excess Returns The table reports the estimates of the model τ i=1 rex t+τ = b0 + b1levt + b2i/yt + b3l/yt + εt+τ, τ = {1,5,10,15,20}, where the dependent variable is the cumulative excess returns r ex computed over the future horizon of τ = {1,5,10,15,20} years; the independent variables are the current level of the aggregate leverage ratio Lev, the investment-share I/Y and the labor-share L/Y. The Newey-West corrected t-statistics are reported in parenthesis. The symbols *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels. Data are yearly on the sample 1932:2012 from NIPA tables and 1945:2012 from Flow of Funds. Cumulative Excess Returns Horizon Lev Coeff 0.378** 0.243** 0.189*** 0.095** t R adj-r I/Y Coeff * *** ** t R adj-r L/Y Coeff *** *** t R adj-r Lev Coeff 0.549*** 0.447*** 0.273*** 0.102** t I/Y Coeff *** *** t L/Y Coeff *** *** *** ** t R adj-r Back to the text. 60

62 Panel A Sign of compensations transient risk Table VI Premia on dividend strips non-transient long-run risk a Y t = X t, µ t = µ > 0 { ERU & ISE e Y t = X t Z t > 0 > 0 if non-transient short-run risk b Y t = X t > 0 > 0 if LRU & WE c Y t = Z t > 0 < 0 if ERU d Y t = X t Z t, µ t = µ > 0 { < 0 if ERU ERU & ISE LRU & WE < 0 if ERU Panel B Sign of compensations slopes transient risk non-transient long-run risk non-transient short-run risk a Y t = X t, µ t = µ { b Y t = X t > 0 if ERU & ISE LRU & WE c Y t = Z t < 0 if ISE < 0 if ISE d Y t = X t Z t, µ t = µ < 0 if ISE { < 0 if ISE e Y t = X t Z t < 0 if ISE > 0 if ERU & ISE LRU & WE < 0 if ISE Panel A reports the sign of the components of the premia on dividend strip s returns associated respectively to transient risk, non-transient long-run risk and non-transient short-run risk as defined in Section V.A. Panel B reports the sign of the components of the slope of term-structure of the premia on dividend strip s returns associated respectively to transient risk, non-transient long-run risk and non-transient short-run risk as defined in Section V.A. ISE and WE stay respectively for intertemporal substitution effect and wealth effect. ERU and LRU stay respectively for preference for the early and the late resolution of uncertainty. Back to the text. 61

63 Table VII Model parameters Baseline calibration Cash-flows volatility of log permanent shock σ x.045 steady-state of long-run growth µ.042 speed of reversion of long-run growth λ µ.644 volatility of long-run growth σ µ.027 steady-state of log transitory shock z.000 speed of reversion of log transitory shock λ z.279 volatility of log transitory shock σ z.037 Labor relations labor-share Walrasian benchmark α.857 preference heterogeneity h Preferences time-discount rate β.037 shareholders relative risk aversion γ s 10 elasticity of intertemporal substitution ψ 1.5 Implied parameters workers relative risk aversion γ w bargaining power ϑ.914 optimal degree of income insurance φ.425 steady-state labor share ω z.915 dividends leverage coefficient d z 5.61 Back to the text. 62

64 Table VIII Cash-flows moments Data Model φ = 0 φ = φ consumption growth g Y t, consumption volatility σ Y t, years variance ratio of consumption V R Y t, years variance ratio of consumption V R Y t, dividends growth g D t, dividends volatility σ D t, years variance ratio of dividends V R D t, years variance ratio of dividends V R D t, wages growth g W t, wages volatility σ W t, years variance ratio of wages V R W t, years variance ratio of wages V R W t, unconditional dividend-share D/Y dividend-share volatility σ D/Y t, dividend-share autocorrelation AC D/Y t, log consumption and dividend-share correlation ρ d logy,d/y log consumption and labor-share correlation ρ d logy,w/y φ =.00 φ =.15 φ = φ φ =.60 σ D t, σ W t, V R D t, V R W t, σ D/Y t, ρ d logy,d/y Steady-state values of model cash-flows moments are compared with the data see Section II and Table I. The symbol denotes moment conditions used in the calibration of model parameters in Table VII. The cases of φ equal to 0,.15 and.60 obtain by setting h equal to respectively 1, 1.19 and All unreported parameters are from Table VII. Back to the text. 63

65 Table IX Term-structures of cash-flows and equity returns Panel A Volatilities φ = φ φ = 0 τ σ Y t,τ σ D t,τ σ W t,τ Panel B Spreads φ = φ φ = 0 τ σ Y t,τ σ Y t, σ D t,τ σ D t, σ W t,τ σ W t, Panel C Equity risk, premia and excess-volatility φ = φ φ = 0 τ σ P t,τ µ P rt,τ σ P t,τ σ D t,τ σ P t,τ σ Y t,τ Panel D Spreads on bond and equity yields φ = φ φ = 0 τ εt, τ εt, pt, τ pt, ρt, τ ρt, Model statistics are computed at the steady-state µ t = µ, z t = z. The case φ = 0 obtains by setting h = 1. All unreported parameters are from Table VII. Back to the text. 64

66 Table X Steady-state moments of the market asset returns Data Sample r σ r µ P r σ P SR logp/d σ logp/d Model γ s ψ β φ r σ r µ P r σ P SR logp/d σ logp/d a b φ φ φ φ φ φ φ φ φ φ φ φ Steady-state values for the risk-free rate r and its volatility σ r, the excess market return µ P r and volatility σ P, the market Sharpe ratio SR, the log price-dividend ratio logp/d and its volatility σ logp/d are compared with empirical moments from Constantinides and Ghosh a and b denote the sign of respectively the market asset excess-volatility over dividend and consumption at 10 year horizon: σ P σ D t,10 and σ P σ Y t,10. For each pair γ s,ψ, β is set to match if possible a 1% risk-free rate. The case φ = 0 obtains by setting h = 1. All unreported parameters are from Table VII. Back to the text. 65

67 Online Appendix of Income Insurance and the Equilibrium Term-Structure of Equity Roberto Marfè Online Appendix A. Empirical Support: the Non-Financial Corporate Sector This Appendix provides further support to the empirical Section II and focuses on the non-financial corporate sector. Data are from the Flow of Funds, Integrated macroeconomic accounts, table S.5.a from 1945 to 2013 at yearly frequency. The main quantities of interest are reported in the following table. % share a Gross value added 100 a.1 - Capital depreciation 11.6 b = Net value added 88.4 b.1 - Compensations to employees 62.9 b.2 - Taxes on production and imports less subsidies 8.7 c = Net operating surplus 16.8 c.1 - net interest paid 2.5 c.2 - net dividends paid 4.0 c.3 - net reinvestment of earnings -0.8 d = Net national income 11.1 d.1 - Current taxes on income, wealth, and other transfers 5.8 e = Net disposable income 5.3 e.1 - Capital transfers 0.0 f = Net saving 5.3 Summary of Integrated macroeconomic accounts, table S.5.a The term structures of variance ratios VR of growth rates of net value added, wages, operating surplus, interest rates and dividends are reported in the left panel of the next figure. 2.5 Term structure of risk: non financial corporate sector interest Term structure of risk: from gross value added a to net saving f wages 2.0 a 1.5 net value added 1.5 b operating surplus dividends 0.5 c e, f d Similarly to the aggregate data Figure 3, wages and dividends feature VR s which are respectively increasing and decreasing. Interest rates have VR s similar to those of wages. The right panel of the above figure reports the term structures of VR s of the various quantities in the table. We observe that both gross and net value added feature VR s increasing and larger than unity, whereas the operating surplus, as well as the disposable income and the net saving, have VR s strongly decreasing below unity. In particular, the shape of the VR s of dividends and operating surplus is quite similar. In order to understand whence the change in the slope of the term-structure of risk comes from, the eight panels of the next figure show the term-structure effects due to each variable from the table. 66

68 Term structure effect of capital depreciation a Term structure effect of wages b.1 a a 1.5 b 1.5 b c 0.5 c e, f e, f d d Term structure effect of teaxes on production b Term structure effect of service of debt c.1 a a 1.5 b 1.5 b c 0.5 c e, f e, f d d Term structure effect of shareholders remuneration c Term structure effect of reinvestment of earnings c.3 a a 1.5 b 1.5 b c 0.5 c e, f e, f d d Term structure effect of taxes on income and wealth d Term structure effect of capital transfers e.1 a a 1.5 b 1.5 b c 0.5 c e, f e, f d d Namely, each shadow area represents the increase blue or decrease red in the slope at a given horizon. Capital depreciation explains a limited decrease between the slopes of term-structures of gross and net value added. Then, wages do explain the large gap between the slope of net value added and essentially the slope of all other variables from the table. For instance, the term-structure effect due to interest payments is quite negligible and is positive up to about 6 years and then negative. Therefore, even if wages and interest payments have similar upward-sloping VR s, the former has a big impact whereas the latter does not affect the timing of risk. 67

69 Given co-integration, we can understand the heterogeneity in the slopes of the VR s as heterogeneity in the exposition of a given variable with the transitory component of value added. Wages and dividends load respectively more and less than value added on such a transitory shock and, hence, feature increasing and decreasing term-structures. This interpretation is consistent with the idea that income insurance within the firm from shareholders to workers is the main determinant of the timing of dividend risk. Such an hypothesis is confirmed by i the strong negative correlation between the wage and dividend shares of value added, which is about -68.5% p-value:.000; and ii the counter-cyclical dynamics of the wage share its correlation with the de-trended component of value added is about -33.1%, with p-value of.006. Instead, the fraction of value added imputed to interest payments that is the bondholders remuneration is essentially uncorrelated with the dividend-share about 2.4%, with p-value of.842. The left panel of the next figure shows the scatter plot of the wage- and the dividend-shares; the right panel shows the scatter plot of the de-trended increments of value added and the increments of the wage-share. dividend share wage share wage share value added de trended To provide further support to the interpretation of the data as a result of income insurance, I regress the dividend-share D/V on its lag and the lags of the interest-share I/V and the wage-share W/V t-statistics are reported in parentheses: D/V t = D/V t I/V t 1.16 W/V t 1 + ε t, adj-r 2 = 64.1%, D/V t = D/V t D/V t 2.05 I/V t I/V t 2.18 W/V t W/V t 2 + ε t, adj-r 2 = 68.8% By testing for Granger causality, the wage-share Granger causes the dividend-share p-value:.002 or.031 with 1 or 2 lags with negative sign, whereas the interest-share does not p-value:.538 or.544 with 1 or 2 lags. Similarly to the analysis of Section II, also the non-financial corporate sector data suggest that income insurance is the main determinant of the timing of risk. Online Appendix B. Results Under Countercyclical Heteroscedasticity Recall that total resources are characterized by Y t = X t Z t with logx t = x t, logz t = z t = z ẑ t and Wages and dividends satisfy dx t =µ t σ 2 x/2dt + σ x db x,t, OA.1 dµ t =λ µ µ µ t dt + σ µ db µ,t, OA.2 dẑ t =λ z z ẑ t dt + ˆσ z ẑt db z,t. OA.3 W t =X t α 1 φ e 1 φ z ẑ t, D t =X t e z ẑ t α 1 φ e 1 φ z ẑ t. OA.4 OA.5 The linearization of the logarithm of the dividend process implies: logd t logx t + log D + z logd z=0 z t = x t + d 0 + d z ẑ t, OA.6 where log D = d 0 = log 1 α 1 φ dz z with d z = 1 + φ α. OA.7 α φ α 68

70 Proposition A The shareholders utility process under preferences in Eq. 3 and dynamics in Eq. OA.1-OA.2-OA.3-OA.6 is given by where u 0, JX t,µ t,ẑ t = 1 1 γ s X 1 γ s t expu γ s d 0 + u µ µ t + u z + 1 γ s d z ẑ t, OA.8 u µ = 1 γ s e cq, u z = λ zγ s 1 φ α + λ µ e cq λ z α φ e φ z, α and cq = E[cq t ] are endogenous constants depending on the primitive parameters. The shareholders consumption-wealth ratio is equal to cq t = logc s,t /Q s,t = logβ χ 1 u 0 + u µ µ t + u z ẑ t. OA.9 Proposition B The equilibrium state price density has dynamics given by dξ 0,t =ξ 0,t d f C fc + ξ 0,t f J dt = rtξ 0,t dt θ x tξ 0,t db x,t θ µ tξ 0,t db µ,t θ z tξ 0,t db z,t, OA.10 where the instantaneous risk-free rate satisfies and the instantaneous prices of risk are given by rt = r 0 + r µ µ t + r z ẑ t, OA.11 θ x t =γ s σ x, θ µ t = γ s 1/ψ e cq + λ µ σ µ, θ z t = ẑ t d z γ s ˆσ z ψγ s 1e cq Θ+λ z +d z γ s 1 ˆσ 2 z ˆσ z ψγ s 1 OA.12 OA.13. OA.14 Proof of Proposition A and B: Under the infinite horizon, the utility process J satisfies the following Bellman equation: DJX,µ,ẑ + f C s,j = 0, OA.15 where D denotes the differential operator. Eq. OA.15 can be written as 0 =J X µx J X,Xσ 2 xx 2 + J µ λ µ µ µ J µ,µσ 2 µ + J z λ z z ẑ J z,zσ 2 z ẑ + f C s,j. Guess a solution of the form JX,µ,ẑ = 1 1 γ s X 1 γ s gµ,ẑ. The Bellman equation reduces to 0 =µ 1 2 γ sσ 2 x + g µ λ µ µ µ g 1 γ s + 1 g µ,µ 2 g σ 2 µ 1 γ s + g z λ z z ẑ g 1 γ s g z,z g ˆσ 2 z ẑ 1 γ s + Under Assumption 2 and stochastic differential utility, the pricing kernel has dynamics given by β 1 1/ψ g 1/χ e 1 1/ψd 0+d z ẑ 1. OA.16 dξ 0,t = ξ 0,t d f C fc + ξ 0,t f J dt = rtξ 0,t θ x tξ 0,t db x,t θ µ tξ 0,t db µ,t θ z tξ 0,t db z,t, OA.17 where, by use of Itô s Lemma and Eq. OA.16, we get rt = X f C f C µx 1 2 θ x t = X f C f C σ x X, θ µ t = µ f C f C σ µ, θ z t = ẑ, z f C f C ˆσ z X,X f C f C σ 2 xx 2 µ f C f C λ µ µ µ 2 1 µ,µ f C f C σ 2 µ z f C f C λ z z z 1 z,z f C 2 f C ˆσ 2 z ẑ f J, 69

71 An exact solution for gµ,ẑ satisfying Eq. OA.16 does not exist for ψ 1. Therefore, I look for a solution of gµ,ẑ around the unconditional mean of the consumption-wealth ratio. Aggregate wealth is given by Q s,t = E t [ t ] ξ t,u C s,u du, and, applying Fubini s Theorem and taking standard limits, the consumption-wealth ratio satisfies [ ] C s,t Q s,t = rt dt 1 E dq t Q dt 1 E t [ dξ ξ ] dq Q. OA.18 Guess Q s,t = C s,t β 1 gµ t,ẑ t e γ s 1d 0 +d z ẑ t 1/χ and apply Itô s Lemma to get dq Q. Then, plug in the wealth dynamics, the risk-free rate and the pricing kernel into Eq. OA.18: after tedious calculus you can recognize that the guess solution is correct. Notice that the consumption-wealth ratio approaches to β when ψ 1 as usual. Denote cq = E[logC s,t logq s,t ], hence, a first-order approximation of the consumption-wealth ratio around cq produces C s,t Q s,t = βgµ t,ẑ t 1/χ e 1 1/ψd 0+d z ẑ t e cx 1 cx + logβ 1 χ loggµt,ẑ t + γ s 1d 0 + d z ẑ t. Using such approximation in the Bellman equation OA.16 leads to 0 =µ 1 2 γ sσ 2 x + g µ /ψ g λ µ µ µ 1 γ s g µ,µ g σ 2 µ 1 γ s + g z λ z z ẑ g 1 γ s + 1 g z,z 2 g ˆσ 2 z ẑ 1 γ s e cx 1 cx + logβ 1 χ loggµ,ẑ + 1 1/ψd 0 + d z ẑ which has exponentially affine solution gµ,ẑ = e u 0+1 γ s d 0 +u µ µ+u z +1 γ s d z ẑ where u 0 = 1 2 e cq 1 γ s 2βψ 1 ψ + 2ecq ψcq 1 logβ 1 ψ σ 2 xγ s + 2 µλ µ e cq +λ µ 2 zλ z u µ = 1 γ s e cq +λ µ, u z = ecq +λ z +d z γ s 1 ˆσ 2 z e 2cq +λ 2 z +2e cq λ z +d z γ s 1 ˆσ 2 z ˆσ 2 z, β, e cq +λ z e 2cq +λ 2 z +2e cq λ z +d z γ s 1 ˆσ 2 z γ s 1 ˆσ 2 z + 1 γ sσ 2 µ, e cq +λ µ 2 and the endogenous constant cq satisfies cq = logβ χ 1 u 0 + u µ µ + u z z. The risk-free rate and the prices of risk take the form: r 0 = 2βψ+2d z zλ z 1+ψγ s σ 2 x 2ψ + 1 ψψγ s 1σ 2 µ 2ψ 2 e cq +λ µ 2, r µ = 1 ψ, r z = 1 2ψ 2 1 γ s 2 ˆσ 2 z θ x t =γ s σ x, +d z γ s 1 ˆσ 2 z 2e 2cq ψ 1ψγ s 1 + 2ψ 1ψγ s 1λ 2 z 2ψ 1λ z Θψγ s 1 + d z γ s 1 ˆσ 2 z 2Θψγs 1 + d z γ s 1 ˆσ 2 z + 2e cq ψγ s 1 Θ Θψ + 2ψ 1λ z + ψ 2d z γ s 1 ˆσ 2 z, θ µ t = γ s 1/ψσ µ e cq +λ µ, θ z t = ẑ t d z γ s ˆσ z ψγ s 1e cq Θ+λ z +d z γ s 1 ˆσ 2 z ˆσ z ψγ s 1, where Θ = e 2cq + λ 2 z + 2e cq λ z + d z γ s 1 ˆσ 2 z. q.e.d. Proposition C The following conditional expectation has exponential affine solution: M t,τ c = E t [e c 0+c 1 logξ 0,t+τ +c 2 x t+τ +c 3 µ t+τ +c 4 ẑ t+τ ] = ξ c 1 0,t X c 2 t e l 0τ, c+l µ τ, cµ t +l z τ, cẑ t, OA.19 70

72 where c = c 0,c 1,c 2,c 3,c 4, model parameters are such that the expectation exists finite and l 0,l µ and l z are deterministic functions of time. Proposition D The moment generating function of the logarithm of dividends has the following approximation: D t τ,n = E t [D n t+τ] e nx t+b 0 n,τ+b µ n,τµ t +B z n,τẑ t, OA.20 where n and model parameters are such that the expectation exists finite, the approximation makes use of Eq. OA.6 and B 0,B µ and B z are deterministic functions of time. Proof of Proposition C: Consider the following conditional expectation: M t,τ c = E t [e c 0+c 1 logξ 0,t+τ +c 2 logx t+τ +c 3 µ t+τ +c 4 ẑ t+τ ] OA.21 where c = c 0,c 1,c 2,c 3,c 4 is a coefficient vector such that the expectation exists. Guess an exponential affine solution of the kind: M t,τ c = e c 1 logξ 0,t +c 2 logx t +l 0 τ, c+l µ τ, cµ t +l z τ, cẑ t, OA.22 where l 0 τ, c,l µ τ, c and l z τ, c are deterministic functions of time. Feynman-Kac gives that M has to meet the following partial differential equation 0 =M t M ξ r 0 + r µ µ + r z ẑ M ξ,ξθ x t 2 + θ µ t 2 + θ z t 2 +M X µx M X,Xσ 2 xx 2 +M µ λ µ µ µ M µ,µσ 2 µ +M z λ z z ẑ M z,z ˆσ 2 z ẑ M ξ,x θ x tσ x X M ξ,µ θ µ tσ µ M ξ,z θ z t ˆσ z ẑ, 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 a linear function of the states µ and ẑ: 0 =l 0τ, c + l µτ, cµ + l zτ, cẑ c 1 r 0 + r µ µ + r z ẑ c 1c 1 1θ x t 2 + θ µ t 2 + θ z t 2 + c 2 µ c 2c 2 1σ 2 x + l µ τ, cλ µ µ µ l µτ, c 2 σ 2 µ + l z τ, cλ z z ẑ l zτ, c 2 ˆσ 2 z ẑ c 1 c 2 θ x tσ x c 1 l µ τ, cθ µ tσ µ c 1 l z τ, cθ z t ˆσ z ẑ. Hence, we get three ordinary differential equations for l 0 τ, c,l µ τ, c and l z τ, c: 0 =l 0τ, c c 1 r c 1c 1 1θ x t 2 + θ µ t 2 + θ z t c 2c 2 1σ 2 x + l µ τ, cλ µ µ l µτ, c 2 σ 2 µ + l z τ, cλ z z c 1 c 2 θ x tσ x c 1 l µ τ, cθ µ tσ µ, 0 =l µτ, c c 1 r µ + c 2 l µ τ, cλ µ 0 =l zτ, c c 1 r z l z τ, cλ z l zτ, c 2 ˆσ 2 z c 1 l z τ, cθ 0 z ˆσ z with θ 0 z = θ z t/ ẑ and initial conditions: l 0 0, c = c 0,l µ 0, c = c 3 and l z 0, c = c 4. Explicit solutions are available. q.e.d. Proof of Proposition D: The conditional moment generating function D t τ,n of Eq. OA.20 obtains as a special case of M t,τ c with c = nd 0,0,n,0,nd z. Therefore, it is given by D t τ,n = X n t e l 0τ, c+l µ τ, cµ t +l z τ, cẑ t with l 0 0, c = nd 0,l µ 0, c = 0 and l z 0, c = nd z and B 0 n,τ,b µ n,τ and B z n,τ are implicitly defined. q.e.d. Proposition E The equilibrium price of the market dividend strip with maturity τ is given by P t,τ = E t [ξ t,t+τ D t+τ ] X t e A 0τ+A µ τµ t +A z τẑ t, OA.23 71

73 where model parameters are such that the expectation exists finite, the approximation makes use of Eq. OA.6 and A 0,A µ and A z are deterministic functions of time. The instantaneous volatility and premium on the dividend strip with maturity τ are given by σ P t,τ = σ 2 x + 1 e λ µτ 2 r µ 1 2 λ 2 σ 2 µ + λz ˆσ z + θ 0 z + ˆσ z tan τ /2 arctanθ 0 z ˆσ z + λ z d z ˆσ 2 z / 2 ẑt, µ OA.24 µ P rt,τ =θ x tσ x + 1 e λ µτ 1 r µ θ µ tσ µ λ µ + θ 0 z λz ˆσ z + θ 0 z + ˆσ z tan τ /2 arctanθ 0 z ˆσ z + λ z d z ˆσ 2 z / ẑ t. OA.25 with θ 0 z = θ z t/ ẑ t and = λ 2 z 2θ 0 z λ z ˆσ z θ 0 z + 2r z ˆσ 2 z. Proof of Proposition E: The equilibrium price of the market dividend strip with maturity τ of Eq. OA.25 obtains as a special case of M t,τ c with c = d 0,1,1,0,d z. Therefore, it is given by P t,τ = ξ 1 0,t M t,τ c = X t e l 0τ, c+l µ τ, cµ t +l z τ, cẑ t with l 0 0, c = d 0,l µ 0, c = 0 and l z 0, c = d z and A 0 τ,a µ τ and A z τ are implicitly defined. The dynamics of the market dividend strip price obtains by applying Itô s Lemma to P t,τ : Therefore the return volatility is given by and the premium is given by dp t,τ = [ ]dt + x P t,τ σ x db x,t + µ P t,τ σ µ db µ,t + z P t,τ ˆσ z ẑt db z,t. σ P t,τ = Pt,τ 1 x P t,τ σ x 2 + µ P t,τ σ µ 2 + z P t,τ ˆσ z ẑt 2 = σ 2 x + A µ τσ µ 2 + A z τ ˆσ z ẑt 2, µ P rt,τ = 1 dt dξ0,t ξ 0,t, dp t,τ P t,τ = θ x tσ x + θ µ ta µ τσ µ + θ z ta z τ ˆσ z ẑt. q.e.d. Proposition F The equilibrium price of the market asset is given by P t = E t [ t ξ t,u D u du] X t β 1 e u 0χ 1 +d 0 +u µ χ 1 µ t + u z χ 1 +d z ẑ t OA.26 where model parameters are such that the expectation exists finite, the approximation makes use of Eq. OA.6, and u 0,u µ and u z are endogenous constants depending on the primitive parameters. The instantaneous volatility and premium on the market asset are given by where θ 0 z = θ z t/ ẑ t. σ P t = σ 2 x + 1 1/ψ2 u 2 µσ 2 µ + 1 γ s 2 µ P rt =θ x tσ x + 1 1/ψu µθ µ tσ µ 1 γ s + d z ˆσ z + 1 1/ψu z ˆσ 2 z 1 γ ẑt s, OA.27 θ 0 z d z ˆσ z + θ0 z 1 1/ψu z ˆσ z 1 γ s ẑ t, OA.28 Proof of Proposition F: Under Assumption 2, the shareholders act as a representative agent on the financial markets and, hence, the equilibrium price of the market asset is equal to the shareholders wealth. Therefore, using the results of Proposition 1, the market asset price can be written as P t = Q s,t = C s,t e cq t = X t e logβ+u 0χ 1 +d 0 +u µ χ 1 µ t +u z χ 1 +d z ẑ t. The dynamics of the market asset price obtains by applying Itô s Lemma to P t : dp t = [ ]dt + x P t σ x db x,t + µ P t σ µ db µ,t + z P t ˆσ z ẑt db z,t. 72

74 Therefore the return volatility is given by and the premium is given by σ P t = Pt 1 x P t σ x 2 + µ P t σ µ 2 + z P t ˆσ z ẑt 2 = σ 2 x + u µ χ 1 σ µ 2 + u z χ 1 + d z ˆσ z ẑt 2, µ P rt = 1 dt dξ0,t ξ 0,t, dp t P t = θ x tσ x + θ µ tu µ χ 1 σ µ + θ z t u z χ 1 + d z ˆσz ẑt. q.e.d. Online Appendix C. Results under Power Utility Proposition A The equilibrium state price density satisfies ξ 0,t = e βt Y γ s t 1 + α γs = e βt Y γ s α φ e φ zt α t 1 ωz t γ s, OA.29 and has dynamics: dξ 0,t = ξ 0,t rtdt θx tdb x,t θ z tdb z,t, where the instantaneous risk-free rate satisfies rt = r 0 + r µ µ t + r z z t, OA.30 with r 0 = β γ s1+γ s σ 2 x 2, r µ µ t = γ s µ t, r z z t = λ z γ s 1 + αφ α φ e φ zt α z zt γ2 s e z t φ 2+γs α φ +αφ 1 e z t φ α 1+φ φ 2 σ 2 2α e z t φ α φ 2 z, and the instantaneous prices of risk are given by θ x t = γ s σ x, θ z t = γ s σ z 1 + αφ α φ e φ zt α Proposition B The equilibrium price of the market dividend strip with maturity τ is given by. P t,τ = E t [ξ t,t+τ D t+τ ] = X t e βτ P 0 z t P 1 τ,µ t,z t, OA.31 where model parameters are such that the expectation exists finite, P 0 z t = e z t α 1 φ e 1 φ z t γs, P 1 τ,µ t,z t = j=0 B1 γ s, j α 1 φ j e G 0τ, j+g z τ, jµ t +G z τ, jz t, OA.32 OA.33 r Γr with Br,k = r kk!γr k, and G 0,G µ and G z are deterministic functions of time. The instantaneous volatility and premium are given by σ P t,τ = σ 2 x + µ logp 1 τ,µ t,z t 2 σ 2 µ + z logp 0 z t + z logp 1 τ,µ t,z t 2 σ 2 z, µ P rt,τ =γ s σ 2 x + γ s z logp 0 z t + z logp 1 τ,µ t,z t 1 + αφ σ 2 α φ e φ zt α z. OA.34 OA.35 Proof of Proposition A: Under Assumption 2, the marginal rate of substitution of market participants equals the marginal utility of shareholders evaluated at their optimal consumption: C s,t = D t t. Therefore, we have that the state price density is given by ξ 0,t = V D t = c 0 e βt D γ s t, 73

75 where c 0 is a positive constant which can be standardized to one. Using the resource constraint Y t = W t + D t, the dividend process can be written as D t = Y t 1 α 1 φ e φ z t and Eq. OA.29 follows. Consequently, the state price density has dynamics given by dξ 0,t ξ 0,t = te βt Dt γs e βt Dt γs dt + X e βt D t γs e βt Dt γs dx t + ze βt D t γs e βt Dt γs dz t X,X e βt Dt γs e βt Dt γs d X t z,z e βt Dt γs e βt Dt γs d z t. The results about rt,θ x t and θ z t automatically follow. q.e.d. Proof of Proposition B: The equilibrium price of the market dividend strip with maturity τ of Eq. OA.31 is defined as Provided z t > 1 φ φ [ ] P t,τ =E t e βτ D 1 γs t+τ Dt γs = e βτ X γ [ s t e z t α 1 φ e 1 φ z t γs E t logα t, using the generalized binomial formula we have [ P t,τ =e βτ X γ s t e z t α 1 φ e 1 φ z t γs E t X 1 γ s t+τ =e βτ X γ s t j=0 X 1 γ s t+τ e z t+τ α 1 φ e 1 φ z t+τ 1 γs ]. B1 γ s, j α 1 φ j e 1 γ s jφ z t+τ ] e z t α 1 φ e 1 φ z t γs j=0b1 γ s, j α 1 φ j E t [ X 1 γ s t+τ e 1 γ s jφ z t+τ ], [ r Γr where Br,k = r kk!γr k. For each j = 0,1,..., the conditional expectation E t X 1 γ s t+τ e 1 γ s jφ z t+τ ], is a special case of M t,τ c with c = 0,0,1 γ s,0,1 γ s jφ. Therefore, it is given by E t [ X 1 γ s t+τ e 1 γ s jφ z t+τ ] = X 1 γ s t e G 0τ, j+g µ τ, jµ t +G z τ, jz t, where G 0 τ, j = 1 4λ z λ 3 µ 3 γ s 1 2 λ z + e 2τλ µ 4e τλ µ 2τλ µ σ 2 µ 2τλ 3 µσ 2 x µ1 γ s +2τ1 γ s λ z λ 3 µσ 2 x + 4λ z λ 2 µ + 1 e τλ µ τλ µ + e 2τλ z 1 λ 3 µσ 2 z γ s 1 + jφ 2 e τλ z 1 zλ µ 1 γ s jφ, G µ τ, j = 1 e τλµ 1 γ s γ s λ µ, G z τ, j =e τλ z 1 γ s jφ. The result of Eq. OA.31 automatically follows. Eq. OA.34 and OA.35 obtain by an application of Itô s Lemma. q.e.d. 74