The Equity Premium and the One Percent∗ Alexis Akira Toda†

Kieran Walsh‡

This version: August 31, 2015

Abstract We find that when the income share of the top 1% income earners in the U.S. rises above trend by one percentage point, subsequent one year market excess returns decline on average by about 3–6%. This negative relation remains strong and significant even when controlling for classic return predictors such as the price-dividend and the consumption-wealth ratios. To explain this stylized fact, we build general equilibrium asset pricing models with heterogeneity in wealth and risk aversion across agents. We analytically show in a two-period model with CRRA utility as well as in an infinite horizon model with log utility and limited market participation that an increase in the wealth share of the risk tolerant agents reduces the equity premium. Intuitively, when wealth shifts into the hands of rich and risk tolerant agents, average risk aversion falls, pushing down the risk premium. Cross-country panel regressions suggest that the inverse relation between inequality and returns is smaller but still significant outside of the U.S. Keywords: equity premium; heterogeneous risk aversion; return prediction; wealth distribution; international equity markets. JEL codes: D31, D53, D58, F30, G12, G17.

1

Introduction

Does the wealth distribution matter for asset pricing? Common sense tells us that it does: as the rich get richer, they buy risky assets and drive up prices. Indeed, over a century ago prior to the advent of modern mathematical finance, Fisher (1910) argued that there is an intimate relationship between prices, the heterogeneity of agents in the economy, and booms and busts. He contrasted (p. 175) the “enterpriser-borrower” with the “creditor, the salaried man, or the laborer,” emphasizing that the former class of society accelerates fluctuations in prices and production. Central to his theories of fluctuations were differences in preferences and wealth across people. Following the seminal work of Lucas (1978), however, the “representative agent” consumption-based asset pricing models—which seem to allow no role ∗ We benefited from comments by Dan Cao, Vasco Carvalho, Peter Debaere, Graham Elliott, John Geanakoplos, Jim Hamilton, Gordon Hanson, Fumio Hayashi, Toshiki Honda, Jiasun Li, Larry Schmidt, Frank Warnock, and seminar participants at Boston College, Cambridge-INET, Carleton, Darden, Federal Reserve Board of Governors, HEC Lausanne, Hitotsubashi ICS, Kyoto, Tokyo, UCSD, Vassar, Yale, Yokohama National University, 2014 Australasian Finance and Banking Conference, 2014 Northern Finance Association Conference, 2015 Econometric Society World Congress, 2015 ICMAIF, 2015 Midwest Macro, 2015 SED, and 2015 UVa-Richmond Fed Jamboree. Earlier drafts of this paper were circulated with the title “Asset Pricing and the One Percent.” † Department of Economics, University of California San Diego. Email: [email protected]. ‡ Darden School of Business, University of Virginia. Email: [email protected].

1

for agent heterogeneity—have dominated the literature, at least until recently. Yet agent heterogeneity may (and is likely to) matter even if a representative agent exists: unless agents have very specific preferences that admit the Gorman (1953) aggregation (a knife-edge case, which is unlikely to hold in reality), the preferences of the representative agent (in the sense of Constantinides (1982)) will in general depend on the wealth distribution, as pointed out by Gollier (2001). Indeed, even with complete markets, the preferences of the representative agent are typically nonstandard when individual utilities do not reside within quite particular classes. To see the intuition as to why the wealth distribution affects asset pricing, consider an economy consisting of people with different attitudes towards risk or beliefs about future dividends. In this economy, equilibrium risk premiums and prices balance the agents’ preferences and beliefs. If wealth shifts into the hands of the optimistic or less risk averse, for markets to clear, prices of risky assets must rise and risk premiums must fall to counterbalance the new demand of these agents. In this paper, we establish both the theoretical and empirical links between income/wealth inequality and asset prices. This paper has two main contributions. First, we theoretically explore the asset pricing implications of general equilibrium models with heterogeneous agents and derive testable implications linking asset returns and inequality across risk aversion types. Using a two-period model, we show that the stochastic discount factor depends both on market returns and average risk tolerance, which depends on the wealth distribution. In the special case with heterogeneous CRRA utility, we prove that there exists an equilibrium in which wealth concentration leads to a decline in the equity premium (If the equilibrium is unique, the inverse relationship between inequality and excess returns is unambiguous). Although the connection between the heterogeneity of agents’ risk aversion and asset prices has been recognized at least since Dumas (1989) and recently emphasized by Gˆ arleanu and Panageas (2015), the literature presents few testable implications that can be easily examined in financial data. We then explore the robustness of this link between prices and inequality by turning to an infinite horizon model with two assets, stocks and bonds, and a stark form of heterogeneity in risk aversion: one type of agent does not participate in risky stock markets. Exploiting log utility, we solve the model in closed-form (up to a single nonlinear equation with one unknown). Moreover, we prove that wealth redistribution from the bond holders to the stock and bond investors leads to a decline in the subsequent equity premium. We also illustrate the effect of the wealth distribution on asset prices with numerical examples. Two agents inhabit the model and trade a riskless bond in zero net supply and a risky asset in positive supply. We consider two examples, one in which agents have constant but heterogeneous relative risk aversion preferences, and another in which agents have identical but decreasing relative risk aversion preferences. Note that with declining relative risk aversion the wealthy are endogenously less risk averse. We perform comparative statics with respect to the initial endowment share of the more risk tolerant agent and find an inverse relationship between his income and the subsequent equity premium. In line with intuition, as the risk tolerant rich get richer, they buy risky assets, increasing their relative price. Subsequent excess returns thus fall. Second, we empirically explore the theoretical predictions. We find that when the income share of the top 1% income earners in the U.S. is above trend, 2

the subsequent one year U.S. stock market equity premium is below average. That is, current inequality appears to forecast the subsequent risk premium of the U.S. stock market. Many heterogeneous agent general equilibrium models in both macroeconomics and finance predict a relationship between the concentration of income and asset prices (see Section 1.1). We thus provide empirical support for a literature which has been subject to relatively little direct testing. Furthermore, the patterns we uncover are intuitive. In short, if one believes top earners are all else equal more willing to trade risk for return, then it should not be surprising that in the data asset returns suffer as the rich get richer. More specifically, we employ regression analysis to establish the correlation between inequality and returns. Regressions of the year t to year t + 1 excess return on the year t top 1% income share indicate a strong and significant negative correlation: when the top 1% income share rises above trend by one percentage point, subsequent one year market excess returns decline on average by 3–6%, depending on the detrending method. This relation is strongly statistically significant and admits an R-squared of 9%. We show that estimated top wealth share series and the .1% income share also negatively predict subsequent returns. Furthermore, the top 1% income share predicts asset returns even after we control for some classic return forecasters such as the price-dividend ratio (Shiller, 1981) and the consumption-wealth ratio (Lettau and Ludvigson, 2001). It appears that the top 1% income share is not simply a proxy for the relative price level, which previous research shows correlates with subsequent returns. This is perhaps surprising because one imagines the rich being disproportionately exposed to stock price fluctuations. Our findings are also robust to the exclusion of capital gains in the income share series, to the inclusion of macro control variables, and to a large variety of detrending methods. We uncover a similar pattern in international data on inequality and financial markets: post-1969 cross-country fixed-effects panel regressions suggest that when the top 1% income share rises above trend by one percentage point, subsequent one year market returns significantly decline on average by 2%. This relationship is particularly strong for relatively “closed” economies such as poorer countries and emerging markets, as predicted by theory. In countries with low levels of investing home bias (“small open economies”), we find a large and significant inverse relationship between the U.S. 1% share (a proxy of the global 1% share) and subsequent domestic excess returns.

1.1

Related literature

For many years after Fisher, in analyzing the link between individual utility maximization and asset prices, financial theorists either employed a rational representative agent or considered cases of heterogeneous agent models that admit aggregation, that is, cases in which the model is equivalent to one with a representative agent. Extending the portfolio choice work of Markowitz (1952) and Tobin (1958), Sharpe (1964) and Lintner (1965a,b) established the Capital Asset Pricing Model (CAPM).1 These original CAPM papers, which concluded that an asset’s covariance with the aggregate market determines its return, actually allowed for substantial heterogeneity in endowments and risk preferences across investors. However, their form of quadratic or mean-variance preferences 1 See

Geanakoplos and Shubik (1990) for a general and rigorous treatment of CAPM theory.

3

admitted aggregation and obviated the role of the wealth distribution. The seminal consumption-based asset pricing work of Lucas (1978), Breeden (1979), and Hansen and Singleton (1983) also abstracted from investor heterogeneity. They and others derived and tested analytic relationships between the marginal rate of substitution of a representative agent (with standard preferences) and asset prices. Despite the elegance and tractability of the representative agent/aggregation approach, it has failed to adequately explain the fluctuations of asset prices in the economy. Largely inspired by the limited empirical fit of the CAPM (in explaining the cross section of stock returns), the equity premium puzzle (Mehra and Prescott, 1985), and excess stock market volatility and related price-dividend ratio anomalies (Shiller, 1981), since the 1980s theorists have extended macro/finance general equilibrium models to consider non-standard utility functions and meaningful investor heterogeneity. These models can be categorized into two groups. In the first group, agents have identical standard (constant relative risk aversion) preferences but are subject to uninsured idiosyncratic risks.2 Although the models of this literature have had some quantitative success, the empirical results (based on consumption panel data) are mixed and may even be spuriously caused by the heavy tails in the cross-sectional consumption distribution (Toda and Walsh, 2015). In the second group, markets are complete and agents have either heterogeneous CRRA preferences3 or identical but non-homothetic preferences.4 In this class of models the marginal rates of substitution are equalized across agents and a “representative agent” in the sense of Constantinides (1982) exists, but aggregation in the sense of Gorman (1953) fails. Therefore there is room for agent heterogeneity to matter for asset pricing. However, this type of agent heterogeneity is generally considered to be irrelevant for asset pricing because in dynamic models the economy is dominated by the richest agent (the agent with the largest expected wealth growth rate) in the long run (Sandroni, 2000; Blume and Easley, 2006).5 One notable exception is Gˆarleanu and Panageas (2015), who study a continuous-time overlapping generations endowment economy with two agent types with Epstein-Zin constant elasticity of intertemporal substitution/constant relative risk aversion preferences. Even if the aggregate consumption growth is i.i.d. (geometric Brownian motion), the risk-free rate and the equity premium are time-varying, even in the long run. The intuition is that when the risk tolerant agents have a higher wealth share, they drive up asset prices and the interest rate. The effect of preference heterogeneity persists since new agents are constantly born. Consistent with our empirical findings and model, the calibration of Gˆarleanu and Panageas (2015) suggests that increasing the consumption share of more risk tolerant agents pushes down the equity premium. All of the above works are theoretical, and our paper seems to be the first in the literature to empirically test the asset pricing implications of models with preference heterogeneity. 2 Examples are Mankiw (1986), Constantinides and Duffie (1996), Heaton and Lucas (1996), Krusell and Smith (1998), Brav et al. (2002), Cogley (2002), Balduzzi and Yao (2007), Kocherlakota and Pistaferri (2009), among many others. See Ludvigson (2013) for a review. 3 Examples are Dumas (1989), Wang (1996), Chan and Kogan (2002), Hara et al. (2007), Longstaff and Wang (2012), and Bhamra and Uppal (2014). 4 Examples are Gollier (2001) and Hatchondo (2008). 5 Guvenen (2009) studies cases with incomplete markets and heterogeneous Epstein-Zin preferences.

4

Although the wealth distribution theoretically affects asset prices, there are few empirical papers that directly document this connection. To the best of our knowledge, Johnson (2012) is the only one that explores this issue using income/wealth distribution data. However, his analysis is quite different from ours: his model relies on a “keeping up with the Joneses”-type consumption externality with incomplete markets. In contrast, we employ a standard general equilibrium model (a plain vanilla Arrow-Debreu model). Moreover, Johnson (2012) does not explore the ability of top income shares to predict market excess returns (our main result), and he detrends inequality differently from the way we do. Lastly, our study is related to the empirical literature on heterogeneity in risk preferences. A number of recent papers have found that the wealthy have portfolios more heavily skewed towards risky assets, and many of these studies have concluded that the wealthy are relatively more risk tolerant, either due to declining relative risk aversion or innate heterogeneity in relative risk aversion. See, for example, Carroll (2002), Vissing-Jørgensen (2002), Campbell (2006), Bucciol and Miniaci (2011), or Calvet and Sodini (2014). This literature lends credibility to our premise that the rich are relatively more tolerant.

2

Asset pricing implications of preference heterogeneity

In this section we present two models in which the heterogeneity in agents’ attitude towards risk matters for asset pricing and derive testable implications. In Section 2.1 we use a two period model to derive a moment condition one could take to the data. In the CRRA case, we prove that wealth concentration leads to declines in excess returns. We solve this model numerically in Section 2.3 to illustrate this relationship between inequality and asset prices. In the infinite horizon, log utility model of Section 2.2 we analytically prove that wealth concentration pushes down the subsequent equity premium.

2.1

Asset pricing with heterogeneous risk aversion in a two period model

Consider a two period model with time indexed by t and t + 1. There are I agents indexed by i = 1, . . . , I. Agent i has the expected utility over final wealth wi,t+1 , Et [ui (wi,t+1 )], where ui is von Neumann-Morgenstern utility function with u′i > 0 and u′′i < 0. There are J assets indexed by j = 1, . . . , J. Asset j trades at price qj per share (to be determined in equilibrium) at t and pays dividend Dj at t + 1. Agent i PJ is endowed with nij shares of asset j at t. Let wit = j=1 qj nij be the initial wealth of agent i. Letting n′ij be the number of shares agent i holds after trade,

5

the optimal portfolio problem is Et [ui (wi,t+1 )]

maximize {n′ij }

J X

subject to

qj n′ij = wit , wi,t+1 =

J X

Dj n′ij .

(2.1)

j=1

j=1

Assuming no trade frictions, the first-order condition for optimality with respect to n′ij is Et [u′i (wi,t+1 )Dj ] = λi qj , where λi > 0 is the Lagrange multiplier for the budget constraint. Dividing by qj and letting Rj,t+1 = Dj /qj be the gross return on asset j and assuming the existence of a risk-free asset (with gross risk-free rate Rf,t ), we obtain Et [u′i (wi,t+1 )(Rj,t+1 − Rf,t )] = 0. Using the Taylor approximation u′i (x) ≈ u′i (ai ) + u′′i (ai )(x − ai ) around the expected future wealth ait = Et [wi,t+1 ], letting x = wi,t+1 , we obtain Et [(u′i (ait ) + u′′i (ait )(wi,t+1 − ait ))(Rj,t+1 − Rf,t )] = 0,

(2.2)

where we have written = instead of ≈. Dividing both sides by −u′′i (ait ) > 0 and using the definition of the relative risk tolerance (reciprocal of the Arrow-Pratt measure of relative risk aversion) τi = −

u′i (ait ) , ait u′′i (ait )

we obtain Et [(ait τi − (wi,t+1 − ait ))(Rj,t+1 − Rf,t )] = 0. (2.3) PI Adding across all agents, letting Wt+1 = i=1 wi,t+1 be the aggregate wealth PI at t + 1, and dividing by Et [Wt+1 ] = i=1 ait , we obtain

Et [(¯ τ − Wt+1 / Et [Wt+1 ] + 1)(Rj,t+1 − Rf,t )] = 0, P P where τ¯ = i ait τi / i ait is the weighted average risk tolerance. Now since every asset must be held by some agent in equilibrium and there is no consumption at t, adding individual budget constraints, the growth rate of aggregate wealth must be equal to the market return Rm,t+1 . Therefore Wt+1 = Rm,t+1 Wt . Taking expectations, we obtain Et [Wt+1 ] = Et [Rm,t+1 ]Wt . Therefore Wt+1 / Et [Wt+1 ] = Rm,t+1 / Et [Rm,t+1 ]. Putting all the pieces together, we obtain Et [((¯ τ + 1) Et [Rm,t+1 ] − Rm,t+1 )(Rj,t+1 − Rf,t )] = 0, (2.4) which shows that Mt+1 = (¯ τ + 1) Et [Rm,t+1 ] − Rm,t+1

6

is an approximate scaled stochastic discount factor.6 Since (2.4) holds for any asset return Rj,t+1 , it also holds for the market return Rm,t+1 . Using E[XY ] = Cov[XY ]+E[X] E[Y ] for X = (¯ τ +1) Et [Rm,t+1 ]− Rm,t+1 and Y = Rm,t+1 − Rf,t , after some algebra we obtain Rf,t = Et [Rm,t+1 ] −

Vart [Rm,t+1 ] . τ¯ Et [Rm,t+1 ]

(2.5)

(2.5) shows that fixing the risk-free rate, if we increase the average risk tolerance τ¯, then either the expected stock return Et [Rm,t+1 ] must go down or the stock volatility Vart [Rm,t+1 ] must go up. If rich agents are on average more risk tolerant, then the stock return should suffer when the rich become richer, as wealth concentration increases the average risk tolerance of the economy. While we derived this negative relationship between inequality and stock returns using a Taylor approximation, we can formally prove this result in a special case. Assume that agent i has a CRRA utility ui (w) =

1 w1−γi . 1 − γi

Suppose that there are two assets, a risky asset in unit supply with dividend D and a risk-free asset in zero net supply. Let P = 1 be the price of the risky asset, R = D/P = D be the gross return, and Rf be the risk-free rate, which is endogenous.7 Let wi = P ni = ni be the initial wealth of agent i, where ni is the number of shares of the risky asset agent i is endowed with. Theorem 2.1. Consider an economy with CRRA agents and two assets. If there are multiple equilibria, consider the one with the highest risk-free rate. Suppose that γi > γj . If we shift some wealth from agent i to j, then the equity premium decreases. If in addition γi ≤ 1 for all i, the equilibrium is unique. The proof is in Appendix A. Theorem 2.1 states that if we compare the sets of equilibria before and after the wealth transfer, then the lowest equity premium decreases (so it unambiguously decreases if the equilibrium is unique). However, if there are multiple equilibria, the comparative statics can go in the opposite direction for some equilibria because the sign of the slope of the aggregate demand function alternates across equilibria. 6A

similar moment condition holds exactly with mean-variance utility: 1 Var[R(θ)] 2τi   J J J X X X R(θ) = Rj θj + Rf 1 − θj  = Rf + (Rj − Rf )θj ,

vi (θ) = E[R(θ)] −

j=1

j=1

j=1

where θ = (θ1 , . . . , θJ ) is the portfolio, θj is the fraction of wealth of a typical agent invested in asset j, and Rj is the gross return on asset j. 7 With just one consumption period, only the ratio of the stock and bond price is determined in equilibrium. We have fixed the stock return, but we could have instead fixed the interest rate and let the stock price be an endogenous variable. Regardless, the equity premium is endogenous.

7

2.2

Asset pricing with limited market participation in an infinite horizon model

In this section we exploit limited asset market participation and log utility to derive analytically the relationship between wealth concentration and excess returns in an infinite horizon heterogeneous agent model with stocks and bonds. Consider an infinite horizon economy consisting of two agent types, 1 and 2. Each type has log utility with discount factor 0 < β < 1. There are two assets, a stock in unit supply and a one period risk-free bond in zero net supply. We consider an extreme form of heterogeneity in risk aversion: while agent 1 may hold either asset, agent 2 may not buy or sell stock. That is, while the bondholder has curvature in his utility function and is willing to substitute consumption across time periods, he only invests in safe assets with nonrandom payoffs. We can interpret agent 2 as infinitely risk averse. ∞ The dividend process for the stock is exogenous and is denoted by {Dt }t=0 . Let wit be the wealth of agent i at time t, Wt = w1t + w2t be aggregate wealth, and xt = w1t /Wt be the wealth share of agent 1. A sequential equilibrium consists of allocations (consumption and portfolio) and prices (stock price Pt and risk-free rate Rf,t ) such that agents behave optimally taking prices as given and markets clear. The following theorem characterizes the equilibrium in closed-form, up to a single state-by-state equation (part 2 of the theorem) for the interest rate. See Appendix A for the proof. Theorem 2.2. Suppose that 0 < β < 1 and supt E[log(Dt+1 /Dt )] < ∞. Then there exists a unique equilibrium. The equilibrium has the following properties. 1. The stock price is given by Pt = Pt+1 +Dt+1 Pt

=

β 1−β Dt

and the stock return is Rt+1 =

Dt+1 βDt .

2. The risk-free rate Rf,t satisfies   (Rt+1 − Rf,t )xt = 0. Et Rt+1 + Rf,t (xt − 1) The equity premium Et [Rt+1 ] − Rf,t is a decreasing function of agent 1’s wealth share xt . 3. Each agent’s wealth evolves according to w1,t+1 = β(Rt+1 /xt + Rf,t (1 − 1/xt ))w1t , w2,t+1 = βRf,t w2,t . Agent 1’s wealth share evolves according to xt+1 = 1 −

Rf,t Rt+1 (1

− xt ).

Theorem 2.2 confirms our intuition that concentration of wealth in the hands of the risk tolerant drives down the equity premium. In particular, Theorem 2.2 implies that a one-time, unanticipated shock redistributing from risky investors to bond holders would increase subsequent excess returns on average. One caveat for this example is that while the interest rate depends on the wealth distribution, the expected stock return does not. That is, the wealth distribution affects the equity premium only through the interest rate. Note, however, that 8

this property is driven by the assumption of log utility, which ensures that the savings rate out of wealth is independent of the state and implies c = (1 − β)w. With general homothetic preferences, the savings rate depends on the state, which includes the wealth distribution. A final caveat is that in this analysis we assume that risk aversion heterogeneity takes the form of stock market exclusion for some agents. This yields closed-form solutions but is a stark (though perhaps realistic) way to introduce portfolio choice heterogeneity and non-aggregation. In Section 2.3 we provide numerical examples with full asset market participation in which the concentration of wealth in the hands of the relatively risk tolerant rich drives down the equity premium.

2.3

Numerical example

In this section we numerically solve two examples of the model in Section 2.1, one with agents with constant but heterogeneous relative risk aversion (CRRA) and another with identical decreasing relative risk aversion (DRRA) agents with heterogeneous wealth. Appendix C discusses the numerical algorithm in detail. 2.3.1

Two CRRA agents with heterogeneous risk aversion

Assume that there are two agent types, i = 1, 2. Agent 1 has high risk tolerance τH and agent 2 has low risk tolerance τL . For numerical values, we set γH = 1/τH = 0.5 and γL = 1/τL = 2. There is only one risky asset (stock) and a risk-free asset in zero net supply. Fraction α of stocks are initially held by agent 1 and fraction 1 − α by agent 2. There are two states with equal probability, and the dividend of the stock is 1 + µ ± σ, where µ = 0.07 and σ = 0.2. To see the accuracy of the approximation, we both solve the exact model numerically as well as the approximate model semi-analytically using (2.4). The results are shown in Figure 1. According to Figure 1a, the optimal portfolio of the exact and the approximate model are close, at least when the wealth share of the risk tolerant agent 1 is not too small. As the risk tolerant agent gets richer, the risk averse agent’s portfolio share of stock declines. Essentially agent 1 is providing insurance to agent 2. With respect to the equity premium, the approximation is virtually indistinguishable from the exact model. As the risk tolerant agent gets richer, there is more demand for borrowing, and therefore the risk-free rate increases in order to clear the market. In this example since the expected stock return is fixed at 7%, the equity premium shrinks as the risk tolerant agent gets richer. These results are consistent with Theorem 2.1. 2.3.2

Two agents with identical DRRA utilities

Consider the same example as above except that preferences are identical and exhibit decreasing relative risk aversion (DRRA). It is natural to assume that the Arrow-Pratt measure of relative risk aversion is a decreasing power function, RRA(x) = −

 x −η xu′′ (x) , = γ u′ (x) c

9

7 Agent 1 (Exact) Agent 1 (Approximate) Agent 2 (Exact) Agent 2 (Approximate)

4

Exact Approximate

6

Equity premium (%)

Portfolio share of stocks

5

3

2

1

5 4 3 2

0

1 0

0.2

0.4

0.6

0.8

1

0

Wealth share of agent 1

0.2

0.4

0.6

0.8

1

Wealth share of agent 1

(a) Portfolio share of stock.

(b) Equity premium.

Figure 1: Numerical example with heterogeneous CRRA preferences. Exact: numerical solution of exact model; Approximate: semi-analytical solution of approximate model using (2.4). where γ, η, c > 0 are parameters.8 The economic interpretation of the parameters is that c is a reference point for wealth, γ is the relative risk aversion coefficient at this reference point, and η governs the speed (elasticity) at which RRA decreases. Solving the ordinary differential equation, it follows that the von Neumann-Morgenstern utility function is Z x γ y −η u(x) = A e η ( c ) dy + B, c

where A > 0 and B are some constants. Since A and B merely define an affine transformation, they do not affect agents’ behavior. Therefore, without loss of generality we may assume A = 1 and B = 0, so the utility function is Z x γ y −η (2.6) u(x) = e η ( c ) dy. c

For a numerical example, we normalize the aggregate wealth at t = 0 to be W0 = 1 and set γ = 2, η = R1, and c = 1/2 (the reference point is equal distribux tion of wealth), so u(x) = 1/2 e1/y dy and RRA(x) = 1/x. Figure 2 shows the numerical solution. According to Figure 2a, as agent 1 gets richer, he becomes less risk averse and invests more in stocks. However, when he is too rich agent 2 is too poor to lend, and agent 1’s portfolio share of stocks eventually decreases. Consistent with empirical evidence discussed in Section 1.1, the wealthy choose risker portfolios. According to Figure 2b, the equity premium is highest when the wealth is equally distributed. As the wealth distribution becomes more skewed, the richer and more risk tolerant agent leverages and drives down the equity premium. As in the previous example, the approximation is excellent.

3

Empirical link between inequality and equity premium

Thus far, we have theoretically analyzed some models and examples in which the extent of inequality across agents with heterogeneous risk aversion is key 8 This specification has essentially two parameters, since only η and γcη are identified. η = 0 corresponds to the CRRA case.

10

7

1.2

6.5

Equity premium (%)

Portfolio share of stocks

1.4

1 0.8 0.6 0.4 Agent 1 (Exact) Agent 1 (Approximate) Agent 2 (Exact) Agent 2 (Approximate)

0.2 0 0

0.2

0.4

0.6

6 5.5 5

Exact Approximate

4.5 4 3.5

0.8

1

0

Wealth share of agent 1

0.2

0.4

0.6

0.8

1

Wealth share of agent 1

(a) Portfolio share of stock.

(b) Equity premium.

Figure 2: Numerical example with identical DRRA preference. Exact: numerical solution of exact model; Approximate: semi-analytical solution of approximate model using (2.4). in predicting returns. We found not only that the wealth distribution affects the relative prices of risky assets but also that the extent of inequality may determine an economy’s overall risk premium (and thus the equity premium). But, are macroeconomic and financial data consistent with the implications of this paper’s model and those in the above literature? In this section, we show that there is a strong and robust negative relationship between the top income/wealth share and subsequent medium-term excess stock market returns in the U.S. That is, current inequality appears to forecast the risk premium of the U.S. stock market (in Section 4 we uncover a similar pattern in the international data). The negative sign of the relationship is consistent with the above example, and the inequality measures do not seem to merely be proxying for either of two leading predictors of excess returns, price-dividend ratio (Shiller, 1981) and the consumption-wealth ratio (Lettau and Ludvigson, 2001).

3.1

Data

We employ the Piketty and Saez (2003) inequality measures for the U.S., which are available in spreadsheets on the website of Emmanuel Saez.9 In particular, we consider top income and wealth share measures. The income measures (with or without capital gains) are at the annual frequency and are based on tax return data, and cover the period 1913–2012. The wealth series (the top 1% wealth share, considered for robustness), covers 1916–2000 at the annual frequency and is based on estate tax data.10 The income series reflect in a given year the percent of income earned by the top earners pretax. Similarly, the wealth series is the percent of wealth owned by the richest 1%. See Piketty and Saez (2003) and Kopczuk and Saez (2004) for further details on the construction of these 9 http://elsa.berkeley.edu/

~ saez/ opposed to the income data, many years are missing in the 50s, 60s, and 70s for the wealth data, so we complete the series with cubic interpolation. In view of the theoretical results in Section 2, the relevant data is the wealth data. However, since the wealth data have many missing years, we focus on the income data and use the wealth data only as a robustness check. This choice can be justified because the major income source of the rich is capital income, which should be proportional to wealth. 10 As

11

Top income share (%)

20

Relative income share within top 10%

series. In Appendix B, we also consider .1% shares and the wealth share series of Saez and Zucman (2014). Figure 3a shows the top income share for each group, the richest 0–0.5%, 0.5–1%, 1–5%, and 5–10%. All groups seem to share a common trend, which is similar to the highest marginal tax rate in Figure 4. However, the behavior of these series around the trend is quite different. First, the top 0.5–1% share is very smooth. Second, the top 0–0.5% share seems procyclical (move in the same direction as booms and busts), which is most apparent in the 1920s, 1960s, 1990s, and mid-2000s. On the other hand, the behavior of the top 1–5% and 5–10% resemble each other and seems countercyclical (move in the opposite direction as booms and busts). Figure 3b shows the relative income share of each group within the top 10%. We can see that the top 0.5–1% is stable, the top 0–0.5% moves in the same direction as booms and busts, and the top 1–10% moves in the opposite direction. Top 0-0.5% Top 0.5-1% Top 1-5% Top 5-10%

15

10

5

0 1920

1940

1960

1980

2000

30

Top 0-0.5% Top 0.5-1% Top 1-5% Top 5-10%

20

10

0 1920

Year

(a) Absolute shares.

40

1940

1960

1980

2000

Year

(b) Relative shares within the top 10%.

Figure 3: Top income shares including capital gains (1913–2012). Within the context of the model in Section 2, this behavior can be explained if the richer agents are more risk tolerant. Consider, for example, the meanvariance model. Then the mutual fund theorem holds and agents invest more or less than 100% in stocks according as whether they are more or less risk tolerant than the average. If we assume that the top 10% hold the entire stock market, Figure 3b tells us that the top 0.5–1% roughly hold the market portfolio, the top 0–0.5% are more risk tolerant and leverage (borrow from the poor), and the top 1–10% are more risk averse and lend to the richest 0.5%. Thus, in bringing our theory to the data, we take the top 1%, rather than say the top 5% or 10%, as our dividing line between more and less risk tolerant agents. Below, we use not the raw Piketty-Saez series but rather detrended, stationary versions. Specifically, we detrend the inequality measures using either the Hodrick-Prescott (HP) filter with a smoothing parameter of 100, which is standard for annual frequencies, or the Kalman filter with an AR(2) cyclical component (see Appendix D for details). Stochastically detrending asset return predictors is in the tradition of Campbell (1991), for example, who removes a trend in the short-term interest rate before including it in stock return vector autoregressions. Indeed, the Piketty-Saez series appear to exhibit a U-shaped trend over the century, which might be due to the change in the marginal income tax rates. According to Figure 4, the marginal tax rate for the highest income earners increased from about 25% to 90% over the period 1930–1945 12

and started to decline in the 1960s, reaching about 40% in the 1980s.11 Thus the marginal tax rate exhibits an inverse U-shape that coincides with the trend in the Piketty-Saez series. Imposing stationarity in this way helps ensure the validity of standard error calculations and inference and prevent spurious regressions. Figure 5 plots the top 1% series (with capital gains) and their estimated trends. In Appendix B we employ different filtering methods. Our results appear robust to using a smoothing parameter of 10, the one-sided HP filter, the AR(1) Kalman filter (which is also one-sided), the moving average filter, or linear detrending. 25

100 90 Top Tax Rate

80

20 70

50

Percent

Percent

60 15

40 30 10 20 Top 1% Share 5

1920

1930

1940

1950

1960

1970

10

1980

1990

2000

0 2010

Figure 4: Top 1% income share including capital gains (left axis) and top marginal tax rate (right axis), 1913–2012. 45 40

Income Trend Wealth Trend

35

Percent

30 25 20 15 10 5

1920

1940

1960

1980

2000

Figure 5: Top 1% income share including capital gains (1913–2012) and top 1% wealth share (1916–2000). The thin lines are the HP filter trends. We calculate annual one year U.S. stock market excess returns using the annual stock market spreadsheet from the website of Robert Shiller.12 The spreadsheet contains historical one year interest rates and price, dividend, and earnings series for the S&P 500 index, which are all put into real terms using the consumer price index (CPI). These data are also used to calculate the series P/E10 and P/D10, which are the price-dividend and price-earnings ratios (in 11 The

tax rate data is from the Tax Foundation (http://taxfoundation.org/ ). ~ shiller/data.htm

12 http://www.econ.yale.edu/

13

real terms) for the S&P 500 based on 10 year moving averages of earnings and dividends. For the Lettau-Ludvigson consumption-wealth ratio, commonly referred to as CAY, we use 100 times the annual version of this series from the website of Amit Goyal (the spreadsheet for Welch and Goyal (2008)).13 It spans the period 1945–2012. We also include as controls GDP growth and, inspired by Lettau et al. (2008) and Bansal et al. (2014), consumption volatility. Annual data for real GDP and real consumption are from the website of the Federal Reserve Bank of St. Louis14 and span 1930–2012. We estimate consumption volatility using a standard GARCH(1,1) model for consumption growth.

3.2

Regression analysis

Table 1 shows the results of regressions of one year (t to t+1) excess stock market returns on top share measures (time t) and some classic return predictors (time t). We find that when the top 1% income share (January to December of year t) rises above trend by one percentage point, subsequent one year market excess returns (January to December of year t + 1) decline on average by 5.6% with the HP filter and by 3.46% with the Kalman filter. These coefficients are significant at the 1% and 5% levels respectively (using Newey-West standard errors), and the R-squared statistics are .09 and .05. Figure 6 shows the corresponding scatter plot for five year returns. It is clear, at least in sample, that the detrended top 1% share series has substantial power in forecasting the subsequent overall excess return on the stock market. This relationship also holds for the top 1% wealth share. With respect to one year returns, the top wealth share is strongly significant and yields an R-squared of .24, which is greater than the income share R-squared. Given the strength of the relationship, a question immediately arises. Is there some mechanical, non-equilibrium explanation for the relationship between inequality and subsequent excess returns? For example, might stock returns somehow be determining the top share measures? For a few reasons, the answer is likely no. First, the relationship is between initial inequality and subsequent returns. Returns could affect contemporaneous top shares but likely not lagged top shares. One might still worry that our results are driven by the HP filter, which uses the past, current, and future data to obtain a smooth trend, thereby introducing a look-ahead bias.15 However, as we see in column (2) of Table 1, our main result is only marginally mitigated when we use the Kalman filter, which does not use future information to estimate the trend. Furthermore, as mentioned above, in Appendix B we also detrend the top income share by the AR(1) Kalman filter, the one-sided HP filter, and the moving average filter, all of which use only past information and obtain similar results. Finally, as we see in regression (3) from Table 1, when excluding capital gains, the top 1% income 13 http://www.hec.unil.ch/agoyal/ 14 http://research.stlouisfed.org/fred2/ 15 For example, since the rich are likely to be more exposed to the stock market, when the stock market goes up at year t + 1, the rich will be richer than usual. But then the trend in the top income share will shift upwards, and the year t deviation of the top income share will be lower. Therefore the low income share at year t may spuriously predict a high stock return at t + 1.

14

share coefficient is only slightly smaller. If returns were strongly affecting lagged inequality, excluding capital gains would likely mitigate the regression results. But, one might say, we have known at least since Shiller (1981) that when prices are high relative to either earnings or dividends, subsequent market excess returns are low. The current price could indeed affect current inequality (see Section 3.3). Are the top shares series simply proxying for the price-dividend or price-earnings ratios, which are known to predict returns? Again, the answer seems to be no for two reasons. First, excluding capital gains from income does not significantly mitigate the relationship, and capital gains are the main avenue through which prices would determine inequality. Second, as we see in regressions (7) and (8) from Table 1, top shares predict excess returns even when controlling for the log price-dividend or price-earnings ratio. Including these controls does decrease the top shares coefficients slightly, but they remain large and significant. In the case of one year returns, the P/D and P/E ratios are not significant after controlling for top income shares. In regressions (5), (6), (9), and (10) from Table 1, we also control for real GDP growth, consumption volatility (Lettau et al. (2008) and Bansal et al. (2014)), and CAY, which Lettau and Ludvigson (2001) show forecasts market excess returns. In the case of one year returns, including these controls has little impact on the relationship between the top income share and subsequent returns. Our empirical analysis thus far has relied on detrending, which requires the researcher to take a stand on smoothing parameters and the underlying trend model. Do the raw data indicate a relationship between asset prices and the one percent? Figure 7 suggests that the answer is the yes. Over 1913-2012, both overall and within subsamples, there is a clear positive correlation between the top 1% income share (not detrended) and the price-dividend ratio. Of course, this scatter plot does not establish causation, but it is more evidence in favor of our theory and suggests that our empirical results are not simply artifacts of detrending. Indeed, as we have shown, above trend inequality predicts subsequent excess returns even when using a simple, one-sided trend estimation method like the ten year moving average. In summary, the data appear consistent with our theory that an increasing concentration of income or wealth decreases the market risk premium.

15

Year t to t+5 Excess Market Return (%)

250 200 150 100 50 0 −50 −100 −4

−2 0 2 Year t Detrended Top 1% Income Share (%)

4

Figure 6: Year t to year t + 5 excess stock market return vs. year t detrended (HP) top 1% income share including capital gains (1913–2008).

Top 1% Income Share (not detrended)

22 20 18 16 14 12 10 8 0

20

40

60

80

100

P/(D10)

Figure 7: Top 1% income share (not detrended) vs. price-dividend ratio (in real terms) for the S&P 500 based on 10 year moving averages of dividends. 1913-1945 (*), 1946-1978 (o), and 1979-2012 (+).

16

Table 1: Regressions of one year excess stock market returns on top income shares and other predictors Regressors (t) Constant Top 1% Top 1% (Kalman, p = 2) Top 1% (no cg) Top 1% (wealth) 17

Real GDP Growth Cons. Growth Volatility

(1) 6.66 (1.77) -5.61*** (1.60)

(2) 7.07 (1.79)

Dependent Variable: t to t + 1 Excess Stock Market Return (3) (4) (5) (6) (7) (8) 6.69 7.09 3.90 8.41 21.60 21.83 (1.76) (1.83) (2.36) (3.03) (11.65) (12.86) -7.63*** -7.67*** -5.02*** -4.86*** (1.82) (1.68) (1.63) (1.71)

(10) 32.28 (19.96) -5.53*** (1.61)

-3.46** (1.50) -5.16** (2.25) -7.93*** (1.91) 0.73* (0.37)

0.34 (0.59) -1.46 (2.34) -6.91 (4.89)

-0.46 (0.69)

log(P/D10)

-4.52 (3.40) -5.62 (4.66)

log(P/E10) CAY 191319131913191619301931-2012 -2012 -2012 -2000 -2012 -2012 R2 .09 .05 .05 .24 .14 .15 Newey-West standard errors in parentheses (k = 4) 1% series detrended with HP filter unless otherwise noted Consumption growth volatility from GARCH(1,1) model ***, **, and * indicate significance at 1%, 5%, and 10% levels (suppressed for constants) Sample

(9) 6.28 (1.64) -5.50*** (1.56)

1913-2012 .10

1913-2012 .11

2.39*** (0.58) 1945-2012 .22

2.34*** (0.68) 1945-2012 .24

3.3

Relationship with return predictors

As we saw in Section 3.2, controlling for the price-dividend (or price-earnings ratio) or CAY mitigates to a small degree the estimated effect of inequality on subsequent excess returns. Furthermore, because the rich hold more stock than do the poor, high prices and the resulting capital gains likely have some direct impact on the top income shares. To what extent then are the top income shares correlated with classic return predictors? In Table 2, we regress the top 1% share on two series known to predict or explain asset returns. For the top 1% share, the correlation with the log price-dividend ratio is significant, but the R-squared is only .07. Therefore, while correlation with the price-dividend ratio likely explains some of the relationship between inequality and subsequent returns, it is not all or even most of the story. CAY, however, is not significantly correlated with the top 1% share. Overall, the top 1% income share appears to represent a component of the equity premium orthogonal to CAY and only slightly related to the price-dividend ratio. Table 2: Regressions of top income share on factors Dependent Variable: Top 1% Share (1) (2) -1.74 -1.07 Constant (0.82) (0.93) 0.53** 0.31 log(P/D10) (0.25) (0.28) -0.02 CAY (0.05) 19131945Sample -2012 -2012 R2 .07 .03 Newey-West standard errors in parentheses (k = 4) ***, **, and * indicate significance at 1%, 5%, and 10% levels (suppressed for constants) Regressors

4

International Evidence

Thus far, we have shown that in the U.S. shocks to the concentration of income are associated with large and significant declines in excess returns on average. We have also provided a theoretical explanation for this pattern: if the rich are relatively more risk tolerant, when their wealth share rises relative aggregate demand for risky assets increases, which in equilibrium leads to a decline in the equity premium. Our theoretical argument, however, is not specific to the U.S. Therefore, we can test our theory by seeing whether or not this pattern holds internationally. In this section, we employ cross country fixed effects panel regressions and show that outside of the U.S. there also appears to be an inverse relationship between inequality and subsequent excess returns.

18

4.1

Data

We consider twenty-nine countries, for the time period 1969-2013, spanning the continents : Americas (Argentina, Canada, Colombia, U.S.), Europe (Denmark, Finland, France, Germany, Ireland, Italy, Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, and U.K.), Africa (Mauritius and South Africa), Asia (China, India (INI), Japan, Singapore, South Korea, Malaysia, and Taiwan), and Oceania (Australia, Indonesia (INO), and New Zealand). Due to missing data points for some countries, we have around 100-800 observations, depending on the regions included. In the regressions below, we divide the countries into the following groups: Rich (AUS, CAN, DNM, FIN, FRA ,GER, IRE, JPN, NET, NOR, SIN, SWE, SWI, TAI, UNK, and USA), Not Rich (ARG, CHN, COL, INI, INO, ITA, KOR, MAL, MAU, NZL, POR, SPA, and SAF), IIPS (IRE, ITA, POR, and SPA), and EME (ARG, CHN, COL, INI, INO, KOR, MAL, MAU, and SAF). Our panel data on inequality are from Alvaredo et al. (2015). To be consistent across countries, we use top 1% income shares excluding capital gains. See Appendix E for country-specific details on top income shares. To calculate annual stock returns (EOP) we acquire from Datastream the MSCI total return indexes in local currency. To convert returns into local real terms, we deflate the stock indexes by local CPI (or GDP deflator when CPI is unavailable), which we obtain from Haver’s IMF data. See Appendix E for country-specific details on stock market and price indexes. Given the liquidity and safety of U.S. Treasuries, T-bill returns provide a standard and relatively uncontroversial measure of the risk-free rate in the U.S. In markets outside of the U.S., especially emerging ones where government and private sector default are not uncommon, it is not immediately obvious how to measure the risk-free rate. To make the definition of excess returns relatively consistent across countries, we use the Haver/IMF “deposit rate” series (in most cases), which is, depending on the country, the savings rate offered on one to twenty-four month deposits. Specifically, we take the year t safe return to be the average of annualized rates quoted in January to September of that year. Local nominal rates are converted into real terms by local CPI (or GDP deflator when CPI is unavailable). See Appendix E for more details.

4.2

International regression results

In Section 3, we showed that income/wealth concentration is inversely related to subsequent excess returns. However, quantitatively, this result was really about stock returns. Indeed, redoing column (1) of Table 1 with stock returns instead of excess returns, the 1% coefficient is -5.89 with a Newey-West p-value of .001. Also, there does not appear to be a significant relationship between inequality and risk-free rates in the U.S. Furthermore, due to the limited availability of similar interest rates across countries, using stock returns instead of excess returns substantially expands the sample size. In light of these facts and because of the nebulous nature of international risk-free rates, we first present the international results for stock market returns without netting out an interest rate. Another difference from our U.S. analysis in Section 3 is that in the post1969 sample there is no obvious U-shape for top income shares, which simplifies

19

Table 3: Country fixed effects panel regressions of one year stock market returns on top income shares Dependent Variable: t to t + 1 Stock Market Return All Rich Not Rich IIPS EME -1.99** −1.38+ -4.64** -7.16* -6.14** Top 1% (t) (0.92) (0.91) (1.95) (3.01) (1.92) Time Trend Yes Yes Yes Yes Yes Country FE Yes Yes Yes Yes Yes Obs. 790 573 217 106 109 (.01,.08) (.00,.12) (.03,.21) (.05,.16) (.04,.14) R2 (w,b) Clustered standard errors in parentheses, ***1%, **5%, *10%, + 15% R2 (w,b): Within and between R-squared Constants suppressed

Regressors

handling the potentially nonstationary nature of inequality. In this section, we simply include a linear time trend as one of regressors. Table 3 presents the panel regression results for both the whole sample and different regions. First, we see in the column “All” that when including all countries a one percentage point increase above trend in the top income share is associated with a subsequent decline in stock market returns of 2% on average. The coefficient is significant at the 5% level with standard errors clustered by country. Columns “Not Rich,” “IIPS,” and “EME” show that this inverse relationship is even stronger and on the order of the full U.S. results when we restrict the sample to the lower GDP countries, the “GIIPS” (without Greece), or the emerging market economies. The pattern is weaker in the richer countries.16 As a robustness check, Table 7 in Appendix F shows the panel regressions without time trends. The results are similar to the case with the linear time trend.17 Table 4 is the same as Table 3 except with excess returns (using real deposit rates) as the dependent variable. For “Not Rich” and “EME” countries, the results are essentially unchanged. Including all countries, the 1% coefficient falls in magnitude slightly to -1.49 but remains significant at the 10% level without clustering standard errors (with country clustering, the p-value is .16). In Tables 1, 3, 4, and 7 we see that the relationship between inequality and returns is most apparent in the U.S., emerging markets, and less rich countries. One potential explanation for this finding is variation in the degree of stock market home bias. In either very large markets (such as the U.S.) or relatively closed ones (such as emerging markets), our theory suggests that local inequality should impact domestic stock markets. In small and open markets, however, foreigners own a substantial fraction of the domestic stock markets and mitigate the role of local inequality. Indeed, according to measures in Mishra (2015), 16 Does including the time trend mitigate potential nonstationarity? The answers appears to be yes: the Phillips-Perron test (Phillips and Perron, 1988) rejects the presence of a unit root in the fitted residuals for each country (at least at the 5% level) except in Argentina (p-value of .31), Indonesia (p-value of .31), and South Africa (p-value of .052), all three of which have low sample size (≤ 12). 17 And, somewhat surprisingly, the unit root tests on the residuals have the same results as with the inclusion of the time trend: we only fail to reject a unit root in Argentina, Indonesia, and South Africa, all of which are short time series.

20

Table 4: Country fixed effects panel regressions of one year excess returns on top income shares Dependent Variable: t to t + 1 Excess Market Return All Rich Not Rich IIPS EME −1.49+ -0.55 -4.73* -1.42 -6.07** Top 1% (t) (1.04) (1.00) (2.19) (4.56) (2.35) Time Trend Yes Yes Yes Yes Yes Country FE Yes Yes Yes Yes Yes Obs. 660 475 185 72 109 (.00,.01) (.00,.27) (.03,.12) (.00,.20) (.04,.08) R2 (w,b) Clustered standard errors in parentheses, ***1%, **5%, *10% + : p-value = .16, significant at 10% level without clustering R2 (w,b): Within and between R-squared Constants suppressed Regressors

many of our “Not Rich” and “EME” countries (such as India, Indonesia, Colombia, and Malaysia) exhibit some of the highest degrees of home bias, while most of our “Rich” and “IIPS” members are in the bottom half of countries ranked by home bias. Averaging his measures, Italy, the Netherlands, Singapore, Portugal, and Norway have the lowest home bias, and the Philippines, India, Turkey, Indonesia, and Pakistan have the highest (with Colombia and Malaysia close behind). While local inequality appears less important in small and open financial markets, inequality amongst global investors should still impact excess returns in these markets. Table 5 repeats the regressions of Table 4 but also includes the U.S. 1% share as a proxy for world investor inequality. As conjectured, the U.S. 1% share has a large and significant inverse correlation with subsequent excess returns for the “Rich” and “IIPS” groups. Table 5: Country fixed effects panel regressions of one year excess returns on local and U.S. top income shares Dependent Variable: t to t + 1 Excess Market Return All+ Rich+ Not Rich IIPS EME -1.62 -0.64 -4.61* -2.53 -5.85** Top 1% (t) (1.17) (1.25) (2.15) (4.20) (2.49) -3.37*** -2.08** -8.42** -6.44** -9.72 U.S. Top 1% (t) (1.02) (0.82) (3.42) (1.74) (7.19) Time Trend Yes Yes Yes Yes Yes Country FE Yes Yes Yes Yes Yes 616 431 185 72 109 Obs. R2 (w,b) (.02,.01) (.01,.13) (.06,.10) (.04,.00) (.07,.08) Clustered standard errors in parentheses, ***1%, **5%, *10% + : excluding U.S. R2 (w,b): Within and between R-squared Constants suppressed Regressors

21

5

Concluding remarks

In this paper we found that the income/wealth distribution is closely connected with stock market returns. When the rich are richer than usual the stock market subsequently performs poorly. To explain this stylized fact, we built general equilibrium models with agents that are heterogeneous in both wealth and attitudes towards risk. Both analytically and in numerical examples, the concentration of wealth/income drove down subsequent excess returns on average. Our model is a mathematical formulation of Irving Fisher’s narrative that booms and busts are caused by changes in the relative wealth of the rich (the “enterpriser-borrower”) and the poor (the “creditor, the salaried man, or the laborer”). Overall, we found that the predictions of our model are consistent with U.S. and international data on excess stock market returns and inequality. Could one exploit the predictive power of top income shares to beat the market on average? The answer is probably no since the top income share—which comes from tax return data—is calculated with a substantial lag. One would receive the inequality update too late to act on its asset pricing information. However, our analysis provides a novel positive explanation of market excess returns over time. We conclude, as decades of macro/finance theory have suggested, that stock market fluctuations are intimately tied to the distribution of wealth, income, and assets.

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A A.1

Characterization of Equilbrium (Proof ) Proof of Theorem 2.1

The proof of Theorem 2.1 requires further notation and several lemmas. Let θ be the fraction of wealth invested in the risky asset. Let R(θ) = Rθ + Rf (1 − θ) be the gross return on the portfolio. Suppressing individual subscript, a typical agent’s optimal portfolio problem is max E[u(R(θ)w)], θ

where w is initial wealth. The following lemma is basic. Lemma A.1. Consider an agent with initial wealth w and utility function u, where u′ > 0 and u′′ < 0. Let θ be the optimal portfolio. Then the following is true. 1. θ is unique. 2. θ ≷ 0 according as E[R] ≷ Rf . 3. Suppose E[R] > Rf . If u exhibits decreasing relative risk aversion (DRRA), so −xu′′ (x)/u′ (x) is decreasing, then ∂θ/∂w ≥ 0, i.e., the agent invests comparatively more in the risky asset as he becomes richer. The opposite is true if u exhibits increasing relative risk aversion (IRRA). Proof. 1. Let f (θ) = E[u(R(θ)w)]. Then f ′ (θ) = E[u′ (R(θ)w)(R−Rf )w] and ′′ f (θ) = E[u′′ (R(θ)w)(R− Rf )2 w2 ] < 0, so f is strictly concave. Therefore the optimal θ is unique (if it exists). 2. Since f ′ (θ) = 0 and f ′ (0) = u′ (Rf w)w(E[R] − Rf ), the result follows. 3. Dividing the first-order condition by w, we obtain E[u′ (R(θ)w)(R−Rf )] = 0. Let F (θ, w) be the left-hand side. Then by the implicit function theorem we have ∂θ/∂w = −Fw /Fθ . Since Fθ = E[u′′ (R(θ)w)(R − Rf )2 w] < 0, it suffices to show Fw ≥ 0. Let γ(x) = −xu′′ (x)/u′ (x) > 0 be the relative risk aversion coefficient. Then Fw = E[u′′ (R(θ)w)(R − Rf )R(θ)] 1 = − E[γ(R(θ)w)u′ (R(θ)w)(R − Rf )]. w Since E[R] > Rf , by the previous result we have θ > 0. Therefore R(θ) = Rθ + Rf (1 − θ) ≷ Rf according as R ≷ Rf . Since u is DRRA, γ is decreasing, so γ(R(θ)w) ≤ γ(Rf w) if R ≥ Rf (and reverse inequality if R ≤ Rf ). Therefore γ(R(θ)w)(R − Rf ) ≤ γ(Rf w)(R − Rf ) 26

always. Multiplying both sides by −u′ (R(θ)w) < 0 and taking expectations, we obtain wFw = − E[γ(R(θ)w)u′ (R(θ)w)(R − Rf )] ≥ − E[γ(Rf w)u′ (R(θ)w)(R − Rf )] = 0, where the last equality uses the first-order condition. Lemma A.2. Suppose E[R] > Rf and consider two agents indexed by i = 1, 2. Let wi , ui (x), γi (x) = −xu′′i (x)/u′i (x), and θi be the initial wealth, utility function, relative risk aversion, and the optimal portfolio of agent i. If γ1 (w1 x) > γ2 (w2 x) for all x, then θ1 < θ2 , i.e., the more risk averse agent invests comparatively less in the risky asset. Proof. Since γ1 (w1 x) > γ2 (w2 x), we have   w2 u′′2 u′1 − u′2 w1 u′′1 d u′2 (w2 x) 1 u′2 = = (γ1 (w1 x) − γ2 (w2 x)) > 0, dx u′1 (w1 x) (u′1 )2 x u′1 so u′2 (w2 x)/u′1 (w1 x) is increasing. Letting θ = θ1 > 0 (which follows from E[R] > Rf ), since R(θ) ≷ Rf according as R ≷ Rf and u′2 (w2 x)/u′1 (w1 x) is increasing (and positive), we have u′2 (R(θ)w2 ) u′2 (Rf w2 ) (R − R ) > (R − Rf ) f u′1 (R(θ)w1 ) u′1 (Rf w1 ) always (except when R = Rf ). Multiplying both sides by u′1 (R(θ)w1 ) > 0 and taking expectations, we get   ′ u2 (R(θ)w2 ) ′ ′ u (R(θ)w1 )(R − Rf ) E[u2 (R(θ)w2 )(R − Rf )] = E ′ u (R(θ)w1 ) 1  ′1  u2 (Rf w2 ) ′ >E ′ u (R(θ)w1 )(R − Rf ) = 0, u1 (Rf w1 ) 1 where the last equality uses the first-order condition for agent 1. Letting f2 (θ) = E[u2 (R(θ)w2 )], the above inequality shows that f2′ (θ1 ) > 0. Since f2 (θ) is concave and f2′ (θ2 ) = 0, we have θ2 > θ1 . Lemma A.3. Consider an agent with initial wealth w, utility function u(x) and relative risk aversion γ(x) = −xu′′ (x)/u′ (x). Let Rf be the risk-free rate and θ the optimal portfolio. If γ(x) ≤ 1, then ∂θ/∂Rf < 0. Proof. By the first-order condition, we have E[u′ (R(θ)w)(R − Rf )] = 0. Let F (θ, Rf ) be the left-hand side. Then by the implicit function theorem we have ∂θ/∂Rf = −FRf /Fθ . Since Fθ = E[u′′ (R(θ)w)(R − Rf )2 w] < 0, it suffices to show FRf < 0. Differentiating F with respect to Rf and using the definition of the relative risk aversion, we get FRf = E[u′′ (R(θ)w)(1 − θ)(R − Rf )w] − E[u′ (R(θ)w)]   γ(R(θ)w) ′ u (R(θ)w)(1 − θ)(R − Rf ) − E[u′ (R(θ)w)] =E − R(θ)  ′  u (R(θ)w) = −E (γ(1 − θ)(R − Rf ) + R(θ)) . R(θ) 27

Since (1 − θ)(R − Rf ) = R − Rθ − Rf (1 − θ) = R − R(θ), it follows that   ′ u (R(θ)w) (γR − γR(θ) + R(θ)) FRf = − E R(θ)  ′  u (R(θ)w) = −E (γR + (1 − γ)R(θ)) . R(θ) Since u′ , R, R(θ) > 0, if γ ≤ 1, then FRf < 0. Note that the conclusion of Lemma A.3 need not hold without the restriction γ(x) ≤ 1. In fact, in Figure 2, when the wealth share of agent 1 is between 0.5 and 0.7, the risk-free rate is increasing but the portfolio share of the risky asset is also increasing. Theorem A.4. Consider and economy with I agents indexed by i = 1, . . . , I. Let ui (x) and γi (x) = −xu′′i (x)/u′i (x) be the utility function and relative risk aversion of agent i. Suppose that 1. γ1 (x) is constant (agent 1 is CRRA), 2. γ2 (x) is decreasing (agent 2 is DRRA), and 3. γ1 (x) > γ2 (x) (agent 1 is more risk averse than 2). If we shift some wealth from agent 1 to 2, then there exists an equilibrium in which the risk-free rate increases. The same conclusion holds if agent 1 is IRRA and 2 is CRRA. If in addition γi (x) ≤ 1 for all i, then the equilibrium is unique (and hence the risk-free rate unambiguously increases). Proof. Let θi be the optimal portfolio of agent i. By Lemma A.1, θi ≷ 0 according as E[R] ≷ Rf . Since the risky asset is in positive supply, in equilibrium we must have E[R] > Rf . Suppose that we transfer some wealth ǫ from agent 1 to 2. Let θi′ be the new portfolio of agent i. The change in agent 1 and 2’s demand in the risky asset is ∆ = (w1 − ǫ)θ1′ + (w2 + ǫ)θ2′ − (w1 θ1 + w2 θ2 ) = w1 (θ1′ − θ1 ) + w2 (θ2′ − θ2 ) + ǫ(θ2′ − θ1′ ). Suppose that the risk-free rate does not change. Since agent 1 is CRRA, we have θ1′ = θ1 . Since agent 2 is DRRA and has become richer, by Lemma A.1 we have θ2′ ≥ θ2 . Since agent 1 is more risk averse than 2, by Lemma A.2 we have θ2′ > θ1′ . Therefore ∆ > 0. Since agents i > 2 are unaffected unless the risk-free rate changes, there is positive excess demand in the risky asset. Regard θi as a function of the risk-free rate Rf . By the maximum theorem, θi is continuous, and so is the aggregate demand. Since θi → 0 as Rf → E[R] by Lemma A.1, the excess demand becomes negative as Rf → E[R]. Therefore by the intermediate value theorem, there exists an equilibrium risk-free rate higher than the original one. If γi (x) ≤ 1 for all i, then by Lemma A.3 we have ∂θi /∂Rf < 0, so the individual (hence aggregate) demand is downward sloping. Therefore the equilibrium is unique. Clearly Theorem 2.1 is a special case of Theorem A.4. 28

A.2

Proof of Theorem 2.2

First we characterize the equilibrium, assuming existence. t+1 be the gross return on stock (to Individual problem Let Rt+1 = Pt+1P+D t be determined in equilibrium). Let Vit (w) be the value function of agent i at time t. Since utility is logarithmic, we can guess that

Vit (w) = ait +

β log w 1−β

for some bounded random variable ait (that does not affect the agent’s behavior). The Bellman equation for agent 1 is V1t (w) = max {log c + β Et [V1,t+1 ((Rt+1 θ1 + Rf,t (1 − θ1 ))(w − c))]} , c,θ1

where θ1 is the fraction of agent 1’s wealth invested in the stock. Substituting the guess into the Bellman equation and taking the first-order condition with respect to consumption, we get c = (1 − β)w. The optimal portfolio problem becomes max Et [log(Rt+1 θ1 + Rf,t (1 − θ1 ))]. θ1

Suppressing the t subscript and using prime (′ ) for time t + 1 variables, the first-order condition is   R′ − Rf E = 0. R′ θ1 + Rf (1 − θ1 ) For agent 2, there is no portfolio choice (θ2 = 1 always) and the consumption rule is c = (1 − β)w. Market clearing By the market clearing for the good, we have D = c1 + c2 = (1 − β)(w1 + w2 ) = (1 − β)W . Since the only asset in positive supply is the β stock, we have W = P + D. Therefore D = (1 − β)(P + D) ⇐⇒ P = 1−β D: the stock price depends only on the current dividend. Hence the stock return R′ =

P ′ + D′ = P

β ′ ′ 1−β D + D β 1−β D

=

D′ βD

is exogenous. Since agents save at rate β out of wealth, agent 1 invests a dollar amount of (1 − β)w1 (1 − θ1 ) in the bond. Similarly, agent 2 invests (1 − β)w2 . Since the bond is in zero net supply, by market clearing we have (1 − β)w1 (1 − θ1 ) + (1 − β)w2 = 0 ⇐⇒ x(1 − θ1 ) + (1 − x) = 0 ⇐⇒ θ1 = 1/x. Substituting into the first-order condition of agent 1’s portfolio problem, we obtain     (R′ − Rf )x R′ − Rf = E . (A.1) 0=E R′ /x + Rf (1 − 1/x) R′ + Rf (x − 1)

29

Equilibrium Given the history up to t, the risk-free rate is determined by (A.1), where R′ = D′ /βD is the stock return. By the budget constraints, we obtain w1′ = β(R′ /x + Rf (1 − 1/x))w1 , w2′ = βRf w2 . Since W ′ = βR′ W , it follows that x′ =

R′ /x + Rf (1 − 1/x) Rf w1′ = x = 1 + ′ (x − 1), ′ W R′ R

which are the equations of motion. Next, we show the existence and uniqueness of equilibrium. To this end, it hsuffices to ishow that (A.1) has a unique solution Rf . Let F (x, Rf ) = E

(R′ −Rf )x R′ +Rf (x−1)

. Then F (x, 0) = x > 0, and letting Rf →

inf R′ 1−x ,

we have

F (x, Rf ) = −∞. Clearly F (x, Rf ) is continuous in Rf . Therefore there exists an equilibrium.18 To show uniqueness, note that   ∂ −x(R′ + Rf (x − 1)) − (R′ − Rf )x(x − 1) F (x, Rf ) = E ∂Rf (R′ + Rf (x − 1))2   −R′ x2 =E < 0, (R′ + Rf (x − 1))2

so F (x, Rf ) is decreasing in Rf . Finally, we show that the equity premium Et [Rt+1 ] − Rf,t is decreasing in t+1 is exogenous, it suffices to show that Rf is increasing xt . Since Rt+1 = DβD t in x. By the implicit function theorem, we have ∂Rf /∂x = −Fx /FRf . Since FRf < 0 by the previous proof, it suffices to show that Fx > 0. Now   ′ ∂ (R − Rf )(R′ + Rf (x − 1)) − (R − Rf )xRf F (x, Rf ) = E ∂x (R′ + Rf (x − 1))2   (R′ − Rf )2 =E > 0, (R′ + Rf (x − 1))2 so the claim is proved.

B

Robustness of predictability

Table 6 explores the robustness of the result that when the top income or wealth share is above trend, subsequent one year excess returns are significantly below average. Column (1) shows the regression for the .1% income share from Piketty and Saez (2003). Compared with the 1%, the relationship is actually 18 The technical condition sup E[log(D t+1 /Dt )] < ∞ guarantees the transversality condit β tion lim supt→∞ β t E[Vit (wit )] ≤ 0. To see this, since Wt = Pt + Dt and Pt = 1−β Dt , 1 Dt , so in particular Wt+1 /Wt = Dt+1 /Dt . Taking logs, expectawe have Wt = 1−β P tions, and summing over t, it follows that E[log Wt ] − log W0 = E[log(Dt+1 /Dt )]. If β log w, supt E[log Dt+1 /Dt ] ≤ M < ∞, then E[log Wt ] ≤ log W0 +M t. Since Vit (w) = ait + 1−β ait is bounded, 0 < β < 1, and wit ≤ Wt , the transversality condition holds.

30

much stronger. Column (2) uses the Piketty and Saez (2003) 1% series in which the income rank includes capital gains. This mitigates the relationship but only slightly. Columns (3) and (4) use the estimated top wealth share series of Saez and Zucman (2014). The impact of inequality on asset returns is smaller than with the Kopczuk and Saez (2004) wealth series, but the coefficients remain significant. In the other columns, we detrend the 1% series but without using the HP filter with a smoothing parameter of 100. First, using a smoothing parameter of 10 strengthens the relationship between inequality and asset returns. In column (5), we use the one-sided HP filter with a smoothing parameter of 100. The one-sided HP filter detrends each data point by applying the filter only to the previous data. This method gives slightly weaker results than our baseline regression. In column (7), we estimate and remove two linear trends, a downward one pre-1977 and an upward one post-1977. Doing so weakens the relationship somewhat, but the inequality coefficient remains significant. We also detrend using the Kalman filter method outlined in Appendix D. Column (8) uses and AR(1) model for the cyclical component (we used AR(2) in the main text). We also tried white noise (p = 0) for the cyclical component, but then the trend becomes almost identical to the raw series. Letting p ≥ 3 is similar to AR(2). In this regression, the 1% series remains significant. Since the Kalman filter uses only current and past data, these results show that the look-ahead bias of the HP filter is not severe. Finally, in column (9) we estimate the trend using a ten year moving average. This method, which is also one-sided, yields a slightly weaker but still significant relationship inequality and subsequent excess returns.

31

Table 6: Regressions of one year excess stock market returns on top income and wealth shares Regressors (t) Constant Top .1% Top 1% (rank with cg) Top 1% (SZ wealth) Top .1% (SZ wealth) 32

Top 1% (one-sided HP) Top 1% (HP, λ = 10)

(1) 6.67 (1.70) -15.32*** (4.09)

Dependent Variable: t to t + 1 Excess Stock Market Return (2) (3) (4) (5) (6) (7) (8) 6.67 6.71 6.71 6.68 6.62 6.73 7.15 (1.77) (1.69) (1.69) (1.95) (1.74) (1.72) (1.81)

-4.85*** (1.29) -2.87** (1.14) -3.84*** (1.29) -5.48** (2.41) -8.13*** (1.97)

Top 1% (linear detrending)

-3.82*** (1.18) -3.76** (1.83)

Top 1% (Kalman, p = 1) Top 1% (10 year MA) 191319131913191319361913-2012 -2012 -2012 -2012 -2012 -2012 R2 .09 .12 .05 .05 .06 .11 Newey-West standard errors in parentheses (k = 4) ***, **, and * indicate significance at 1%, 5%, and 10% levels (suppressed for constants) Sample

(9) 7.45 (1.90)

1913-2012 .09

1913-2012 .03

-2.54** (1.13) 1922-2012 .05

C

Numerical algorithm

This appendix explains how to compute the equilibrium of the numerical examples in Section 2.3 in the general case. Suppose that there are I agents and J risky assets. Interpret the risky assets as constant-returns-to-scale, stochastic savings technologies; let R = (R1 , . . . , RJ ) be the vector of gross returns with expected return µ = E[R] and variance-covariance matrix Σ = E[(R − µ)(R − µ)′ ]. I The equilibrium objects are the portfolios {θi∗ }i=1 and the risk-free rate Rf . Consider the approximation (2.4). Using the approximation of the first-order condition (2.3) with a = ai = E[wi1 ] and noting that ai E[wi1 ] = = Rf + hµ − Rf 1, θi , wi0 wi0 we obtain E[(τi (Rf + hµ − Rf 1, θi) − hR − µ, θi)(R − Rf 1)] = 0 ⇐⇒ (E[(R − Rf 1)(R − µ)′ ] − τi (µ − Rf 1)(µ − Rf 1)′ )θ = τi Rf (µ − Rf 1) ⇐⇒ θi∗ = τi Rf [Σ − τi (µ − Rf 1)(µ − Rf 1)′ ]−1 (µ − Rf 1). Given the risk-free rate Rf , for each agent we compute the optimal portfolio θi∗ by the P market portfolio θm = P above formula. Then we compute the P ∗ w . The market clearing condition is w θ / j θmj = 1, which we can i i0 i i0 i solve by a nonlinear equation solver. Solving the exact model is similar, except that for each agent we have to numerically solve the optimal portfolio problem max E[ui ((Rf + hR − Rf 1, θi)wi0 )]. θ

Letting V (θ) be the objective function, if the functional form of u′i and u′′i are explicitly known, we can solve this problem by the Newton algorithm since the gradient and the Hessian can be computed as ∇V (θ) = E[u′i ((Rf + hR − Rf 1, θi)wi0 )(R − Rf 1)], ∇2 V (θ) = E[u′′i ((Rf + hR − Rf 1, θi)wi0 )(R − Rf 1)(R − Rf 1)′ ], respectively.

D

Kalman filter

This appendix explains how we detrend the top income/wealth share using the Kalman filter. Let yt be the observed top income/wealth share data at time t. Let yt = g t + u t ,

(D.1)

where gt is the trend and ut is the cyclical component. We conjecture that the trend is an I(2) process and the cycle is an AR(p) process, so (1 − L)2 gt = ǫt , φ(L)ut = wt ,

ǫt ∼ i.i.d. N (0, σǫ2 ),

(D.2a)

2 N (0, σw ),

(D.2b)

wt ∼ i.i.d. 33

where L is the lag operator and φ(z) = 1 − φ1 z − · · · − φp z p is the lag polynomial for the autoregressive process. For concreteness, assume p = 1 so φ(z) = 1 − φ1 z. Then (D.1) and (D.2) can be written as        gt 2 −1 gt−1 ǫ = + t , (D.3a) 1 0 gt−1 gt−2 0 yt = φ1 yt−1 + gt − φ1 gt−1 + wt .  2 Letting ξt = (gt , gt−1 )′ , vt = (ǫt , 0)′ , xt = yt−1 , A = φ1 , F = 1   H = 1 −φ1 , (D.3) reduces to

(D.3b)  −1 , and 0

ξt = F ξt−1 + vt ,

(D.4a)

yt = Axt + Hξt + wt .

(D.4b)

(D.4a) is the state equation and (D.4b) is the observation equation of the state 2 space model. We can then estimate the model parameters φ1 , σǫ2 , σw as well as the trend {gt } by maximum likelihood: see Chapter 13 of Hamilton (1994) for details. The extension to general AR(p) model is straightforward.

E

International Data

Unless otherwise noted, the top income share series is the “Top 1% income share” excluding capital gains from Alvaredo et al. (2015) (see also their documentation), the price index is the Haver/IMF CPI, and the interest rate is the Haver “Deposit Rate” series. 1. Argentina (ARG) Coverage 1998-2005. Local Currency Deposit Rate 30-59 day deposit rate. 2. Australia (AUS) Coverage 1970-2011. Local Currency Deposit Rate 1972-2011. 3. Canada (CAN) Coverage 1970-2011. 1% Income Share LAD series post-1995. Local Currency Deposit Rate 90 day deposit rate. 1971-2011. 4. China (CHN) Coverage 1993-2004. Local Currency Deposit Rate 1 year deposit rate. 5. Colombia (COL) Coverage 1994-2011. 6. Denmark (DNM) Coverage 1971-1973, 1975-2011.

34

1% Income Share “Adults” series. Local Currency Deposit Rate 1980-2002. 7. Finland (FIN) Coverage 1988-2010. 1% Income Share “Tax data” series pre-1993 and “IDS” 1993-. We average the two for 1990-1992. Local Currency Deposit Rate 23 month deposit rate, 1988-2005. 8. France (FRA) Coverage 1970-2010. 9. Germany (GER) Coverage 1972, 1975, 1978, 1981, 1984, 1986, 1990, 1993, 1996, 1999, 2002-2009. Local Currency Deposit Rate 3 month deposit rate, 1978-2003. Price Index GDP deflator pre-1991. 10. India (INI) Coverage 1993-2000. Local Currency Deposit Rate Bank discount rate from Haver. 11. Indonesia (INO) Coverage 1988, 1991, 1994, 1997, 1999-2005. Local Currency Deposit Rate 3 months deposit rate. 12. Ireland (IRE) Coverage 1988-2010. Local Currency Deposit Rate 1988-2006. Price Index http://www.cso.ie 13. Italy (ITA) Coverage 1975-1996, 1999-2010. Local Currency Deposit Rate 1983-2004. 14. Japan (JPN) Coverage 1970-2011. Local Currency Deposit Rate 3 month deposit rate. 15. South Korea (KOR) Coverage 1996-2013. Local Currency Deposit Rate 1 year deposit rate. 16. Malaysia (MAL) Coverage 1989, 1994-1996, 2001-2004, 2006, 2010-2013. Local Currency Deposit Rate 3 month deposit rate. Price Index blabla 17. Mauritius (MAU) Coverage 2003-2009, 2011-2012. Local Currency Deposit Rate 3 month deposit rate. 18. Netherlands (NET) Coverage 1971, 1974, 1976, 1978, 1982, 1986, 1990-2013. 19. New Zealand (NZL)

35

Coverage 1988-2012. 1% Income Share “Adults” series. Local Currency Deposit Rate 6 month deposit rate, 1990-2012. 20. Norway (NOR) Coverage 1970-2012. Local Currency Deposit Rate 1979-2010. 21. Portugal (POR) Coverage 1990-2006. Local Currency Deposit Rate 180-360 day deposit rate, 1990-2000. 22. South Africa (SAF) Coverage 1993-1994, 2003-2012. 1% Income Share Pre-1990, “Married Couples and Single Adults” series. Post-1990, “Adults” series. Local Currency Deposit Rate 88-91 day deposit. 23. Singapore (SIN) Coverage 1970-1992, 1994-2013. Local Currency Deposit Rate 3 month deposit rate, 1977-2013. Price Index blabla 24. Spain (SPA) Coverage 1982-2013. Local Currency Deposit Rate 6-12 month deposit rate, 1982-2013. 25. Sweden (SWE) Coverage 1970-2013. Local Currency Deposit Rate 1970-2006. 26. Switzerland (SWI) Coverage 1970, 1972, 1974, 1976, 1978, 1980, 1982, 1984, 1986, 1988, 1990, 1992, 1994, 1996-2011. Local Currency Deposit Rate 3 month deposit rate, Price Index 1982-2011. 27. Taiwan (TAI) Coverage 1988-2014. Local Currency Deposit Rate Missing. Price Index CPI, Datastream. 28. United Kingdom (UNK) Coverage 1970-2013. 1% Income Share Pre-1990, “Married Couples and Single Adults” series. Post-1990, “Adults” series. Local Currency Deposit Rate 90 day t-bill rate. Price Index GDP deflator pre-1988. 29. United States (USA) Coverage 1970-2013. Local Currency Deposit Rate 3 month t-bill rate.

F

Additional International Results 36

Table 7: Country fixed effects panel regressions of one year stock returns on top income shares Dependent Variable: t to t + 1 Stock Market Return All Rich Not Rich IIPS EME -1.49** -0.97 -3.93** -5.42** -4.48* Top 1% (t) (0.67) (0.57) (1.68) (1.47) (2.14) No No No No No Time Trend Country FE Yes Yes Yes Yes Yes 790 573 217 106 109 Obs. R2 (w,b) (.00,.11) (.00,.10) (.02,.23) (.05,.06) (.03,.18) Clustered standard errors in parentheses, ***1%, **5%, *10% + : significant at 10% level without clustering R2 (w,b): Within and between R-squared Constants suppressed Rich: AUS,CAN,DNM,FIN,FRA,GER,IRE,JPN,NET,NOR,SIN Rich cont’d: SWE,SWI,TAI,UNK,USA IIPS: IRE,ITA,POR,SPA EME: ARG,CHN,COL,INI,INO,KOR,MAL,MAU,SAF Other: NZL,KOR

Regressors

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