The Equity Premium and the One Percent

Abstract

Does the wealth distribution matter for asset pricing? Intuition tells us that it does: as the rich get richer, they buy risky assets and drive up prices. Indeed, over a century ago, Fisher (1910) argued for the 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 theory of fluctuations were differences in preferences and wealth across people.

To see the intuition for why the wealth distribution affects asset pricing, consider an economy consisting of investors with different attitudes toward 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. Motivated by this intuition, we establish both the theoretical and empirical links between inequality and asset prices, which constitute two main contributions of this paper.

First, we theoretically study the asset pricing implications of general equilibrium models with heterogeneous agents. In a two-period economy populated by Epstein-Zin agents with arbitrary risk aversion, belief, and wealth heterogeneity, we prove there exists a unique equilibrium, in which increasing wealth concentration in the hands of stockholders leads to a decline in the equity premium. Although the inverse relationship between wealth concentration and risk premiums under heterogeneous risk aversion has been recognized at least since Dumas (1989) and recently emphasized by Gârleanu and Panageas (2015), testing the existing theory requires identifying the preference types, which is challenging. In contrast, we show that it is sufficient to identify the portfolio types (⁠|$\textrm{e.g.}$|⁠, the type of agents that have larger portfolio shares of stocks). It does not matter why some agents hold more stocks: although we prove that high risk tolerance or optimism is sufficient for investing more in stocks, increased investment also could be due to other reasons, such as low participation costs. Furthermore, we calibrate this two-period model as well as an infinite horizon extension and show that the wealth distribution can have a quantitatively large effect on the equity premium.

Second, we empirically test our theoretical predictions. Given the existing empirical evidence that the rich invest relatively more in stocks,¹ theory implies that rising inequality should negatively predict subsequent excess stock market returns. Consistent with our theory, we find that when the income share of the top 1% income earners in the United States rises, the subsequent 1-year excess stock market return falls on average; therefore, current inequality appears to forecast the subsequent risk premium of the U.S. stock market.

Specifically, for a number of reasons including the apparent high persistence of top income and wealth shares, we employ a stationary component of inequality, “|${\mathrm{KGR}}$|” (capital gains ratio), defined as the difference between the top 1% income share with and without realized capital gains income, divided by the bottom 99% income share. We use data and theory to argue that |${\mathrm{KGR}}$| is a reasonable proxy for capital wealth and income inequality. Regressions of the year |$t$| to |$t+1$| excess return on the year |$t$| top 1% income share indicate a strong and significant negative correlation. Our evidence suggests that the top 1% income share is not simply a proxy for the price level, which existing research shows correlates with subsequent returns, or for aggregate consumption factors: the top 1% income share predicts excess returns even after we control for some classic return predictors, such as the price-dividend ratio (Fama and French 1988) and the consumption-wealth ratio (Lettau and Ludvigson 2001). Our findings are also robust to the inclusion of macro control variables, such as gross domestic product (GDP) growth. Using 5-year excess returns or the top 0.1% or 10% income share also yields similar results, although the predictability is really due to the top 1% (Table 4).

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 by 1 percentage point, subsequent 1-year market returns significantly decline on average by 1%. However, this effect is not uniform across countries. Our theory suggests that for relatively “closed” economies, such as emerging markets, the domestic top 1% share should matter for asset pricing because domestic agents account for a substantial proportion of the universe of investors. However, for small open economies, the inequality among global investors (proxied by U.S. |${\mathrm{KGR}}$|⁠) should matter because domestic agents compose only a small fraction of investors. Consistent with our theory, we find that the interaction terms between top income shares and home bias measures significantly predict stock returns. In an economy with complete home bias, a 1-percentage-point increase in the top 1% income share is associated with a subsequent 2.8% decline in stock market returns. In a small open economy (no home bias), a 1-percentage-point increase in U.S. |${\mathrm{KGR}}$| is associated with a subsequent decline in stock market returns of 4.7%.

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. Although the original capital asset pricing model (CAPM; see Sharpe 1964; Lintner 1965a, 1965b; Geanakoplos and Shubik 1990 for a general and rigorous treatment) allows for substantial heterogeneity in endowments and risk preferences across investors, the quadratic or mean-variance preferences admit aggregation and obviate the role of the wealth distribution. Inspired by the limited empirical fit of the CAPM and asset pricing puzzles that arise in representative agent models, since the 1980s theorists have extended macrofinance models to consider meaningful investor heterogeneity. Such heterogeneous agent models fall into two groups.

In the first group, agents have identical standard (constant relative-risk aversion, CRRA) preferences but are subject to uninsured idiosyncratic risks.³ Although the models in this literature have had some success in explaining returns in calibrations, 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, 2017b). In the second group, markets are complete and agents have either heterogeneous CRRA preferences or identical but nonhomothetic preferences. 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, although aggregation in the sense of Gorman (1953) fails. Therefore, agent heterogeneity should matter in discussions about asset pricing.

Gollier (2001) studies the asset pricing implication of wealth inequality among agents with identical preferences. He shows that more inequality increases (decreases) the equity premium if and only if agents’ absolute risk tolerance is concave (convex). In particular, wealth inequality has no effect on asset pricing when agents have hyperbolic absolute risk aversion (HARA) preferences, for which the absolute risk tolerance is linear. Both he and Hatchondo (2008) also calibrate the model and find that the effect of wealth inequality on the equity premium is small.

Dumas (1989) solves a dynamic general equilibrium model with constant-returns-to-scale production and two agents (one with log utility and the other CRRA). He shows (proposition 17) that when the wealth share of the less risk-averse agent increases, then the risk-free rate goes up and the equity premium goes down. Although this prediction is similar to ours, he imposes an assumption on endogenous variables (see his equation (8)).

Following Dumas (1989), a large theoretical literature has studied the asset pricing implication of preference heterogeneity under complete markets.⁴ All these papers characterize the equilibrium and asset prices by solving a planner’s problem. However, this approach is not suitable for conducting comparative statics exercises of changing the wealth distribution, for two reasons. First, although by the first welfare theorem, for each equilibrium we can find Pareto weights such that the consumption allocation is the solution to the planner’s problem, because in general the Pareto weights depend on the initial wealth distribution, changing the wealth distribution will change the Pareto weights and, consequently, the asset prices. However, in general it is difficult to predict how the Pareto weights change. Second, even if we can predict how the Pareto weights change, the possibility of multiple equilibria must be considered. In such cases the comparative statics often go in the opposite direction depending on the choice of the equilibrium. Thus, our results are quite different, because we prove the uniqueness of equilibrium and derive comparative statics with respect to the initial wealth distribution.

Gârleanu and Panageas (2015) study a continuous-time overlapping generations endowment economy with two agent types with Epstein-Zin preferences. Unlike other papers on asset pricing models with heterogeneous preferences, all agent types survive in the long run because of birth and death, and they also solve the model without appealing to a planner’s problem. As a result, all endogenous variables are expressed as functions of the state variable, the consumption share of one agent type. They find that the concentration of wealth to the more risk-tolerant type (“the rich”) tends to lower the equity premium. When the preferences are restricted to additive CRRA, then the relation between the consumption share and equity premium (more precisely, market price of risk) is monotonic (see their discussion on p. 10). Thus, our results are closely related to theirs, although different, because we prove more general comparative statics results (though in two-period models).

Our model is also related to the work on limited asset market participation, such as Basak and Cuoco (1998), Guvenen (2009), Chien, Cole, and Lustig (2011, 2012), and Chabakauri (2013, 2015). In these papers some agents do not participate in certain asset markets or face portfolio constraints, which affects the asset prices beyond the heterogeneity in preferences or beliefs. In our model we also have hand-to-mouth laborers, but because they do not participate in any asset market, we prove that their presence affects only the risk-free rate, not the equity premium (Theorem 2).

Although the wealth distribution theoretically affects asset prices, few empirical papers directly document this connection. To the best of our knowledge, Johnson (2012) and Campbell et al. (2016) are alone in exploring this issue. Using incomplete markets models, they show that top income shares or top income growth innovations are cross-sectional asset pricing factors. However, they do not investigate the ability of top income shares to predict excess market returns (our main empirical result).⁵

Lastly, our study is related to the findings of Greenwald, Lettau, and Ludvigson (2014), who identify innovations to wealth (⁠|$e_{a,t}$|⁠) that explain much of the variation in the stock market and significantly predict low subsequent excess returns. In an equilibrium model, they show that |$e_{a,t}$| captures the risk tolerance of a representative stockholder. Interestingly, there is substantial correlation between |$e_{a,t}$| and our inequality predictor variable |${\mathrm{KGR}}$|⁠.⁶ In heterogeneous risk aversion models without aggregation, rising wealth concentration can effectively decrease the risk aversion of the corresponding representative stockholder/planner, so an interpretation of |$e_{a,t}$| is that it reflects the wealth share of relatively risk-tolerant stockholders versus more risk-averse ones.

1. Wealth Distribution and Equity Premium

In this section we present a theoretical model in which the wealth distribution across heterogeneous agents affects the equity premium.

1.1 Uniqueness of equilibrium

First, we consider a static model with agents that have heterogeneous but homothetic preferences and prove the uniqueness of equilibrium.

Consider a standard general equilibrium model with incomplete markets consisting of |$I$| agents and |$J$| assets (Geanakoplos 1990). Time is denoted by |$t=0,1$|⁠: agents trade assets at |$t=0$| and consume only at |$t=1$|⁠. At |$t=1$|⁠, there are |$S$| states denoted by |$s=1,\dots,S$|⁠. Let |$A=(A_{sj})\in \mathbb{R}^{SJ}$| be the |$S\times J$| payoff matrix of assets, |$U_i:\mathbb{R}_+^S\to\mathbb{R}$| be agent |$i$|’s utility function, and |$n_i\in \mathbb{R}^J,e_i\in \mathbb{R}_+^S$| be agent |$i$|’s endowment vectors of asset shares at |$t=0$| and consumption goods in each state. By removing redundant assets, without loss of generality we may assume that the matrix |$A$| has full column rank.

We make the following assumptions.

Under Assumption 2, because there exists |$y_i\in \mathbb{R}^J$| such that |$e_i=Ay_i$|⁠, by redefining |$e_i$| to be zero and |$n_i$| to be |$n_i+y_i$|⁠, without loss of generality we may assume |$e_i=0$|⁠, that is, agents are endowed only with assets.

Because |$e_i=0$| by assumption, the aggregate endowment of goods is |$An$|⁠. Hence the assumption |$An\gg 0$| simply says that aggregate endowment is positive. While the collinearity assumption is strong, it is indispensable to guarantee the uniqueness of equilibrium.⁷ With multiple equilibria, comparative statics may go in opposite directions, depending on the choice of equilibrium.

Under these assumptions, we can prove the uniqueness of GEI and obtain a complete characterization.

Theorem 1 is essentially an incomplete-market version of the aggregation result in Chipman and Moore (1979). Equilibrium uniqueness is important for our purposes because it rules out unstable equilibria and thus allows for the below unambiguous comparative statics regarding the wealth distribution.⁸

1.2 Comparative statics for asset prices

To derive implications for asset pricing, we specialize the economy in Section 1.1 as follows.

In Section 1.1, we assumed that there is no consumption at |$t=0$|⁠, but we can obtain similar results to Theorem 1 with consumption at |$t=0$| by interpreting |$t=0$| as a new “state” denoted by |$s=0$|⁠.

Furthermore, we introduce an agent type that does not participate in the asset market. Let |$e_s$| (⁠|$s=0,1,\dots,S$|⁠) be the aggregate endowment in state |$s$|⁠. Suppose a hand-to-mouth agent |$i=0$| (whom we call the laborer) is endowed with goods but does not trade assets. Let |$1-\alpha_t$| be the fraction of aggregate income earned by the laborer at time |$t$|⁠. Then the endowment of agent |$i\ge 1$| (whom we call the capitalist) in state |$s$| is |$\alpha_tw_ie_s$|⁠, where |$t=0$| if |$s=0$| and |$t=1$| if |$s\ge 1$|⁠.

Regarding the financial structure, suppose that there are only two financial assets, a stock (a claim to the aggregate endowment) and a risk-free bond. By interpreting |$t=0$| as state |$s=0$|⁠, there are effectively three assets, the other being a claim to |$t=0$| consumption. Therefore, the asset structure is as follows.

Under these assumptions, we can show that a redistribution of wealth from a bondholder to a stockholder reduces the equity premium, while redistribution between laborers and capitalists does not affect the equity premium. To make the statement precise, we introduce some additional notation.

Now we can state our main theoretical result, which characterizes the equilibrium allocation and derives implications to asset pricing.

Intuitively, in an economy with financial assets, the equilibrium prices and risk premiums balance the agents’ preferences and beliefs. Because the stock is the only saving vehicle in the aggregate (because the risk-free asset is in zero net supply), shifting wealth to a patient agent increases the demand for stocks, and, hence, its price rises. If wealth shifts into the hands of the natural stockholder (either the risk-tolerant or optimistic agent), everything else fixed, the aggregate demand for the stock increases. Hence for markets to clear, the risk premium must fall to counterbalance the new demand of these agents.

A surprising aspect of Theorem 2 is that the equity premium is independent of the capitalist/laborer income shares |$\alpha_0,\alpha_1$| and only depends on the wealth distribution among capitalists, |$\left\{ {w_i} \right\}_{i=1}^I$|⁠. The intuition is that while |$\alpha_0,\alpha_1$| affect the overall level of asset prices and the risk-free rate by changing the relative income between |$t=0,1$|⁠, they do not affect the equity premium because assets are only held by capitalists; the equity premium balances the relative demand of stocks and bonds, which only depends on the capitalists’ wealth distribution.

The following propositions show that when agents have heterogeneous risk aversion or beliefs, the portfolio share of the risky asset |$\theta_i$| is ordered as risk tolerance or optimism. To define optimism, we take the following approach. First, by relabeling states if necessary, without loss of generality we may assume that states are ordered from bad to good ones: |$e_1<\dots<e_S$|⁠. Consider two agents |$i=1,2$| with subjective probability |$\pi_{is}>0$|⁠. We say that agent 1 is more pessimistic than agent 2 if the likelihood ratio |$\lambda_s:=\pi_{1s}/\pi_{2s}>0$| is monotonically decreasing: |$\lambda_1\ge \dots \ge \lambda_S$|⁠, with at least one strict inequality.

Combining Theorem 2 together with either Proposition 1 or 2, provided that two agents have the same discount factor (hence the same |$\phi_{i1}=1-\beta_i$|⁠), shifting wealth from a more risk-averse or pessimistic agent to a more risk-tolerant or an optimistic agent reduces the equity premium. In particular, if the rich are relatively more risk tolerant, optimistic, or simply more likely to buy risky assets (⁠|$\textrm{e.g.}$|⁠, because of fixed stock market participation costs), rising inequality should forecast declining excess returns.

1.3 Numerical example

Theorem 2 tells us that the wealth distribution qualitatively affects asset prices, but does it matter quantitatively? To address this issue, we compute a numerical example calibrated at annual frequency. For simplicity we ignore the laborers, so |$\alpha_0=\alpha_1=1$|⁠. We specialize the above economy to one with two agents denoted by |$i=A,B$|⁠. For preference parameters, let |$\rho_i>0$| be the discount rate of agent |$i$| and define the discount factor by |$\beta_i=\frac{\mathrm{e}^{-\rho_i}}{1+\mathrm{e}^{-\rho_i}}$|⁠. We set |$(\rho_A,\rho_B)=(0.015,0.06)$| and |$(\gamma_A,\gamma_B)=(1,5)$|⁠, so we can interpret type |$A$| as the “rich” (patient, risk tolerant) and type |$B$| as the “poor” (impatient, risk averse). The log dividend growth takes |$S=3$| values |$g=(g_s)=(-0.3584,0.0094,0.2779)$| with probability |$\pi=(\pi_s)=(0.0549,0.8552,0.0899)$|⁠.⁹ Given these parameters, we can easily solve for the equilibrium by numerically solving the planner’s problem (8). Figures 1a and 1b show the log equity premium and the log risk-free rate, respectively. Consistent with Theorem 2, increasing the wealth share of type |$A$| agents monotonically decreases the equity premium. The equity premium ranges between about 8% and 1%, so the wealth distribution has a quantitatively large effect.

Figure 1

Wealth distribution and asset prices

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Our two-period model is arguably highly stylized. However, the advantage of a simple two-period model over a more complicated one is that we can prove theorems. In Appendix B, we extend our two-period model to infinite horizon with parameters calibrated to match various asset pricing moments and find that all of our qualitative results also hold in the more realistic setting.

In Appendix C we simulate the calibrated infinite horizon model and run the same predictive regressions that we perform with actual data below in Section 2. The results from data and simulations are quantitatively similar. However, the model makes a prediction not evident in the data about one aspect. In the model, the type |$A$| stock wealth share |$w_A\theta_A/(w_A\theta_A+w_B\theta_B)$| negatively predicts returns, whereas in the data the relationship is near zero (or even positive). Therefore, inspired by (Fagereng et al. 2016a, 2016b) and Kacperczyk, Nosal, and Stevens (2019), in Appendix F we consider heterogeneous belief extensions to both our two-period and infinite horizon models: dividend growth is Markovian and type |$A$| is sophisticated and knows conditional probabilities of dividend growth, whereas type |$B$| is less sophisticated and uses unconditional probabilities. Because type |$A$| adjusts the portfolio based on the dividend growth state independent of the wealth distribution, the type |$A$| stock wealth share is no longer closely related to excess returns, as in the data. However, even in this extension, the wealth distribution channel (Theorem 2) survives.

2. Predictability of Returns with Inequality

In Theorem 2, we have theoretically shown that shifting wealth from an agent who holds comparatively fewer stocks to one who holds more reduces the subsequent equity premium. Many empirical papers show that the rich hold relatively more stocks than do the poor and argue that the rich are relatively more risk tolerant (see footnote 1). Therefore, rising inequality should negatively predict subsequent excess stock market returns. In this section we construct a stationary measure of inequality and show that it predicts subsequent returns. (We address causality in Section 3.)

2.1 Connecting theory to empirics

2.1.1 Empirical motivation

We employ the Piketty and Saez (2003) income inequality measures for the United States, which are available in updated form in a spreadsheet on Emmanuel Saez’s Web site.¹⁰ In particular, we consider top income share measures based on tax return data, which are at the annual frequency and cover the period 1913–2015. These series reflect in a given year the percentage of income earned by the top 1% of earners pretax. We also employ the top 0.1% share, the top 10% share, and the corresponding series that exclude realized capital gains income. Figure 2 shows these series, both including realized capital gains (Figure 2a) and excluding capital gains (Figure 2b). We can immediately see that all series share a common U-shaped trend over the century, and the series including capital gains are more volatile than those without capital gains.

Figure 2

U.S. top income shares (1913–2015)

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Using Taylor approximation, we can connect this quantity to other measures of inequality, as described below.

2.1.2 Constructing the inequality measure

The left-hand side of (14), |${\mathrm{KGR}}$|(1) (we will explain the acronym shortly), is a quantity that can be calculated from top 1% income shares including/excluding realized capital gains. (14) says that it has three components: |$\alpha=Y^k/Y$| (the capital share of aggregate income), |$\rho_A$| (the fraction of realized capital gains income to total capital income for top earners), and |$Y^k_A/Y^k$| (the capital income share of top earners).

It is possible to approximate all quantities in the right-hand side of (15) from the data of Saez and Zucman (2016), at least for the period 1916–2012.¹¹ To evaluate the accuracy of the approximation (14), inspired by (15), in Columns 1–3 of Table 1 we regress |$\log({\mathrm{KGR}})$| on the logarithm of the three components for the top 0.1%, 1%, and 10% group. In each case |$R^2$| is above 0.9, which suggests that the three components |$\alpha$|⁠, |$\rho_A$|⁠, and |$Y_A^k/Y^k$| explain almost all of the variation in |${\mathrm{KGR}}$|⁠. Furthermore, consistent with (15), with top 0.1% and 1% the constant term is insignificant and the other three coefficients are statistically not different from 1. This result suggests that the approximation (14) is indeed accurate.

Second, we construct the two terms inside the parenthesis in (13). For the top 0.1% and 1%, the sample means for term 1 are 0.279 and 0.154, respectively, which are much larger than the term 2 means of 0.006 and 0.009. Additionally, the term 1 standard deviations are 0.165 and 0.091, respectively, while the term 2 standard deviations are 0.005 and 0.006. Therefore, for the top 0.1% and 1%, term 2 is indeed negligible relative to term 1, consistent with the accuracy of the approximation.¹²

Intuitively, what does |${\mathrm{KGR}}$|(1) measure? The first component, |$\alpha=Y^k/Y$|⁠, is the capital share of income. The second component, |$\rho_A$|⁠, reflects two factors. On the one hand, because capital gains realization is more likely when prices are high and because the rich disproportionately hold capital, |$\rho_A$| should positively correlate with wealth inequality. On the other hand, |$\rho_A$| could simply be capturing the timing of capital gains realization. We provide evidence against this second interpretation in Section 3. The last component, |$Y^k_A/Y^k$|⁠, is capital income inequality.

To see which of the three components mainly determines |${\mathrm{KGR}}$|⁠, in Columns 4–6 of Table 1, we regress |$\log({\mathrm{KGR}}(1))$| on the components one at a time. The |$R^2$| takes a large value 0.78 when we use only |$\rho_A$|⁠, an intermediate value of 0.14 when we use |$Y_A^k/Y^k$|⁠, and 0.04 when we use |$\alpha$|⁠, whose coefficient is also insignificant. Therefore, |${\mathrm{KGR}}$| is mostly reflecting variation in |$\rho_A$|⁠, the fraction of realized capital gains income to total capital income for top earners. This is why we chose the name |${\mathrm{KGR}}$|⁠: capital gains ratio. To some extent |${\mathrm{KGR}}$| moves with capital income inequality |$Y^k_A/Y^k$|⁠. The capital share |$\alpha$| does not appear to drive |${\mathrm{KGR}}$|⁠. In summary, provided the timing component of |$\rho_A$| is not dominating (see Section 3), |${\mathrm{KGR}}$| measures capital wealth and income inequality.¹³.

An advantage of |${\mathrm{KGR}}$| is its stationarity (the Phillips and Perron 1988,p-values are less than .01 for the top 0.1%, 1%, and 10%). In contrast, the raw top wealth and income series appear nonstationary, or at least highly persistent, and thus introduce econometric problems when used to predict stationary returns (Granger 1981). |${\mathrm{KGR}}$|(1) looks very much like the detrended top 1% income share series. Figure 3 shows the |${\mathrm{KGR}}$|(1) series and the detrended versions of the raw top 1% series using the Kalman filter with an AR(1) cyclical component (see Appendix D for details) or subtracting the 10-year moving average.

Figure 3

Time-series plot of inequality measures The graph shows the stationary component of the top 1% income share using |${\mathrm{KGR}}$|(1), using the AR(1) Kalman filter (both demeaned), and subtracting the 10-year moving average. The units of all series are percentage points.

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In light of Figures 2a and 2b and the capitalist/laborer share irrelevance in Theorem 2, one can argue for a noneconometric explanation of detrending as well. Because in the figures both the top 0.1% and 10% (and the shares in between) appear to have a common U-shaped trend, the slow-moving component of inequality is plausibly due to redistribution between the poor and rich uniformly rather than from intrarich redistribution. Assuming the poor/nonrich are less likely to participate in financial markets, then the trends in inequality correspond to changes in the capitalist/laborer income shares (⁠|$\alpha_0, \alpha_1$|⁠) in Theorem 2, which are irrelevant for the equity premium according to our model. In this case, we would want to strip out the trends prior to predictive regressions. Intuitively, long-term trends in the capitalist and laborer income share should affect the overall level of asset markets, but not the equity premium per se, which only depends on the intracapitalist wealth distribution.

In the remainder of this section, we focus on the ability of |${\mathrm{KGR}}$|(1) defined by (14) to predict excess market returns in and out of sample.

2.2 In-sample predictions

We obtain the U.S. stock market returns, risk-free rates, and other financial variables from the spreadsheet of Welch and Goyal (2008).¹⁴ Before 1926, stock returns are calculated from the S&P 500 index. After 1926, we use CRSP volume weighted average returns. We put returns into real terms using consumer price index (CPI) inflation, and returns are log returns.¹⁵

The series P/D and P/E are the price-dividend and price-earnings ratios for the S&P 500 index. The spreadsheet also contains the Lettau and Ludvigson (2001) consumption-wealth ratio, commonly referred to as CAY, which spans the period 1945–2015. For presentation, we multiply CAY by 100. Our other controls are GDP growth and, inspired by Lettau, Ludvigson, and Wachter (2008) and Bansal et al. (2014), consumption growth variance. Annual data for GDP and consumption come from the Web site of the Federal Reserve Bank of St. Louis (FRED)¹⁶ and span 1930–2016. We estimate consumption growth variance using an AR(1)-GARCH(1,1) model for log consumption growth.

Table 2 shows the results of regressions of 1-year (⁠|$t$| to |$t+1$|⁠) excess stock market returns on |${\mathrm{KGR}}$|(1) (time |$t$|⁠), some classic return predictors (time |$t$|⁠), and macro factors (time |$t$|⁠). Column 1 shows that a 1-percentage-point increase in year |$t$||${\mathrm{KGR}}$|(1) predicts a 2.7% decline in year |$t$| to |$t+1$| excess market returns.¹⁷

Table 3 shows that all three versions of |${\mathrm{KGR}}$|(⁠|$p$|⁠) also significantly predict 5-year excess returns. Figures 4a and 4b show the corresponding scatterplots and time-series plots for 5-year returns. |${\mathrm{KGR}}$|(1) forecasts subsequent 5-year excess returns well except around 1986.¹⁸ Overall, a 1-percentage-point increase in |${\mathrm{KGR}}$|(1) is associated with, roughly, a 2%–4% decline in subsequent excess returns.

Figure 4

Year |$t$| to |$t+5$| excess stock market return (annualized) versus year |$t$||${\mathrm{KGR}}$|(1), 1913–2015

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The finding that inequality predicts future returns is consistent with our theory and robust to the inclusion of controls and the construction of |${\mathrm{KGR}}$|⁠. However, does more inequality cause lower returns, and is |${\mathrm{KGR}}$| actually reflecting inequality? To address causality, we use tax rate changes as an instrument. Because contemporaneous and lagged changes in top estate tax rates explain a substantial portion of the variation in |${\mathrm{KGR}}$| (Table 6), we estimate the effect of inequality on returns using generalized method of moments (GMM) with instrumental variables (Table 7). Including |${\mathrm{KGR}}$|⁠, industrial production growth, and the log price-earnings ratio as endogenous explanatory variables and using lags of top estate tax rate changes and the log price-earnings ratio as instruments, top income shares are still significant in predicting excess returns. To address the concern that part of the variation in |${\mathrm{KGR}}$| is not due to inequality but rather the timing of realizing capital gains, we include 1-year-ahead changes in capital gains tax rates as an additional instrument to separately identify how the timing and inequality components predict returns (Table 9). The coefficient on the inequality component is negative and significant, while the timing coefficient is insignificant.

The empirical literature on return prediction is controversial. While many papers find evidence for return predictability,²Ang and Bekaert (2007) are more skeptical, and others point out econometric issues, such as small sample bias when regressors are persistent (Nelson and Kim 1993; Stambaugh 1999) and problems with overlapping data (Boudoukh, Richardson, and Whitelaw 2008; Valkanov 2003). In an influential study, Welch and Goyal (2008) show that excess return predictors suggested in the literature perform poorly out of sample. We apply the methodologies of McCracken (2007) and Hansen and Timmermann (2015) to show that including the top 1% as a predictor significantly decreases out-of-sample forecast errors relative to using the historical mean excess return (Table 5).

Across specifications throughout the paper, |${\mathrm{KGR}}$|(10) performs relatively well in forecasting excess returns. This is surprising at first glance: the top 10% hold most financial wealth, and we argued in Section 2.1 that intracapitalist inequality is what should matter for excess returns. A straightforward explanation, however, is that |${\mathrm{KGR}}$|(10) overlaps with |${\mathrm{KGR}}$|(1) and |${\mathrm{KGR}}$|(0.1) and contains the predictive power of higher income shares. In Table 4, we predict returns with intra-10% |${\mathrm{KGR}}$| analogs, for example, |${\mathrm{KGR}}$|(5–10) corresponding to the top 5–10% income share. These new variables form a decomposition of |${\mathrm{KGR}}$|(10), because regressing it on |${\mathrm{KGR}}(0.1), \dots, {{\mathrm{KGR}}}(5-10)$| yields an |$R^2$| of over .99 (with all regressors strongly significant). Consistent with our theory, it is the highest share series (top 0.1–0.5% and top 0.1%) with the largest coefficients (regressors are standardized in Table 4) and |$R^2$|⁠. The top 1–5% |${\mathrm{KGR}}$| is not significant, whereas |${\mathrm{KGR}}$|(5–10) has a positive coefficient significant at the 10% level. Hence, while the |${\mathrm{KGR}}$|(0.1) component of |${\mathrm{KGR}}$|(10) inversely forecasts excess returns, the |${\mathrm{KGR}}$|(5–10) component does the opposite, consistent with our story that redistribution to poorer capitalists increases excess returns.

Given the strength of the relationship between inequality and subsequent excess returns, a question immediately arises: is there some mechanical, nonequilibrium explanation? For example, might stock returns be determining the top share measures? For a few reasons, the answer is no. First, the relationship is between initial inequality and subsequent returns. Returns could affect contemporaneous top shares, but not lagged top shares. One might still worry that our results are driven by our transformation of the top share series into |${\mathrm{KGR}}$|(1). However, as we see in Appendix D, we obtain similar results with other methods of creating stationary series.

However, one may argue that we have known at least since Campbell and Shiller (1988) and Fama and French (1988) that when prices are high relative to either earnings or dividends, subsequent excess market returns are low. The current price could affect current inequality. Are the |${\mathrm{KGR}}$| series simply proxying for the price-dividend or price-earnings ratios, which are known to predict returns? Columns 4 and 5 from Table 2 show top shares predict excess returns even when controlling for the log price-dividend or price-earnings ratio. Including these controls barely affects the |${\mathrm{KGR}}$|(1) coefficient, which is large and significant. The P/D and P/E ratios, however, are not significant after controlling for top capital shares.

In Columns 2, 3, and 6 from Table 2, we also control for real GDP growth, consumption growth variance (Bansal et al. 2014; Lettau, Ludvigson, and Wachter 2008), and CAY, which Lettau and Ludvigson (2001) show forecasts excess market returns. Including these controls, we still see a strong relationship between the top income share and subsequent returns. Similar results hold for different percentiles of the top income share and detrending methods (Appendix D).

How do the components of |${\mathrm{KGR}}$|(⁠|$p$|⁠)—|$\alpha$|⁠, |$\rho_p$|⁠, and |$Y^k_p/Y^k$|—perform in forecasting excess returns? We see in Table D4 (Appendix D) that |$\rho_p$|⁠, the primary driver of |${\mathrm{KGR}}$|(⁠|$p$|⁠) according to Table 1, significantly predicts lower excess returns, while |$\alpha$| and |$Y^k_p/Y^k$| are insignificant. Given that the realized capital gains component of |${\mathrm{KGR}}$|⁠, |$\rho$|⁠, is driving return prediction, a question arises. Are realized capital gains and not inequality per se predicting excess returns? We address this issue in Appendices C and E. First, we solve and calibrate an infinite horizon version of our model and simulate model versions of both |${\mathrm{KGR}}$| and a purely mechanical measure of aggregate realized capital gains. Our Monte Carlo experiment shows that in our calibrated model (1) |${\mathrm{KGR}}$|⁠, mechanical capital gains, and wealth inequality all inversely forecast excess returns, (2) all three variables are substantially correlated, and (3) |${\mathrm{KGR}}$| and mechanical capital gains have small sample properties better than those of wealth inequality.

In short, both |${\mathrm{KGR}}$| and mechanical realized capital gains are quantitatively reasonable proxies for wealth inequality, which our model shows inversely forecasts excess returns. Second, we run a horse race between |${\mathrm{KGR}}$| in the data and empirical measures of mechanical realized capital gains. We find that they perform similarly but provide evidence that the distribution of realized capital gains across income groups matters beyond the overall level in forecasting returns.

In summary, we advocate using |${\mathrm{KGR}}$| as our proxy for capitalist wealth inequality in testing our theory for three reasons. First, it is a reasonable proxy according to our calibrated model. Second, it is easily computed from frequently updated, publicly available data, and its computation requires no arbitrary parameters. Third, |${\mathrm{KGR}}$| is stationary and closely resembles detrended versions of income inequality, which has a very persistent component perhaps driven by forces we argue are less relevant for the equity premium. We elaborate on these points in Appendices C and E.

2.3 Out-of-sample predictions

So far, we have seen that the current top income share predicts future excess stock market returns in sample. However, Welch and Goyal (2008) have shown that the predictors suggested in the literature perform poorly out of sample, possibly because of model instability, data snooping, or publication bias. In this section, we explore the ability of the top income share (⁠|${\mathrm{KGR}}$| in particular) to predict excess stock market returns out of sample.

For the regressors in the |$\texttt{ALT}$| model, following Welch and Goyal (2008), we consider the simplest possible case in which |$x_{1t}\equiv 1$| (constant) and |$x_{2t}$| consists of a single predictor. For the predictor |$x_{2t}$|⁠, we consider |${\mathrm{KGR}}$|(⁠|$p$|⁠) for |$p=0.1,1,10$| and valuation ratios (log(P/D) and log(P/E)). The reason is that (1) because the top income series is at annual frequency, the sample size is already small at around 100 (1913 to 2015), so we cannot afford to use variables that are available only in shorter samples (⁠|$\textrm{e.g.}$|⁠, CAY) for performing out-of-sample predictions, and (2) because Welch and Goyal (2008) find that most predictor variables suggested in the literature are poor, comparing many variables is pointless.

The choice of the proportion of the training sample, |$\rho$|⁠, is necessarily subjective. Small |$\rho$| leads to imprecise initial estimates of |$\beta$|⁠, and large |$\rho$| leads to the loss of power. Hence, we simply report results for |$\rho=0.2,0.3,0.4$|⁠. According to Table 5, we can see that the out-of-sample |$F$| statistic is positive and significant when we use |${\mathrm{KGR}}$|(⁠|$p$|⁠), while it is insignificant for log(P/D) and weakly significant for log(P/E).¹⁹

To see this result graphically, in the spirit of Welch and Goyal (2008), we plot the difference in the cumulative sum of squared errors (the numerator of Equation (17)) over the prediction period in Figure 5. The vertical axis is the cumulative sum for the |$\texttt{NULL}$| model minus the |$\texttt{ALT}$|⁠, so a positive value favors the |$\texttt{ALT}$|⁠. We can see that for all |${\mathrm{KGR}}$|(⁠|$p$|⁠) specifications, the plots roughly monotonically increase up to 1980, decrease until 1990, and then increase again. This result is not surprising, since the 1980s were a time when income inequality increased but the stock market did not suffer (Figure 2a). On the other hand, the log(P/D) and log(P/E) specifications deteriorate after 1970, especially so for log(P/D). This finding is consistent with Welch and Goyal (2008), who document that most of the prediction gains stem from the 1973–1975 Oil Shock.

Figure 5

Annual performance in predicting subsequent excess returns The figures plot the out-of-sample performance of annual predictive regressions. The vertical axis is the cumulative squared prediction errors of the |$\texttt{NULL}$| model minus the cumulative squared prediction error of the |$\texttt{ALT}$| model (hence a positive value favors the |$\texttt{ALT}$|⁠). The |$\texttt{NULL}$| model uses only a constant. The |$\texttt{ALT}$| model includes the predictor variables specified in each panel. Predictions start at |$t=\left\lfloor {\rho T} \right\rfloor$|⁠, where |$T$| is the sample size and |$\rho = 0.2, 0.3, 0.4$|⁠.

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In summary, the top income series seem to predict returns out of sample.

3. Tax Instruments and Returns

The top 1% income share is an endogenous variable in the macroeconomy. While in Section 2.2 we showed that top income shares are not merely proxying for GDP growth, volatility, the consumption-wealth ratio, or the level of the stock market in explaining subsequent returns, it is difficult to rule out the possibility that omitted variables are leading to endogeneity bias.

To identify |$\beta_1$|⁠, suppose that there is an instrument |$z_{1t}$| for |$x_{1t}$|⁠, so (1) |$z_{1t}$| is exogenous (uncorrelated with |$\varepsilon_{t+1}$|⁠), (2) |$z_{1t}$| is correlated with |$x_{1t}$|⁠, and, furthermore, (3) |$z_{1t}$| is uncorrelated with |$x_{2t}$|⁠.

To estimate the moment condition (19), we need an instrument for inequality. Research on inequality suggests that increases (decreases) in tax rates reduce (exacerbate) inequality (Kaymak and Poschke 2016; Roine, Vlachos, and Waldenström 2009). 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 6, the marginal tax rate for the highest income earners increased from about 25% to 90% over the period 1930–1945 and started to decline in the 1960s, reaching about 40% in the 1980s. Thus, the marginal tax rate exhibits an inverse U shape that seems to coincide with the trend in the Piketty-Saez series.

Figure 6

Top 1% income share including capital gains (left axis) and top marginal tax rate (right axis), 1913–2014 Source: IRS.

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Both Piketty and Saez (2003) and Piketty (2003) argue that income inequality should decline in response to expansion of progressive estate taxation: capital gains compose a substantial portion of the income of the rich, and high estate taxes decrease the ability and incentive to amass wealth in financial assets. Thus, increasing the top estate tax rate should disproportionately reduce the wealth of the very rich and subsequently mitigate capital gains income inequality, which is driven by inequality in asset holdings. On the other hand, estate taxes apply to both realized and unrealized capital gains, so estate taxes unlikely affect the timing of realizing capital gains beyond their incentive effects. Therefore, current and lagged changes in the estate tax rates are a good candidate for an instrument.

Tax rate changes are the result of Congressional bills, which generally take years to pass and usually stem from wars or pro-long-term growth or antideficit ideologies (de Rugy 2003a, 2003b; Jacobson, Raub, and Johnson 2007; Romer and Romer 2010; Weinzierl and Werker 2009). Therefore, while alterations in top tax rates affect inequality, their timing and justification are not the result of financial market fluctuations. Provided top tax rate changes have a muted effect on returns, except via inequality, they can serve as an instrument for top income shares.

The first-stage regressions in Table 6 confirm this hypothesis: contemporaneous and lagged changes in the top estate tax rate significantly explain a substantial portion of the variation in |${\mathrm{KGR}}$|(1) (and the 10% and 0.1% analogs).

Whether this instrument can test causation depends on the excludability of lagged changes in estate tax rates. One concern is that estate tax cuts stimulate the economy and thus stock market returns. Another concern is that even if estate tax rates only affect inequality, inequality may simply be proxying for the level of stock market, which we already know predicts returns. To control for these possibilities, we allow |${\mathrm{KGR}}$|⁠, industrial production growth, and log(P/E) to be endogenous and instrument all three with contemporaneous and three lags of the change in the top estate tax rate (⁠|$\Delta\mathrm{ETR}$| for |$t,t-1,t-2,t-3$|⁠) as well as the lagged price-earnings ratio (⁠|$\log(\mathrm{P/E})_{t-1}$|⁠).²⁰Table 7 shows the results of GMM estimation of the moment condition (19) (including industrial production growth and log(P/E) as controls). |${\mathrm{KGR}}$| is significant at the 5% level in predicting subsequent excess returns regardless of whether we use the top 0.1%, 1%, or 10% income share in constructing |${\mathrm{KGR}}$|⁠.

In summary, our finding that rising top income shares lead to low subsequent excess returns is robust to instrumenting inequality with changes in estate tax rates, even when controlling for economic growth and the level of the stock market. Introducing one-period-ahead capital gains tax rate changes as an additional instrument, we are able to separately identify how the inequality and timing components of |${\mathrm{KGR}}$| affect returns. The predictive power of |${\mathrm{KGR}}$| established in Section 2 appears driven by the inequality component.

4. International Evidence

To show that the negative correlation between wealth inequality and returns is not specific to the United States, in this section we employ cross-country fixed effects panel regressions with unbalanced panel data for twenty-nine countries and the time period 1969–2015 as described in Appendix G.

Here, our analysis differs from our U.S. analysis in Section 2 in two main ways. The first is the availability of international risk-free rates. While U.S. Treasury returns provide a standard measure of the risk-free rate in the United States, in international markets, especially emerging ones where government and private sector default are not rare, the risk-free rate measure is controversial. Furthermore, because of the limited availability of similar interest rates across countries, using stock returns instead of excess returns substantially expands the sample size. Quantitatively, the predictability of excess returns in Section 2 was really about stock returns.²¹ In light of these facts, we perform the international analysis using stock market returns without netting out an interest rate. The second difference is that the post-1969 sample shows no obvious U shape for top income shares. Furthermore, for many countries we cannot calculate |${\mathrm{KGR}}$|(1)²² and the samples are short or missing years. Therefore, we use the raw top 1% income share data rather than remove time trends.

Table 10 presents the panel regression results for both the whole sample and different groups. We see in Column 1 that when including all countries, a 1-percentage-point increase in the top income share is associated with a subsequent decline in stock market returns of 0.94% on average. Column 2 uses the same regression, but for only advanced economies, and we obtain similar results.

Is the predictive power of the top income share uniform across countries? In either very large markets (such as the U.S. market) or relatively closed ones (such as emerging markets), our theory suggests that domestic inequality should affect domestic stock markets. In small open markets, however, foreign investors own a substantial fraction of the domestic stock markets and should mitigate the role of domestic inequality. However, even if domestic inequality is less important in small and open financial markets, inequality among global investors should still affect returns in these markets.

5. Concluding Remarks

This paper builds a general equilibrium model with agents that are heterogeneous in wealth, risk aversion, and belief. We show that increasing inequality drives down the subsequent equity premium. 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”). Consistent with our theory, we find that in the United States, the wealth distribution is closely connected with stock market returns. When the rich are richer, the stock market subsequently performs poorly, both in and out of sample. The inverse relationship between returns and inequality is robust to controlling for standard return predictors and instrumenting with changes in estate taxes. It also holds internationally, although in relatively open economies with low home bias it is United States, not domestic, inequality that matters.

Could one exploit the predictive power of top income shares to beat the market on average? The answer is probably no, because the top income share based on 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 explanation of excess market returns over time. We conclude, as decades of macroeconomics and finance theory have suggested, that stock market fluctuations are intimately tied to the distribution of wealth and income.

A. Proofs

A.1 Proof of Theorem 1

The proof proceeds as follows: we first solve the planner’s problem, then show that the planner’s solution is a GEI, and finally prove that this equilibrium is unique.

A.2 Proof of Theorem 2

Now we prove Theorem 2.