Asset pricing with return extrapolation

Abstract

We present a new model of asset prices in which a representative agent has extrapolative beliefs about stock market returns and Epstein-Zin preferences. The model quantitatively explains facts about asset prices, return expectations, and cash-flow expectations. When the agent’s beliefs about stock market returns are calibrated to survey expectations of investors, the model generates excess volatility and predictability of stock market returns, a high equity premium, a low and stable risk-free rate, and a low correlation between stock market returns and consumption growth. Moreover, the model has implications for expectations about future cash flows that are consistent with empirical findings.

Introduction

In financial economics, there is growing interest in “return extrapolation,” the idea that investors’ beliefs about an asset’s future return are a positive function of the asset’s recent past returns. Models with return extrapolation have two appealing features. First, they are consistent with survey evidence on the beliefs of real-world investors.¹ Second, they show promise in matching important asset pricing facts, such as volatility and predictability in the aggregate market, momentum and reversals in the cross-section, and bubbles (Hong, Stein, 1999, Barberis, 2018, Barberis et al., 2015, Barberis, Greenwood, Jin, Shleifer, 2018, Liao, Peng, Zhu, 2021). To study the asset pricing implications of return extrapolation, a researcher must also make an assumption about investor preferences. The most prominent preference specification in recent research on asset prices is arguably Epstein-Zin utility.

In this paper, we propose a new model of aggregate stock market prices based on return extrapolation and Epstein-Zin preferences. The goal of the paper is to provide a new behavioral model that simultaneously explains facts about asset prices, return expectations, and cash-flow expectations. We show that, when a representative agent’s beliefs about stock market returns are calibrated to match survey expectations of real-world investors, return extrapolation and Epstein-Zin preferences together allow the model to quantitatively explain facts about stock market prices. Moreover, we find that return extrapolation has direct implications for expectations about future cash flows that are consistent with recent empirical findings.

We consider a Lucas economy in continuous time with a representative agent. The Lucas tree is a claim to an aggregate consumption process which follows a geometric Brownian motion. There are two tradeable assets in the economy: the stock market and an instantaneous riskless asset. The stock market is a claim to an aggregate dividend process whose growth rate is positively correlated with consumption growth. The riskless asset is in zero net supply with its interest rate determined in equilibrium. The representative agent has extrapolative beliefs and Epstein-Zin preferences. She perceives that the expected growth rate of stock market prices is governed by a switching process between two regimes. If the recent price growth of the stock market has been high, the agent thinks it is likely that a high-mean price growth regime is generating prices and therefore forecasts high price growth in the future. Conversely, if the recent price growth has been low, the agent thinks that it is likely that a low-mean price growth regime is generating prices and therefore forecasts low price growth in the future.

We calibrate the agent’s beliefs about stock market returns to match the survey expectations of investors studied in Greenwood and Shleifer (2014). Specifically, we set the belief-based parameters so that, in a regression of the agent’s expectations about future stock market returns on past 12-month returns, the model produces a regression coefficient and a $t$-statistic that match the empirical estimates from surveys. Our parameter choice also allows the agent’s beliefs to match the survey evidence on the relative weight investors put on recent versus distant past returns when forming beliefs about future returns. With the agent’s beliefs disciplined by survey data in this way, the model quantitatively matches important facts about the aggregate stock market: it generates significant excess volatility and predictability of stock market returns, a high equity premium, a low and stable interest rate, as well as a low correlation between stock market returns and consumption growth.

We now explain the model’s implications, starting with excess volatility. The model generates average return volatility of 24.8% for the stock market, much higher than the volatility of dividend growth, which is set to 11%. The excess volatility of stock market returns comes from the interaction between return extrapolation and Epstein-Zin preferences. Suppose that the stock market has had high past returns. Return extrapolation then leads the agent to forecast high future returns. Under Epstein-Zin preferences, and consistent with the prior literature, we set risk aversion to be higher than the reciprocal of the elasticity of intertemporal substitution, giving rise to a strong substitution effect. As such, the agent’s forecast of high future returns leads her to push up the current price significantly, generating excess volatility.²

The mechanism described above for generating excess volatility, together with a strong degree of mean reversion in the agent’s expectations about stock market returns, produces the predictability of stock market returns from the price-dividend ratio that we observe in the data. The agent’s beliefs mean-revert for two reasons. First, by assumption, the agent believes that the expected growth rate of stock market prices tends to switch over time from one regime to the other; as a result, she perceives that her expectations about stock market returns will mean-revert. Second, the agent’s return expectations actually mean-revert faster than she thinks they will: when the agent thinks that the future price growth is high, future price growth tends to be low endogenously, causing her return expectations to decrease at a pace that exceeds what she anticipated. As a result, following periods with a high price-dividend ratio—this is when the agent’s expectation about future returns is high—the agent’s return expectation tends to revert back to its mean, giving rise to low subsequent returns and hence the predictability of stock market returns using the price-dividend ratio.

Next, we turn to the model’s implications for the equity premium. When measured as the rational expectation of stock market returns in excess of the interest rate, the model generates an average equity premium of 7.36%.³ Two factors contribute to the high equity premium. First, the agent’s risk aversion causes her to demand a substantial equity premium in the face of excess return volatility. Second, return extrapolation gives rise to perceived persistence of the aggregate dividend process, which, under Epstein-Zin preferences, is significantly priced, pushing up the equity premium. Note that both return extrapolation and Epstein-Zin preferences are important for generating a high equity premium: in the absence of return extrapolation, Epstein-Zin preferences with i.i.d. dividend and consumption growth give rise to a very small equity premium of 0.84%; in the absence of Epstein-Zin preferences—that is, with power utility—return extrapolation alone generates an equity premium of only 1.93%.

Finally, the model generates low interest rate volatility and a low correlation between stock market returns and consumption growth. In the model, the agent forms different beliefs about the dividend growth of the stock market and about aggregate consumption growth. Here, we assume that the bias in the agent’s beliefs about consumption growth derives only from the bias in her beliefs about dividend growth. Given the low observed correlation between consumption growth and dividend growth, the derived bias in the agent’s beliefs about consumption growth is small, consistent with the fact that there is no direct empirical evidence for extrapolative beliefs about consumption growth. The agent’s approximately correct beliefs about consumption growth allow the model to generate low interest rate volatility. They also imply that the agent’s beliefs about stock market returns, which comove strongly with the agent’s beliefs about dividend growth, are not significantly affected by fluctuations in consumption growth, giving rise to a low correlation between stock market returns and consumption growth.

Although our model is based on return extrapolation, it also has direct implications for cash-flow expectations. When the past price growth of the stock market has been high, this has a positive effect not only on the agent’s beliefs about future returns, but also on her beliefs about future dividend growth; indeed, her expectations about dividend growth rise more quickly than her expectations about future returns.⁴ Given this, a Campbell-Shiller decomposition using the agent’s subjective expectations about stock market returns and dividend growth shows that changes in subjective expectations about future dividend growth explain 110% of the variance of the price-dividend ratio, while changes in subjective return expectations explain $-$10% of the variance of the price-dividend ratio. This quantitatively matches the recent empirical findings of De la O and Myers (2021), who show that changes in investors’ subjective expectations of future dividend growth explain the majority of stock market movements. In this way, our model simultaneously accounts for the empirical findings of Greenwood and Shleifer (2014) on return expectations and the empirical findings of De la O and Myers (2021) on cash-flow expectations.

Our model also points to some challenges for return extrapolation: when calibrated to the survey expectations data, the model predicts a persistence of the price-dividend ratio that is lower than its empirical value. To match the empirical persistence of the price-dividend ratio, investors need to form beliefs about future returns based on many years of past returns. However, the available survey evidence suggests that they focus on just the past year or two. Section 4 discusses a potential resolution of this issue.

After presenting the model, we compare it to alternative models of the aggregate stock market, including models with cash flow extrapolation and models with rational expectations. Our model is developed in a Lucas economy with a representative agent. This allows for a quantitative comparison between our model and many alternative models of the stock market. We discuss their distinct implications in Section 4.

Our paper adds to a growing theoretical literature that studies the role of return extrapolation in explaining asset pricing facts. Early models, such as Cutler et al. (1990) and De Long et al. (1990), highlight the conceptual importance of return extrapolation, but they are not designed to match asset pricing facts quantitatively. Barberis et al. (2015) present a dynamic consumption-based model that tries to make sense of both survey expectations and aggregate stock market prices. However, the simplifying assumptions in their model make it difficult to evaluate the model’s fit with the empirical facts.⁵ Our model overcomes this limitation and allows for a quantitative comparison with the data. Moreover, our model makes a methodological contribution to the literature: it provides a numerical procedure for solving asset pricing models that assume biased beliefs about endogenous equilibrium outcomes.

Our model is related to, but fundamentally different from, several recent models that use biased beliefs to quantitatively match empirical facts about the stock market. The models of Choi and Mertens (2013), Hirshleifer et al. (2015), and Nagel and Xu (2021) assume a representative agent with extrapolative beliefs about fundamentals, in which the agent’s expectation of future cash-flow growth depends positively on a weighted average of past cash-flow growth. Our model, on the other hand, assumes return extrapolation. Both assumptions, fundamental extrapolation and return extrapolation, are supported by survey expectations of real-world investors, and both types of models aim to make sense of facts about asset prices. As we explain in Section 4, while our model assumes only return extrapolation, it also produces patterns of investor beliefs that are consistent with fundamental extrapolation. However, the models that assume fundamental extrapolation typically do not give rise to return extrapolation; for example, the model of Nagel and Xu (2021) is not able to match the observed positive correlation between return expectations and past 12-month returns. This contrast highlights a key difference between these two types of models. Adam et al. (2017) present a model that explains the data on both return expectations and asset prices. The mechanism in their model is quite distinct from the mechanism in our model. In their model, agents have fully rational beliefs about future dividend growth, but do not observe the exact mapping between dividends and stock prices. As such, their model generates extrapolative beliefs about returns but nonextrapolative beliefs about dividend growth. In our model, however, beliefs about returns and beliefs about dividend growth are closely tied to each other and both extrapolative; therefore, our model makes sense of both facts about return expectations (Greenwood and Shleifer, 2014) and facts about cash-flow expectations (De la O, Myers, 2021, Nagel, Xu, 2021). More broadly, our model is related to asset pricing models under biased or nontraditional beliefs in the form of natural expectations (Fuster et al., 2011), diagnostic expectations (Bordalo et al., 2018), memory retrieval (Wachter and Kahana, 2020, Wachter and Kahana, 2021), and investor overconfidence (Daniel et al., 1998, Daniel et al., 2021).

The paper proceeds as follows. In Section 2, we lay out the basic elements of the model and characterize its solution. In Section 3, we parameterize the model and examine its implications for asset prices, return expectations, and cash-flow expectations. Section 4 discusses some additional aspects of our analysis. Section 5 concludes and suggests directions for future research. Additional details are in the Appendix.