**Risk-Factor Irrelevance∗**Alex Chinco†, Samuel M. Hartzmark‡, and Abigail B. Sussman§ August 29, 2019 Abstract Standard asset-pricing models assume investors a) recognize correlations with risk factors such as aggregate consumption growth, b) prefer assets with lower risk-factor correlations, and c) adjust their portfolio holdings accordingly. We survey finance professionals and a general population of MTurkers and find no evidence to support these assumptions. Participants do not adjust their demand in response to changes in correlation with consumption growth. Participants state that they do not consider correlation with consumption growth in a manner consistent with standard finance theory. And, participants are not able to recognize economically large differences in correlations when presented graphically. Our paper suggests risk-based explanations for asset-pricing facts should be paired with evidence that investors recognize, think about, and trade on this risk. We provide a road map showing how to do this. Keywords: Expected Returns, Correlation Neglect, Asset Pricing ∗We would like to thank Arpit Gupta, Marina Niessner, David Solomon, and Kelly Shue for useful comments and suggestions. We would also like to thank Nicholas Herzog and Kari Greenswag for excellent research assistance. †University of Chicago, Booth School of Business. alex.chinco@chicagobooth.edu ‡University of Chicago, Booth School of Business. samuel.hartzmark@chicagobooth.edu §University of Chicago, Booth School of Business. abigail.sussman@chicagobooth.edu1 The consumption capital asset-pricing model (CCAPM; Lucas, 1978) is the backbone of standard finance theory. The key insight in this model is that the price of an asset should be related to its correlation with consumption growth. CCAPM investors worry about not having enough money when the economy falters and their consumption falls. If an asset’s returns are less correlated with consumption growth, then the asset is more likely to have positive returns during bad times. Since bad times are when they most want spare cash, CCAPM investors should be willing to pay more for assets that are less correlated with consumption growth, giving these assets lower expected returns. Although the CCAPM embodies an elegant line of reasoning, we have long known it does not fit key aspects of the data (e.g., Mehra and Prescott, 1985). Researchers have responded to these empirical shortcomings by introducing new risk factors to the model, arguing that we should be using correlations with consumption growth and other state variables (Merton, 1973) to quantify investors’ fear of bad times. According to Cochrane (2017), the way to understand why markets fluctuate “is to find concrete, quantitative, and theoretically explicit measures of fearful outcomes...that quantitatively account for asset-pricing facts.” Examples of this approach include habit formation (Campbell and Cochrane, 1999), long-run risk (Bansal and Yaron, 2004), and rare disasters (Rietz, 1988; Barro, 2006). But, if the goal is to understand why markets actually fluctuate, then this approach only makes sense if people actually invest based on these risk factors. To figure out whether this is likely, we survey people (including finance professionals who trade securities) about their investment decisions in an experimental setting that allows us to manipulate an asset’s correlation with consumption growth. We document that a) participants do not adjust their portfolio holdings in response to changes in correlation with consumption growth, b) 91% of participants report either ignoring these correlations or thinking about them in a manner inconsistent with standard finance theory, and c) participants are unable to recognize economically large differences in correlations when presented graphically. In short, we find no evidence that people trade the way textbook investors are assumed to trade, think the way textbook investors are assumed to think, or recognize the correlations textbook investors are assumed to recognize. While our paper suggests that macro risk-factors are not important aspects of investor decision making, we are not arguing that the tradeoff between risk and expected returns is irrelevant. Our data suggest that participants strongly respond to changing risk, just not risk related to consumption growth correlations. Survey participants adjust their demand in response to changes in average returns and return volatility. They also think about these statistics in the way standard finance theory would suggest, reporting that they increase their demand when average returns are higher and when return volatility is lower. Moreover, they are able to recognize differences in the mean and Introduction 1

volatility of returns when shown graphically. It is only risk-factor correlations that are irrelevant. The underlying assumption in standard finance theory is that the way to model investors’ risk-return tradeoff is to use risk-factor correlations. We are casting doubt on this specific assumption. If risk-factor correlations were the right way to model this tradeoff, then people should be able to recognize them in the data. When asked, they should report liking assets with lower risk-factor correlations. And, their portfolio positions should reflect this preference. There is no evidence of this in our data. To explain why one restaurant is more popular than another, you could write down a model in which diners prefer nutrients that are more prevalent in dishes of the first restaurant (Ang, 2014, §6.2). If you asked people though, you would quickly realize that this is not a good model of how diners choose where to eat. We ask people about their investment decisions and find that risk-factor correlations are not a good basis for understanding why markets move. Markets are populated by intelligent forward-looking investors. As a result, prices today can change due to subtle shifts in investors’ perception of future market conditions. This idea is a cornerstone of modern finance and we do not call it into question. However, if a researcher wants to argue that a particular risk factor is causing prices to change, then he should be able to provide evidence that investors are following the logic in his model. Proposing an asset-pricing model that would “quantitatively account for asset-pricing facts” if investors were worried about his preferred risk factor is not enough. To support such a model, a researcher needs to demonstrate that these intelligent forward-looking investors a) recognize the risk factor, b) think about the risk factor in a way that is consistent with the model, and c) adjust their demand based on this reasoning. We hope our paper provides a road map showing how to do this. Main Results. While observational data is limited to the single historical time series, in a laboratory setting we can experimentally manipulate parameters of the data-generating process to see how participants respond. We show participants data on the relationship between stock returns and economic growth and we ask them to allocate an endowment between stocks and bonds. Our participant pools include both finance professionals, some of whom are professional traders, and Amazon Mechanical Turkers. For economic growth, all participants see the historical US gross domestic product (GDP) time series from 1980 until 2018. But, each participant sees a stock-return time series based on simulated data. We vary the mean, volatility, and correlation of these simulated returns both across questions and between participants. For each question, participants see a graph with GDP and cumulative returns as well as numeric values of the mean, variance, and correlation between these series (see Figure 1). While someone would have to work to collect all this information in the real world, our goal is to make it as easy as possible to follow the textbook asset-pricing logic. We find that participants respond to experimental manipulation. They invest more in stocks when average stock returns are higher (p-value < 0.001), and they invest less in stocks when stock returns are more volatile (p-value < 0.001). But, they do not change their demand based on the 2

correlation between stock returns and consumption growth (p-value = 0.778). Standard asset-pricing investment objectives. This is true of both the CCAPM and the more complex models built on top of it. Yet, in our setting, there is no significant change in holdings when stock returns move from being uncorrelated with consumption growth to having a correlation of 0.45, even though standard calibrations suggest a CCAPM investor would demand an 11% risk premium to accommodate such a large increase in correlations. To a first approximation, participants completely ignore risk-factor correlations when making their investment decisions. While these revealed-preference results suggest that participants ignore risk-factor correlations, perhaps participants are thinking about risk-factor correlations in a manner not captured by our experimental setup. To address this concern, we ask participants how they made their investment decisions after they completed this task. Participants report caring about the mean and volatility of stock returns, but most participants (58%) state that correlations did not play any role in their decision making. Among the 42% of participants who did think about correlations, three out of four report increasing their demand when stock returns were more correlated with consumption growth— the exact opposite of what textbook investors are assumed to do. Only around 10% of participants report thinking about risk-factor correlations in a manner consistent with standard finance theory. If investors were primarily interested in risk-factor correlations, then it would be trivial for financial-news outlets to offer this information. The fact that the Wall Street Journal does not routinely report risk-factor correlations should give us pause. However, it could be that everyone already knows these numbers, leaving no need to report them. To explore this idea, we examine mutual-fund prospectuses to see what funds explicitly name as their key risks and objectives. Even if everyone is aware of a fund’s risk-factor correlations, the fund would still need to name them in its prospectus for legal reasons if they were important to investors. For example, presumably most investors are aware that the market can go up or down, but nearly every mutual fund discusses this risk in its prospectus. Not a single fund we looked at discussed its correlation with consumption growth as a major risk. In fact, while each prospectus contained hundreds of numerical values, no fund reported a correlation with any macro risk factor. Given that correlations are not explicitly reported in the financial media or in fund documents, the most likely way that an investor would acquire this information is by looking at graphs. To ascertain whether participants can identify differences in correlations in such a setting we show each participant two charts side-by-side (see Figure 2) and ask them to identify the figure with the higher parameter. To create these charts, we simulate stock-return data using different input parameters: mean, volatility, and correlation. The two charts always use the same value for two parameters and a different value for the third. For example, both figures might have stock returns with the same average return and return volatility, but one figure would have a correlation with consumption models all assume that investors view these correlations as having primary importance for their 3

growth of 0.00 while the other would have a correlation of 0.45. When shown charts with different means or volatilities, participants easily identify which chart has the higher value. However, when examining charts with different correlations, they are unable to identify which has the higher value at a rate better than chance, correctly doing so precisely 50% of the time. To the extent that these results reflect how investors perceive the market, they suggest that differences in risk-factor correlations are unknown and unpriced in the real world. Discussion. We specifically designed the experiment to make it as easy as possible for participants to follow the logic of standard asset-pricing models. We removed all superfluous information. We presented difficult-to-find information about correlations in a straightforward manner together with the relevant definitions. We only examined participants who passed a comprehension test demonstrating that they understood these definitions. And, we repeatedly asked about and emphasized the variables from standard finance theory in the materials we provided to our participants. While experimenter demand is a concern in any lab setting, if anything, experimenter-demand effects should have made participants more likely to follow the textbook asset-pricing logic than they otherwise would have been. Were participants simply confused? This is unlikely in our setting as participants respond to changes in the mean and volatility of returns just as standard finance theory says they should. Further, we study their behavior using multiple methods: we use a revealed-preference approach, we ask participants about their reasoning, and we investigate their ability to identify a parameter value graphically. All three of these methods point to the same conclusion—participants are not investing based on risk-factor correlations. This consistency should ameliorate any concerns about participants not understanding what is being asked. Are we asking the right people? We partially address this concern by examining a broad range of participants. Our participant pool includes finance professionals, some of whom trade financial securities for a living. That said, our sample clearly misses some important market participants, such as super high net-worth individuals or finance-industry executives. However, worries about representativeness need to provide a plausible explanation as to why risk-factor correlations are of primary importance for these specific individuals but not for the broad group of investors that we examine. None of our participants behaves like the representative investor in a standard asset-pricing model. They are wholly unresponsive to changes in the key parameter that a representative investor should care about. If we are going to use such a model to describe how assets are priced, then the model needs to explain why the views of these participants should be discarded. Do participants have the right incentives? The stakes are clearly much higher in real-world financial markets than in our experiment. However, it is unclear why this would explain our participants’ indifference to risk-factor correlations. If this were the explanation, then why are participants providing the right answers to questions about the mean and volatility of returns? 4

Participants increase their demand for stocks when the mean return is higher and when the return volatility is lower. We pay participants for accurately identifying graphs with differences in parameter values, but they are only able to identify differences in the mean and volatility of returns. Participants demonstrate that they are thinking about the tradeoff between risk and expected returns; they are just not thinking about this tradeoff in terms of risk-factor correlations. Can we ignore these findings because of the well-known Friedman (1953) ‘as if’ critique? No. Under this view, the fact that our participants do not seem to care about risk-factor correlations is unimportant as long as a model involving investors who do fits the data. We feel that this critique misses the point of looking for the right asset-pricing model. We know that many models can match certain moments of the data (Lewellen et al., 2010). We care about finding the right one of these models precisely because understanding why asset prices fluctuate requires understanding the mechanism behind why a model fits the data. This means that a model must reflect how real-world investors attend to, think about, and act on market information. It is the mechanism that allows us to understand what will happen and why when market conditions change. Contribution to the Literature. This paper adds to the literature on failure to perceive correlations (Jennings et al., 1982; Ungeheuer and Weber, 2018; Matthies, 2018). We contribute to this literature by showing that meaningful shifts in correlation are not graphically observable by most people. In addition, researchers have examined how misunderstanding correlations can lead to biased decision making in non-financial settings (Enke and Zimmermann, 2017; Levy and Razin, 2015). Researchers have also demonstrated that when combining risky assets into a portfolio, participants do not appropriately account for the correlation structure between the assets (Eyster and Weizsacker, 2016; Kallir and Sonsino, 2009; Matthies, 2018). And, they have experimentally examined whether people behave like classic mean-variance investors (Kroll et al., 1988; Kroll and Levy, 1992; Bossaerts and Plott, 2004; Huber et al., 2017). These patterns are primarily about the cross-section of stock returns, while our paper is more closely related to the time-series. This paper also contributes to the literature examining how people actually make financial decisions. Prior work has examined how people frame financial decisions—e.g., over individual positions (Odean, 1998), across positions (Frydman et al., 2017), and over portfolios (Hartzmark, 2014)—and how this framing influences portfolio decisions. Other research has examined which attributes investors are drawn to—e.g., saliency (Barber and Odean, 2007), sustainability (Hartzmark and Sussman, 2018), and dividend payments (Hartzmark and Solomon, 2017; Harris et al., 2015). A separate line of papers has examined more macro questions about consumption (Di Maggio et al., 2018) and investors’ agreement with particular lines of economic reasoning (Giglio et al., 2019; Choi and Robertson, 2018). This paper contributes by showing that portfolio choices do not seem to be influenced by the underlying logic behind standard asset-pricing models. 5

2 Our main results come from surveying people about their portfolio decisions in an experimental setup where we can manipulate the correlation between an asset’s returns and consumption growth. Our goal is to give participants the best chance possible to react to risk-factor correlations in a way that is consistent with standard asset-pricing theory. So, we directly provide participants with the information that a textbook investor would want even though such information is typically not so easy to ascertain in real-world settings. We give participants this information in both numeric and graphical form. We also do not distract participants by including any of the extraneous information that is present in real-world settings, such as expense ratios, style tilts, fund companies, etc. Survey Design 2.1 Participant Populations We conducted our survey using two participant populations. The first was recruited by Prime Panel, a service with access to more than 30 million online panelists worldwide that specializes in connecting researchers with unique and hard-to-reach samples.1This panel was comprised of people who work in the finance or banking industries. We present summary statistics for the 529 finance professionals who completed our survey in Table 1. This participant pool is fairly wealthy, with 59% making more than $100k per year in income. Furthermore, these participants tend to be active investors, with 90% reporting that they own either individual stocks or mutual funds. We also ask participants about their job function, and 27% of this population states that their job involves investing in financial securities. The second population we examine is drawn from Amazon’s Mechanical Turk marketplace. Research examining this platform finds that participants recruited through MTurk, referred to as “MTurkers”, tend to perform similarly on tasks (Casler et al., 2013) and better in attention checks (Hauser and Schwarz, 2016) than traditional participant pools recruited through labs. MTurkers also represent a more diverse set of participants (Paolacci and Chandler, 2014). Our sample consists of 322 participants who completed our survey. MTurkers have lower incomes when compared to the finance professionals. 87% of MTurkers report earning less than $100k per year. MTurkers also tend to be younger with 72% under 40 years of age. A lower fraction of MTurkers owns financial securities. But, even in this sample, 65% still report owning either stocks or a mutual fund. 2.2 Experiment After reading the instructions and completing a comprehension check, there were three key parts of the survey: investment decisions, economic reasoning, and parameter recognition. 1See https://www.turkprime.com/Service/ConnectWithParticipants#prime-panels-nav. 6

2.2.1 Instructions Upon entering the survey and completing a consent form, all participants were given instructions explaining that they will be making investment decisions based on economic growth and stock market performance. The instructions include intuitive and straightforward definitions of key terms: Economic growth refers to how well the economy as a whole is doing. It is commonly reported as Gross Domestic Product (GDP) which is a measure of the goods and services produced in the US economy. The information about the stock market is for a mutual fund that passively invests in a broad blue-chip stock market index, such as the S&P 500 or the Dow Jones Index. The value of the mutual fund reflects the value of its investments, so when the stocks it invests in have a higher price, the value of the mutual fund will be higher. Each participant was informed that they would be seeing annualized numeric values for the mean, variance, and correlation between the two time series as well as a graph showing the cumulative performance of both stock returns and the economy as a whole. Again, to make sure that they are completely clear as to what this meant, we provide each participant with the definition of mean, variance, and correlation: When the average per year is higher you should expect greater increases in value in a given year, corresponding to steeper increases in the line displayed. When uncertainty is higher, you should expect greater swings, for example higher highs and lower lows are more likely than if uncertainty is low. When a correlation is higher, this means that if one series goes up, the other is more likely to go up too, and if it goes down, the other is also more likely to go down. It is important that each participant understood the information about stock returns being presented to them. To ensure that this was the case, after reading the instructions, participants answer several multiple-choice comprehension questions asking them to define economic growth, average growth, uncertainty, and correlation. Participants drawn from the pool of finance professionals continued to the experiment only if they answered all of the comprehension questions correctly on the first try. MTurkers were allowed to complete the full survey irrespective of their responses to the comprehension check. However, we only include data from the subset of people who correctly answered these questions correctly on their first try in our analysis. 2.2.2 Investment Decisions The first portion of the experiment asked participants to allocate an $1,000 initial endowment between stocks and bonds based on information we provided about economic growth and stock returns. Specifically, we asked them to allocate money to a “mutual fund that passively invests in a broad blue-chip stock market index, such as the S&P 500 or the Dow Jones Index.” See Figure 1 for 7

a sample question. Each participant saw ten questions of this type. To ensure that participants were not responding to specific characteristics of these ten questions, these questions were randomly selected from a larger set of 36 possible questions for each participant. For each question, the participant observed a time series of cumulative stock returns and aggregate consumption as well as summary statistics describing the mean, variance, and correlation between these two time series. For consumption growth, all participants saw the same historical US GDP time series from 1980 until 2018 in every question. But, we simulated the stock-return time series and varied the mean, volatility, and correlation both across questions and between participants. For each question, participants saw a graph with GDP and cumulative returns as well as numeric values of the mean, variance, and correlation between these two time series. Participants were told that the numeric values provided to them are stable predictors of the future returns for that particular fund and that each round of questioning was related to a completely separate mutual fund from the last. 2.2.3 Economic Reasoning After each participant completed his ten investment-decision questions, they were next asked a series of questions about the economic reasoning they used to make these decisions. He was first asked whether he considered the mean of stock returns, the mean of economic growth, the volatility of stock returns, the volatility of economic growth, or the correlation between the stock market and economic growth, and could select all options that apply in a series of binary responses. Whenever a participant reported that he considered a particular variable, we then asked him precisely how he acted on this information. For example, if a participant reported that he considered the correlation between returns and consumption growth, we then followed up by asking if he invested more in the stock market when this correlation was higher or lower. 2.2.4 Parameter Recognition Finally, information on risk-factor correlations is not generally directly given to investors when looking at financial news stories. So, in the third part of our experiment, we investigate which parameter values people are able to recognize when shown as graphs. For each question in this task, we presented participants with a pair of graphs. Both graphs displayed the same consumption growth data; however, the cumulative-return time series in each graph was different. These data could vary on three dimensions: mean, variance, and correlation with consumption growth. Each pair of graphs always holds two of these three parameters at their median values and differs on the third. e.g., a participant might see a pair of graphs with the same average returns, E[rTop] = E[rBottom] = 0.06, and the same correlation between returns and consumption growth, Cor[rTop,∆c] = Cor[rBottom,∆c] = 0.15, but with different return volatilities, Sd[rTop] = 0.10 8

while Sd[rBottom] = 0.20. We ask participants which of the two graphs has the higher parameter value that varies across the two charts. See Figure 2 for a sample question. To ensure that participants were not responding to specific characteristics of particular image pairs, each participant answered two questions for each parameter value that varied for a total of six questions, and each question was randomly selected from one of three possible variants. The order of these questions was randomized. We incentivized all participants by offering a $0.20 bonus for each question answered correctly, for a maximum of $1.20. As a point of comparison for this payment amount, it gave Mturk participants a chance to double their base payment of $1.20. value, the one on the top or the one on the bottom. This question is only asked for the parameter 2.3 Data Simulation Our experiment involves time-series data on economic growth and stock returns. Every participant saw the same economic-growth time series in every question. This time series represents seasonally-adjusted quarterly US GDP, ∆ct to 2018Q4 (i.e., T = 156 observations in total). Average annualized GDP growth during this sample this macroeconomic time series, it appears fairly smooth. Since larger swings in the data might make it easier for participants to graphically recognize co-movement between consumption growth and stock returns, we showed some participants a version of this economic growth time series with noise added. In the Internet Appendix we show that this step has no influence on our main results. We simulate stock returns using various combinations of the following parameter values: def= log(GDPt)−log(GDPt−1), from 1980Q1 period is µ∆c= 5.2% while the annualized volatility of GDP growth is σ∆c= 1.6%. When we graph µr∈ {4%,6%,8%} ρr,∆c∈ {0.00,0.15,0.30,0.45} σr∈ {10%,15%,20%} Above, µrdenotes the mean annualized stock return, σrdenotes annualized stock-return volatility, and ρr,∆cdenotes the correlation between stock returns and consumption growth. Each participant answers ten investment-decision questions in part #1 of our experiment and six parameter-recognition questions in part #3 of our experiment. The typical stock-market return is 6% per year, and annualized return volatility is roughly 15% (Cochrane, 2001). The CCAPM says an asset’s expected excess return should be µr= γ ×(ρr,∆c⋅σr⋅σ∆c) where γ denotes investors’ coefficient of relative risk aversion. To match the equity premium, we need to assume γ ≈ 100 (Campbell, 2003), and the annualized volatility of consumption is about 9

σ∆c= 1.6% in our sample period. So, according to the CCAPM, investors should view a mutual ρr,∆c= 0.00; whereas, they should see this same 6%-per-year fund as overpriced when ρr,∆c= 0.45: 100×(0.00⋅15%⋅1.6%) = 0% in low correlation setting (underpriced) fund with a 6% per year average returns as underpriced and have high demand for its shares when < 6% per year < 10.8% = 100×(0.45⋅15%⋅1.6%) in high correlation setting (overpriced) ·ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ‚ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ¶ In other words, moving from ρr,∆c= 0.00 to ρr,∆c= 0.45 should cause a CCAPM investor to double the sample average that we observe. For each set of parameter values, we first draw T = 156 iid realizations ∆zt growth shocks,̃ by our finite sample period, which contains 156 quarters. If we skip this step, the resulting error-in-variables problem is not large. But, since we can control everything about our laboratory setting, we follow best practices and try to remove all avoidable sources of error from our results. Finally, we simulate stock returns using the formula below: ·ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ‚ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ¶ CCAPM-implied expected excess return CCAPM-implied expected excess return increase the expected excess return he would demand for holding an asset from zero to roughly iid∼ N(0,1). Then, we orthogonalize these 156 random draws with respect to the realized set of 156 consumption ∆zt denotes the consumption-growth shocks. We do this to avoid the error-in-variables problem caused ∆ct])/̂ def= (∆ct−̂E[∆ct])/̂ def= (∆zt−̂E[∆zt∣̃ Sd[∆zt∣̃ ∆ct], wherẽ Sd[∆ct] ∆ct √ rt= µr+σr×(ρr,∆c⋅̃ r,∆c⋅̃ ∆ct+ 1−ρ2 ∆zt) Because of the orthogonalization step, the resulting stock-return time series has a mean of exactly µr, a volatility of exactly σr, and a correlation with consumption growth of exactly ρr,∆c. We show participants a line labeled “stock market” representing the cumulative returns to investing $1 in this portfolio starting the first quarter of 1980 on a log scale. See Figure 1. For each set of parameter values, we run the simulation five using five different random-number seeds. 3 According to standard finance theory, markets fluctuate because investors are worried about not having enough money during particularly bad future states of the world. There are different kinds of bad states and each one is associated with a different risk factor, such as aggregate consumption growth. If an asset’s returns are less correlated with a risk factor, then the asset is more likely to have positive returns during that particular bad future state of the world. As a result, investors should be willing to pay more for this asset today. Because all investors recognize this risk-factor correlation and adjust their demand accordingly, in equilibrium this asset will then have lower expected returns going forward. Main Results 10

We know there are many such models that can match certain moments of the data (Lewellen et al., 2010), but for this sort of model to explain why markets fluctuate, investors must be following the above logic. To do so, people must be able to recognize an asset’s correlation with important risk factors. When asked, they should report a preference for assets with lower risk-factor correlations. Further, there should be evidence that they adjust their asset demand based on this information. This is not what we find in our survey results. We find no evidence that people trade the way textbook investors are assumed to trade, think the way textbook investors are assumed to think, or recognize the correlations textbook investors are assumed to recognize. 3.1 Investment Decisions We begin by exploring how investment decisions change as we vary the mean, volatility, and correlation with consumption growth of stock returns. To do so, we use data from the first part of our experiment to estimate the following regression: stockFractions,q= ˆ α +ˆβ ×means,q + ˆ γ ×volatilitys,q +ˆδ ×correlations,q+ˆ ?s,q The dependent variable, stockFractions,q, is the fraction of the $1,000 initial endowment that participant s invests in the stock market when answering the question q ∈ {1,...,10}. ˆ α regression residual. The variables means,q, volatilitys,q, and correlations,qrepresent the mean µr∈ {0.04,0.06,0.08}, volatility σr∈ {0.10,0.15,0.20}, and correlation with consumption our survey answers ten investment-decision questions, so our regressions involving financial professionals contain 529×10 = 5,290 participant×question observations, and our regressions observations. We estimate all t-statistics and p-values using standard errors clustered by participant. Tables 2a and 2b report the results of these regressions. Column (1) in each table regresses the fraction invested in the stock market on just the mean stock return and we find a coefficient of 2.61 for finance professionals and 3.62 for MTurkers. Both of these coefficient estimates are statistically significant at the 1% level. A coefficient of 2.61 implies that finance professionals increase the fraction they invest in stocks by 10.44% = (8%−4%)×2.61 in response to a move from the corresponds to a demand increase of 14.48% in response to the same shift. Finance professionals invested 55% of their endowment in the stock on average while MTurkers invested 58% of their denotes an estimated intercept;ˆβ, ˆ γ, andˆδ are estimated slope coefficients; and, ˆ ?s,qis the growth ρr,∆c∈ {0.00,0.15,0.30,0.45} used to simulate the stock returns. Each participant who takes involving the general population of MTurkers contain 322×10 = 3,220 participant×question low-mean regime, µr= 4%, to the high-mean regime, µr= 8%. The 3.62 coefficient for MTurkers 11

endowment. Thus, a 4% change in expected stock returns increases the proportion allocated to stocks by about 20% of the average for both participant pools. Column (2) in each table repeats this same regression using return volatility rather than average returns as the sole right-hand-side variable. Finance professionals have a coefficient of −0.41 while significant at the 1% level. These two regression coefficients both imply roughly a 5% increase in the fraction invested in the stock market in response to a 10% drop in annualized volatility—i.e., in response to a move from the high-volatility regime, σr= 20%, to the low-volatility regime, σr= 10%. changes in the mean and volatility of stock returns when making investment decisions. Moreover, they do so in a manner consistent with an investor who likes higher means and dislikes volatility. When examining correlations in Column (3), the results are quite different. This column repeats the regression using the correlation with consumption growth as the right-hand-side variable and shows that there is no measurable change in our participants’ behavior in response to a change in risk-factor correlations. For finance professionals, the coefficient on correlation is 0.01 with a t-statistic of 0.47. For MTurkers, the coefficient it −0.01 with a t-statistics of 0.51. The p-values statistically insignificant but also economically insignificant. In response to a shift in correlation of 0.45, the regression implies financial professionals would decrease their allocation to stocks by 0.45% = (0.45− 0)×−0.01∗ 100. For MTurk participants, the sign of the coefficient is in the wrong participant are simply not adjusting their demand in response to changes in the correlation between stock returns and consumption growth, the canonical macro risk factor found in the CCAPM and all models built on top of it. In column (4), we include all three of these right-hand-side variables in the same regression specification as shown above. The results are nearly identical to before. They show that participants adjust their demand in response to changes in the mean and volatility of returns but not to changes in the correlation of stock returns with economic growth. To examine whether these results are being driven by participant-specific attributes, we add participant fixed effects in column (5). Our point estimates on each of the three variables hardly changes from column (4). Another concern might be that participants are gradually changing their behavior over the course of our experiment as they get more familiar with the setup. To account for this, we introduce question-order fixed effects in column (6). Again, we find nearly identical results. Our point estimates on each of the three variables hardly changes from column (4) to column (6). Finally, column (7) adds both participant and question-order fixed effects. Again, the point estimates are almost unchanged. The results show that on average both participant pools strongly respond to means and MTurkers have a coefficient of −0.60. Again, both of these coefficient estimates are statistically Taken together, the results in columns (1) and (2) suggest that our participants do respond to for these two coefficients are 0.64 and 0.61, respectively. These point estimates are not only direction and suggests they would invest 0.45% in response to a 0.45 increase in correlation. Our 12

variances and ignore covariances. It remains possible that this obfuscates the behavior of a specific subset of participants that behaves differently. To address this concern, we re-estimate the coefficients in column (4) on subsets of our participant pools. We report these investment-decision results cut by participant characteristics in Figures 3a and 3b. Every subgroup of investors we examine exhibits similar behavior. Old and young participants; male and female participants; participants with incomes greater than $100k and those with incomes less than $100k; participants who think they invest wisely and those who do not think they invest wisely all ignore changes in correlation. We repeat the analysis for all participants who state that they own either stocks or mutual funds as for those who are likely more financially savvy. We also repeat the analysis on the subset of 144 finance professionals in our sample who stated that their job involved trading financial securities. These participants do not adjust their demand in response to changes in correlations, either. No matter how we cut our data, the same pattern emerges. Participants react to changes in average returns; they react to changes in return volatility; they ignore changes in correlations. We would like to emphasize that the correlation changes we explore in this paper, from ρr,∆c= 0.00 to 0.45, should be sufficient to generate large differences in participant behavior. As underpriced and have high demand for its shares when ρr,∆c= 0.00; whereas, they should see this ρr,∆c= 0.00 to ρr,∆c= 0.45 should cause a CCAPM investor to demand an additional 11% per year per year. While it would be possible to further extend the range of correlations in our study, to our knowledge, there is not a general belief that stock returns and consumption growth have a correlation close to one or that they are negatively correlated. Another benchmark worth considering is the magnitude of the shift in correlations required to ‘solve’ the well-known equity-premium puzzle (Mehra and Prescott, 1985). Standard measures of consumption growth based on nondurables and services have a correlation with the excess market returns of about 0.38. Savov (2011) measures consumption growth using garbage and finds a correlation of 0.54. This shift in correlations of 0.16 is large enough “to formally resolve the joint equity premium-risk-free rate puzzle.” In our paper, we show that participants are not adjusting their demand in response to a change in correlations that is roughly three times as large, 3×0.16 = 0.48 ≈ 0.45. Thus, investors in a textbook asset-pricing model should view the changes in discussed earlier, investors should view a mutual fund with a 6% per year average return as same 6%-per-year fund as overpriced and have low demand when ρr,∆c= 0.45. Moving from risk premium—a change that is roughly double the average excess return on the market, µr= 6% correlations we are showing to our participants as large and economically significant. 3.2 Economic Reasoning The results above suggest that participants are not adjusting their demand in response to changes in correlations with consumption growth, the canonical aggregate risk factor in standard asset-pricing 13

models. But, perhaps these results do not tell the whole story. Maybe participants are considering risk-factor correlations in a manner that is not captured by our experimental setup. To address this concern, we follow up on our investment-decision questions by asking participants about the economic reasoning behind their choices. We first ask participants whether they considered the relation between consumption growth and the stock-market returns when making their investment decisions. Figures 4a and 4b show that 64% of finance professionals and 84% of MTurkers said that they did. Next, we ask participants who said “Yes” to this question about how they were thinking about this relationship. e.g., were they comparing mean consumption growth to mean returns? Were they comparing the volatility of each time series? Were they looking at the correlation between consumption growth and stock returns like the CCAPM says they should? We find that only 37% of finance professionals and 48% of MTurkers consider the relationship between consumption growth and stock returns as a correlation. Finally, for those participants who report thinking about the correlation between consumption growth and stock returns, we ask about the way in which they were using this information to make their investment decisions. Only 11% of finance professionals and 8% of MTurkers report that they were investing more in the stock market when it was less correlated with consumption growth. These findings are stark. Most participants did not even consider the relation between the stock market and consumption growth, even after being given this precise numeric value and after being asked directly about it. And, of those who did consider this correlation, most did so in the opposite direction of what a standard asset-pricing model would suggest. If you were to randomly select a particpiant who thought about correlations, then 3 out of 4 times you would select a participant who tried to buy more stocks when stocks were more correlated with consumption growth. i.e., 3 of the 4 participants who claimed they were thinking about correlation with consumption growth would tell you that they were trying to hold more stocks precisely when stocks were a worse hedge against bad economic times. These results strongly suggest that the non-responsiveness of participants’ demand to changes in risk-factor correlations is not a statistical fluke. It simply reflects how they are thinking about their investment decisions. 3.3 Reporting Correlations If an investor from your favorite asset-pricing model were to peruse a popular financial-news source, he would be puzzled by the lack of information related to correlations with macroeconomic variables. An investor who cared about risk-factor correlations would clearly like this information to be presented as clearly as possible. Suppose that investors did not care only about correlations with consumption growth. Or, suppose instead that they cared about both correlations with consumption growth as well as correlations with slow moving habit or with estimates of long-run risks. There is no reason why such a correlation could not be widely reported. From a revealed preference 14

perspective, it seems likely that investors do not demand such information. While puzzling, it could be that real-world investors do care about risk-factor correlations in spite of the fact that they are not widely reported. We analyzed mutual-fund prospectuses to systematically document that this does not appear to be the case. A fund is required to report its investment objectives and risks in its prospectus regardless of how newsworthy these objectives and risks are. Funds also have discretion to highlight a variety of potential aspects of the fund. For example, Vanguard funds include a “plain talk” section in its prospectus that attempts to explain investing concepts or strategies using straightforward language. Thus, if a fund thought that its correlation with an aggregate risk factor was an important component of investors’ decision making, it could and should present information about this statistic in its prospectus. If a fund had correlations with macroeconomic risks that would make investors want to buy more of the fund, then it should say so its prospectus to drive flows. Yet, mutual funds do not report these numbers or discuss their correlations with macro risk factors in their prospectuses. e.g., the Vanguard 500 (VFIAX), which has nearly $500b in assets under management, stated its investment objective as tracking a benchmark index. It did not discuss any aggregate risk factors or its correlation with such variables. Under principle risks, the fund lists stock-market risk, “which is the chance that stock prices overall will decline”, as well as investment-style risk, “which is the chance that returns from large-capitalization stocks will trail returns from the overall stock market.” But, it never talks about exposure to aggregate risk factors. There is a further discussion of risks later in the prospectus, but this is largely related to volatility: “stock markets tend to move in cycles, with periods of rising prices and periods of falling prices.” Funds report a variety of statistics about their past performance such as fees, taxes, distributions, performance. e.g., the Vanguard 500 fund’s prospectus reports 171 numeric values in various just the tables and figures, with even more values in the text. None of these numbers corresponds to the correlation between the fund’s returns and anything that could be construed as a macro risk factor. We systematically reviewed the mutual-fund prospectuses of the the largest 25 US mutual funds, which jointly held about $3.7t at the time and summarize the results in Table 3. Funds were chosen based on share-class asset value from among all US open-ended funds listed on Morningstar Direct as of July 30, 2019. We examined each prospectus for five characteristics related to risk-factor correlations. Did a fund report a numerical value for the correlation between the fund’s performance and any macroeconomic variable? Did a fund graph its performance together with any macroeconomic variable? In the section on risks, did a fund list its return correlation with any macroeconomic variable other than the stock market itself? In the section on objectives, did a fund list an objective related to its correlation with a macroeconomic variable, such as hedging exposure to a macroeconomic variable? Finally, we search the text of each prospectus for the keywords “covary”, “covariance”, “correlate”, and “correlation”, counting the total number of times that any of 15

these words appears in the document. Table 3 confirms our that mutual-fund prospectuses lack any relevant information for understanding how a fund’s returns covary with aggregate risk factors. None of the funds report any numeric or graphical information relating their performance to macroeconomic fundamentals. In addition, no fund lists its correlation with macroeconomic outcomes as an investment risk. Similarly, no fund lists hedging an aggregate risk factor as an investment objective. Perhaps the closest is that some funds list reasons why the market may be volatile. For example, the Fidelity Contra Fund (FCNTX) warns its investors that “stock markets are volatile and can decline significantly in response to adverse issuer, political, regulatory, market, or economic developments.” Notably, these are descriptions of why there may be volatility in returns not of the fund’s correlation with the market and such variables. Our word search reveals that 22 of 25 funds fail to even use a word related to correlation or covariance in their prospectus. The three funds with prospectuses that do contain one of these words use them in a way that is wholly unrelated to macroeconomic risk, as they reference the fund’s tracking error or its relationship to derivative securities. Mutual fund prospectuses contain no evidence of correlation with aggregate risk factors being an important investment objective to their investors. 3.4 Parameter Recognition We have just seen that investors cannot find information about risk-factor correlations in the financial news or standard fund documents. So, if they are acting on risk-factor correlations, then they must be gathering these statistics from somewhere else. While clearly possible, it seems unlikely that many investors are downloading the relevant data and calculating the correlations themselves. If investors know the relevant risk-factor correlations, then this knowledge likely comes from viewing information in a graph. So, in the last part of our survey experiment, we explore whether our participants can identify such variation in correlation when presented graphically. We show participants pairs of graphs and ask them to identify which one was simulated using the higher parameter value. The macroeconomic data is the same across figures. Just as before, it is seasonally adjusted US GDP from 1980Q1 to 2018Q4. But, the stock returns shown in each graph are simulated using three different input parameters: the mean return, µr; the volatility of returns, σr; and, the correlation of returns with consumption growth, ρr,∆c. Two of these three parameters values are always the same in each pair of graphs that a participant sees. We set these common parameter values to the median value in our experiment. For the one parameter that is different, one of the graphs has the minimum value used in our experiment while the other has the maximum value. e.g., when examining differences in correlations, both graphs would have 6% per year mean returns and 15% per year return volatility, but one graph would have a correlation with consumption growth of 0.00 while the other would have a correlation of 0.45. See Figure 2. We ask 16

each participant two questions about each parameter: µr, σr, and ρr,∆c. Figure 5 summarizes our participants’ responses to these parameter-recognition questions. The y-axis in both panels depicts the probability that a participant correctly guesses which figure has the higher parameter value. The left-most bar shows that participants are proficient at recognizing differences in average returns. Finance professionals correctly identify the figure with higher mean 74% of the time, while MTurkers identify the figure with the higher mean 78% of the time. Similarly, the middle bar shows that participants are also proficient at recognizing differences in return volatility. Finance professionals choose the correct figure 77% of the time, and MTurkers do so 78% of the time. In both instances, participants significantly outperform the random-guess benchmark of getting the answer right 50% of the time. This suggests that people are generally able to identify differences in means and volatilities when presented this information in a graph. The right-most bar in Figure 5, however, shows that the same cannot be said for correlations. Both finance professionals and MTurkers guess which figure has the higher correlation exactly 50% of the time. i.e., it is almost as if they were just randomly guessing. Recall that one of the figures shows a return time series that is uncorrelated with consumption growth, ρr,∆c= 0.00, while the This is an economically large change in correlations. The fact that our participants cannot recognize such a change suggests that investors are generally unaware of risk-factor correlations. Furthermore, this result calls into question the idea of solving the equity-premium puzzle by showing that correlations are being mis-measured. Existing proposals of this sort, such as Savov (2011), suggest measurement errors on the order of 0.16. It is unlikely that we have been mis-measuring the correlation by an amount larger than 0.45. We repeat this analysis for various subsets of investors in Figures 6a and 6b. While there are subtle variations across groups, these differences seem to be noise rather than a systematic difference in perceptual abilities across groups. Not a single one of the subsets that we examine is able to recognize which figure has the higher correlation at a rate better than chance. Finance professionals are no better at recognizing correlations than the general population of MTurkers, and Figure 6a shows that the subset of finance professionals whose job involves trading financial securities are also unable to identify such correlations. We would like to emphasize that this last part of our survey experiment investigates participants’ visual perceptiveness not their financial literacy. Put differently, asking participants which figure has the higher correlation does not test their ability to make financial decisions. It tests their ability to perceive a pattern in a graph. After seeing the results in parts #1 and #2 of our experiment, you might still think it is the case that the marginal investor in real-world markets has better financial acumen than the high-income finance professionals in our survey. But, it is unclear why this marginal investor would have better visual acuity. other figure shows a return time series that has ρr,∆c= 0.45 correlation with consumption growth. 17

The underlying assumption in standard finance theory is that the right way to model investors’ risk-return tradeoff is to use risk-factor correlations. However, correlations almost never get reported in the financial news or in important fund documents. And, participants cannot identify large differences in correlations when they see them in a graph. Together, these two facts call into question whether correlations should be used as the key input to standard asset-pricing models. Researchers usually assume that investors study historical data and use this information to infer risk-factor correlations. Our results suggest that this is not likely to be the case. If a researcher wants to propose a model where market fluctuations are driven by investors’ perceptions of risk-factor correlations, then to support this model a researcher needs to show that investors a) recognize the right risk factors, b) think about these risk factors in the way the model suggests, and c) adjust their demand based on this reasoning. Just proposing the model and showing that it could “quantitatively account for asset-pricing facts” if investors were worried about these risk factors is not enough. 4 Our experiment suggests that people do not adjust their portfolio holdings in response to changes in risk-factor correlations, they do not think about an asset’s risk-factor correlations in a manner consistent with standard asset-pricing theory, and they cannot recognize economically large differences in correlations when shown graphically. In this section, we address several potential concerns that naturally arise when using survey data to study financial markets. We relate our results to recent work on survey evidence about investor beliefs. And, we outline what these results imply about how financial economists should be extending asset-pricing models. Discussion 4.1 Potential Concerns There are several common concerns about using survey evidence to study financial markets. And, we designed our experiment to mitigate these concerns. One concern is that participants behave differently in the lab than they do in other settings. Maybe they are ignoring risk-factor correlations in our survey experiment but acting on them in real life? There are two reasons why this concern is unlikely to explain our results. First, we designed our experiment to make it as easy as possible for the participants to follow the standard asset-pricing logic. We removed any superfluous information. We presented difficult-to-find information about correlations in a straightforward manner. We only examined participants who passed a comprehension test. If anything, experimenter-demand effects should make participants more likely to follow the logic of standard asset-pricing models (Schwarz, 1999). Second, when we examine our results, we see evidence that participants follow the standard logic when it comes to means and volatilities. They only deviate from what a textbook investor would do when it comes to risk-factor correlations. When examining their investment decisions, we find that 18

participants make choices that reflect the mean and variance of returns. They just do not respond to changes in risk-factor correlations. Similarly, when we ask them about their thought process, participants report trying to increase their demand for stocks when the mean return is higher or when the variance is lower. They just do not report thinking about risk-factor correlations. And, participants are able to spot the time series with the higher mean return and the time series with the higher return volatility. They are just unable to identify differences in correlations. We want to be very clear about how far you should take our results. We are not trying to claim that our precise point estimates apply to every real-world investor. It is not clear that every finance professional you meet will react to a 1% increase in average stock returns by tilting his portfolio 2.61%-points more towards stocks. However, due to the extreme nature of participants’ deviations from textbook asset-pricing logic, we strongly believe that our main qualitative finding—i.e., that participants wholly disregard risk-factor correlations—does apply to real-world investors. You might worry that our participants do not look like real-world investors. We directly address this concern by soliciting responses from finance professionals. Some of the participants in this pool even report that their job involves trading financial securities. Thus, our participant pool cannot be entirely dissimilar from investors in the real-world. Even if you believe our participants are not important traders, it is still noteworthy that unimportant traders think nothing like the representative investor in a standard asset-pricing model. If we are going to use a model based on risk-factor correlations to describe how assets are priced, then the model needs to explain why the views of such investors are not reflected in the representative investor. Our participant pool is missing some important market participants, such as ultra high net-worth individuals or finance-industry executives. While it is true that ultra high net-worth individuals or finance-industry executives might play a special role in real-world markets, they do not play any such role in standard asset-pricing models. The CCAPM (Lucas, 1978) studies a representative investor, and so do popular models built on top of the CCAPM framework (Campbell and Cochrane, 1999; Bansal and Yaron, 2004; Rietz, 1988; Barro, 2006). If we are not surveying the right subset of investors, then financial economists are not modeling the right subset of investors, either. The fact that participants act on, think about, and recognize changes in the mean and volatility of stock returns also addresses concerns about whether our survey participants have been properly incentivized. It is certainly true that the financial stakes are higher in real-world financial markets than in our survey experiment. But, if low-powered incentives are distorting participants’ responses to questions about risk-factor correlations, then why are they not distorting participants’ responses to questions about the mean and volatility of returns? The facts that participants do respond to changes in the mean and volatility of returns indicates that our measurement apparatus is working; our experimental setup would be capable of capturing participant responses to risk-factor correlation if they existed. 19

A final concern is that the Friedman critique makes knowing what investors are thinking irrelevant. In its original form, the Friedman critique was not a critique at all. It was an explanation for why markets might move in more complicated ways than a researcher could understand.2 However, this logic has been broadly interpreted to imply that a researcher can ignore how market participants recognize, think about, and act on information when forming their models. The central purpose of writing down such models is to explain the part of market fluctuations that we do understand to allow for counterfactual analysis. This requires accurately modeling which information real-world investors use. 4.2 For a risk-based explanation to price movements to be true, market participants must be aware of the risk. Thus, when a researcher proposes a model for why markets fluctuate based on a new risk factor, he should be able to provide evidence that investors recognize, think about, and act on an asset’s return correlation with this risk factor. Conducting a survey experiment like we have done for the consumption-growth risk factor used by the CCAPM is a straightforward way to do this and our paper provides a road map for doing so. We do not argue that investors never use financial markets to hedge their exposure to particular risks. Such a claim is obviously false. For example, “commodity futures markets have had a long history of assisting commodity producers to hedge their commodity price risks (Cheng and Xiong, 2014).” Further, “exchange rates are a major source of uncertainty for multinationals (Jorion, 1990)” and FX forward markets exist so firms can hedge this risk. Likewise, “sovereign CDS contracts function as insurance contracts that allow investors to buy protection against the event that a sovereign defaults on or restructures its debt (Longstaff et al., 2011).” In these examples, changes in future risk exposure affect the price of each contract today. In all of these examples, if someone were to ask an investor about his decisions, he would likely be able to state the specific risk he was concerned about, consistent with the logic of standard asset-pricing models. For example, investors trading oil-futures are aware that the price is determined by exposure to oil-price risk. When Southwest airlines was asked about why they were active in the oil-futures market, they stated that the company had “loaded up years ago on hedges against higher fuel prices.”3That is why oil-price risk is priced and market participants agree on this risk. There is a growing literature demonstrating the usefulness of using survey evidence to examine asset-pricing problems. As argued by Adam et al. (2018), “measurement of expectations allows researchers to consider alternatives to the rational expectations assumption in an empirically disciplined way.” Recent examples of this approach include, Amromin and Sharpe (2013) studies Identifying Relevant Risks 2He states: “In the same way that a pool shark can make trick shots even if he does not understand the underlying physics, markets can behave in ways that no researcher has yet modeled.” 3Bailey, J. “Southwest Airlines gains advantage by hedging long-term oil contracts.” New York Times. Nov 28, 2007. 20

how investors use economic conditions to inform their expectations about the stock market. Armona et al. (2018) looks at how home buyers form expectations about house-price appreciation based on past experience. Greenwood and Shleifer (2014) demonstrate that investors form expectations by over-extrapolating past market performance. And, Giglio et al. (2019) studies how the trading behavior of institutional investors is related to these investors’ expectations about future returns as well as their past market experience. These papers measure expectations of economic variables and show the relationship between these variables and financial decisions.4 While valuable for a number of important questions, researchers who want to test a specific asset pricing model should not stop here. Therefore, it is incumbent on a researcher who proposes a new model to provide evidence that investors consider the relevant mechanism. Risk based models all include the testable assumption that participants are aware of and react to a specific risk. While each model will require a tailored approach, our paper provides a case study for testing this assumption. A key feature of our approach is that we use multiple methodologies to provide converging evidence, as any given method could be flawed. For example, directly asking participants about their agreement with certain statements may be problematic because participants may not have direct access to their thoughts (Nisbett and Wilson, 1977) or because their responses may be subject to experimenter demand in which participants respond to the survey based on beliefs about how the experimenter would want them to respond rather than based on their true beliefs (Orne, 1962). In particular, we examine revealed preferences in addition to asking direct questions about thought processes. Furthermore, we demonstrate that investors are aware of and can identify model parameters. If multiple methodologies yield consistent evidence, this should be viewed as fairly strong support for the underlying mechanism.5 4.3 Implications for Model Development We have known for a long time that the CCAPM does not fit key asset-pricing facts (e.g., Mehra and Prescott, 1985). The results in this paper shed light on how financial economists should respond to these failings. The current approach to addressing these problems is based on introducing new risk factors to the CCAPM. The basic idea is that, if exposure to consumption growth cannot entirely explain why markets fluctuate, then maybe exposure to consumption growth and an additional state variables can. These theories work by amplifying the importance of seemingly small differences in risk-factor correlations. For example, Bansal and Yaron (2004) give the representative investor Epstein-Zin 4Choi and Robertson (2018) is a noteworthy exception which surveys individuals, asking them about the importance of the intuition underlying leading academic theories for financial decisions. 5While this is specific to risk-based models, there are implications for other models as well. You cannot ask someone directly whether they have a specific bias, if a behavioral model is built on such a bias, the paper should provide direct evidence that investors exhibit such a bias. 21

preferences so that he cares a lot about an asset’s exposure to small slow-moving changes in average consumption growth. Markets are populated by intelligent forward-looking investors, so it is not unreasonable to think that investors might be noticing subtle changes in future market conditions that researchers are unaware of. The key insight in this paper is that this is a falsifiable claim. If small changes in risk-factor correlations are the reason why markets fluctuation, then investors must be able to notice them. Investors should think about these risk-factor correlations as dictated in the model and their portfolio holdings should reflect this reasoning. We find no evidence to support these requirements. Suppose our experiment had produced the exact opposite results—that participants cared deeply about the correlation between stock returns and consumption growth when making their investment decisions. Further, suppose they reported thinking about an asset’s correlation with consumption growth as suggested by the CCAPM, and that they were able to identify graphical differences in correlations. These results would not solve the CCAPM’s empirical shortcomings, but they would support the current approach to addressing them. They would suggest that investors care about correlations, but in a more complicated way than is captured by the CCAPM. Thus, models introducing new risk-factor correlations to the CCAPM, such as habit and long-run risk, would have a strong foundation to build on. This is not what we find in our study which means that even if models like habit formation and long-run risk are able to “quantitatively account for asset-pricing facts”, they are unlikely to be the explanation for why markets actually fluctuate. Importantly, such models can still make important normative prescriptions. For example, it is likely the case that more people should be hedging idiosyncratic shocks to their own consumption stream, even if they do not. Demonstrating to investors a behavior that they should adopt could have important welfare implications. However, the results in our paper suggest a standard asset-pricing model in the spirit of the CCAPM cannot explain how assets are actually priced. 5 Standard asset-pricing models study a representative investor whose chief concern is not having enough money in bad times. An asset’s equilibrium price in one of these models is determined by its suitability as a hedge against this outcome. Thus, an asset that is less correlated with consumption growth should have a higher price today because holding this asset provides better insurance against being poor in bad future states of the world. The canonical CCAPM contains just this one risk factor—aggregate consumption growth. But, a similar logic holds in more complex models with multiple risk factors. e.g., in the Campbell and Cochrane model of habit formation, agents like assets that are less correlated with consumption growth due to both the original hedging motive and the knowledge that they will be more risk averse in bad times, too. Standard finance theory assumes that investors a) recognize an asset’s correlation with important risk factors, b) prefer assets with Conclusion 22

lower risk-factor correlations, and c) adjust their portfolio holdings accordingly. In this paper, we survey people about their portfolio decisions using an experiment where we can manipulate the correlation between asset’s returns and consumption growth. We found no evidence to support any of these three assumptions. Participants do not adjust their portfolio holdings in response to changes in correlation with consumption growth. 91% of participants report ignoring correlations or using them in a manner inconsistent with standard finance theory. Participants are simply unable to recognize economically large differences in correlations when presented graphically. In short, we find no evidence that our participants trade the way textbook investors are assumed to trade, think the way textbook investors are assumed to think, or recognize the correlations textbook investors are assumed to recognize. Our findings present a challenge for standard finance theory. They suggest that we, as a field, have been trying to explain asset prices by adding complexity to the wrong benchmark model. Going forward, when a researcher proposes a new risk-factor based explanation for asset-pricing facts, we should ask for evidence that investors think along the lines of the representative investor in the model. Showing that the model can “quantitatively account for asset-pricing facts” assuming that investors worry about the particular risks is not enough. Financial economists typically apply a strict hierarchy to asset-pricing explanations. If there is a risk-based explanation for an asset-pricing phenomenon, this explanation is viewed as likely correct even if there are other more parsimonious models. Our results cast doubt on this hierarchy. They suggest that researchers should be much more skeptical of risk-based explanations in the absence of supporting evidence that investors follow the logic of the model. 23

References Adam, K., D. Matveev, and S. Nagel (2018). Do survey expectations of stock returns reflect risk-adjustments? Working Paper. Amromin, G. and S. Sharpe (2013). From the horse’s mouth: economic conditions and investor expectations of risk and return. Management Science. Ang, A. (2014). Asset management: a systematic approach to factor investing. Oxford University Press. Armona, L., A. Fuster, and B. Zafar (2018). Home-price expectations and behavior: evidence from a randomized-information experiment. Review of Economic Studies. Bansal, R. and A. Yaron (2004). Risks for the long run: a potential resolution of asset-pricing puzzles. Journal of Finance. Barber, B. and T. Odean (2007). All that glitters: the effect of attention and news on the buying behavior of individual and institutional investors. Review of Financial Studies. Barro, R. (2006). Rare disasters and asset markets in the twentieth century. Quarterly Journal of Economics. Bossaerts, P. and C. Plott (2004). Basic principles of asset-pricing theory: evidence from large-scale experimental financial markets. Review of Finance. Campbell, J. (2003). Consumption-based asset pricing. Handbook of the Economics of Finance. Campbell, J. and J. Cochrane (1999). By force of habit: a consumption-based explanation of aggregate stock-market behavior. Journal of Political Economy. Casler, K., L. Bickel, and E. Hackett (2013). Separate but equal? a comparison of participants and data gathered via Amazon’s MTurk, social media, and face-to-face behavioral testing. Computers in Human Behavior. Cheng, I.-H. and W. Xiong (2014). Financialization of commodity markets. Annual Review of Financial Economics. Choi, J. and A. Robertson (2018). What matters to individual investors? Evidence from the horse’s mouth. Working Paper. Cochrane, J. (2001). Asset pricing. Princeton University Press. Cochrane, J. (2017). Macro-finance. Review of Finance. Di Maggio, M., A. Kermani, and K. Majlesi (2018). Stock-market returns and consumption. Working Paper. Enke, B. and F. Zimmermann (2017). Correlation neglect in belief formation. Review of Economic Studies. 24

Eyster, E. and G. Weizsacker (2016). Correlation neglect in portfolio choice: lab evidence. Working Paper. Friedman, M. (1953). Essays in positive economics. University of Chicago Press. Frydman, C., S. Hartzmark, and D. Solomon (2017). Rolling mental accounts. Review of Financial Studies. Giglio, S., M. Maggiori, J. Stroebel, and S. Utkus (2019). Five facts about beliefs and portfolios. Working Paper. Greenwood, R. and A. Shleifer (2014). Expectations of returns and expected returns. Review of Financial Studies. Harris, L., S. Hartzmark, and D. Solomon (2015). Juicing the dividend yield: mutual funds and the demand for dividends. Journal of Financial Economics. Hartzmark, S. (2014). The worst, the best, ignoring all the rest: the rank effect and trading behavior. Review of Financial Studies. Hartzmark, S. and D. Solomon (2017). The dividend disconnect. Journal of Finance. Hartzmark, S. and A. Sussman (2018). Do investors value sustainability? a natural experiment examining ranking and fund flows. Working Paper. Hauser, D. and N. Schwarz (2016). Attentive Turkers: MTurk participants perform better on online attention checks than do subject-pool participants. Behavior Research Methods. Huber, J., S. Palan, and S. Zeisberger (2017). Does investor risk perception drive asset prices in markets? Experimental evidence. Working Paper. Jennings, D., T. Amabile, and L. Ross (1982). Informal covariation assessment: data-based vs. theory-based judgements. Working Paper. Jorion, P. (1990). The exchange-rate exposure of US multinationals. Journal of Business. Kallir, I. and D. Sonsino (2009). The neglect of correlation in allocation decisions. Southern Economic Journal. Kroll, Y. and H. Levy (1992). Further tests of the separation theorem and the capital asset-pricing model. American Economic Review. Kroll, Y., H. Levy, and A. Rapoport (1988). Experimental tests of the separation theorem and the capital asset-pricing model. American Economic Review. Levy, G. and R. Razin (2015). Correlation neglect, voting behavior, and information aggregation. American Economic Review. Lewellen, J., S. Nagel, and J. Shanken (2010). A skeptical appraisal of asset-pricing tests. Journal of Financial Economics. 25

Longstaff, F., J. Pan, L. Pedersen, and K. Singleton (2011). How sovereign is sovereign credit risk? American Economic Journal: Macroeconomics. Lucas, R. (1978). Asset prices in an exchange economy. Econometrica. Matthies, B. (2018). Biases in the perception of covariance. Working Paper. Mehra, R. and E. Prescott (1985). The equity premium: a puzzle. Journal of Monetary Economics. Merton, R. (1973). An intertemporal capital asset-pricing model. Econometrica. Nisbett, R. and T. Wilson (1977). Telling more than we can know: verbal reports on mental processes. Psychological Review. Odean, T. (1998). Are investors reluctant to realize their losses? Journal of Finance. Orne, M. (1962). On the social psychology of the psychological experiment: with particular reference to demand characteristics and their implications. American Psychologist. Paolacci, G. and J. Chandler (2014). Inside the Turk: understanding Mechanical Turk as a participant pool. Current Directions in Psychological Science. Rietz, T. (1988). The equity risk-premium: a solution. Journal of Monetary Economics. Savov, A. (2011). Asset pricing with garbage. Journal of Finance. Schwarz, N. (1999). Self-reports: how the questions shape the answers. American Psychologist. Ungeheuer, M. and M. Weber (2018). The perception of dependence, investment decisions, and stock prices. Working Paper. 26

A Figures Figure 1. Sample Investment-Decision Question. This figure shows a sample question, which was given to participants in the first part of our survey experiment. The question investigates how participants’ portfolio choices are affected by an asset’s correlation with economic growth. 27

Figure 2. Sample Parameter-Recognition Question. This figure shows a sample question, which was given to participants in the third part of our survey experiment. The question investigates whether participants can recognize large differences in correlations when shown graphically. 28

Finance Professionals Age < 40 Income < $100k “I invest wisely” Owns a stock or mut fund “I’m a trader” Is male 5 4 3 mean FALSE FALSE 2 TRUE TRUE 1 0 0.00 volatility Coefficient -0.25 -0.50 -0.75 0.06 29 correlation 0.04 0.02 0.00 -0.02 Figure 3a. Investment Decisions by Participant Characteristics. This figure reports regression results corresponding to column (4) in Table 2a. From top to bottom, each set of three bars represents the slope coefficientsˆβ, ˆ γ, andˆδ in the regression stockFractions,q = ˆ α +ˆβ × means,q+ ˆ γ × volatilitys,q+ˆδ × correlations,q+ ˆ ?s,q. professionals. Opaque bars are significant at the 5% level using standard errors clustered by participant. Transparent bars are insignificant. Blue bars denote positive values. Red bars denote negative values. The horizontal dotted gray lines correspond to coefficient values from Table 2a column (4). Regressions are estimated using data on the participant pool containing 529 finance

MTurkers Age < 40 Income < $100k “I invest wisely” Owns a stock or mut fund “I’ve worked in finance” Is male 4 3 mean 2 FALSE FALSE TRUE TRUE 1 0 0.0 volatility Coefficient -0.2 -0.4 -0.6 -0.8 30 correlation 0.02 0.00 -0.02 -0.04 -0.06 Figure 3b. Investment Decisions by Participant Characteristics, Ctd. This figure reports regression results corresponding to column (4) in Table 2b. From top to bottom, each set of three bars represents the slope coefficientsˆβ, ˆ γ, andˆδ in the regression stockFractions,q = ˆ α +ˆβ × means,q+ ˆ γ × volatilitys,q+ˆδ × correlations,q+ ˆ ?s,q. Opaque bars are significant at the 5% level using standard errors clustered by participant. Transparent bars are insignificant. Blue bars denote positive values. Red bars denote negative values. The horizontal dotted gray lines correspond to coefficient values from Table 2b column (4). Regressions are estimated using data on the participant pool containing 322 MTurkers.

Finance Professionals “I tried to invest more when average returns were higher.” 0.52 (277) “I thought about ave- rage stock returns.” 0.73 (384) “I tried to invest more when returns were less volatile.” 0.35 (184) “I thought about the vol- atility of stock returns.” 0.50 (263) “I tried to invest more when returns were less correlated with growth.” 0.11 (58) “I thought about this rela- tionship as a correlation.” 0.37 (195) “I considered the relationship between returns and growth.” 0.64 (338) 0 529 Figure 4a. Economic Reasoning. This figure tallies the answers given by the 529 finance professionals to our economic-reasoning questions in the second part of the survey. The fraction to the right of each bar represents the fraction of participants who responded “yes”. The number to the right of each bar in parentheses represents the corresponding number of participants. e.g., 37% of financial professionals said that the correlation between stock returns and economic growth played a role in their investment decisions, which corresponds to 0.37×529 = 195 participants. 31

MTurkers “I thought about ave- rage stock returns.” 0.77 (249) “I thought about the vol- atility of stock returns.” 0.53 (170) “I tried to invest more when returns were less correlated with growth.” 0.08 (27) “I thought about this rela- tionship as a correlation.” 0.48 (156) “I considered the relationship between returns and growth.” 0.84 (272) 0 322 Figure 4b. Economic Reasoning, Ctd. This figure tallies the answers given by the 322 MTurkers to our economic-reasoning questions in the second part of the survey. The fraction to the right of each bar represents the fraction of participants who responded “yes”. The number to the right of each bar in parentheses represents the corresponding number of participants. e.g., 48% of MTurkers said that the correlation between stock returns and economic growth played a role in their investment decisions, which corresponds to 0.48×322 = 156 participants. 32

Finance Professionals 0.75 0.77 0.74 Pr[correctGuess] 0.50 0.50 0.25 0.00 mean volatility correlation MTurkers 0.75 0.78 0.78 Pr[correctGuess] 0.50 0.50 0.25 0.00 mean volatility correlation Figure 5. Parameter Recognition. This figure tallies the answers given to our parameter-recognition questions in the third part of the survey. The top panel reports results for the participant pool containing 529 finance professionals. The bottom panel reports results for the participant pool containing 322 MTurkers. Height of each bar corresponds to fraction of participants who correctly guessed which of two time series presented to them had the higher value of the parameter listed on the x-axis. The exact fraction is reported in white at the top of each bar. e.g., finance professionals were able to correctly identify the time series with the higher average stock return 74% of the time. Random guessing would result in Pr[correctGuess] = 0.50. 33

Finance Professionals Age < 40 Income < $100k “I invest wisely” Owns a stock or mut fund “I’m a trader” Is male 0.75 mean 0.50 FALSE TRUE 0.25 0.00 Pr[correctGuess] volatility 0.75 0.50 0.25 0.00 34 correlation 0.75 0.50 0.25 0.00 Figure 6a. Parameter Recognition by Participant Characteristics. This figure tallies the answers given by different subsets of the finance-professionals participant pool to our parameter-recognition questions in the third part of the survey. Height of each bar corresponds to fraction of participant who correctly guessed which of two time series presented to them had the higher value of the parameter listed on the right side. Random guessing would result in Pr[correctGuess] = 0.50. Horizontal dotted gray lines are pooled sample estimates from top panel of Figure 5.

MTurkers Age < 40 Income < $100k “I invest wisely” Owns a stock or mut fund “I’ve worked in finance” Is male 0.75 mean 0.50 FALSE TRUE 0.25 0.00 Pr[correctGuess] volatility 0.75 0.50 0.25 0.00 35 correlation 0.75 0.50 0.25 0.00 Figure 6b. Parameter Recognition by Participant Characteristics, Ctd. This figure tallies the answers given by different subsets of the MTurker participant pool to our parameter-recognition questions in the third part of the survey. Height of each bar corresponds to fraction of participants who correctly guessed which of two time series presented to them had the higher value of the parameter listed on the right side. Random guessing would result in Pr[correctGuess] = 0.50. Horizontal dotted gray lines are pooled sample estimates from bottom panel of Figure 5.

B Tables Finance Professionals 529 Participants # Avg Sd Min Med Max Stock Fraction Age < 40 221 0.55 0.23 0.00 0.50 1.00 0.42 0.43 0.41 Is male 228 Income < $100k 217 Owns stock 404 Owns a mutual fund 334 Owns either 478 “I invest wisely” 389 0.74 0.76 0.63 0.90 “My job specifically involves... investing in financial securities” forecasting financial outcomes” analyzing financial statements” 144 111 170 0.27 0.21 0.32 MTurkers 322 Participants # Avg Sd Min Med Max Stock Fraction Age < 40 232 0.58 0.27 0.00 0.60 1.00 0.72 0.65 0.87 Is male 210 Income < $100k 281 Owns stock 160 Owns a mutual fund 112 Owns either 209 “I invest wisely” 155 0.48 0.50 0.35 0.65 Table 1. Summary Statistics. This table presents summary statistics describing the two participant pools in our survey experiment. The top panel shows results for the 529 finance professionals. The bottom panel shows results for the 322 MTurkers. The “Stock Fraction” row in each panel is computed at the participant×question level. This means computed at the participant level, which means using 529 and 322 observations for the finance-professionals and MTurker participant pools, respectively. “#” represents the number of participants who answered “Yes.” e.g., 404 of 529 finance professionals said they owned at least one stock. using 529×10 = 5,290 observations for the finance-professionals participant pool and 322×10 = 3,220 observations for the MTurker participant pool. All remaining rows are 36

Finance Professionals Dependent Variable: stockFraction (1) (22.09) (9.25) (2) (48.26) −0.41⋆⋆⋆ (3) (68.56) (4) (23.08) (9.27) (5.79) 0.01 (0.38) (5) 2.72⋆⋆⋆ −0.38⋆⋆⋆ 0.00 (0.08) (6) 2.62⋆⋆⋆ −0.42⋆⋆⋆ 0.01 (0.42) ✓ 0.04 (7) 2.73⋆⋆⋆ −0.38⋆⋆⋆ 0.00 (0.04) ✓ 0.41 0.40⋆⋆⋆ 2.61⋆⋆⋆ 0.62⋆⋆⋆ 0.55⋆⋆⋆ 0.46⋆⋆⋆ 2.62⋆⋆⋆ −0.42⋆⋆⋆ intercept mean (9.61) (5.39) ✓ 5,290 0.41 (9.30) (5.72) (9.65) (5.32) ✓ 5,290 volatility (5.67) correlation 0.01 (0.47) Participant FE Question FE 37 # Obs Adj. R2 5,290 0.03 5,290 0.01 5,290 0.00 5,290 0.04 5,290 Table 2a. Investment Decisions. This table examines the sample of finance professionals and shows how participants’ investment decisions vary with average stock returns, stock-return volatility, and stock correlation with consumption growth. The dependent variable is the fraction of a participant’s initial $1,000 endowment that was invested in stocks, stockFractions,q∈ [0,1]. Columns (5) and (7) include participant fixed effects, t-stats based on standard errors clustered by participant.⋆,⋆⋆, and⋆⋆⋆indicate statistically significant coefficient estimates at the 10%, 5%, and 1% levels, respectively. and columns (6) and (7) include question fixed effects. The numbers in parentheses are

MTurkers Dependent Variable: stockFraction (1) (13.87) (9.00) (2) (34.19) −0.60⋆⋆⋆ (3) (48.37) (4) (15.39) (8.94) (5.12) (0.28) (5) 3.71⋆⋆⋆ −0.61⋆⋆⋆ 0.01 (0.25) (6) 3.58⋆⋆⋆ −0.55⋆⋆⋆ (0.22) ✓ 0.06 (7) 3.72⋆⋆⋆ −0.59⋆⋆⋆ 0.01 (0.32) ✓ 0.50 0.36⋆⋆⋆ 3.62⋆⋆⋆ 0.67⋆⋆⋆ 0.58⋆⋆⋆ 0.45⋆⋆⋆ 3.58⋆⋆⋆ −0.56⋆⋆⋆ intercept mean (9.39) (6.02) ✓ 3,220 0.49 (9.03) (5.03) (9.50) (5.91) ✓ 3,220 volatility (5.35) −0.01 −0.01 −0.01 correlation (0.51) participant FE Question FE 38 # Obs Adj. R2 3,220 0.05 3,220 0.01 3,220 0.00 3,220 0.06 3,220 Table 2b. Investment Decisions, Ctd. This table examines the sample of MTurkers and shows how participants’ investment decisions vary with average stock returns, stock-return volatility, and stock correlation with consumption growth. The dependent variable is the fraction of a participant’s initial $1,000 endowment that was invested in stocks, stockFractions,q∈ [0,1]. Columns (5) and (7) include participant fixed effects, t-stats based on standard errors clustered by participant.⋆,⋆⋆, and⋆⋆⋆indicate statistically significant coefficient estimates at the 10%, 5%, and 1% levels, respectively. and columns (6) and (7) include question fixed effects. The numbers in parentheses are

TNA ($bil) Class Global Broad Category Group Mentions related to macro variables §Investment Risks Other macro correlation info? Mentions of correlat(ion|e) covar(iance|y) Share §Investment Objectives Fund Num. Graph Vguard 500 Idx, Adm Equity Equity Equity Equity Equity Equity Equity Equity Equity Equity 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 3 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No VFIAX VTSAX FXAIX VSMPX VGTSX VITSX VTSMX VIIIX VINIX VTPSX VTBIX FxdInc VBTLX FxdInc FCNTX AGTHX VWENX VTBNX FxdInc VTIAX AMECX ABALX RERGX DODGX VFFSX PIMIX FxdInc CAIBX VWIUX FxdInc 276 225 198 170 146 140 139 115 114 112 107 100 95 91 89 75 75 74 72 71 71 70 67 65 65 2821 483 814 198 814 382 814 814 229 229 382 182 229 122 196 105 182 382 111 150 162 71 483 128 105 68 3734 Vguard Tot Stock Mkt Idx, Adm Fidelity 500 Idx Vguard Tot Stock Mkt Idx, Instl Pl Vguard Tot Intl Stock Idx, Inv Vguard Tot Stock Mkt Idx, I Vguard Tot Stock Mkt Idx, Inv Vguard Instl Idx, Instl Pl Vguard Institutional Idx, I Vguard Tot Intl Stock Idx, Instl Pl Vguard Tot Bond Mkt II Idx, Inv Vguard Tot Bond Mkt Idx, Adm Fidelity Contrafund Amer Funds Gr Fund of Amer, A Vguard Wellington, Adm Vguard Tot Bond Mkt II Idx, I Vguard Tot Intl Stock Idx, Adm Amer Funds Incm Fund of Amer, A Amer Funds American Bal, A Amer Funds Europa Gr, R6 Dodge & Cox Stock Vguard 500 Idx, Instl Select PIMCO Income Instl Amer Funds Cap Income Bldr, A Vguard Interm-Term Tx-Ex, Adm Equity Equity Alloc 39 Equity Alloc Alloc Equity Equity Equity Alloc Table 3. Mutual-Fund Prospectuses. This table describes how the 25 largest US mutual funds talk about risk-factor correlations in their prospectuses. “Mentions of correlat(ion|e) covar(iance|y)” counts the number of times “correlation”, “correlate”, “covariance”, or “covary” appear in a fund’s prospectus. “Mentions related to macro variables” counts the number of times a fund mentions exposure to a macroeconomic variable in either the Investment Risks or Investment Objectives section of its prospectus. “Other macro correlation info?” is an indicator variable for whether a prospectus addressed the fund’s correlation with macroeconomic variables in some other way.