Maxing out: Stocks as lotteries and the cross-section of expected returns☆
Introduction
What determines the cross-section of expected stock returns? This question has been central to modern financial economics since the path breaking work of Sharpe (1964), Lintner (1965), and Mossin (1966). Much of this work has focused on the joint distribution of individual stock returns and the market portfolio as the determinant of expected returns. In the classic capital asset pricing model (CAPM) setting, i.e., with either quadratic preferences or normally distributed returns, expected returns on individual stocks are determined by the covariance of their returns with the market portfolio. Introducing a preference for skewness leads to the three-moment CAPM of Kraus and Litzenberger (1976), which has received empirical support in the literature as, for example, in Harvey and Siddique (2000) and Smith (2007).
Diversification plays a critical role in these models due to the desire of investors to avoid variance risk, i.e., to diversify away idiosyncratic volatility, yet a closer examination of the portfolios of individual investors suggests that these investors are, in general, not well-diversified.3 There may be plausible explanations for this lack of diversification, such as the returns to specialization in information acquisition (Van Nieuwerburgh and Veldkamp, 2010), but nevertheless this empirical phenomenon suggests looking more closely at the distribution of individual stock returns rather than just co-moments as potential determinants of the cross-section of expected returns.
There is also evidence that investors have a preference for lottery-like assets, i.e., assets that have a relatively small probability of a large payoff. Two prominent examples are the favorite-longshot bias in horsetrack betting, i.e., the phenomenon that the expected return per dollar wagered tends to increase monotonically with the probability of the horse winning, and the popularity of lottery games despite the prevalence of negative expected returns (Thaler and Ziemba, 1988). Interestingly, in the latter case, there is increasing evidence that it is the degree of skewness in the payoffs that appeals to participants (Garrett and Sobel, 1999, Walker and Young, 2001), although there are alternative explanations, such as lumpiness in the goods market (Patel and Subrahmanyam, 1978). In the context of the stock market, Kumar (2009) shows that certain groups of individual investors appear to exhibit a preference for lottery-type stocks, which he defines as low-priced stocks with high idiosyncratic volatility and high idiosyncratic skewness.
Motivated by these two literatures, we examine the role of extreme positive returns in the cross-sectional pricing of stocks. Specifically, we sort stocks by their maximum daily return during the previous month and examine the monthly returns on the resulting portfolios over the period July 1962–December 2005. For value-weighted decile portfolios, the difference between returns on the portfolios with the highest and lowest maximum daily returns is −1.03%. The corresponding Fama-French-Carhart four-factor alpha is −1.18%. Both return differences are statistically significant at all standard significance levels. In addition, the results are robust to sorting stocks not only on the single maximum daily return during the month, but also the average of the two, three, four, or five highest daily returns within the month. This evidence suggests that investors may be willing to pay more for stocks that exhibit extreme positive returns, and thus, these stocks exhibit lower returns in the future.
This interpretation is consistent with cumulative prospect theory (Tversky and Kahneman, 1992) as modeled in Barberis and Huang (2008). Errors in the probability weighting of investors cause them to overvalue stocks that have a small probability of a large positive return. It is also consistent with the optimal beliefs framework of Brunnermeier, Gollier, and Parker (2007). In this model, agents optimally choose to distort their beliefs about future probabilities in order to maximize their current utility. Critical to these interpretations of the empirical evidence, stocks with extreme positive returns in a given month should also be more likely to exhibit this phenomenon in the future. We confirm this persistence, showing that stocks in the top decile in one month have a 35% probability of being in the top decile in the subsequent month and an almost 70% probability of being in one of the top three deciles. Moreover, maximum daily returns exhibit substantial persistence in firm-level cross-sectional regressions, even after controlling for a variety of other firm-level variables.
Not surprisingly, the stocks with the most extreme positive returns are not representative of the full universe of equities. For example, they tend to be small, illiquid securities with high returns in the portfolio formation month and low returns over the prior 11 months. To ensure that it is not these characteristics, rather than the extreme returns, that are driving the documented return differences, we perform a battery of bivariate sorts and re-examine the raw return and alpha differences. The results are robust to sorts on size, book-to-market ratio, momentum, short-term reversals, and illiquidity. Results from cross-sectional regressions corroborate this evidence.
Are there alternative interpretations of this apparently robust empirical phenomenon? Recent papers by Ang et al., 2006, Ang et al., 2009 contain the anomalous finding that stocks with high idiosyncratic volatility have low subsequent returns. It is no surprise that the stocks with extreme positive returns also have high idiosyncratic (and total) volatility when measured over the same time period. This positive correlation is partially by construction, since realized monthly volatility is calculated as the sum of squared daily returns, but even excluding the day with the largest return in the volatility calculation only reduces this association slightly. Could the maximum return simply be proxying for idiosyncratic volatility? We investigate this question using two methodologies, bivariate sorts on extreme returns and idiosyncratic volatility and firm-level cross-sectional regressions. The conclusion is that not only is the effect of extreme positive returns we find robust to controls for idiosyncratic volatility, but that this effect reverses the idiosyncratic volatility effect shown in Ang et al., 2006, Ang et al., 2009. When sorted first on maximum returns, the equal-weighted return difference between high and low idiosyncratic volatility portfolios is positive and both economically and statistically significant. In a cross-sectional regression context, when both variables are included, the coefficient on the maximum return is negative and significant while that on idiosyncratic volatility is positive, albeit insignificant in some specifications. These results are consistent with our preferred explanation—poorly diversified investors dislike idiosyncratic volatility, like lottery-like payoffs, and influence prices and hence future returns.
A slightly different interpretation of our evidence is that extreme positive returns proxy for skewness, and investors exhibit a preference for skewness. For example, Mitton and Vorkink (2007) develop a model of agents with heterogeneous skewness preferences and show that the result is an equilibrium in which idiosyncratic skewness is priced. However, we show that the extreme return effect is robust to controls for total and idiosyncratic skewness and to the inclusion of a measure of expected skewness as in Boyer, Mitton, and Vorkink (2010). It is also unaffected by controls for co-skewness, i.e., the contribution of an asset to the skewness of a well-diversified portfolio.
The paper is organized as follows. Section 2 provides the univariate portfolio-level analysis, and the bivariate analyses and firm-level cross-sectional regressions that examine a comprehensive list of control variables. Section 3 focuses more specifically on extreme returns and idiosyncratic volatility. Section 4 presents results for skewness and extreme returns. Section 5 concludes.
Section snippets
Data
The first data set includes all New York Stock Exchange (NYSE), American Stock Exchange (Amex), and Nasdaq financial and nonfinancial firms from the Center for Research in Security Prices (CRSP) for the period from January 1926 through December 2005. We use daily stock returns to calculate the maximum daily stock returns for each firm in each month as well as such variables as the market beta, idiosyncratic volatility, and various skewness measures; we use monthly returns to calculate proxies
Idiosyncratic volatility and extreme returns
While arguably MAX is a theoretically motivated variable, there is still a concern that it may be proxying for a different effect. In particular, stocks with high volatility are likely to exhibit extreme returns of both signs. Moreover, stocks with high maximum daily returns in a given month will also have high realized volatility in the same month, measured using squared daily returns, almost by construction. Ang et al., 2006, Ang et al., 2009 show that idiosyncratic volatility has a
Skewness and MAX
Our final empirical exercise is to examine the link, if any, between extreme positive returns and skewness in terms of their ability to explain the cross-section of expected returns. The investigation of the role of higher moments in asset pricing has a long history. Arditti (1967), Kraus and Litzenberger (1976), and Kane (1982) extend the standard mean-variance portfolio theory to incorporate the effect of skewness on valuation. They present a three-moment asset pricing model in which
Conclusion
We find a statistically and economically significant relation between lagged extreme positive returns, as measured by the maximum daily return over the prior month or the average of the highest daily returns within the month, and future returns. This result is robust to controls for numerous other potential risk factors and control variables. Of particular interest, inclusion of our MAX variable reverses the anomalous negative relation between idiosyncratic volatility and returns in Ang et al.,
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We would like to thank Yakov Amihud, Xavier Gabaix, Evgeny Landres, Orly Sade, Jacob Sagi, Daniel Smith, Jeff Wurgler, and seminar participants at the Cesaerea 6th Annual Conference, Arison School of Business, IDC; HEC, Paris; INSEAD; New York University; and Simon Fraser University for helpful comments.
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