Forecasting crashes: trading volume, past returns, and conditional skewness in stock prices

Abstract

We develop a series of cross-sectional regression specifications to forecast skewness in the daily returns of individual stocks. Negative skewness is most pronounced in stocks that have experienced (1) an increase in trading volume relative to trend over the prior six months, consistent with the model of Hong and Stein (NBER Working Paper, 1999), and (2) positive returns over the prior 36 months, which fits with a number of theories, most notably Blanchard and Watson's (Crises in Economic and Financial Structure. Lexington Books, Lexington, MA, 1982, pp. 295–315) rendition of stock-price bubbles. Analogous results also obtain when we attempt to forecast the skewness of the aggregate stock market, though our statistical power in this case is limited.

Introduction

Aggregate stock market returns are asymmetrically distributed. This asymmetry can be measured in several ways. First, and most simply, the very largest movements in the market are usually decreases, rather than increases – that is, the stock market is more prone to melt down than to melt up. For example, of the ten biggest one-day movements in the S&P 500 since 1947, nine were declines.¹ Second, a large literature documents that market returns exhibit negative skewness, or a closely related property, “asymmetric volatility” – a tendency for volatility to go up with negative returns.² Finally, since the crash of October 1987, the prices of stock index options have been strongly indicative of a negative asymmetry in returns, with the implied volatilities of out-of-the-money puts far exceeding those of out-of-the-money calls; this pattern has come to be known as the “smirk” in index-implied volatilities. (See, e.g., Bates, 1997; Bakshi et al., 1997; and Dumas et al., 1998.)

While the existence of negative asymmetries in market returns is generally not disputed, it is less clear what underlying economic mechanism these asymmetries reflect. Perhaps the most venerable theory is based on leverage effects (Black, 1976; Christie, 1982), whereby a drop in prices raises operating and financial leverage, and hence the volatility of subsequent returns. However, it appears that leverage effects are not of sufficient quantitative importance to explain the data (Schwert, 1989; Bekaert and Wu, 2000). This is especially true if one is interested in asymmetries at a relatively high frequency, e.g., in daily data. To explain these, one has to argue that intraday changes in leverage have a large impact on volatility – that a drop in prices on Monday morning leads to a large increase in leverage and hence in volatility by Monday afternoon, so that overall, the return for the full day Monday is negatively skewed.

An alternative theory is based on a “volatility feedback” mechanism. As developed by Pindyck (1984), French et al. (1987), Campbell and Hentschel (1992), and others, the idea is as follows: When a large piece of good news arrives, this signals that market volatility has increased, so the direct positive effect of the good news is partially offset by an increase in the risk premium. On the other hand, when a large piece of bad news arrives, the direct effect and the risk-premium effect now go in the same direction, so the impact of the news is amplified. While the volatility-feedback story is in some ways more attractive than the leverage-effects story, there are again questions as to whether it has the quantitative kick that is needed to explain the data. The thrust of the critique, first articulated by Poterba and Summers (1986), is that shocks to market volatility are for the most part very short-lived, and hence cannot be expected to have a large impact on risk premiums.

A third explanation for asymmetries in stock market returns comes from stochastic bubble models of the sort pioneered by Blanchard and Watson (1982). The asymmetry here is due to the popping of the bubble – a low-probability event that produces large negative returns.

What the leverage-effects, volatility-feedback, and bubble theories all have in common is that they can be cast in a representative-investor framework. In contrast, a more recent explanation of return asymmetries, Hong and Stein (1999), argues that investor heterogeneity is central to the phenomenon. The Hong-Stein model rests on two key assumptions: (1) there are differences of opinion among investors as to fundamental value, and (2) some – though not all – investors face short-sales constraints. The constrained investors can be thought of as mutual funds, whose charters typically prohibit them from taking short positions; the unconstrained investors can be thought of as hedge funds or other arbitrageurs.

When differences of opinion are initially large, those bearish investors who are subject to the short-sales constraint will be forced to a corner solution, in which they sell all of their shares and just sit out of the market. As a consequence of being at a corner, their information is not fully incorporated into prices. For example, if the market-clearing price is $100, and a particular investor is sitting out, it must be that his valuation is less than $100, but one has no way of knowing by how much – it could be $95, but it could also be much lower, say $50.

However, if after this information is hidden, other, previously more-bullish investors have a change of heart and bail out of the market, the originally more-bearish group may become the marginal “support buyers” and hence more will be learned about their signals. In particular, if the investor who was sitting out at a price of $100 jumps in and buys at $95, this is good news relative to continuing to sit on the sidelines even as the price drops further. Thus, accumulated hidden information tends to come out during market declines, which is another way of saying that returns are negatively skewed.

With its focus on differences of opinion, the Hong-Stein model has distinctive empirical implications that are not shared by the representative-investor theories. In particular, the Hong-Stein model predicts that negative skewness in returns will be most pronounced around periods of heavy trading volume. This is because – like in many models with differences of opinion – trading volume proxies for the intensity of disagreement. (See Varian, 1989; Harris and Raviv, 1993; Kandel and Pearson, 1995; and Odean, 1998a for other models with this feature.)

When disagreement (and hence trading volume) is high, it is more likely that bearish investors will wind up at a corner, with their information incompletely revealed in prices. And it is precisely this hiding of information that sets the stage for negative skewness in subsequent rounds of trade, when the arrival of bad news to other, previously more-bullish investors can force the hidden information to come out.

In this paper, we undertake an empirical investigation that is motivated by this differences-of-opinion theory. We develop a series of cross-sectional regression specifications that attempt to forecast skewness in the daily returns to individual stocks. Thus, when we speak of “forecasting crashes” in the title of the paper, we are adopting a narrow and euphemistic definition of the word “crashes,” associating it solely with the conditional skewness of the return distribution; we are not in the business of forecasting negative expected returns. This usage follows Bates (1991), Bates (1997), who also interprets conditional skewness – in his case, inferred from options prices – as a measure of crash expectations.

One of our key forecasting variables is the recent deviation of turnover from its trend. For example, at the firm level, we ask whether the skewness in daily returns measured over a given six-month period (say, July 1–December 31, 1998) can be predicted from the detrended level of turnover over the prior six-month period (January 1–June 30, 1998). It turns out that firms that experience larger increases in turnover relative to trend are indeed predicted to have more negative skewness; moreover, the effect of turnover is strongly statistically and economically significant.

In an effort to isolate the effects of turnover, our specifications also include a number of control variables. These control variables can be divided into two categories. In the first category are those that, like detrended turnover, capture time-varying influences on skewness. The most significant variable in this category is past returns. We find that when past returns have been high, skewness is forecasted to become more negative. The predictive power is strongest for returns in the prior six months, but there is some ability to predict negative skewness based on returns as far back as 36 months. In a similar vein, glamour stocks – those with low ratios of book value to market value – are also forecasted to have more negative skewness. (Harvey and Siddique (2000) also examine how skewness varies with past returns and book-to-market.) These results can be rationalized in a number of ways, but they are perhaps most clearly suggested by models of stochastic bubbles. In the context of a bubble model, high past returns or a low book-to-market value imply that the bubble has been building up for a long time, so that there is a larger drop when it pops and prices fall back to fundamentals.

The second category of variables that help to explain skewness are those that appear to be picking up relatively fixed firm characteristics. For example, it has been documented by others (e.g., Damodaran, 1987; Harvey and Siddique, 2000) that skewness is more negative on average for large-cap firms – a pattern that also shows up strongly in our multivariate regressions. We are not aware of any theories that would have naturally led one to anticipate this finding. Rather, for our purposes a variable like size is best thought of as an atheoretic control – it is included in our regressions to help ensure that we do not mistakenly attribute explanatory power to turnover when it is actually proxying for some other firm characteristic. Such a control might be redundant to the extent that detrending the turnover variable already removes firm effects, but we keep it in to be safe.

In addition to running our cross-sectional regressions with the individual-firm data, we also experiment briefly with analogous time-series regressions for the U.S. stock market as a whole. Here, we attempt to forecast the skewness in the daily returns to the market using detrended market turnover and past market returns. Obviously, this pure time-series approach entails an enormous loss in statistical power – with data going back to 1962, we have less than 70 independent observations of market skewness measured at six-month intervals – which is why it is not the main focus of our analysis. Nevertheless, it is comforting to note that the qualitative results from the aggregate-market regressions closely parallel those from the cross-sectional regressions in that high values of both detrended turnover and past returns also forecast more negative market skewness. The coefficient estimates continue to imply economically meaningful effects, although that for detrended turnover is no longer statistically significant.

While both the cross-sectional and time-series results for turnover are broadly consistent with the theory we are interested in, we should stress that we do not at this point view them as a tight test. There are several reasons why one might wish to remain skeptical. First, beyond the effects of turnover, we document other strong influences on skewness, such as firm size, that are not easily rationalized within the context of the Hong-Stein model, and for which there are no other widely accepted explanations. Second, even if innovations to trading volume proxy for the intensity of disagreement among investors, they likely capture other factors as well – such as changes in trading costs – that we have not adequately controlled for. Finally, and most generally, our efforts to model the determinants of conditional skewness at the firm level are really quite exploratory in nature. Given how early it is in this game, we are naturally reluctant to declare an unqualified victory for any one theory.

The remainder of the paper is organized as follows. In Section 2, we review in more detail the theoretical work that motivates our empirical specification. In Section 3, we discuss our sample and the construction of our key variables. In Section 4, we present our baseline cross-sectional regressions, along with a variety of sensitivities and sample splits. In Section 5, we consider the analogous time-series regressions, in which we attempt to forecast the skewness in aggregate-market returns. In Section 6, we use an option-pricing metric to evaluate the economic significance of our results. Section 7 concludes.