The authors examine the unique predictions of the frog-in-the-pan (FIP) hypothesis, which postulates a link between investor attention and momentum. The foundation of their hypothesis is a two-period illustrative model that portrays two types of investors: rational and FIP. The authors demonstrate that FIP investors underreact to information that arrives continuously and in small amounts because of the presence of a lower attention constraint, k, which represents a threshold that must be met before information is factored into a stock’s price. This delay results in a reduced short-term price impact but a stronger long-term effect. Over a six-month period, momentum increased from –2.07% in the discrete information portfolio to 5.94% in the continuous information portfolio, even after the authors adjusted for risk.

The view of the economy as being composed of two types of agents is a blow to the idea of efficient markets. Rational investors do not have an attention constraint and process all signals immediately, whereas FIP investors are influenced by the FIP hypothesis. For FIP investors, signals that are initially apparent and whose absolute values are below k are processed later, at a point when the signals are of sufficient magnitude (i.e., above k) that they provoke an action. Not surprisingly, the result is that each agent is valuing the stock differently at different points in time, with his or her respective demands determining the stock price. Although the existing literature recognizes the presence of an upper limit on the quantity of company information that investors can absorb, the introduction of a lower boundary is novel.

In the FIP model, momentum originates from the truncation of small signals below the k threshold, provided the signals are of the same sign (positive/negative) as the total period return. Furthermore, a higher k parameter implies that FIP investors are more likely to truncate signals and delay their incorporation into the stock price, leading to increased momentum. Encouragingly, information discreteness (ID) is equally adept at explaining the momentum of past winners and that of past losers. But the authors do find momentum to be weaker among stocks with high media coverage, where information is large and discrete (i.e., low ID) and thus promptly accounted for.

The authors’ work adds to the growing limited-attention literature that recognizes the need for information to attract investor attention. Although prior researchers have acknowledged that the quantity of information influences stock prices, they have not distinguished between the continuous and discrete arrival of information, which is the focus of the authors’ work. Finally, the authors’ conclusions complement the nascent literature in which the media’s role in asset pricing is explored.

The authors obtain return data for active companies from the Center for Research in Security Prices (CRSP) and firm-level accounting data from Compustat. Firms with negative book values are eliminated. Starting dates for the sample period range from 1927 to 1992 (the analysis covers firms through 2007), depending on the availability of firm data. Data on media coverage are obtained from Factiva.

To create a proxy for information discreteness, the authors construct a two-period descriptive model that assumes the existence of two types of investors: rational and FIP. FIP investors are assumed to have a minimum attention threshold, k, below which firm information is processed gradually over time. It is this truncation of a firm’s signals that is the source of the momentum.

The base-case ID proxy for a firm is determined by subtracting the percentage of positive daily returns from the percentage of negative daily returns for the sample period multiplied by the sign of a firm’s cumulative period return over the previous 12 months. The authors control for the magnitude of returns by equally weighting each return. Scores range from +1 to –1, with negative ID scores representing continuous information and positive ID scores representing discrete information. Derivatives of the base-case model are used to control for the magnitude of daily returns and the impact of zero-return days.

The authors present exhaustive evidence to confirm that the FIP hypothesis links investor attention to momentum. For example, they conduct tests for risk, tax-motivated transactions, turnover, idiosyncratic volatility, analyst coverage, and return consistency as well as several regressions that control for an array of firm characteristics (e.g., size, book to market). All tests confirm that intra-period returns remain an important predictor of price momentum.

I am impressed with the authors’ capacity for developing quantitative measures that proxy for investor behavior. The obvious one is the thrust of the research—the k factor proxy for investor limited attention—but there are others: investors’ indifference to disconfirming company information, which proxies for prolonged momentum, and investors who are more likely to initiate sell orders for previous winners than previous losers, which proxies for the disposition effect. To me, investor underreaction to select stimuli is an area with much potential.