Why Does Cross-Sectional Analyst Coverage Incorporate Market-Wide Information?

1. Introduction

Sell-side analysts are among the most influential information producers in financial markets, playing an crucial role for both investors and managers. Numerous empirical studies have documented the impact of analyst attention (i.e., coverage) on investors’ decisions (e.g., [1,2]), as well as on corporate activities such as investment and financing, innovation (e.g., R&D), acquisition, and earnings management (e.g., [3,4,5,6]).

However, contrary to the conventional wisdom that security analysts specialize in collecting and disseminating firm-specific information, empirical findings indicate that analysts predominantly generate market-wide information for 25 countries [7,8]. It remains unclear why analyst reports reflects market-wide information rather than firm-specific information. To deeply understand the underling mechanism, we examine the empirical distribution of cross-sectional analyst coverage.

This paper illustrates that the empirical distribution of analyst coverage for China’s listed companies maintains an exponential structure in a given month from 2011 to 2020. Our findings are consistent in both the Shanghai and Hong Kong stock markets. Since the system information of exponential distribution can be fully characterized by the mean value, we expect that aggregate analyst coverage can help reduce total uncertainty for both investors and managers. This paper provides evidences that aggregate (or say average) analysts coverage, denoted by ${\lambda }^{-1}$, exhibits a strong negative predictive power for stock-market uncertainty.

Given that cross-sectional analyst coverage is exponentially distributed, a natural question is how the exponential structure occurs. Motivated by [9], who draw on the maximum entropy principle (MEP) to interpret the asymmetric Laplace-shaped distribution of Tobin’ Q. We utilize the MEP to derive the exponential distribution of analyst coverage. The central idea is that when analysts, who are constrained by limited average ability in producing information (i.e., the first-order moment), aim to maximize the amount of market-wide information, this objective can yield an exponential distribution. To the best of our knowledge, it has not been documented in the previous literature that cross-sectional analyst coverage follows an exponential law. Our study is the first to provide an explanation for why analyst coverage predominantly incorporates market-wide information.

Our findings have some important implications. Firstly, sell-side analysts primarily generate market-wide information. Secondly, almost all relevant studies claim that coverage proxies should be used in logarithmic form to mitigate the influence of outliers, however, our results suggest that the exponential structure would be distorted when logarithmic transformation is performed. Thirdly, because more than half of the firms have the similar coverage, future research should be extra cautious when using coverage proxies in cross-sectional and particularly in portfolio analyses.

The rest of this paper is organized as follows. Section 2 reports the empirical distribution of cross-sectional analyst coverage in Chinese stock markets. Section 3 provides a potential mechanism for the exponential law. In Section 4, we make concluding discussions.

2. Exponentially Distributed Cross-Sectional Analyst Coverage

Assume a random variable X is exponentially distributed, it can be written as,

$$\mathrm{Exp}\left[\lambda \right]:f\left[x\right]=\lambda {\mathrm{e}}^{-\lambda x}$$

(1)

where $x>0,\lambda >0$, $f\left[x\right]$ is the probability density function (PDF) with $\mathbb{E}\left[X\right]=\sqrt{\mathbb{D}\left[X\right]}={\lambda }^{-1}$. Empirically, it is better to work with the complementary cumulative distribution function i.e., CCDF: $\mathrm{P}[X>x]={\mathrm{e}}^{-\lambda x}$. The parameter $\lambda$ can be obtained by the slope of straight line on a semi-log scale: $\log \left(\mathrm{CCDF}\right)$ vs. x.

We examine the empirical distribution of cross-sectional analyst coverage in China’s stock markets. The sample consists of all common stocks listed on the Shanghai Stock Exchange (SSE) and the Hong Kong Stock Exchange (HK) from January 2011 to January 2020. The data are all from the Wind database.

2.1. Evidence from the Shanghai and Hong Kong Stock Markets

The monthly results are shown in Figure 1. Two things are evident. First, the CCDFs show the highly skewed structure of cross-sectional analyst coverage, and we can observe that the percent of firms with coverage below 10 is about 80%, i.e., a very high proportion of firms are covered by a small number of analysts, while certain firms have substantial coverage. Second, and most importantly, the empirical distributions are well fitted by Exp $\left[\lambda \right]$ on both linear (CCDF vs. x) and semi-log (log CCDF vs. x) scales, where exponential fits are denoted as blue and red lines, respectively.

One might be intrigued by the goodness of fit, we thus present the regression R² in each semi-log subplot. On average, the R² is as high as 98% for the period 2011-2020. Notably, we do not exclude any coverage data based on any firm characteristics, which ensures that our findings provide a clean and complete picture of attention allocation structure of sell-side analysts.

Furthermore, as shown in Figure 2, a consistent exponential pattern of cross-sectional analyst coverage is evident for the Hong Kong stock market (HK), indicating that our findings are consistent in both emerging and developed stock markets.

This paper argues that using coverage proxy to measure the amount of firm-specific information poses a serious inference problem. It is impossible that the amounts of information for firms with zero-coverage are exactly equal. After excluding firms with no coverage, $80\%$ of firms have almost the same small amount of analyst coverage. In any case, one cannot state that the majority of firms with approximate amount of coverage have similar firm-specific information environments.

Given the finding that sell-side analyst mainly generates market-wide information (see [7,8]), we go one step further and directly examine whether aggregate analyst coverage is related to stock-market uncertainty. In this paper, aggregate analyst coverage is denoted by the inverse of parameter $\lambda$, since ${\lambda }^{-1}$ fully characterizes the mean value of cross-sectional analyst coverage. The finding has many implications, and the most critical one is that aggregate analyst coverage exhibits a negative predictive power for stock-market uncertainty. If analysts have no advantage over insiders in generating firm-specific information, focusing on providing market-wide information becomes an inevitable choice. Next, we test the hypothesis that aggregate analyst coverage (denoted by ${\lambda }^{-1}$) can help reduce the expected total uncertainty.

2.2. Predicting Stock-Market Uncertainty

We examine the forecasting power of aggregate analyst coverage denoted by ${\lambda }^{-1}$ for stock-market uncertainty based on the following time-series predictive regression,

$$U_{t,t+h}^j=\alpha +\delta {\lambda }_t^{-1}+\mathrm{Controls}+{\epsilon }_{t,t+h}$$

(2)

where $U_{t,t+h}^j$ is the stock-market uncertainty over the prediction horizon h, where $h=1,6$ and 12 months, and j denotes two uncertainty proxies, which are market-level cash-flow volatility (i.e., CFV) and investor search volume (i.e., Search). ${\lambda }_t^{-1}$ is the inverse of the parameter of exponentially distributed analyst coverage in month t, which captures continuous-varying marker information flows. We also control for a linear time trend and lagged stock-market uncertainty up to 5 lags.

Our analysis in Table 1 is motivated by [10], who theoretically show that the expected amount of information generated equals the expected reduction in uncertainty. The idea is consistent with the foundational work in information theory [11]. Concretely, we construct two proxies for stock-market uncertainty that are closely related to managers and investors, respectively. One is market-level cash flow volatility i.e., CFV_t (e.g., [12,13]), which directly measures the uncertainty of operating management. The other is investor search volume denoted as Search_t. Because theories of rational information acquisition predict that investors’ information-search demand increases in the uncertainty about asset payoffs (e.g., [14,15]).

Table 1 suggests that an increase in aggregate analyst coverage is associated with an expected decrease in manager uncertainty (Panel A) and investor uncertainty (B), respectively. The results remain robust after controlling for a linear time trend and lagged stock-market uncertainty up to five lags in models (2), (4), and (6). In summary, we conclude that aggregate analyst coverage can alleviate total uncertainty for both investors and managers. Moreover, a higher aggregate analyst coverage also precedes lower market-wide cash holdings and the absolute magnitude of unexpected earnings, as well as greater capital expenditures and long-term debt. For more comprehensive and detailed evidence, see [16,17].

Table 1. Diminishing the expected stock-market uncertainty.

Note: This table reports time-series predictive regressions of expected stock-market uncertainty on aggregate analyst coverage denoted as ${\lambda }_t^{-1}$. The dependent variables are the future market-level cash-flow volatility i.e., ${\mathrm{CFV}}_{t,t+h}$ (Panel A) and investor search volume i.e., ${\mathrm{Search}}_{t,t+h}$ (Panel B), which measure the manager and investor uncertainty, respectively. Controls in models (2), (4), and (6) include a linear time trend and lagged market-uncertainty up to 5 lags. To address concerns about spurious regression, we utilize the ADF tests. The sample period is 2011−2020. The t-statistics in brackets are adjusted for heteroskedasticity [18]. *, **, and *** indicate significance at the $10\%$, $5\%$, and $1\%$ levels, respectively.

Table 1. Diminishing the expected stock-market uncertainty.

Note: This table reports time-series predictive regressions of expected stock-market uncertainty on aggregate analyst coverage denoted as ${\lambda }_t^{-1}$. The dependent variables are the future market-level cash-flow volatility i.e., ${\mathrm{CFV}}_{t,t+h}$ (Panel A) and investor search volume i.e., ${\mathrm{Search}}_{t,t+h}$ (Panel B), which measure the manager and investor uncertainty, respectively. Controls in models (2), (4), and (6) include a linear time trend and lagged market-uncertainty up to 5 lags. To address concerns about spurious regression, we utilize the ADF tests. The sample period is 2011−2020. The t-statistics in brackets are adjusted for heteroskedasticity [18]. *, **, and *** indicate significance at the $10\%$, $5\%$, and $1\%$ levels, respectively.

3. MEP Generates the Exponential Distribution

Finally, this paper attempts to provide a potential generation mechanism for the exponentially distributed analyst coverage. Motivated by [9], who draw on the maximum entropy principle (MEP) to interpret the asymmetric Laplace distribution of Tobin’ Q. In what follows we utilize the MEP to derive the exponential law. The idea of MEP is that in making inferences on the basis of partial information we must use the probability distribution that achieves maximum entropy [19].

Suppose that our only knowledge of analysts coverage distribution is the mean value (i.e., limited average-ability in producing information). Mathematically, the maximum entropy problem subject to the first-order moment constraint is as follows,

$$\underset{\{f\left[x\right]\ge 0|x\in {\mathbb{R}}^{+}\}}{max}h\left(f\left[x\right]\right)=-{\int }_{x}f\left[x\right]\log f\left[x\right]dx\mathrm{s}.\mathrm{t}.{\int }_{x}f\left[x\right]dx=1,\mathbb{E}\left[X\right]={\int }_{x}xf\left[x\right]dx={\mu }_x$$

(3)

The Lagrangian associated with this programming problem is,

$$J\left(f\left[x\right]\right)=-{\int }_{x}f\left[x\right]\log f\left[x\right]dx+{\lambda }_1({\int }_{x}f\left[x\right]dx-1)+{\lambda }_2({\int }_{x}xf\left[x\right]dx-{\mu }_x)$$

(4)

Taking the first-order condition and solving for $f\left[x\right]$ yields,

$$f^{*}[x;\lambda ]=\lambda {\mathrm{e}}^{-\lambda x},\mathbb{E}\left[X\right]={\lambda }^{-1}$$

(5)

The solution is an exponential distribution fully characterized by unique parameter $\lambda$. Note that this entropy maximization property is perhaps the main reason why we encounter exponential distributions so frequently in mathematics and physics. Additionally, the concept of maximum-entropy optimization has been advocated in economic analysis (see [20,21]).

The first-order moment constrain condition is natural due to the fact that analysts have limited average capability in providing information. One might ask what entropy i.e., $h\left(f\right[x\left]\right)$ means in our case. In the Shannon theory, this answer is clear. The average amount of information is given by the information entropy i.e., $h\left(f\right[x\left]\right)$, as Shannon pointed out [11]. Because analyst coverage can be viewed as a standard information source, maximizing the objective function is equivalent to maximizing the amount of market-wide information. Similarly, suppose we do not have any knowledge about distribution, the solution of maximum-entropy optimization is the Uniform distribution. In this case, when each firm has a same coverage (no firm-specific information), the amount of market-wide information attains the maximum value. In both cases, sell-side analysts generate market-wide information, not firm-specific information.

4. Conclusions and Discussion

Contrary to the conventional wisdom that sell-side analysts specialize in collecting and disseminating firm-specific information, it is found that demonstrate that analysts predominantly generate market-wide information for 25 countries [7,8]. However, it is still unclear why analyst coverage reflects market-wide information rather than firm-specific information. To understand the underling reasons, we examine the empirical distribution of cross-sectional analyst coverage.

This paper shows the following key findings: (i) the empirical frequency distribution of cross-sectional analyst coverage follows an exponential law, observed in both the Shanghai and Hong Kong stock markets; (ii) an increase in aggregate analyst coverage is associated with and an expected decrease in total uncertainty; (iii) when analysts, who are constrained by limited average ability in producing information (i.e., the first-order moment), strive to maximize the amount of market-wide information, this objective yields the exponential distribution.

In summary, our findings offer two main insights. First, sell-side analysts in China’s stock markets predominantly contribute to market-wide information, illuminating their role in mitigating total uncertainty. Second, future research should be extra cautious when using coverage proxies in cross-sectional and portfolio analyses due to potential inference issues. Strictly speaking, it is advisable to carefully re-evaluate previous cross-sectional findings, if feasible.