Round number reference points and irregular patterns in reported gross margins

Abstract

We find irregular patterns in the distribution of firms’ reported quarterly gross margin percentages. Specifically, there is significant bunching around percentage integers that are highly round (e.g., multiples of 10, such as 30%, 40%, etc.) or are neatly divisible (e.g., 25%, 75%), compared to what is predicted by counterfactual distributions. Further investigation reveals that highly round gross margin firms are smaller, exert higher effort, achieve higher productivity, have more difficult goals, and pay their CEOs with a higher portion of fixed income. We also find that highly round gross margins are associated with superior performance. Additionally, we do not find consistent evidence that highly round gross margin reference points are linked to external rewards. Collectively, our evidence is consistent with reference-dependent preferences for highly round gross margins likely being driven by intrinsic (rather than extrinsic) motivations.

1 Introduction

Psychologists note that round numbers, such as those ending in 0 or 5, are more accessible and hence are more easily remembered and frequently processed in thoughts (Schindler and Wiman 1989; Hornik et al. 1994; Hultsman et al. 1989; Huttenlocher et al. 1990; Tarrant et al. 1993). The high cognitive accessibility of round numbers makes them candidates for intrinsic reference points, especially when there is no explicit benchmark. Prior research documents that athletes and students have strong preferences for using round numbers as intrinsic reference points (Pope and Simonsohn 2011; Allen et al. 2017). Studies also find discontinuities near round numbers (i.e., bunching) in the distributions of accounting numbers (Carslaw 1988; Thomas 1989; Dyreng et al. 2017; Stice et al. 2022).

Relatively less explored, however, is why firms bunch their reported accounting metrics near round numbers. In studies examining irregular distributional patterns, the most frequently investigated accounting numbers are earnings (including earnings per share) and revenues. The setting of earnings or revenues makes it difficult to disentangle the intrinsic and extrinsic motivations of preferences for round numbers because managers have strong incentives to manage these numbers (Burgstahler and Dichev 1997; Das and Zhang 2003; Stice et al. 2022). In this study, we examine the distribution of gross margins, a performance metric that, compared to earnings and revenues, is less associated with explicit external benchmarks but still an important performance metric that management evaluates. Therefore, gross margin provides a unique setting in which to examine the motivations of preferences for round numbers in the context of financial reporting.

We find that the distribution of firms’ gross margin percentages calculated using reported quarterly financial statement data exhibits irregular patterns.¹ Comparing the distribution of gross margins to counterfactual distributions by fitting a polynomial (as in Chetty et al. 2011 and Allen et al. 2017), we find significant bunching near highly round integer percentages—that is, multiples of 10 (e.g., 40%, 50%, 60%, etc.) or numbers easily divided from 100%, such as 25% and 75%.² These findings indicate that significantly more firm-quarters have gross margins that are at or just above highly round cutoffs than what is predicted by counterfactual distributions or by chance. Because explicit gross margin benchmarks are relatively absent, such irregular patterns in gross margin distributions suggest that these highly round cutoffs are likely being used as implicit psychological reference points, and managers might report revenue or cost of goods sold (COGS) to produce gross margins just above these points. Researchers refer to such behavior as “reference-dependent preferences” (Kőszegi and Rabin 2006).

We investigate characteristics of firm-quarters that report highly round gross margins and attempt to provide insights on why reported gross margins bunch around such numbers.³ Psychology research demonstrates that so-called “mere” goals—which are psychologically based and not accompanied by external rewards—serve as reference points and divide outcomes into regions of gains and losses (Heath et al. 1999; Von Rechenberg et al. 2016).⁴ The goal-setting literature establishes three features of such goals or reference points.⁵ First, goals motivate individuals to exert effort to be in the region of gains (Locke and Latham 1990, 2002). Second, the difficulty of such goals is positively related to exerted effort. For example, Soetevent (2022) finds that marathon runners are more likely to finish just under a round time when it represents a more ambitious goal. Third, Corgnet et al. (2018) show that labor contracts with both monetary incentives and wage-invariant goals can be optimal.

Applying goal-setting theory to the context of gross margins, we expect that reporting highly round gross margins is positively related to effort and goal difficulty and negatively related to the proportion of performance-based pay in executive compensation. We measure firm effort using the level of the firm’s total investment, and employee effort using sales per employee productivity. We assume that higher levels of investment and productivity require higher effort. Because we do not observe firms’ actual goals, we measure goal difficulty based on an estimated gross margin benchmark and the industry environment. Goals are considered more difficult if 1) the difference between realized gross margins and the average of gross margins in the past four quarters is larger; or 2) industry sales growth is lower. To measure the degree of performance-based pay in executive compensation, we use the percentage of total executive compensation that is comprised of base (i.e., fixed) salary. In addition, we expect firm size to be related to reference-dependent preferences because smaller firms likely have fewer explicit benchmarks. We use total assets to measure firm size. We find that firm-quarters with highly round gross margins are smaller, have higher levels of investment and productivity, and have goals that are more difficult to achieve. We also find some evidence that CEOs in the highly round group have a higher portion of compensation that is fixed. Our evidence is stronger when we restrict our analyses to just those firms which had at least one highly round firm-quarter in our sample period. These findings are consistent with the notion that highly round gross margins are more likely to be intrinsic reference points than extrinsically motivated.

Next, we investigate the performance of the highly round gross margin group. In cross-sectional analyses, we do not find consistent evidence that firm-quarters with highly round gross margins have higher financial performance across all metrics than non-highly round gross margin firm-quarters. However, when we restrict our analyses to highly round firm-quarters, we find strong evidence that highly round firm-quarters have superior financial performance when compared to the prior four quarters and when compared to the other three quarters in the same fiscal year. We also find that the positive association between current period investment (i.e., our proxy for firm effort) and future sales growth is stronger when the company reports a highly round gross margin. Taken together, the evidence suggests that highly round gross margins are internal goals that are associated with higher performance.

We then directly test whether reporting highly round gross margins is associated with external rewards. We find that there is no significant difference in overall executive compensation for firms with or without highly round gross margins, and that investors do not seem to reward firms for jumping over these reference points. This evidence adds support to the notion that highly round gross margins are more likely to be implicit, psychologically based reference points than reference points with external benefits.

Lastly, we consider potential underlying mechanisms of gross margin management. We find that firms in the highly round group have higher abnormal revenue, suggesting that highly round gross margins are achieved through efforts to attain higher revenue levels than what fundamentals would predict. While we use Stubben’s (Stubben 2010) discretionary revenue model to measure abnormal revenue (which can indicate earnings management), prior accounting literature provides evidence that traditional earnings management proxies do not necessarily suggest opportunistic behavior (Subramanyam 1996; Gunny 2010). Our overall evidence suggests that the higher abnormal revenue of the highly round gross margin group is likely reflective of underlying performance rather than accounting manipulations. These findings do not conflict with our earlier evidence that highly round gross margins are likely driven by intrinsic motivation, because reaching psychological reference points does not happen by chance and requires actions.

Our paper makes several contributions. We provide initial descriptive evidence on the patterns of gross margins. We show that the frequency of firm-quarters reporting highly round gross margins is significantly greater than that expected by counterfactual distributions or by mere chance.⁶ Gross margin is an important performance measure, particularly in industries with high inventory turnover (e.g., retail) or where inventory obsolescence is of greater concern (e.g., semiconductors). It is often an indicator of brand equity and provides insights into how much a consumer is willing to pay above the actual costs of production.⁷ However, irregular patterns in gross margin distributions are not necessarily expected under existing theories of earnings management. Forecasts of gross margins (from both analysts and management) are far less frequent than forecasts of revenues or earnings,⁸ and gross margins are also less frequently used in contracts as explicit benchmark metrics (Huang et al. 2014; Cheng et al. 2015; Curtis et al. 2021).⁹ Thus, gross margins do not possess the same financial reporting incentives as one would expect in other accounting numbers such as revenues and earnings. Our findings add to the literature on irregular distributions of reported accounting numbers by showing that reference-dependent preferences for round numbers also exist in accounting numbers that managers have fewer ex ante incentives to manage.

Additionally, our study sheds light on the motivations of reference-dependent preferences for round numbers in the financial reporting context. Allen et al. (2017) provide evidence that “round” finishing times among marathon runners are not driven by external rewards, and Soetevent (2022) suggests that such round times are driven mainly by immediate gratification. However, marathons are a different setting from financial reporting. Marathon runner motivations are inherently intrinsic and not economic, as noted by Allen et al. (2017), while the motivations in the financial reporting setting are more economically driven. This difference in setting, as well as the difference in reporting incentives between gross margins and other accounting numbers (e.g., revenues and earnings), makes gross margins a unique setting in which to examine the motivations of round number reference-dependent preferences in the financial reporting context. Our evidence suggests that preferences for highly round gross margins, in the relative absence of explicit contractual benchmarks, seem to be driven more by intrinsic than by extrinsic motivations.

The remainder of the paper proceeds as follows. In Section 2, we provide a brief review of previous literature (including motivations from the psychology literature) that examines reference-dependent preferences in various settings. In Section 3, we describe the data and sample. In Section 4, we present our main empirical analyses. In Section 5, we perform several sensitivity and robustness tests. In Section 6, we conclude the paper.

2 Literature review

2.1 Intrinsic goal-setting and prospect theory

Psychology research suggests that the ease of accessibility of concepts in memory influences personal perception (Smith and Miller 1979; Rholes and Pryor 1982). More accessible units in memory are given more weight in judgments and are more likely to be easily remembered and frequently processed in thoughts (Higgins et al. 1977; Schindler and Kirby 1997). Prior research finds that numbers ending in “0” or “5” are more accessible because people favor multiples of 10 (or their midpoints) in a base-10 number system (Schindler and Wiman 1989; Hornik et al. 1994; Hultsman et al. 1989; Huttenlocher et al. 1990; Tarrant et al. 1993). The high cognitive accessibility of these round numbers makes them candidates for intrinsic reference points, and prior research provides evidence of this in a variety of settings. Pope and Simonsohn (2011) find discontinuities in baseball batting averages (around .300) and SAT score retake rates (around multiples of 100), while Allen et al. (2017) find that marathon finishing times cluster just before hour and half-hour marks. In these settings, performance is not directly tied to explicit rewards, and hence the findings are consistent with round number reference points being psychologically based.

Prospect theory suggests that a loss from an outcome just below a reference point is perceived as larger than an equivalent gain just above the reference point (Kahneman and Tversky 1979). Such loss aversion is a primary driver for reference-dependent preferences in which individuals make efforts to jump over the reference points to avoid the perceived larger loss (Berger and Pope 2011; Garmaise 2015). Heath et al. (1999) investigate the behavior of mere goals and show that they exhibit properties of a prospect theory value function (e.g., reference points, loss aversion). Von Rechenberg et al. (2016) analyze a gamified question-and-answer community and provide field evidence for Heath et al.’s (Heath et al. 1999) model. Their findings are consistent with the S-shape value function under prospect theory and support the notion that goals serve as reference points. Such properties of goals are also reflected in goal-setting theory, which argues that wage-invariant goals can be implicit benchmarks that motivate employees to work harder (Latham and Locke 1979; Locke 1996; Heath et al. 1999; Gomez-Minambres 2012) and that more difficult goals can serve to increase effort (see Locke and Latham (1990) for a review). While much work in the goal-setting literature analyzes specific tasks, such as logging production, arithmetic problem solving, or toy assembly (Locke 1996), Gomez-Minambres (2012) brings goal-setting theory to standard economic theory and proposes a principal-agent model in which the agent’s motivation to work is driven by both extrinsic incentives (i.e., pay-per-performance wage) and intrinsic motivation (i.e., an internal sense of achievement from accomplishing wage-invariant goals). Also applying a principal-agent framework, Corgnet et al. (2018) develop a model with reference-dependent utility and show that optimal labor contracts can include wage-invariant goals in addition to monetary incentives. These studies suggest that goals, as reference points, provide intrinsic motivation for individuals.

2.2 Round number bunching in accounting research

Particularly related to our study is the examination of distributions of accounting measures near “round” numbers. Carslaw (1988) examines New Zealand firms and finds more (fewer) instances of “0” (“9”) than expected in the second digit of earnings. Thomas (1989) examines US firms and finds similar patterns in positive earnings but opposite patterns for loss firms. He also observes unusually high proportions of earnings per share (EPS) numbers that are multiples of five or 10 cents among profitable firms, but no such patterns for loss firms. Stice et al. (2022) find that firms tend to report revenue numbers just above base-10 thresholds more often than just below. These studies suggest that round numbers are used, perhaps implicitly, as reference points for earnings and revenues. Stice et al. (2022) further find that firms beating the base-10 thresholds for the first time have higher media and analyst coverage, trading liquidity, and institutional ownership, which could suggest that firms reporting revenues just above these thresholds can gain some economic benefits. However, their findings do not preclude the notion that round number preferences are also intrinsically motivated. Revenues, along with earnings, are among the numbers most heavily scrutinized by external parties such as analysts and investors and are among the most frequently contracted performance metrics. Therefore, even if round revenue numbers as reference points are driven by intrinsic motivations, the effect of reporting revenues above these cutoffs may result in external rewards, as Stice et al. (2022) document. In contrast, gross margins provide a setting that allows us to more directly examine the motivations of round number reference-dependent preferences in the financial reporting context.

While there is extensive evidence on discontinuities in the distributions of stand-alone accounting numbers (e.g., revenues and earnings), few studies examine the distributions of financial statement ratios—numbers that are easily calculated on the face of financial statements but may or may not be explicitly reported. Dyreng et al. (2017) examine firms’ working capital and find that the distribution of current ratios exhibits a discontinuity at 1, a natural psychological reference point. They find that firms with more long-term debt and lease liabilities are actually less likely to exhibit this pattern, suggesting that their results do not simply reflect management’s attempts to avoid debt covenant violations. Our paper is related to their study in that both metrics (gross margin and the current ratio) are financial statement ratios that may or may not be directly reported on financial statements. A key difference is that unlike the current ratio, gross margin is less commonly used in contracting.

Rounding has also been noted in numbers that are not prepared by managers. Hermann and Thomas (2005) document that 55% of analyst forecasts end in nickel intervals, and Dechow and You (2012) suggest that this phenomenon is partly explained by analysts who have less incentive to spend efforts researching (and thus providing more precise forecasts for) covered firms that generate less brokerage or investment banking business.¹⁰ In finance, Harris (1991) provides early evidence that pre-decimalization stock prices tend to cluster near round fractions. These settings are different from ours in that such forecasts do not represent reference points or goals to be achieved, and the clustering near round numbers results from reasons unrelated to performance motivation, as documented by Dechow and You (2012).

3 Sample selection and descriptive statistics

3.1 Compustat cost of goods sold data

Our metric of interest—gross margins—is calculated using revenue and COGS from the Compustat Fundamentals Quarterly database. This database makes proprietary adjustments to COGS, which can cause Compustat’s stored COGS value to differ from the amount reported on the firm’s financial statements. In particular, if a firm separately reports the allocation of depreciation and amortization expense, then Compustat will exclude this amount from COGS. Bostwick et al. (2016) find that Compustat COGS is different from 10-K COGS for approximately 79% of their sample firm-years. They further examine three adjustment methods and find that adding Compustat depreciation, depletion, and amortization expense (Compustat variable DP) to Compustat COGS when the corresponding Compustat footnote variable (COGSF) is coded as “BD” most approximates COGS. We adapt their method to adjust the value of COGS for a total of 401,245 firm-quarters whose Compustat footnote variable is coded as “BD.”¹¹

3.2 Sample selection and descriptive analyses

To construct our sample, we begin by selecting all firm-quarter observations in the Compustat Fundamentals Quarterly database from 1976 through 2017. We impose relatively minimal data availability requirements. We require that firm-quarters have positive sales and cost of goods sold, and non-negative gross profit.¹²

We note that the manner in which Compustat records numerical dollar amounts could affect our inferences on the distributions of ratios using such dollar amounts. Typically, Compustat units are millions, with three digits after the decimal. Thus, Compustat dollar amounts are effectively rounded to the nearest thousand. However, if the numbers on the financial statements are reported with greater precision (i.e., dollars, rather than thousands or millions of dollars), then this could induce rounding in the computed gross margin percentage, causing it to appear round when it actually is not. In our sample, we find that 2876 firm-quarter observations have a computed gross margin (based on Compustat sales and our adjusted COGS value) of an exact percentage integer. These observations would otherwise be considered round in our analyses and are instances in which Compustat rounding may have induced the observed gross margin percentage. To investigate whether this is the case, we manually retrieve the 10-Qs, 10-Ks, and 4th quarter earnings press releases (where available) to determine whether the income statement amounts are reported in dollars rather than in thousands or millions of dollars. If they are, we recompute the gross margin using the more precise values in the income statement.¹³ Based on this approach, we determine that in 2197 of these observations, Compustat rounding likely induced the observed exact percentage integer gross margin. We eliminate these observations from our sample.¹⁴

For our multivariate analyses, we further require data on total assets, data on net income, CRSP data (to compute returns), and other variables. Table 1 Panel A outlines our sample selection procedures. The final sample for the main distributional analyses consists of 1,121,119 firm-quarter observations. Panel B provides descriptive statistics for the highly round gross margin group and non-highly round gross margin group. We note that highly round and non-highly round firms do not differ significantly in terms of raw sales, cost of goods sold, gross profit, or market capitalization. Highly round gross margin quarters, on average, have a lower book-to-market ratio and higher leverage, both of which we control for in our later multivariate tests.

Table 1 Sample selection and descriptive statistics

Panel C displays the composition of firm-quarters by Fama-French 12 industry for firm-quarters with and without highly round gross margins. We do not find any obvious industry clustering of highly round gross margin firm-quarters. Based on the differences in percentages between the two groups, we find that highly round firm-quarters are, proportionately, least likely to be in the financial services industry and most likely to be in the business equipment, medical equipment, and wholesale and retail industries, consistent with gross margins being of greater interest in these industries.

4 Empirical analyses

4.1 Patterns of gross margin distributions

We define “highly round” as firm-quarters with gross margins equal to or within 0.1% above 0%, 25%, 75%, or any multiple of 10%, and investigate patterns of gross margin distributions.

Uniform distribution

Under the null hypothesis of no irregular patterns in gross margin distributions, each of the 10 digits (zero to nine) would appear equally likely in the tenths place of a gross margin percentage. We plot the frequency distributions of gross margins to provide a visual sense of the discontinuities in highly round integer cutoffs. Figure 1 is a histogram of the number of observations within 0.1% below (in blue) and above (in red) each percentage integer. The discontinuities are noticeable at the highly round percentage amounts. Figure 2 is a plot of the difference in the number of observations within 0.1% above and below each percentage integer. We compute the difference as “above” minus “below,” so that percentage integers with more (fewer) “above” than “below” observations are plotted above (below) the horizontal axis. The discontinuities are more easily visible here, and when we rank these percentage integers based on the difference between the number of “above” and “below” observations, the top seven integers (40%, 0%, 20%, 50%, 25%, 30%, and 10%) are all among our group of highly round percentage integers. The discontinuities near these highly round percentages suggest that there are more gross margins just above highly round numbers than what would be expected by chance.¹⁵

Fig. 1

Histogram of firm-quarter observations just below and just above whole percentages

Fig. 2

Difference in number of observations just above vs. just below each percentage integer

Bunching analyses

Because the true expected distribution of gross margins is unknown and may not be uniform, we adopt an empirical approach that, like Kleven (2016), “uses bunching around points that feature discontinuities in incentives to elicit behavioral responses.” This “bunching” approach measures the excess mass around an incentive split point (in our setting, highly round gross margins). We follow a methodology developed by Chetty et al. (2011) and utilized by Allen et al. (2017) to estimate a counterfactual distribution of gross margins and quantify the excess mass in the interval (i.e., the bunching region) above a highly round number by comparing the actual distribution with the counterfactual distribution.

We estimate the counterfactual distribution by fitting a polynomial to the local density of gross margins excluding the bunching region. For each highly round number, we measure the excess number of firm-quarters as the density difference in the bunching region between the actual distribution and the counterfactual distribution and calculate the standard error of the excess mass using a bootstrap procedure.¹⁶ We use 0.5% as the local window around each highly round reference point. For example, to test the excess mass around the 10% gross margin cutoff, our local window to estimate the counterfactual distribution is from 9.75% to 10.25%. Our local window is chosen to allow reasonable variations while being short enough to minimize confounding factors that may affect the counterfactual distribution. We restrict the region for bunching tests to be 0.04%, starting from each potential reference point. As in Chetty et al. (2011) and Allen et al. (2017), we choose this window based on a visual inspection of the bunching, and we shift the counterfactual distribution upward to allow equal areas underneath the counterfactual curve and under the actual distribution.¹⁷

Following Allen et al. (2017), we plot the results of the bunching analyses in Fig. 3 and provide descriptive statistics in Table 2. Figure 3 shows the actual gross margin distributions (dotted lines) and counterfactual distributions (smoother lines) for each highly round percentage. The actual distribution lines are plotted in 0.01% intervals for each 0.5% window around the highly round percentages. The bunching tests regions are bordered within two vertical lines. The incidence of bunching is reasonably evident, with the exception of the region around 70%.

Fig. 3

Actual and counterfactual distributions of gross margin observations near highly round percentages. This figure displays plots of our bunching analyses from Table 2, showing actual gross margin distributions (dotted lines) and counterfactual distributions (smoother lines) for our defined highly round percentages. In each display, the horizontal axis is the range of bins around the gross margin percentage integer examined, while the vertical axis is number of observations. We use 0.5% as the local window around each highly round reference point, and restrict the region for bunching tests to be 0.04%

Table 2 Discontinuities around highly round gross margin percentages based on counterfactual distributions

In Table 2, we present descriptive statistics for the bunching areas shown in Fig. 3. We report the number of actual firm-quarters with gross margins in the bunching regions around each highly round percentage and the predicted number from the counterfactual distribution. We calculate the percentage of excess observations indicated in the actual gross margin density function and a t-statistic calculated by bootstrapping with 1000 iterations. In 10 of 11 cases, the actual number of firm-quarters exceeds the number predicted by the counterfactual distribution, and the bunching around nine of these highly round percentages is statistically significant. The largest excess mass in percentage (27.1%) is around 75%, and the largest excess mass in number (143) is around 50%. The excess mass around 20% is positive but insignificant, while the excess mass around 70% is insignificantly negative.¹⁸ Overall, the evidence suggests that more firm-quarters report highly round gross margin percentages than what is predicted by an estimated counterfactual distribution.

4.2 Characteristics of firms with highly round gross margins

We build on goal-setting theory to examine the associations between reporting highly round gross margins and effort, goal difficulty, executive pay structure, and firm size.

Effort

The observed bunching around highly round percentages in gross margin distributions suggests that firms use these highly round numbers as reference points. Psychologists note that an individual’s value function can be altered by the presence of a reference point such that the reference point divides outcomes into regions of gains and losses (Heath et al. 1999; Von Rechenberg et al. 2016); therefore, individuals below reference points will increase efforts to work towards their goals. The goal-setting literature documents consistent evidence that goals motivate individuals to exert efforts and work harder, and that psychological goals (i.e., those not associated with external rewards) are associated with higher goal commitment and more prolonged efforts (Locke and Latham 2002). As gross margins are not frequently used as explicit benchmarks for performance evaluation, highly round gross margin percentages acting as reference points are more likely to be implicit and intrinsic psychological goals that managers use to motivate themselves and their employees for higher efforts.

We examine whether firms that report highly round gross margins are associated with higher effort. Employee (or firm) effort is unobservable and difficult to measure empirically. We consider measures for two dimensions of effort. First, we measure overall firm effort by the level of investment. Higher investment levels imply that firms are undertaking more projects that require more time and effort. We investigate firms’ overall investment, measured by the sum of capital expenditures and R&D, scaled by revenue (INVESTMENT). Second, we measure employee effort by their productivity, as higher effort levels should lead to higher productivity. We measure productivity by sales per employee (SALEEMP).

Goal difficulty

The goal-setting literature has documented that more specific and difficult goals lead to increases in employee productivity (Locke and Latham 1984) and that there is a positive and linear relationship between goal difficulty and levels of effort (Locke and Latham 2002). Locke and Latham (1990) find that setting specific and difficult goals can be more effective in eliciting performance than simply urging people to do their best. Relatedly, Soetevent (2022) finds that marathon runners are more likely to finish within a round time when the round time goal is considered ambitious. We test whether highly round gross margins are more likely to serve as reference points in firms for which attaining such highly round gross margins would be considered difficult or more ambitious. Like Allen et al. (2017) and Soetevent (2022), we do not observe firms’ goal setting processes and do not have direct information about actual gross margin goals. We use the following measures to proxy for goal difficulty. First, we measure goal difficulty as the difference between the firm’s reported gross margin in quarter t and the average of gross margins in the previous four quarters (ABGM). Larger differences imply that firms may have used goals that were more difficult or ambitious.¹⁹ Second, we use industry sales growth (INDGROW) as a proxy for goal difficulty. Firms operating in stagnant industries face more challenges in attaining their goals, so those goals are considered more difficult.

Executive pay structure

As discussed earlier, studies in psychology have documented that non-monetary incentives such as wage-invariant goals are effective in motivating individuals to exert more effort and are associated with better performance (Locke 1996; Corgnet et al. 2015; Gomez-Minambres 2012). This literature also adopts goal-setting theory into a principal-agent framework and shows that the agent can be motivated by both extrinsic and intrinsic motivations, which in turn suggests that labor contracts with both monetary and non-monetary incentives could be optimal (Corgnet et al. 2018). It follows that executives of firms which use highly round gross margins as psychological reference points may be compensated with pay that is less performance-based. We examine this conjecture by investigating the relationship between highly round gross margins and the percentage of total CEO compensation that is salary-based and thus fixed (FIXEDCEO).

Firm size

Smaller firms may have fewer explicit benchmarks than larger firms because larger firms are more likely to receive analysts’ attention, and hence the supply of analyst forecasts is higher (Bhushan 1989). Firms that lack explicit benchmarks may be more likely to show reference-dependent preferences for highly round gross margins. We therefore test whether smaller firms, measured by the log of total assets (SIZE), are more likely to show such preferences.

To examine the characteristics of firm-quarters reporting highly round gross margins, in Table 3 we compare firm-quarters with and without highly round gross margins along the characteristics discussed above for all firm-quarters (Panel A) and only for those firms that report at least one highly round firm-quarter (Panel B). All variables are defined in Appendix A. In Panel A, the effort measure INVESTMENT is significantly higher in the highly round group, but the productivity measure SALESEMP is significantly lower. In terms of goal difficulties, ABGM does not show a significant difference between the two groups, but the industries in which highly round gross margin firms operate exhibit significantly lower industry total sales growth (INDGROW), suggesting more challenging circumstances for these firms. Firm-quarters in the highly round gross margin group are also significantly smaller in size (SIZE) and have CEOs for whom fixed salary comprises a greater proportion of total compensation (FIXEDCEO).

Table 3 Firm characteristics associated with highly round gross margins

In Panel B, we compare highly round firm-quarters with non-highly round firm-quarters only for those firms that reported at least one highly round firm-quarter in our sample period. In general, the results are stronger and more consistent with highly round gross margins acting as implicit reference points. Specifically, the positive difference in INVESTMENT and the negative difference in SIZE both become larger and more significant. Additionally, ABGM is now significantly higher for highly round firm-quarters, and SALESEMP shows an insignificant difference (rather than a significantly negative difference as in the full sample). The only weaker result is INDGROW, which shows no difference for these two groups, possibly due to less variation in industries in this more restricted within-firm sample. In general, the univariate comparison provides some evidence that highly round gross margins might be used as implicit psychological reference points that provide intrinsic motivation to firms.

We then conduct multivariate analyses to further examine the characteristics associated with firms reporting highly round gross margins by estimating the logistic regression (for brevity we omit firm-quarter subscripts):

$$\textit{HR}\mathit=\beta_{\mathit0}\mathit+\beta_{\mathit1}\textit{INVESTMENT}\mathit+\beta_{\mathit2}\textit{SALEEMP}\mathit+\beta_{\mathit3}\textit{ABGM}\mathit+\beta_{\mathit4}\textit{INDGROW}\mathit+\beta_{\mathit5}\textit{SIZE}\mathit+\beta_{\mathit6}\textit{FIXEDCEO}\mathit+\textit{Controls}\mathit+\varepsilon$$

(1)

where HR is a dummy variable equal to one if the firm-quarter reports a highly round gross margin and zero otherwise. Other independent variables of interest are as described above and defined in Appendix A. We include book-to-market and leverage to control for growth and risk. Scaled continuous variables are winsorized at the 1% and 99% levels.

Table 3 Panel C reports the results. In Column (1) we estimate Eq. (1) cross-sectionally for the full sample. The coefficients on SIZE and INDGROW are significantly negative and the coefficient on INVESTMENT is significantly positive, suggesting that highly round gross margin firms are smaller, have a higher level of overall investment, and are in industries with lower industry growth (and thus face greater challenges). The coefficients SALESEMP and ABGM are insignificant. In Column (2) we estimate Eq. (1) for firms that reported at least one highly round quarterly gross margin in our sample period and find stronger results. Specifically, the results for SIZE and INVESTMENT continue to hold and are stronger. In addition, the coefficients on SALESEMP and ABGM are now significantly positive, suggesting that firms exhibit higher productivity in quarters with highly round gross margins and that these highly round gross margin goals seem to be ambitious, as evidenced by the relatively higher gross margins in these quarters compared to the previous four quarters. INDGROW is insignificant for this within-firm sample, similar to in the univariate analyses.

Columns (3) and (4) add the variable FIXEDCEO in Eq. (1). Including executive compensation data (from Execucomp) significantly reduces the sample size, and, as a result, most of the coefficients are insignificant.²⁰ However, the coefficient on FIXEDCEO is significantly positive in Column (3) for the full sample, suggesting that firms reporting highly round gross margins compensate their CEOs with a higher portion of fixed salary, consistent with goal-setting theory (i.e., that firms combine intrinsic goals and monetary rewards in executive compensations). FIXEDCEO is insignificant in Column (4) when we run the regression for highly round firms only, possibly because of not enough variation in executive compensation in different quarters for the same firm. Overall, our evidence is consistent with the notion that reference-dependent preferences for highly round gross margins result from intrinsic goals that motivate firms and their employees to exert more efforts.

4.3 Performance

As discussed earlier, there is extensive evidence, in the goal-setting literature, that mere goals increase performance because such goals serve as reference points and motivate increased effort and persistence (Locke and Latham 1991; Locke and Latham 2002). Additionally, goals that are specific and difficult can be more effective at motivating performance than simple open-ended encouragement (Locke and Latham 1990). In this section, we examine whether firms reporting highly round gross margins exhibit higher performance.

Cross-sectional and within-firm

Similar to our analyses on firm characteristics, we first compare the performance of firm-quarters with and without highly round gross margins across the entire sample. Table 4 Panel A shows that the highly round group has significantly higher sales but lower gross margins and net income. These results suggest that highly round firm-quarters actually do not have performance superior to that of non-highly round firm-quarters overall. However, psychologically based goals motivate individuals to work harder towards their internal targets, but they may not motivate them to achieve better performance than other firms. We therefore perform within-firm comparisons and investigate whether a firm reporting a highly round gross margin in a quarter exhibits better performance than it does in its other, non-highly round quarters.

Table 4 Performance comparisons

Table 4 Panel B reports the comparison in financial performance for highly round firm-quarters only. We report [1] the mean difference between the highly round quarter (t) and the average performance in quarters t − 1 through t − 4, and [2] the mean difference between the highly round quarter and the average of the other three quarters in that quarter’s fiscal year.²¹ The results show that, compared to the prior four fiscal quarters, highly round gross margin quarters have higher sales, gross profit, gross margin, and net income. Additionally, highly round quarters have stronger financial performance than the rest of the fiscal year. The results are consistent with the notion that the highly round integer percentages serve as intrinsic reference points that motivate firms to exert higher effort to achieve better performance.

Future performance

Effort in the current period may lead to higher performance in future periods. To investigate whether current period investment (our proxy for effort) is more strongly associated future revenue growth when firms report highly round gross margins, we estimate Eq. (2):

$$\textit{Future Sales Growth}\mathit=\beta_{\mathit0}\mathit+\beta_{\mathit1}\textit{HR}\mathit+\beta_{\mathit2}\textit{INVESTMENT}\mathit+\beta_{\mathit3}\textit{HR}\mathit\times\textit{INVESTMENT}\mathit+\textit{Controls}\mathit+\varepsilon$$

(2)

Table 4 Panel C reports the results. The dependent variable FutureSalesGrowth is measured as the proportionate change in the sum of sales over the four quarters prior to quarter t to the four quarters after quarter t. A positive interaction between HR and INVESTMENT would suggest that investment efforts are more strongly associated with future revenue growth when firms report highly round gross margins. We include SIZE, B/M, LEVERAGE, and INDGROW as our control variables. The coefficient on the interaction term is significantly positive. The results suggest that firm effort during highly round gross margin quarters is more strongly positively associated with future sales growth over the following four quarters. The findings again are consistent with the goal-setting theory that goals increase effort and performance.²²

4.4 External rewards

Explicit reference points that are linked with external incentives can also motivate people to exert higher effort. If highly round gross margins are driven by extrinsic motivations, firms reporting highly round gross margins should gain some sort of rewards. We directly test whether highly round gross margins are associated with external benefits by investigating the association between highly round gross margins and executive compensation and investor responses to such gross margins.

Executive compensation

The earnings management literature has long documented that just meeting or beating a certain performance target is considered a way of managing earnings to improve short term performance in response to a variety of incentives, including ones arising from executive compensation.²³ If firms meet or beat highly round gross margin benchmarks in order to improve their short-term financial performance, their executives may benefit by receiving higher overall compensation. Note that gross margin does not have to be the explicit metric for performance evaluation in executive compensation contracts. As long as compensation contracts include some sort of financial performance metrics for evaluation, reporting higher gross margins will likely be associated with higher numbers for the metrics that such contracts are based on. As these performance metrics can be accounting numbers or non-accounting numbers, we investigate the relationship between total compensation of CEOs and of all executives and the incidence of reporting highly round gross margins.

In Table 5, we follow the model developed by Graham et al. (2012) and regress the log of total compensation (for both the CEO and all executives) on current and lag ROA, returns, and volatility (σ); controls of SIZE, B/M, and LEVERAGE; and the HighlyRound (HR) gross margin firm-quarter indicator.²⁴

Table 5 External rewards (compensation)

$$\textit{Log }\mathit{(\text{TOTAL COMP})}\mathit=\beta_{\mathit0}\mathit+\beta_{\mathit1}\textit{HR}\mathit+\beta_{\mathit2}\textit{ROA}\mathit+\beta_{\mathit3}\textit{Lag}\mathit{(\text{ROA})}\mathit+\beta_{\mathit4}\textit{Return}\mathit+\beta_{\mathit5}\textit{Lag}\mathit{(\text{Return})}\mathit+\beta_{\mathit6}\sigma\mathit+\beta_{\mathit7}\textit{Lag}\mathit{(\sigma)}\mathit+\beta_{\mathit8}\textit{SIZE}\mathit+\beta_{\mathit9}\textit{B}\mathit/\textit{M}\mathit+\beta_{\mathit{10}}\textit{LEVERAGE}\mathit+\varepsilon$$

(3)

In all specifications, HR is insignificant, indicating that executives of firms that report highly round gross margins do not have higher overall compensation than executives of firms that do not report highly round gross margins.

Market response

Another form of possible external reward is a price premium to firms that report highly round gross margins. Behavioral finance suggests that investors can exhibit psychological biases and cognitive limitations (Bhattacharya et al. 2012); Johnson et al. 2008; Bagnoli et al. 2006; Kuo et al. 2015). If highly round gross margins serve as extrinsic goals, then firms reporting these gross margins may take advantage of investor preferences for round numbers to gain price premiums. We investigate this by regressing three-day cumulative abnormal returns (CAR) centered on the earnings announcement date on changes in quarterly net income (ΔNI), along with controls and the highly round gross margin indicator²⁵:

$$\mathit3\mathit-\textit{day}\mathit\;\textit{CAR}\mathit=\beta_{\mathit0}\mathit+\beta_{\mathit1}\mathit\;\Delta\textit{NI}\mathit+\beta_{\mathit2}\mathit\;\textit{HR}\mathit+\beta_{\mathit3}\mathit\;\Delta\textit{NI}\mathit\times\textit{HR}\mathit+\textit{Controls}\mathit+\varepsilon$$

(4)

We also decompose change in net income to ΔGP and ΔOI, which are the change in the gross profit and non-gross profit components of net income, respectively. We use year-over-year quarterly changes to represent unexpected earnings.²⁶

Table 6 reports the results. In Column [1], ∆NI is highly significant and positive, consistent with the prior earnings response coefficient literature (e.g., Lipe 1986). The coefficient on the interaction term between ∆NI and HR is insignificant, suggesting that there is no difference in market premium for firm-quarters with and without highly round gross margins. In Column [2], we disaggregate changes in net income into its gross profit and non-gross profit components. We find that while the main effects of ΔGP and ΔOI are both significantly positive, only the interaction of ΔOI × HR is significantly positive; the interaction ΔGP × HR is insignificant, indicating that investors do not reward gross profits that result from highly round gross margins. In summary, we do not find consistent evidence of market-based or executive compensation rewards to firms reporting highly round gross margins, suggesting that these reference points more likely serve as intrinsic rather than external targets.

Table 6 External rewards (market response)

4.5 Mechanisms underlying highly round gross margins

Increasing gross margin to highly round percentages can be achieved by increasing revenue or decreasing COGS numbers to obtain the desired margins. We investigate these potential mechanisms and examine firms’ abnormal revenues and production costs.²⁷

We employ proxies for discretionary revenues developed by Stubben (20062010) to measure abnormal revenue as the difference between the actual change in receivables and the predicted change based on models in which the change in receivables is regressed on lagged revenues.²⁸ We use Roychowdhury’s (Roychowdhury 2006) production cost model to estimate firms’ abnormal production costs. Higher than normal production costs indicate potential overproduction, which can lead to lower COGS. We estimate logistic model (5) to investigate the potential channels that firms use to reach highly round gross margin reference points.²⁹

$$\textit{HR}\mathit=\alpha_{\mathit0}\mathit+\beta_{\mathit1}\textit{AbRev}\mathit+\beta_{\mathit2}\textit{AbProd}\mathit+\textit{Controls}\mathit+\varepsilon$$

(5)

AbRev is abnormal revenue and AbProd is abnormal production costs. We include SIZE, B/M, and LEVERAGE as controls, and the dependent variable is our indicator for the highly round group. Table 7 reports the results. The coefficient on AbRev in Column [1] is positive and significant, suggesting that firms reporting highly round gross margins have higher abnormal revenues. In Column [2], the coefficient on AbProd is insignificant. When we put the two variables in one regression, the coefficient on AbRev is still significantly positive, and AbProd is insignificant. The results suggest that highly round gross margins are more likely to be achieved via higher abnormal revenues than via COGS. While these findings could be consistent with earnings management, prior accounting research documents that earnings management proxies do not necessarily capture manipulation and can instead reflect performance. Subramanyam (1996) points out that managerial discretion could enhance earnings’ informativeness by allowing the communication of private information. He finds that discretionary accruals contain information about future profitability that is priced by the capital market. Gunny (2010) documents that firms engaging in real earnings management to meet earnings benchmarks have better future performance, which is consistent with their using real earnings management to signal future performance. Our earlier evidence that highly round gross margins are associated with higher effort and higher performance (including future performance) suggests that highly round gross margins are achieved through exerting more effort to attain higher abnormal revenues than what is predicted by the fundamental factors in Stubben’s (Stubben 2010) model. These findings are consistent with managers taking actions to achieve implicit psychological goals.

Table 7 Mechanisms underlying highly round gross margins

5 Sensitivity and robustness tests

The bunching analyses conducted in Section 4 use a fifth-order polynomial with a bunching region of 0.04% and a local window of 0.5%. We conduct additional robustness tests (untabulated) to check if the bunching identified in our main test is sensitive to alternative parameters. Specifically, we use different combinations of bunching regions (0.01%–0.05%), local windows (0.40% and 1%), and different-order polynomials (2 to 7). The results for the 10%, 25%, 50%, 60%, and 75% reference points are highly robust, indicating that these points are likely the most important psychological reference points. The amount of excess mass for 20%, 30%, 40%, 80%, and 90% is positive and mostly significant, with a few specifications being insignificant. Only the bunching for the 70% point is usually negative. We rerun our main tests on the characteristics and performance of firms reporting highly round gross margins using a refined HR indicator variable by excluding firm-quarters reporting gross margins equal to or just above 70%. The results (untabulated) are qualitatively similar.

To further mitigate the concern that the pattern of gross margin bunching near highly round percentages may still be driven by Compustat rounding algorithms even after the adjustment to Compustat COGS and the elimination of observations via manual checking of financial statements, we conduct falsification tests in which we examine whether similar patterns exist for six other income statement-based measures scaled by revenue: revenue minus operating expenses, revenue minus nonoperating expenses, revenue minus SG&A expenses, revenue minus R&D expense, revenue minus income tax expense, and revenue minus pension expense. Because many firms have zero R&D, tax, and pension expenses and because SG&A and nonoperating expenses typically account for a small percentage of sales, the distributions of these ratios are highly skewed, which may bias our bunching results for these ratios. We tabulate the results in the online appendix. In general, we do not observe the broad patterns in these artificial ratios using the bunching analysis, although there is some isolated bunching around certain percentages.³⁰ These results suggest that the irregular patterns of gross margins are unlikely to be driven by the Compustat rounding algorithms.

Although gross margins are less frequently forecasted than revenues and earnings, we observe 114,666 (10%) firm-quarters of our sample that have either management guidance or an analyst forecast of gross margins. To address the concern that the existence of these gross margin targets confounds our results, we remove these observations and rerun our tests in Tables 3 through 6, and our results hold. We also add an indicator variable for gross margin forecasts in our firm characteristic analyses and find no evidence that the presence of a gross margin forecast influences the likelihood of a highly round gross margin.

Our findings that revenue management is a potential channel for achieving highly round gross margins raise a concern that our results regarding gross margins may be driven by managers’ tendencies to report revenues just above base-10 thresholds, as documented by Stice et al. (2022). As in Stice et al. (2022), we define “round” revenues as those within 0.25% above a base-10 threshold. The Venn diagram and cross-tabulation in Fig. 4 show that only 1.46% of firm-quarters with round revenues overlap with our highly round gross margin group, which is similar to the percentage (1.32%) of non-round revenues that overlap with our highly round gross margin group. The result suggests that the irregular patterns of gross margins we find in this study are unlikely to be driven by managers’ tendencies to report round revenues.

Fig. 4

Intersection of highly round gross margins and round revenues. These Venn diagram and cross-tabulation frequency figures display the intersection of highly round gross margins and round total revenue numbers. As in Stice et al. (2022), we define “round” revenues as those equal to or within 0.25% above a base-10 threshold (i.e., second digit is zero)

Our sample includes firm-quarters with gross margins at or just above the zero threshold. Discontinuities around the zero threshold are well documented in the literature (Burgstahler and Dichev 1997). To address the concern that our results are driven by firms reporting gross margins just above zero, we exclude firm-quarters with gross margins smaller than 0.1% and, in untabulated analyses, find that our main results are qualitatively unchanged.

Finally, gross margin as a performance metric can be viewed as more important for certain industries (e.g., manufacturers) than for others (e.g., services). We control for industry effects by including Fama-French 12 industry fixed effects in our regressions, and our main results hold (untabulated). Additionally, when using two-digit SIC codes, business services (#73) accounts for the largest percentage of our sample. We again find that our main results (untabulated) are qualitatively unchanged when eliminating firms from this industry.

6 Conclusion

In this study, we find that distributions of reported gross margins exhibit irregular patterns. Specifically, we document that firms are more likely to report gross margins that are equal to or slightly above highly round percentage integers, compared to what would be predicted by counterfactual distributions. These results suggest that managers exhibit reference-dependent preferences for highly round gross margins and that highly round percentage cutoffs function as reference points, even though gross margins are typically not used as performance measures in formal contracting and have fewer explicit external benchmarks.

We then explore the motivations of these reference-dependent preferences in the context of financial reporting. Building on the goal-setting literature, we investigate whether firms reporting highly round gross margins exhibit characteristics consistent with these reference points being used as intrinsic goals. We find that firms reporting highly round gross margins show higher effort and productivity, tend to be smaller, and use highly round gross margin thresholds as ambitious goals, relative to their past performance and industry situation. We also find some evidence that firms with highly round gross margins compensate their CEOs with a higher portion of fixed income. Moreover, while we find that the highly round gross margin group has higher financial performance, these highly round gross margin reference points are not consistently linked with external rewards. Collectively, the evidence is consistent with the literature of goal-setting and suggests that these irregular patterns of gross margin distributions are reflective of highly round numbers serving as implicit psychological reference points and are driven by intrinsic motivations to improve performance.

Our study is subject to some important caveats. The irregular patterns of gross margins documented in this study are observed using gross margins calculated from sales and COGS numbers in Compustat, the latter of which uses proprietary adjustments. Although we have undertaken a number of approaches to mitigate these concerns, some gross margin values in our tests might still not agree with the precise values reported in financial statements, and it is possible that some of our highly round gross margins are still induced by Compustat rounding algorithms, which may confound our findings. Nevertheless, we believe our study offers important insights on the motivations of reference-dependent preferences for round numbers in the context of financial reporting, and our findings should be of interest to researchers, investors, and other users of financial statements.

Appendix A: Variable definitions

HighlyRound (HR) – An indicator variable taking a value of one if the firm’s gross margin for the quarter was within 0.1% above the zero, half, quarters, fifths, or tenths percentage cutoff marks.

Sales – Total revenue in the firm-quarter (Compustat variable SALEQ). We require firm-quarters to have non-missing and positive sales to be included in the sample.

Cost of goods sold – Cost of goods sold during the firm-quarter (Compustat variable COGSQ). We adjust Compustat’s proprietary measure of COGS using the method recommended in Bostwick et al. (2016). We require firm-quarters to have non-missing and positive cost of goods sold to be included in the sample.

Gross profit (GP) – Sales minus cost of goods sold. In regression specifications, we scale gross profit by market capitalization at the beginning of the quarter.

Gross margin – Gross profit divided by sales for the firm-quarter, expressed as a percentage.

Net income (NI) – Net income for the firm-quarter (Compustat item NIQ). In the regression specifications, we scale net income by market capitalization at the beginning of the quarter.

Other income (OI) – The non-gross profit component of the firm’s net income for the year, computed as the difference between net income (Compustat item NIQ) and gross profit. In regression specifications, we scale other income by market capitalization at the beginning of the quarter.

Market capitalization (MVE) – The firm’s market capitalization at the end of the quarter, computed as the number of shares outstanding at the end of the quarter (Compustat variable CSHOQ) multiplied by the firm’s stock price at the end of the quarter (Compustat variable PRCCQ).

SIZE – The logarithmic transformation of total assets (Compustat variable ATQ).

INVESTMENT – The sum of capital expenditures (calculated as the quarterly difference in Compustat item CAPXY) and research and development expense (Compustat item XRDQ), scaled by total revenue.

SALESEMP – Sales per employee productivity, calculated as total revenue (Compustat item SALEQ) divided by total employees (Compustat item EMP).

ABGM – The difference between gross margin in the quarter and the average gross margin over the last four quarters.

INDGROW – Year-over-year revenue growth for the firm’s Fama-French 12 industry.

FIXEDCEO – The percentage of total CEO compensation that is fixed, calculated as salary (Execucomp item SALARY) divided by total compensation (Execucomp item TDC1).

FutureSalesGrowth – The proportionate growth (change in logs) of total sales from quarter t − 4 through t − 1 to total sales from quarter t + 1 through t + 4.

TOTALCOMP – Log of total compensation for the CEO or all named executives (Execucomp item TDC1).

Book-to-market (B/M) – Total common stockholders’ equity (Compustat variable CEQQ) divided by market capitalization (Compustat variable CSHOQ multiplied by Compustat variable PRCCQ).

Leverage – The firm’s total long-term debt at the end of the quarter (Compustat item LTQ) divided by total assets at the end of the quarter (Compustat variable ATQ).

Return – Firm returns during the fiscal year, calculated using the CRSP daily returns file.

σ – Daily return volatility during the fiscal year, calculated using the CRSP daily returns file.

3-day CAR – Three-day cumulative abnormal (based on the CRSP value-weighted index) return centered on the earnings announcement date (Compustat variable RDQ).

ROA – Net income (Compustat variable NIQ) divided by assets (Compustat variable ATQ).

AbRev – Abnormal revenue, as measured in Stubben (20062010). Measured as the residual from a regression of the change in receivables on changes in revenue in the current and each of the three lagged quarters.

AbProd – Abnormal production costs, measured as the Roychowdhury (2006) production cost model residual.