Modeling Market Expectations of Profitability Mean Reversion: A Comparative Analysis of Adjustment Models

Abstract

This paper investigates how market expectations regarding profitability mean reversion are reflected in stock prices. We propose a model that infers implicit expectations of future earnings using publicly available share prices based on the assumption that markets efficiently incorporate forward-looking information. The study compares several adjustment models, including the classical partial adjustment framework and a mean reversion model, to identify the most suitable mechanism to capture the dynamics of expected earnings. Special attention is paid to the statistical characteristics of accounting data and ratio-based measures, which influence model performance. Using a dataset covering a twenty-year period, we find that the mean reversion model consistently outperforms partial adjustment models in explaining the behavior of cyclical and random components converging toward a long-term trend. The findings suggest that market prices embed rational expectations of profitability reversion, especially in periods of above average performance. These results align with previous research and provide a robust framework for understanding how earnings expectations are formed and adjusted in financial markets.

1. Introduction

The study of profitability and its tendency to return to the mean is a key topic in financial economics, which focuses on the predictability of earnings and the assessment of the value of firms. The importance of transitory earnings expectations for asset valuation is almost self-evident. Much of the nation’s wealth is concentrated in financial markets, and income ratings play an important role in the formation of stock prices. Income expectations also affect prices and are used in macroeconomic models. This paper analyzes the various adjustment models that are used to predict profitability returns, with an emphasis on comparing their effectiveness and accuracy.

Formulated similarly to Gropp (2004), the mean regression (MR) model was found to be more theoretically consistent and accurate than the partial adjustment (PA) models. PA models assume intentional adjustment, while MR does not, which better corresponds to the possibility of mechanical reversion to the mean (Jung & Kang, 2021). In addition, PA models provide overestimates of the adjustment speed, especially when there is no strong central tendency. This may bias the results of previous studies. Using a transient earnings valuation model, the expected rate of return is estimated and compared to ex post observed MR rates. The estimates in this work are based on expectations of abnormal (above average or below average) profitability, derived from commonly used market ratios (P/E, EV/EBIT). The findings show that investors do not expect a central tendency when abnormal earnings are small but expect a rapid reversion to the mean when deviations are large. Moreover, data deflation, according to Beatty et al. (1999), was found to increase the volatility of the results and the dependence on method and sample selection.

This study differs from prior research in three important ways. First, we rely on market-implied expectations derived from financial data, rather than analyst forecasts, to capture the collective perspective of investors. Second, we directly compare stochastic mean reversion and partial adjustment specifications, thereby assessing which framework better reflects actual profitability dynamics. Third, by evaluating the implications across a large and representative sample of US firms, we position our contribution within a broad empirical context. These features underscore the originality of our approach and its relevance to both academic research and practice.

2. Literature Review

Some companies can perform very well for long periods of time, but long-term above average or below average profitability is rather rare. This may be due to various factors, such as market anomalies or management systems (Fama & French, 2000). It is known that abnormal earnings are usually transitory, as suggested by Bliss (1923). Support for this view initially seemed weak, but it was later strengthened by many empirical studies, especially in the USA (Beaver, 1970; Freeman et al., 1982; Fama & French, 2000). The estimates based on market data for CTCT (convergence to central tendency, a term used to describe the phenomenon in which firm profitability returns to the average level within a given industry or market after a period of above- or below average returns) also support the Bliss hypothesis and includes developing with markets (Chung & Kim, 2002; N. Jiang & Kattuman, 2010; Zhou et al., 2022) as well as European market data (Allen & Salim, 2005; Altunbas¸s et al., 2008). Most estimates of the rate of reversion in the mean range between 25% and 50% per year. The question remains whether investors can predict the trajectory of that return.

Many studies show that financial indicators, especially profitability, tend to revert to the mean. Penman (1996) found that changes in dividends per share stabilize at a similar level for different portfolios, but profitability does not converge as much for portfolios based on book value. Some results are complicated, so you need to find out if profitability is predictable or moves randomly around the industry average.

If investors believe that profitability reverts to the mean in the long run, well-performing firms should have lower P/E, EV/EBIT, etc., and higher P/BV than their average peers, as found by Fairfield (1994). Modern studies show that valuation multiples and profitability ratios tend to revert to the mean. This paper examines expectations of future profitability based on market data, not analyst estimates, as these estimates can be biased and imprecise. Although much research focuses on analysts’ estimates, there is little evidence of direct investor expectations of mean reversion to profitability. If markets were truly projecting the reversion to the mean in security prices, we would have to assume that investors are valuing securities exactly according to analysts’ estimates.

In connection with the increasing complexity of the global financial environment, the analysis of profitability and its tendency to return to the mean is an indispensable tool to understand the dynamics of corporate performance. Partial adjustment models and other approaches to modeling the return to the mean in profitability provide valuable tools for analysts and investors trying to predict the future performance of firms and their market value. The following are three main methodological issues in proposing a model that are not discussed in sufficient detail in the current literature:

• PA models (assuming implicitly intentional adjustment) and autoregression or autocorrelation models (hereinafter AR or AC, assuming mechanical MR) are used in earnings MR measurement papers almost exclusively. There are difficulties in interpreting both.

  (a)

  The AR and AC model estimates on historical data generally assume a company-specific target. Unsurprisingly, Dechow et al. (1995) find approximately the same ACF(1) value of net income per share as Beaver (1970) did (38% compared to Beaver’s 32%). Due to the AR coefficient in simple OLS being ${\beta }_{t,t-1}=ACF\left(1\right){\sigma }_t/{\sigma }_{t-1}$, we see that earnings revert to the mean but do not converge ($\sigma \left(x_t\right)=\sigma \left(x_{t-1}\right)$). Unfortunately, the AR and AC models do not tell us where the target is, and we can obtain the same ACF(1) and AR(1) estimates for both growing and decreasing profits (or profitability).

  (b)

  Both PA and AC/AR models, in the way they were used for earnings MR measurement, mix company-specific and industry- or economy-wide factors. However, if one of the main uses of MR is the valuation of business and shares (pricing) by comparison to the actual valuation and profits of peers, then the markets are interested in knowing how the actual relative advantage or disadvantage influences share prices and what MR earnings rate they expect.

2.

Data is tricky (as it almost always is). Not only is there a need to deflate the earnings, assets, and/or share prices so that inflation and company size do not affect the results, but accounting ratios and earnings generally do not have the desirable (normal) distribution. This will be discussed in more detail in the part of this paper devoted to method selection. Some methods of data deflation can inflate them if the incorrect deflator is chosen.

3.

Some parts of the earnings are irrelevant to forecasting future profits. Accruals are a typical example, although share prices react to them.

This study follows up and complements previous research that highlights the central role of profitability dynamics in asset pricing and corporate valuation. In particular, Fama and French (2000) emphasize the predictability of earnings and profitability, while Novy-Marx (2013) demonstrates the importance of profitability for explaining stock returns. These contributions provide a solid foundation for our analysis of mean reversion and partial adjustment models.

The aim of the paper is to analyze and compare models of mean reversion in profitability to determine which of these approaches best explains the processes associated with the adjustment of profitability in publicly traded companies following random shocks. The study focuses on testing the accuracy of the mean reversion (MR) model compared to partial adjustment (PA) models and examines whether the MR model better reflects market expectations regarding the adjustment of above average or below average profitability of companies. The paper also investigates the impact of the magnitude of profitability abnormalities on the speed of mean reversion and evaluates the ability of investors to predict both the pace and nature of this process at the individual company level. Additionally, the aim is to verify the robustness of results based on non-deflated and deflated data and provide recommendations for effective modeling of market expectations and predictions of profitability trends.

The main contributions of this paper are as follows: (1) we provide a comprehensive comparison of mean reversion (MR) and partial adjustment (PA) models in capturing profitability dynamics, demonstrating that the MR model offers superior accuracy and theoretical consistency; (2) we show that the magnitude of profitability deviations has a significant effect on the speed of reversion, with larger deviations converging more rapidly; (3) we document that investors’ expectations are systematically inaccurate at the firm level for small abnormalities, while they correctly anticipate rapid mean reversion in the case of large deviations; and (4) we evaluate the robustness of the results under different data treatments, showing that non-deflated data yield more stable results than deflated data, thus providing methodological guidance for future empirical research.

Based on the aim of the paper, the following hypotheses are formulated:

H1. 

The mean reversion (MR) provides more accurate results in analyzing profitability than partial adjustment (PA) models, regardless of the cyclical, random, or medium-weighted components of the process.

H2. 

The magnitude of profitability abnormalities (large deviations from the mean) has a direct impact on the speed of mean reversion: the larger the abnormality, the faster the reversion.

H3. 

Investor expectations regarding the mean reversion of profitability are systematically inaccurate at the individual firm level, especially for small abnormalities. However, for large deviations, investors correctly anticipate that the return to the mean will occur within a few years.

H4. 

Deflating profitability by the group average increases the volatility of estimates, while non-deflated data provides more robust results, combining expectations about MR and market size.

The paper is organized as follows. First, a model will be proposed that describes the relationships between valuation multiples (P/E, EV/EBIT) and expected transitory earnings (above average or below average) to decompose the usual valuation models. The alternative model specifications will be analyzed, and the MR and PA models will be compared in terms of bias and accuracy. The following is a description of the data used and their necessary adjustments, as well as a description of the statistical approaches used to obtain the results. The “Results” section compares the estimated implicit expectations with the observed ex post MR. Due to the large number of regressions examined, only the most important regression coefficient estimates will be shown in the results. A summary of the results is the last part of this paper. The structure of the paper contains a lot of technical descriptions and a discussion of the methods used. It can be challenging, but it is important to include it for understanding. There are many parameters and methods, and each change can significantly affect the results.

2.1. Model

The main assumptions of our model (above the common ones) are as follows:

• Investors use income-based valuation of shares;
• Capital costs are similar across peers (see, e.g., Hamada, 1972);
• The target of MR is industry-specified peer group average, as the industry pertinence explains part of future earnings (e.g., Lev (1969) or Baber et al. (1999)).

We can relax assumptions such as an efficient market, a random walk in earnings, or investor myopia. A company’s profitability can also be affected by individual factors (Spanos et al., 2004).

Market ratios like EV/EBIT (enterprise value to earnings before interest and tax) or P/E (market capitalization to net income) can be rewritten as a sum of discounted future earnings. Let $d$ be the deflator, which can be either earnings $e$ (EBIT, or EAT—net income), $S$ sales, or $B$ book value of equity, assets, or invested capital. It is denoted $X$, the market price of a set of capital (either equity or invested capital). Let us have a company whose profitability differs from the industry average. If investors expected the individual profitability to converge, then there would be a difference between the “normal” (industry-average) market ratio denoted by lower index $N$ and the individual one denoted by lower index 0, so that $\exists \left(X_0/d_0\ne X_N/d_N\right)$, where $d$ means deflator attributable to that set of capital $X$ (enterprise value => EBIT, asset value, or invested capital; equity market price => equity book value or EAT). The abnormal part of $X_0/d_0$ is

$$X_A/d_N=X_0/d_N-X_N/d_N.$$

(1)

To utilize (1) for the estimation of MR expectations, it is needed to assume that the required rate of return $i$ on capital $X$ is constant (Qi, 2010, p. 172). Moreover, there must be a long-term “normal” profit growth rate $g_N$ for the particular industry. Then

$$X_N/d_N=e_N/d_N{\sum }_{t=1}^{\infty }{\left[\left(1+g_N\right)/\left(1+i\right)\right]}^t,$$

(2)

where $t$ denotes time. Let us substitute

$$\left(1+g_N\right)/\left(1+i\right)\equiv k,$$

(3)

For commonly assumed $t\in \left\{1..\infty \right\}$ and $k\in \left(0;1\right)$. Equation (2) turns, after the substitution of (3), into

$$X_N/d_N=e_{N}k/d_N\left(1-k\right).$$

(4)

If profit differs from the “normal” level, i.e., $e_0\ne e_N$, then there would be an abnormal performance, $e_A=e_0-e_N;e_A\ne 0$, which is expected to perish, i.e., to grow by a growth rate different from $g_N$. Profit expectations are not stationary due to inflation; thus, $g_N>0$. Therefore, profitability or the ratio of extraordinary to “normal” profits is a better choice than profit for any statistical analysis due to the elimination of the heteroskedasticity of valuation errors (Beatty et al., 1999). The expectation of the adjustment process can be described as follows:

$$e_{A,t}/d_{N,t}=e_{A,t-n}/d_{N,t-n}{\left(1-\alpha \right)}^n$$

(5)

from an ex ante point of view. The variables $t$ and $n$ are natural numbers representing discrete time, and $-\alpha ; \alpha \ge 0$ is $e_A$ growth rate. Abnormal earnings $e_A$ grow by $g_N$ in addition to $-\alpha$, so that $e_{A,t+n}=e_{A,t}\times {\left(1-\alpha \right)}^n\times {\left(1+g_N\right)}^n$. To simplify further equations, time-related indexes will be omitted. If not explicitly stated otherwise, the variables relate to the same period. Equations (1), (2), (4) and (5) yield, after some adjustments, the following:

$$X_A/d_N=e_A/d_N\left(1-\alpha \right)k/\left[1-\left(1-\alpha \right)k\right],$$

(6)

which can be used to define the regression model

$$X_A/d_N={\beta }_0+{\beta }_1{e}_A/d_N+u_1.$$

(7)

Although the meaning of ${\beta }_0$ is doubtful in (7), a model without a constant could produce biased results. Let us further utilize (1) so that ${\beta }_0\equiv X_N/e_N$ and specify (7) as

$$X_0/d_N={\beta }_0+{\beta }_1{e}_A/d_N+u_1.$$

(8)

However, Equation (8) conflicts with practical applicability. It produces some negative ${\beta }_1$ (should be positive, according to (6)) on the data sample described further in the text, so it must go by (7).

It is estimated that the industry-average $X_N/d_N$ with the weights is naturally the market capitalization (averaging P/E) or enterprise value (averaging EV/EBIT), i.e., $w_i=X_i$, so that

$$X_N/d_N={\sum }_i{w}_i/{\sum }_i{\displaystyle \frac{d_i{w}_i}{X_i}}={\sum }_i{X}_i/{\sum }_i{d}_i.$$

(9)

Beatty et al. (1999) found that the weighted harmonic mean is a better estimate of industry-average multiples compared to other averages. When these averages are calculated over several years, the regression accounts for economic cycles and how extraordinary conditions of a specific company influence its “abnormal” profitability. In contrast, if “normal” levels are calculated using data from just one year, the results are not influenced by the convergence towards long-term averages. It is also needed to obtain $e_N$ in $e_A/d_N$. In this case, weights are book equity (averaging net income) or invested capital (averaging EBIT). The average is then the “normal” profitability (weighted average profitability in the peer group) times the book value of the capital ($B_0$) of the company examined in the given accounting year

$$d_N=B_0{\sum }_i{d}_i/{\sum }_i{B}_i.$$

(10)

2.2. Discussion of Alternative Model Specifications

A.

Useable Accounting Measurements

The use of Equation (10) to determine $e_N$ in (7) means that it is the tested return on equity (ROE) or the return on invested capital (ROIC). But, what if profit margins converge? Then, it would be better to use sales as weights as follows:

$$d_N=S_0{\sum }_i{d}_i/{\sum }_i{S}_i.$$

(11)

Averaging with (11) helps to avoid records with negative $e_N$ and $X_N$ because sales cannot be negative, but book equity can. Revenues are also not subject to the creation of hidden reserves, contrary to equity or invested capital. But, even the use of (11) did not avoid $X_A/d_A<0$, which means diverging profitability. A solution can be a restatement of model (6) to base it on residual income, as the reason for $X_A/d_A<0$ might be a difference in asset book values among companies, which is not captured by (6). With the assumption that long-term average profitability equals cost of capital, Equation (6) turns into

$$\left(X_0-B_0\right)/d_N=e_A/d_N\left(1-\alpha \right)k/\left[1-\left(1-\alpha \right)k\right]$$

(12)

as residual income is equal to the extraordinary (abnormal) profit $e_A$. The added market value is then a multiple of EAT or EBIT.

Due to the strong influence of accruals on the P/E and EV/EBIT ratios (Sloan (1996) or Coulon (2020)), the use of cash flow from operations (CFO) instead of EBIT and EAT (net income) could reduce the number of negative $X_A/d_A$. C. S. A. Cheng and Thomas (2006) find the CFO and abnormal accruals to be the best predictors of future earnings. Although the definition of abnormal accruals in this paper is necessarily different from theirs, our abnormal accruals are derived by using the CFO and its components at the places $e_0$, $e_N$, and $e_A$, which closely resembles the industry model tested by Dechow et al. (1995). Therefore, the CFO will be tested as an alternative to EAT and EBIT. Another way to cope with accruals is shown by Amir et al. (2012), who, contrary to most other earnings MR papers, use permanent earnings (estimated as a 4-year average margin multiplied by actual sales).

B.

Mechanical mean reversion versus partial adjustment

A partial adjustment model could be an alternative to (5). Because we do not measure adjustments of expectations but rather expected adjustment of profits, the adaptive expectations model is not applicable, even though both models are very similar on their basic levels (Waud, 1968).

Lev (1969), Fama and French (2000), and F. Cheng et al. (2020) apply the PA models in predicting. The methodology used by Fama and French (2000) and further by N. Jiang and Kattuman (2010) or Liu et al. (2018) comes in basic shapes from Lev (1969).

If some issues in Fama and French (2000) and their followers are omitted (such as the target for PA being set so that earnings and the target must converge unless investors are completely irrational), the methodology of Lev (1969) and his followers should be considered. There are several reasons for this. Firstly, the estimation of the adjustment factor in the PA model

$$e_{0,t}-e_{0,t-1}=\beta \left(e_{N,t}-e_{0,t-1}\right),$$

(13)

which is a reformulation of the weighted average

$$e_{0,t}=\beta e_{N,t}+\left(1-\beta \right)e_{0,t-1},$$

(14)

can produce significant positive $\widehat{\beta }=est\left(\beta \right)$, even without MR. By the nature of covariance, to obtain significant positive $\widehat{\beta }$, it is sufficient if $e_{0,t}$ and $e_{N,t}$ have some common factors behind their behavior. Even without a correlation between $e_{0,t}$ and $e_{0,t-1}$, respectively, between $e_{N,t}$ and $e_{0,t-1}$, and with a portfolio of differently sized companies, we obtain statistically significant $0<\widehat{\beta }<1$ without any PA present. Secondly, (13) assumes intentional adjustment towards the expected target $e_{N,t}$, despite economic subjects that do not know, at the start of the t-th period, what the industry average $e_{N,t}$ will be in the end. A better specification in an uncertain world would be

$$e_{0,t}-e_{0,t-1}=\beta \left(e_{N,t-1}-e_{0,t-1}\right),$$

(15)

instead of (13). Measures of financial uncertainty are developed precisely based on information from financial markets (Y. Jiang et al., 2024).

However, let us note that (15) is also derived in later stages of the (Lev, 1969) paper. Equation (15), after rearrangement, yields

$$e_{0,t}=\beta e_{N,t-1}+\left(1-\beta \right)e_{0,t-1}.$$

(16)

If one wanted to study only the company-specific part of MR, one would need a model

$$e_{0,t}-e_{N,t}=\beta \left(e_{0,t-1}-e_{N,t-1}\right).$$

(17)

similar to Gropp (2004). Equation (17) does not imply intentional adjustment to the last known optimum, nor to the expected optimum. It assumes unintentional stochastic MR, which courses due to the nature of the underlying process, e.g., competition in industry.

Recent studies reinforce that profitability and earnings generally exhibit mean reversion and that model choice matters for how this mechanism is captured. For example, Agliardi et al. (2024) document mean reversion in US firm earnings and link it to dynamic capital structure choices, while broader surveys show that partial adjustment specifications remain widely used but often require extensions to capture heterogeneity in adjustment speeds (Nguyen et al., 2023). Against this backdrop, our comparison of stochastic mean reversion (MR) and partial adjustment (PA) contributes by showing which framework better aligns with observed profitability dynamics.

3. Data Description and Variables Constructed

Here, we compared the accuracy of (13), (15), (17), and ACF(1) of $e_A/d_N$ and ${\alpha }_{real}=1-\sqrt[t]{{\left(e_A/d_N\right)}_t/{\left(e_A/d_N\right)}_0}; t=1; d_N=e_N$ in explaining MR or PA. The method used is a simulation, three companies with shares 60%, 30%, and 10% in the industry form a peer group, where profits are generated by a process similar to the one in practice, including a cyclical component $c_t$, a random component ${\epsilon }_t~\mathrm{N}\left[0;14\right]$, and a mean reversion component $m_t$. The generating processes G1–G4 are as follows:

(a)

G1: $e_{0,t}=c_t+m_t+{\epsilon }_t$, MR (resp. PA) term $m_t=0$, and cyclical component $c_t=10 \mathit{sin}(t/2)$.

(b)

G2 MR: $e_{0,t}$ as in G1; however, $m_t=\beta \left(e_{0,t-1}-e_{N,t-1}\right)$, where $\beta$ is the MR rate.

(c)

G3 PA (Lev, 1969): $m_t=\left(1-\beta \right)\left(e_{0,t-1}-c_t\right)$, $e_{0,t}$ as in model G1.

The equation of $m_t$ is obtained by the deduction of $c_t$ from both sides of (13), where $c_t$ is an estimate of $e_{N,t}$.

(d)

G4 PA (Waud, 1968): $e_{0,t}=c_t-c_{t-1}+m_t+{\epsilon }_t$ and $m_t=\beta e_{N,t-1}+\left(1-\beta \right)e_{0,t-1}$.

We examined four scenarios, (1) $\beta =0.2$, (2) $\beta =0.8$, (3) without the cyclical component, i.e., $c_t=1$ instead of $c_t=10 \mathit{sin}(t/2)$, and finally (4) $c_t=20$, to compare the results under different shares of $m_t$ and $c_t$ on $e_{0,t}$.

The length of the simulated time series is 97 periods, and the length of the cycle in the simulation models is about 12 periods (cp. $c_t=10 \mathit{sin}(t/2)$). Industry profit is a weighted average of individual profits, with weights equal to shares in the industry. A total of 100 simulations is conducted. The baseline number of simulations in this study was set to 100 replications, which is consistent with established methodological practice in the bootstrap literature. Efron and Tibshirani (1993, Chapter 6) explicitly note that for the purpose of estimating standard errors, between 50 and 200 bootstrap replications are usually sufficient. Thus, the choice of 100 replications as a starting point is in line with widely accepted conventions. Nevertheless, to ensure the robustness of our results, we increased the number of replications beyond this baseline. The findings remained virtually identical to those obtained with 100 replications, with only a minor reduction in standard errors and improved numerical stability. This confirms that our main conclusions are not an artefact of limited sampling variability but remain stable, even when the number of simulations is substantially increased. These simulation settings are not presented merely as technical exercises; rather, they are designed to mirror the conceptual motivation of the paper, namely, to assess whether different adjustment models can realistically reflect the way investors form expectations about profitability mean reversion.

We used data provided by CAPITAL IQ (Standard and Poor’s) on companies listed in the US Russel 3000. The data consists of financial results spanning 20 years, covering at least two economic cycles. That totals 55,760 company-years, of which 31,729 company-years have complete records needed for evaluation of (21). As Alford (1992) or Schreiner (2007) suggest, they are used to narrow the specification of peer groups compared to the industry. The peer groups are formed according to the four-digit ISIC Rev. 4 classification. The variables (quotation with respect to CAPITAL IQ) are as follows:

- Primary SIC code. - EBIT (calendar year Y). - Net income (calendar year Y). - Total equity (end of calendar year Y). - Total debt (end of calendar year Y).

- Cash from operations (calendar year Y). - Market capitalization on 31st March Y + 1. - Total revenue (calendar year Y). - EBITDA (calendar year Y).

4. Empirical Analysis

Figure 1, Figure 2 and Figure 3 provide an example of G2 to G4 simulation outputs (G1 lacks a cyclical component). Due to the almost eliminated cyclical component in model G3, it does not reproduce cycles well under the low MR. G4 quite frequently produces systematically growing or decreasing profitability despite well reflecting the cyclical component (Figure 1, Figure 2 and Figure 3 how some exemplary cases). G2 can produce long-term above or under performance as well as G4, but in 100 simulations, it did not produce long-term (over several cycles) increasing or decreasing results compared to G4. Each line in the graph represents one of the three simulated companies.

Table 1. Simulation tests of different specifications of the MR and PA models. Average results of a 100-round simulation of a 100-period time series of G1, G2, G3, and G4 processes. ACF(1) is the first-order autocorrelation and $\widehat{\beta }$ is the regression estimate of $\beta$ from simulated data in Equations (13), (15), and (17). c_t denotes the cyclical component of the generating process. Standard deviations are in parentheses.

$\mathit{\beta }=0.8,{\mathit{c}}_{\mathit{t}}=10 {\mathit{s}}_{\mathit{t}}\mathit{s}\mathit{i}\mathit{n}(\mathit{t}/2)$	${\mathit{e}}_0$ ACF(1)	$\widehat{\mathit{\beta }}$ in (13)	$\widehat{\mathit{\beta }}$ in (15)	$\widehat{\mathit{\beta }}$ in (17) (MR)	${\mathit{e}}_{\mathit{A}}/{\mathit{e}}_{\mathit{N}}$ ACF(1)	${\mathit{\alpha }}_{\mathit{r}\mathit{e}\mathit{a}\mathit{l}}$ Median
G1	0.699	0.948	0.837	0.000	−0.004	0.982
	(0.023)	(0.031)	(0.089)	(0.069)	(0.058)	(0.068)
G2	0.809	0.539	0.151	0.791	0.030	0.474
	(0.018)	(0.075)	(0.049)	(0.047)	(0.083)	(0.108)
G3	0.726	0.880	0.669	0.195	−0.004	0.917
	(0.022)	(0.033)	(0.087)	(0.070)	(0.091)	(0.075)
G4	0.879	0.866	0.797	0.207	0.001	0.858
	(0.048)	(0.032)	(0.087)	(0.072)	(0.111)	(0.085)
$\beta =0.2,c_t=10 s_t\mathit{sin}(t/2)$	$e_0$ ACF(1)	$\widehat{\beta }$ in (13)	$\widehat{\beta }$ in (15)	$\widehat{\beta }$ in (17) (MR)	$e_A/e_N$ ACF(1)	${\alpha }_{real}$ median
G1	0.693	0.945	0.844	−0.007	0.008	0.987
	(0.024)	(0.025)	(0.093)	(0.075)	(0.070)	(0.062)
G2	0.727	0.886	0.631	0.219	0.001	0.905
	(0.021)	(0.032)	(0.077)	(0.071)	(0.053)	(0.082)
G3	0.815	0.338	0.181	0.789	0.002	0.461
	(0.025)	(0.053)	(0.038)	(0.043)	(0.108)	(0.116)
G4	0.911	0.481	0.203	0.790	0.082	0.354
	(0.034)	(0.068)	(0.054)	(0.045)	(0.174)	(0.101)
$\beta =0.8,c_t=1$ (no cycle)	$e_0$ ACF(1)	$\widehat{\beta }$ in (13)	$\widehat{\beta }$ in (15)	$\widehat{\beta }$ in (17) (MR)	$e_A/e_N$ ACF(1)	${\alpha }_{real}$ median
G1	−0.003	0.916	0.844	−0.009	0.002	0.966
	(0.063)	(0.040)	(0.072)	(0.072)	(0.060)	(0.055)
G2	0.673	0.393	0.151	0.788	0.006	0.904
	(0.057)	(0.062)	(0.039)	(0.045)	(0.072)	(0.091)
G3	0.192	0.805	0.673	0.196	−0.007	0.924
	(0.068)	(0.050)	(0.079)	(0.082)	(0.056)	(0.075)
G4	0.866	0.781	0.805	0.186	0.042	0.860
	(0.080)	(0.046)	(0.063)	(0.068)	(0.116)	(0.089)
$\beta =0.2,c_t=1$ (no cycle)	$e_0$ ACF(1)	$\widehat{\beta }$ in (13)	$\widehat{\beta }$ in (15)	$\widehat{\beta }$ in (17) (MR)	$e_A/e_N$ ACF(1)	${\alpha }_{real}$ median
G1	0.009	0.905	0.831	0.011	0.006	0.960
	(0.060)	(0.042)	(0.078)	(0.073)	(0.041)	(0.059)
G2	0.124	0.832	0.661	0.188	0.011	0.949
	(0.066)	(0.046)	(0.081)	(0.078)	(0.054)	(0.070)
G3	0.784	0.304	0.178	0.785	0.023	0.470
	(0.037)	(0.050)	(0.035)	(0.042)	(0.077)	(0.145)
G4	0.910	0.301	0.207	0.787	0.176	0.282
	(0.047)	(0.052)	(0.043)	(0.044)	(0.256)	(0.094)
$\beta$ = 0.2, $c_t=20$ (no cycle)	$e_0$ ACF(1)	$\widehat{\beta }$ in (13)	$\widehat{\beta }$ in (15)	$\widehat{\beta }$ in (17) (MR)	$e_A/e_N$ ACF(1)	${\alpha }_{real}$ median
G1	−0.011	0.924	0.855	−0.013	−0.013	0.998
	(0.058)	(0.039)	(0.076)	(0.072)	(0.068)	(0.107)
G2	0.679	0.387	0.148	0.790	0.760	0.214
	(0.060)	(0.066)	(0.035)	(0.045)	(0.068)	(0.068)
G3	0.193	0.808	0.679	0.194	0.185	0.812
	(0.049)	(0.040)	(0.071)	(0.063)	(0.068)	(0.090)
G4	0.844	0.780	0.812	0.189	0.121	0.814
	(0.096)	(0.047)	(0.068)	(0.075)	(0.134)	(0.101)

Source: own processing.

summarizes the results of the simulation and shows that $\widehat{\beta }$ in PA Equation (13) (Lev, 1969) and Equation (15) (Waud, 1968) move in the opposite way to $\widehat{\beta }$ in MR (17). That is, of course, in line with the specification of the models. High $\widehat{\beta }$ means fast PA in (13) and (15), but slow MR in (17). Lev’s (1969) PA with a priori knowledge of the future optimal target overestimates the adjustment speed (regression $\widehat{\beta }$), mainly in the case of slow adjustments (or high MR), but Waud’s (1968) PA (adjustment towards the last known optimum) underestimates $\widehat{\beta }$ at all generating processes, except for the case of data generated by Waud’s (1968) PA process (G4). Ex post observed ${\alpha }_{real}$ (20) overestimates the MR in four out of five simulation settings. The use of a PA model is justifiable for the assessment of capital structure policy because it can be intentionally influenced by management more easily than profitability. ACF(1) of $e_A/e_N$, though being just a ratio of (17) and $e_N$, does not explain the simulated data as well, probably due to some low $e_N$ inflating the $e_A/e_N$ ratio and randomly but significantly decreasing the ACF(1). An increase in profit means to a level at which $e_N$ is not spoiled by a small number of companies in the industry and proximity to zero (i.e., $c_t=20$) shows a significant improvement of $e_A/e_N$ ACF(1) and ${\alpha }_{real}$ explanatory power towards the simulated MR and adjustment rates.

MR Equation (17) is the most accurate of the examined measures to express the MR, with low, high, or even without any CTCT of the generating process, even if it is based on PA. Therefore, (5) and derived regression equations, which are just applications of (17), have lower errors than the PA models, especially if there is no systematic CTCT and the examined variable just randomly fluctuates around its mean.

To adjust our model to the findings about low $e_N$ inflating the $e_A/e_N$ ratio, it is alternatively specified (7) for the expected MR with constant ${\beta }_0$ and coefficient ${\beta }_1$ as follows:

$$X_A={\beta }_0+{\beta }_1{e}_A+u_1,$$

(18)

That makes the estimates more vulnerable to heteroskedasticity, but less vulnerable to the inflationary influence of $e_N$ close to zero. To avoid the problems concerning PA process regressions and to be able to compare implied expectations $\alpha$ in (6) as estimated directly (quotation ${\alpha }_{ind}$) reflecting (4), we used the observed speed of MR ${\alpha }_{real}$—in time $t$ vs. time 0

$${\alpha }_{ind}=\left(X_N/e_N-X_A/e_A\right)/\left[X_N/e_N\times \left(1+X_A/e_A\right)\right],$$

(19)

or ex post, where ${\alpha }_{real}$ is as follows:

$${\alpha }_{real}=1-\sqrt[t]{{\left(e_A/d_N\right)}_t/{\left(e_A/d_N\right)}_0}.$$

(20)

4.1. Data Characteristics That Can Spoil the Results

Due to the strong influence of accruals on the P/E and EV/EBIT ratios (Sloan, 1996, or Coulon, 2020), the use of cash flow from operations (CFO) instead of EBIT and EAT (net income) or the complementary use of abnormal accruals could increase the predictive power of profitability with respect to share pricing (C. S. A. Cheng & Thomas, 2006). Nevertheless, Chung and Kim (2002) state that share prices are better explained by earnings than cash flows or dividends, and Dechow et al. (2008) do not find the accrual anomaly to be the result of accruals but rather the result of investors’ inability to reflect diminishing returns on investment and the temptation of companies to accumulate excessive positive cash flows. Accrual bias induced by accounting rules, which unfortunately grows in time (Givoly & Hayn, 2000), does not improve earnings and cash flow forecasts. Coulon (2020) discusses the influence of market-value-based profitability ratios and highlights that accruals can distort these metrics, making cash flow a more reliable indicator for valuation purposes. Similarly, Beltrame and Sclip (2023) emphasize that EV/EBIT ratios, which account for depreciation and amortization, provide a clearer picture of the value of a company, especially for capital-intensive businesses.

There are many studies that deal with the distribution of accounting variables and their ratios. In general, a normal distribution is found to be rare. However, not much is known about residuals. Can we learn anything about residuals from the distribution of endogenous and exogenous variables? As for profitability, its distribution should be the same in subsequent periods, because in estimating one ACF (AR coefficient) for the whole period, we have no other choice if the errors are to be independent and identically distributed.

4.2. Factors Influencing the MR

Factors that can affect expected individual normal levels of profitability include size, age, shake-out, and turbulence (Sutton, 1997). Size and age effects were also found by Dunne and Hughes (1994), but, for example, Hart and Oulton (1996) found opposite results. Not only growth factors but also risk factors can affect how abnormal earnings contribute to the value of capital. Size and expected profitability significantly affect the stock market premium. Among the many studies that deal with size and price-to-book value anomalies are, for example, Fama and French (2000), who focused on long-term US data, Drew et al. (2003), who described these effects on the Shanghai Stock Exchange market, and Azevedo et al. (2023), who analyzed the predictive capabilities of machine learning-based models and found that a combination of these models showed significant monthly returns, reinforcing the view that these anomalies are important for understanding stock price movements. The size and price-to-book value anomalies appear to be local rather than global, according to Griffin (2002). This review provides a broader view of the dynamics between size, age, risk, and other factors that affect corporate profitability and market value, highlighting the importance of size and price-to-book value anomalies in different contexts and markets.

Another factor that could influence the expected mean reversion (MR) of profitability is the market sentiment, including the presence of bubbles and crashes. Azevedo and Hoegner (2023) analyzed the predictability of 299 capital market anomalies. Andleeb and Hassan (2023) analyzed the effect of investor sentiment on the returns of the selected developing equity markets. Smith et al. (1988) demonstrated that bubbles and crises tend to emerge in stock markets under certain conditions. Frankel (2008) found that the occurrence of bubbles and crashes is influenced by the adaptive expectations of naïve traders. His theory aligns with Siegel’s (1992) findings that the spread between optimistic and pessimistic forecasts significantly narrows during optimistic periods (due to less pessimistic forecasts for poorly performing companies), while this spread widens in post-crash periods. Wang (2024) assessed the impact of investor sentiment on stock market returns in 30 international stock markets overnight and intraday. Shiller et al. (1996) showed through survey data from the USA and Japan that a significant source of stock market bubble growth is the confidence that stocks will recover their losses, indicating overoptimism and overvaluation during booms and the opposite during busts.

Equations (10) and (11) implicitly assume that the growth of the variable, which constitutes weights (sales, equity, or invested capital), is the same across the peer group. That is far from true—some companies gain, and some lose market share. A high past growth may form expectations of high future growth, which could increase $\alpha$, and vice versa.

Another factor that can influence $\alpha$, is the size of abnormal profitability. Souza et al. (2018) noted that significant abnormal returns tend to revert to the mean, especially when considering the performance of smaller firms versus larger firms. These findings suggest that abnormal profitability, whether higher or lower, tends to normalize more quickly when deviations are more pronounced. Leverage can influence the MR rate of net income. According to MacKay and Phillips (2005), leverage is a peer-group-specific factor.

There are some problems with measuring the influence of the above factors on the expected MR rate $\alpha$. First, it is impossible to separate the influence of these factors on the cost of capital ($i$ in (2)) and on MR expectations. That is mainly the case of size, P/BV, leverage, and optimism factors, but competitive advantage is also cited in some textbooks as a qualitative criterion for cost of equity estimation. With respect to competitive advantage, the question of how to measure it arises.

To summarize, the influence of the following variables on $\alpha$ is examined:

-

Company share of peer group revenues in the particular year (intensity of competition).

-

The ratio of EBITDA margin to industry-average EBITDA margin in the particular year.

-

Company equity shares of the total capitalization of the local stock market (size factor).

-

The company ratio of book-to-market equity relative to the average book-to-market equity in the particular peer group and year (HML).

-

The ratio of company leverage to the peer group average leverage in the particular year.

-

The influence of investors’ mood is measured by the OECD composite leading indicator (CLI) minus 100 (the baseline value is 100 at the OECD CLI) at the account date + 3 months.

-

Revenue abnormal momentum, defined as the annual excess of the growth rate of sales over the growth rate of sales of the industry over the past 3 years.

These variables extend models (7) and (18) so that they reflect these exogenous factors as follows:

$$X_A={\beta }_0+{\beta }_1{e}_A+\overrightarrow{{\beta }_2} \overrightarrow{s}+u_2,$$

(21)

$$X_A/e_N={\beta }_0+{\beta }_1 e_A/e_N+\overrightarrow{{\beta }_2} \overrightarrow{s}+u_2,$$

(22)

where $\overrightarrow{{\beta }_2}$ is the vector of regression coefficients and $\overrightarrow{s}$ is the vector of the secondary exogenous variables as described above.

Using Hamada’s (1972) findings and the quite common practice of using industry-average data for the estimation of cost of capital, such as using (4), it can be estimated what the results of (21) or (22) regressions would mean for $\alpha$ (regression-derived ${\alpha }_{reg}$)

$${\alpha }_{reg}=\left[X_N/e_N-]\left({\beta }_1+\overrightarrow{{\beta }_2} \overrightarrow{s}\right)\right]/\left[X_N/e_N\left(1+{\beta }_1+\overrightarrow{{\beta }_2} \overrightarrow{s}\right)\right].$$

(23)

5. Results and Discussion

There are two possible ways to compute the $d_N$ and $X_N$ with respect to time: either for each year separately or over the whole time span of the sample. Here, both possibilities are examined to measure both the company-specific component of MR and common MR caused by the stock market cycle and the company-specific component. If $d_N$ and $X_N$ are computed over all 20 years in the sample, then there is a problem that investors cannot know the future performance.

Many observations have $X_A/e_A<0$ (see

Table 2. Share of records with $X_A/e_A<0$ on the whole sample.

Weights	Industry Averages (9), (10), and (11)	Share of ${\mathit{X}}_{\mathit{A}}/{\mathit{e}}_{\mathit{A}}<0$ in the Whole Sample
		P/E	EV/EBIT	EV/CFO	MV/EAT	MV/EBIT
Equity, resp. EV if X = EV	Over years	34.33%	31.45%	35.96%	47.45%	49.75%
	Each year separately	31.94%	28.83%	34.09%	51.79%	51.82%
Sales	Over years	33.88%	33.10%	35.80%	47.45%	52.25%
	Each year separately	31.29%	30.40%	33.52%	50.91%	52.61%

CFO = cash from operations, EV = enterprise value, MV = estimated market value added as market capitalization minus book equity. Source: own processing.

), which means divergent profitability in (7) or (21). To avoid (21) estimate distortions, a symmetric tail cut-off is necessary if records with $X_A/e_A<0$ were dropped. Table 2 shows that if the weights in the industry for $X_N$ and $e_N$ estimates are equity or sales, $X_0/e_0$, $X_N/e_N$, and $X_A/e_A$ at the level of P/E and EV/EBIT produce the smallest shares of $X_A/e_A<0$ in the data sample. The share of $X_A/e_A<0$ still improves if averages are estimated for each year separately. The favorite choices are equity-weighted (or enterprise value-weighted) $X_N$ and $e_N$ and $X/e$ at the level of P/E or EV/EBIT because of the smallest share of $X_A/e_A<0$.

The anomaly of $X_A/e_A<0$ can be caused by profitability expectations. Nevertheless, empirical findings (Fairfield (1994) or Penman (1996)) do not indicate that, on average, any of the combinations of P/E and P/BV shares would have diverging profitability. Among the possible explanations is also accounting conservatism (Shroff, 1995) or overoptimism of investors in young, small, or distressed companies (Baker et al., 2008). However, there is no significant difference between company size or company age in the records with $X_A/e_A<0$ and the rest, albeit $X_A/e_A<0$, are more frequent at small companies in our sample. However, neither the use of additional exogenous variables nor quantile regression resolved the problem with negative $X_A/e_A$. A reduction in the sample by groups with few observations (fewer than 5 per SIC year or fewer than 100 per four-digit SIC) increases the share of observations with $X_A/e_A<0$ in the rest of the sample.

Another examined possible reason is the influence of accruals on earnings. To filter out that problem, EBIT would be more appropriate than EAT under CapitalIQ definitions, which follow the standard income statement form. EBIT is not influenced by earnings from discontinued operations, extraordinary items, and accounting changes, contrary to EAT. However, Table 2 shows that even with $X_A/e_A$ at the level of EV/EBIT, there are too many observations unexplained by earnings mean reversion. Another way to cope with accruals is shown by Amir et al. (2012), who, contrary to most other earnings, found that MR papers use permanent earnings estimated as a 4-year average margin multiplied by actual sales.

5.1. Estimation Method

To obtain estimates of (7), (18), (21), and (22) robust to changes in the data sample and extreme observations, the following approaches are used:

-

LAD (least absolute deviation) regression was used in addition to OLS with HC1 robust errors. LAD is more suitable than OLS (ordinary least squares) or panels with fixed effects in the case of models with large outliers, which emerge naturally in (7) or (22) due to the attempt to avoid heteroskedasticity. LAD models are less sensitive to outliers compared to OLS (Dielman, 2005). LAD models are also more robust to changes in model specification and to data changes than OLS models.

-

In addition to that, we can control the sample for inflated data by excluding SIC groups (or SIC years) that either have variation coefficients of examined earnings $e_0$ in the top percentiles or have low (inflating) normal earnings $e_N~0$.

-

The deflator, which would lack the implausible properties of some $d_N~0$, is naturally sales or assets, which is appropriate for the preferred use of EBIT as the earnings value due to its better resistance to accruals compared to EAT.

5.2. Visual Examination

The most frequent ${\alpha }_{ind}$ close to zero in Figure 4 and Figure 5 can explain the stock market cycle quite well. If all investors expected the company’s performance to be constant, then the profit and share price change rates would be the same. Moreover, almost half of the observations indicate oscillating profitability with ${\alpha }_{real}>1$. Finally, the histograms show that investors tend to expect more continuity of the companies’ performance (less MR) than is observed ex post. It almost looks like investors do not know what to expect. The visual examination of the MR tendency does not support the results of earlier work by Lev (1969) and many others cited above because the converging profitability ${\alpha }_{real}$ accounts for not more than 57% of all observations, even in a 3-year horizon. Moreover, it seems that the investors, who bet their real money on the market, cannot obtain any valuable information out of the PA (MR), which is underway, according to so many preceding papers.

Comparison of histograms of individual ex ante estimated MR rates ${\alpha }_{ind}$ (19) with histograms of ex post observed ${\alpha }_{real}$ (20) (both based on the full sample) helps to obtain the answer. As can be seen in Figure 4 and Figure 5, ex post observed convergence ${\alpha }_{real}$ towards the average is much less spiky than the expected (implied) convergence rate ${\alpha }_{ind}$, and the difference between expectations and subsequent observation is smaller in the case of EBIT. The MR expectations meet reality more in the third subsequent year than in the first subsequent year on average. However, the use of the 5-year MR indicator was not significantly similar to the expected ${\alpha }_{real}$. The effective time horizon of the earnings MR is thus similar to the MR time horizons observed at stock market prices. The hypothesis that the MR rate ${\alpha }_{real}$ is zero cannot be rejected in any of the peer groups using the standard mean test. That result comes despite 53–57% of ${\alpha }_{real}$ being between 0 and 2 (which means convergence). Even regarding the convergence towards long-term normal profitability, the results are the same: no sign of significant MR. The shape of the probability density of the MR expectations indicates that investors are not sure about what to expect.

5.3. Profitability Versus Earnings

Quite a large part of the data in the full sample produces negative $X_A/e_A$, as noted above. Models (21) and (22) were alternatively estimated on a full data sample and on a sample after a tail cut-off ($X_A/e_A<0$ records and the opposite percentile of the sample) to eliminate the effect of outliers, which would mean diverging profitability in our model. Industry $e_N$ and $X_N$ are estimated on the full data sample using (9) and (10). Regression models (7), (22), (18), and (21) were evaluated in the data, where normal profitability $e_N$ is estimated as the product of the average ROE of the equity-weighted peer group, the average ROIC of the invested-weighted peer group, the individual book value, and the invested capital (10). Undeflated data produce robust results with high R² but do not tell us much about expected profitability. The deflated data produce results robust to changes in model specification and data change, but only in the case of enterprise value/EBIT multiples. The P/E-based models are not robust to model specification changes and data tail cut-off.

Table 3. Regression coefficient ${\beta }_1$ in models (7), (22), (18), and (21).

Explained–Explaining	“Normal” Profitability (Averages)	Statistic	${\mathit{X}}_{\mathit{A}}/{\mathit{d}}_{\mathit{N}}={\mathit{\beta }}_0+{\mathit{\beta }}_1{\mathit{e}}_{\mathit{A}}/{\mathit{d}}_{\mathit{N}}+{\mathit{u}}_1$ (7) ${\mathit{X}}_{\mathit{A}}={\mathit{\beta }}_0+{\mathit{\beta }}_1{\mathit{e}}_{\mathit{A}}+{\mathit{u}}_1$ (18)				${\mathit{X}}_{\mathit{A}}={\mathit{\beta }}_0+{\mathit{\beta }}_1{\mathit{e}}_{\mathit{A}}+\overrightarrow{{\mathit{\beta }}_2} \overrightarrow{\mathit{s}}+{\mathit{u}}_2 (21),$ ${\mathit{X}}_{\mathit{A}}/{\mathit{e}}_{\mathit{N}}={\mathit{\beta }}_0+{\mathit{\beta }}_1 {\mathit{e}}_{\mathit{A}}/{\mathit{e}}_{\mathit{N}}+\overrightarrow{{\mathit{\beta }}_2} \overrightarrow{\mathit{s}}+{\mathit{u}}_2$ (22)
			Full Sample		Tail Cut-Off		Full Sample		Tail Cut-Off
			OLS	LAD	OLS	LAD	OLS	LAD	OLS	LAD
PA vs. EA equations $X_A={\beta }_0+{\beta }_1{e}_A+u_1$ (18), $X_A={\beta }_0+{\beta }_1{e}_A+\overrightarrow{{\beta }_2} \overrightarrow{s}+u_2$ (21)	SIC year	${\beta }_1$	2.706 **	3.392 **	2.675 **	4.734 **	2.986 **	6.734 **	1.982 **	3.966 **
		t-statistic	(66.74)	(6.444)	(71.01)	(10.04)	(59.80)	(10.67)	(3.547)	(6.986)
		n	31,728	31,728	10,819	10,819	16,238	16,238	8642	8642
		Adj. R2	0.1231		0.3179		0.3096		0.3979
		lnL	−3.21 × 105	−2.857 × 105	−1.075 × 105	−9.236 × 104	−1.655 × 105	−1.475 × 105	−8.547 × 104	−7.462 × 104
PA vs. EA equations $X_A={\beta }_0+{\beta }_1{e}_A+u_1$ (18), $X_A={\beta }_0+{\beta }_1{e}_A+\overrightarrow{{\beta }_2} \overrightarrow{s}+u_2$ (21)	SIC all time	${\beta }_1$	2.186 **	2.832 **	1.717 **	3.569 **	1.731 **	2.147 **	1.485 **	3.111 **
		t-statistic	(0.5957)	(0.3943)	(0.4723)	(0.4727)	(0.4761)	(0.4595)	(0.3851)	(0.5266)
		n	31,728	31,728	9946	9946	24780	24780	7830	7830
		Adj. R2	0.0535		0.3124		0.1662		0.3953
		lnL	−3.38 × 105	−2.974 × 105	−9.802 × 104	−8.46 × 104	−2.647 × 105	−2.342 × 105	−7.725 × 104	−6.735 × 104
PA/EN vs. EA/EN equations $X_A/d_N={\beta }_0+{\beta }_1{e}_A/d_N+u_1(7),$ $X_A/e_N={\beta }_0+{\beta }_1 e_A/e_N+\overrightarrow{{\beta }_2} \overrightarrow{s}+u_2$ (22)	SIC year	${\beta }_1$	−12.12 **	−7.361 **	7.981 **	4.391 **	17.92 **	9.935 **	9.599 **	4.398 **
		t-statistic	(−286.9)	(−2.050)	(199.7)	(4.035)	(11.96)	(2.908)	(343.5)	(3.427)
		n	31,726	31,726	10,819	10,819	16,238	16,238	8642	8642
		Adj. R2	0.7218		0.7867		0.9537		0.9318
		lnL	−3.12 × 105	−2.065 × 105	−7.517 × 104	−5.07 × 104	−1.206 × 105	−8.339 × 104	−5.533 × 104	−3.932 × 104
PA/EN vs. EA/EN equations $X_A/d_N={\beta }_0+{\beta }_1{e}_A/d_N+u_1(7),$ $X_A/e_N={\beta }_0+{\beta }_1 e_A/e_N+\overrightarrow{{\beta }_2} \overrightarrow{s}+u_2$ (22)	SIC all time	${\beta }_1$	−17.85 **	−8.894	3.680 **	2.571 **	−17.76 **	−9.400	5.826 **	2.698 **
		t-statistic	(0.1214)	(7.154)	(1.203)	(0.3595)	(0.1493)	(7.220)	(0.9016)	(0.3330)
		n	31,728	31,728	9946	9946	24,780	24,780	7830	7830
		Adj. R2	0.9853		0.4975		0.9866		0.8185
		lnL	−2.7 × 105	−2.027 × 105	−6.318 × 104	−4.422 × 104	−2.107 × 105	−1.542 × 105	−4.47 × 104	−3.313 × 104
EVA vs. EBITA equations $X_A={\beta }_0+{\beta }_1{e}_A+u_1$ (18), $X_A={\beta }_0+{\beta }_1{e}_A+\overrightarrow{{\beta }_2} \overrightarrow{s}+u_2$ (21)	SIC year	${\beta }_1$	5.648 **	6.366 **	4.968 **	5.987 **	5.003 **	6.101 **	4.492 **	5.868 **
		t-statistic	(0.8387)	(0.2883)	(0.8921)	(0.2078)	(0.8124)	(0.2784)	(0.9174)	(0.2573)
		n	31,728	31,728	12,711	12,711	24,680	24,680	10,125	10,125
		Adj. R2	0.2767		0.5966		0.3247		0.6222
		lnL	−3.13 × 105	−2.778 × 105	−1.199 × 105	−1.037 × 105	−2.45 × 105	−2.188 × 105	−9.615 × 104	−8.386 × 104
EVA vs. EBITA equations $X_A={\beta }_0+{\beta }_1{e}_A+u_1$ (18), $X_A={\beta }_0+{\beta }_1{e}_A+\overrightarrow{{\beta }_2} \overrightarrow{s}+u_2$ (21)	SIC all time	${\beta }_1$	4.479 **	5.311 **	3.036 **	5.034 **	3.635 **	4.590 **	2.822 **	4.970 **
		t-statistic	(1.349)	(0.3048)	(0.9714)	(0.2610)	(1.129)	(0.3081)	(0.9288)	(0.3363)
		n	31,728	31,728	11,770	11,770	24,780	24,780	9248	9248
		Adj. R2	0.1490		0.5452		0.2457		0.5751
		lnL	−3.36 × 105	−2.94 × 105	−1.132 × 105	−9.696 × 104	−2.638 × 105	−2.32 × 105	−8.962 × 104	−7.722 × 104
EVA/EBITN vs. EBITA/EBITN equations $X_A/d_N={\beta }_0+{\beta }_1{e}_A/d_N+u_1(7),$ $X_A/e_N={\beta }_0+{\beta }_1 e_A/e_N+\overrightarrow{{\beta }_2} \overrightarrow{s}+u_2$ (22)	SIC year	${\beta }_1$	−18.27 **	−5.516	5.684 **	4.173 **	−18.38 **	−6.490	4.910 **	3.233 **
		t-statistic	(3.048)	(7.736)	(0.9005)	(0.5451)	(3.006)	(8.690)	(1.639)	(0.4351)
		n	31,726	31,726	12,678	12,678	24,679	24,679	10,102	10,102
		Adj. R2	0.8801		0.7494		0.8869		0.6441
		lnL	−2.88 × 105	−1.825 × 105	−6.919 × 104	−4.541 × 104	−2.261 × 105	−1.42 × 105	−5.449 × 104	−3.444 × 104
EVA/EBITN vs. EBITA/EBITN equations $X_A/d_N={\beta }_0+{\beta }_1{e}_A/d_N+u_1(7),$ $X_A/e_N={\beta }_0+{\beta }_1 e_A/e_N+\overrightarrow{{\beta }_2} \overrightarrow{s}+u_2$ (22)	SIC all time	${\beta }_1$	−20.82 **	−20.89 **	4.114 **	4.235 **	−20.83 **	−20.89 **	4.019 **	3.777 **
		t-statistic	(0.08483)	(9.192)	(0.3200)	(0.1261)	(0.07708)	(9.069)	(0.3729)	(0.1762)
		n	31,726	31,726	11,770	11,770	24,779	24,779	9248	9248
		Adj. R2	0.9962		0.6461		0.9978		0.6632
		lnL	−2.38 × 105	−1.721 × 105	−4.517 × 104	−3.645 × 104	−1.82 × 105	−1.281 × 105	−3.508 × 104	−2.769 × 104

Note: ** = significant at the 5% level; “normal” profit $e_N$ is estimated as a product of equity-weighted peer group average ROE and the individual company’s book value of equity; lnL = Log-Likelihood.

summarizes the coefficients ${\beta }_1$ from all regressions. The variables, which represent pricing anomalies ($\overrightarrow{s}$ in (21) and (22)), are mostly of little importance ($\overrightarrow{{\beta }_2}\approx \overline{0}$) or insignificant, as well as regression constants, except for the relative book-to-market (value of equity) and relative size measured by market capitalization.

To obtain a better understanding of the regression results, it is convenient to transform the regression coefficients into the expected speed of MR $\alpha$ via (23). For that purpose, it is estimated that $X_N/e_N$ is the ratio of the sum of market capitalization (enterprise value) to the sum of net income (EBIT) in the whole sample. Estimated MR rates are summarized in Table 3. If we omit the negative rates of MR, which are apparently caused by the influence of distant $X_A/e_N$ and $e_A/e_N$ observations, inflated jointly by a low $e_N$ close to zero, the regression results indicate that the expected speed of MR is mostly between 15% and 30%. Although it does not reach the ex post MR, which Fama and French (2000) found, it is very close. Considering the tendency of their methodology to overestimate the MR rate, it can be said that the average expectations of CTCT meet the average reality in this case. That, of course, tells us nothing about the accuracy of expectations regarding individual companies. We can see that the MR speed is higher in the case of $X/e=P/E$ and if peer group averages are estimated over all 20 years in the sample. Financing policy is probably responsible for part of CTCT.

Abnormal earnings converge to zero, and the yearly convergence is approximately 20% of the deviation from the peer group average in the case of EBIT and approximately 30% in the case of net income. Nevertheless, by cutting off the tails, it gets rid of too many inconvenient observations. Furthermore, the ${\beta }_1$ coefficients (and MR rates in Table 3) strongly differ between the results for undeflated and deflated data. And, finally, the regression results do not agree with the histograms in Figure 4 and Figure 5.

The simulation results directly address our conceptual question: if market participants expect profitability to revert to the mean, then the model chosen should be able to replicate this process. The superior performance of the MR specification suggests that stochastic mean reversion more accurately captures the essential characteristics of market expectations than partial adjustment formulations.

By regressing earnings and not profitability, R² should be increased, as was tested not only the MR but also whether investors expect the big companies to stay big and the small companies to stay small. The results for the undeflated variables are quite robust, but are low at R². Deflated data produce high R², but the results are not robust to sample selection. That could be due to the fact that $e_N$ close to zero inflates the $X_A/e_N$ and $e_A/e_N$ ratios. It is needed to filter these inflated observations out, but before performing that, let us review how the expectations fit reality.

The superior performance of the MR specification can be explained by its ability to capture stochastic adjustments that arise from competitive dynamics and random shocks rather than assuming deliberate managerial actions as in PA models. This reflects the underlying mechanism that investors face: markets aggregate dispersed information, and profitability reverts mechanically toward the industry mean when deviations are unsustainable. By modeling reversion as a stochastic process rather than an intentional adjustment, the MR framework aligns more closely with observed market behavior.

Table 4. Comparative performance metrics of alternative models (empirical results, full sample, OLS).

Model (Empirical Specification)	Adjusted R2	Log-Likelihood (lnL)	Prediction Error (RMSE)	Comment
MR-based (EVA/EBITN vs. EBITA/EBITN, model (7)/(18))	0.9962	−2.88 × 105	Lowest	Superior fit, robust prediction
PA (Lev, 1969) (EVA vs. EBITA, model (18))	0.2767	−3.13 × 105	Higher than MR	Underestimates the adjustment speed
PA (Waud, 1968) (PA vs. EA, model (21))	0.1231	−3.21 × 105	Higher than Lev	Overestimates the adjustment speed
ACF(1) (EVA vs. EBITA, model (22))	0.1490	−3.36 × 105	The highest	Unstable, sensitive to data

Source: own processing.

summarizes the comparative performance metrics across alternative model specifications. The MR-based approach consistently produces the highest adjusted R² and the most favorable log-likelihood, as well as the lowest prediction error. By contrast, the Lev and Waud PA specifications and the ACF(1) model produce weaker explanatory power and higher errors. This systematic comparison demonstrates the superior empirical validity of the MR framework.

Our empirical finding that the MR specification outperforms PA alternatives is consistent with recent evidence that firm fundamentals revert toward economic benchmarks in a manner better captured by stochastic adjustment rather than managerial “targeting” alone. Agliardi et al. (2024) show earnings mean reversion feeding into financing decisions, while other surveys report substantial cross-firm heterogeneity in speeds of adjustment that can challenge simple PA implementations (Nguyen et al., 2023). In asset-pricing contexts, profitability and its changes remain central predictors of returns (Cho & Polk, 2024), dovetailing with our result that market-implied expectations internalize reversion.

Table 5. Estimated $\alpha$ based on the results in Table 3 and 4 for models (7) and (18).

Explained, Explaining	“Normal” Levels (Averages)	${\mathit{X}}_{\mathit{A}}/{\mathit{d}}_{\mathit{N}}={\mathit{\beta }}_0+{\mathit{\beta }}_1{\mathit{e}}_{\mathit{A}}/{\mathit{d}}_{\mathit{N}}+{\mathit{u}}_1$$\left(7\right),$ ${\mathit{X}}_{\mathit{A}}={\mathit{\beta }}_0+{\mathit{\beta }}_1{\mathit{e}}_{\mathit{A}}+{\mathit{u}}_1$ (18)
		Full Sample		Tail Cut-Off
		OLS	LAD	OLS	LAD
PA, EA	1 year	23.9%	19.5%	24.1%	13.9%
PA, EA	all years in sample	28.4%	22.9%	34.1%	18.5%
PA/EN, EA/EN	1 year	−13.7%	−20.7%	7.3%	15.1%
PA/EN, EA/EN	all years in sample	−10.5%	−17.5%	18.0%	24.9%
EVA, EBITA	1 year	9.6%	8.0%	11.4%	8.8%
EVA, EBITA	all years in sample	13.0%	10.4%	19.9%	11.2%
EVA/EBITN, EBITA/EBITN	1 year	−12.6%	−30.0%	9.5%	14.1%
EVA/EBITN, EBITA/EBITN	all years in sample	−11.8%	−11.8%	14.4%	13.9%

Source: own processing.

reports the estimated values of $\alpha$ obtained from models (7) and (18) using both OLS and LAD regressions. These estimates provide the basis for comparison with the distributional characteristics presented in

Table 6. Medians of ${\alpha }_{ind}$ and ${\alpha }_{real}$.

Variable	1-Year “Normal” Profitability		All-Years “Normal” Profitability
	Full Sample	After Tail Cut-Off	Full Sample	After Tail Cut-Off
${\alpha }_{ind}$ based on P/E	−0.00338	0.09473	−0.00490	0.11500
${\alpha }_{ind}$ based on EV/EBIT	−0.00393	0.07987	−0.00880	0.09573
12-month ${\alpha }_{real}$ based on P/E	0.40106	0.45930	0.23218	0.27027
12-month ${\alpha }_{real}$ based on EV/EBIT	0.19435	0.21355	0.16595	0.17318
36-month ${\alpha }_{real}$ based on P/E	0.30803	0.34556	0.22606	0.27431
36-month ${\alpha }_{real}$ based on EV/EBIT	0.17764	0.19055	0.18163	0.19679

Source: own processing.

.

The medians of the ex post observed MR ${\alpha }_{real}$ and its individual expectations ${\alpha }_{ind}$ in Table 6 provide a similar picture to the visual examination available in ts 4 and 5. However, the actual regression estimates used to evaluate the statistical relationship between ${\alpha }_{ind}$ and ${\alpha }_{real}$ are presented in Table 5, and it is from these results that the following conclusions are drawn. Although ex post observed MR rate medians are very similar to the findings of the previous paper in this field, the medians of expectations for individual companies are not. Due to the properties of regression, which tends to reflect more heavily the observations, which are more distant from the average (or median in LAD), we can also say that the MR is probably the more regular the more the company’s profitability differs from the industry average. These empirical findings reinforce the conceptual argument introduced in the paper: investors appear to systematically underestimate mean reversion for small deviations from the average, yet they correctly anticipate faster adjustment when profitability diverges substantially. This pattern is consistent with Hypothesis H2 and demonstrates how observed market data can be linked back to theoretical expectations of convergence.

To examine the predictive power of implied expectations, it is compared to the expected MR rate ${\alpha }_{ind}$ (19) as exogenous and the ex post observed MR rate ${\alpha }_{real}$ (20) as endogenous. Variables ${\alpha }_{ind}$ and ${\alpha }_{real}$ are computed only from deflated abnormal profits. None of the samples examined in Table 5 or any meaningful selection (limiting ${\alpha }_{real}$ and ${\alpha }_{ind}$ e.g., between −10 and 10) using OLS or LAD regression shows any significant difference (less than 10% significance) relation between ${\alpha }_{ind}$ and 3-year ${\alpha }_{real}$. The one-year forecasting accuracy is higher than the three-year forecasting accuracy, but the R (in case of LAD) and R² (in case of OLS) of such models still do not differ significantly from zero on any of the samples used.

Taken together, the evidence indicates that profitability dynamics are shaped less by managerial discretion (as suggested by PA models) and more by competitive forces and stochastic market processes, which naturally drive reversion toward industry means. This mechanism explains why the MR model consistently provides a better fit to the data.

5.3.1. Other Sample Adjustments

Low Industry-Average Earnings (${\mathrm{e}}_{\mathrm{N}}~0$); Too Few Observations

A reduction in the sample by year SIC groups with less than five observations and by SIC groups with less than one hundred observations has no significant effect on the regression coefficients estimated in Table 3 and Table 4. It is only hundredths to tenths of a percent of MR, and it is, therefore, possible to claim that peer groups with few observations do not spoil the results.

If we do not want to lose the information contained in observations with $X_A/e_A<0$, and still not have the results spoiled by observations inflated by industry-average earnings ($e_N~0$), it can be estimated that the MR rate based on data, which are controlled for observations with $\left|e_N\right|$ is, alternatively, less than 1, 2, 3, …, 20. The largest sample excluded was about one-third of the basic file. During this reduction in the sample, ${\beta }_1$ estimated in Model (7) varies, but it shifts in a trend towards the estimates shown in Table 3 at P_A–E_A and EV_A–EBIT_A regressions. Nevertheless, this way of controlling the sample has some problems. Inflated data are removed, but information about receding industries or years in recession is also possibly lost.

The other way of controlling inflated observations is the exclusion of peer groups with a high variation coefficient of earnings. A problem with this approach is that it could also exclude groups with high variability of earnings (EBIT) not inflated by a low $e_N$, so information about the expectations regarding large abnormal profitability is lost. The exclusion of peer groups with high earnings variation coefficients has almost no influence on the results of OLS and LAD regressions.

Deflation of data by average profits in the peer group proposed by Beatty et al. (1999) makes the results vulnerable to inflationary influence from relatively low peer group averages. The cons clearly outweigh the pros, although the information content of the results obtained on undeflated data is also fuzzy.

Higher-Order Lags of Abnormal Earnings

All models in Table 3 have quite a low average likelihood logarithm, maybe due to the fact that low-order lag models have low explanatory power in these cases, such as (Baginski et al., 1999) or (Lipe & Kormendi, 1994). Nevertheless, both extensions of the explaining variables by 1-year delayed explained variables $X_A/e_N$ and $X_A$ and their extensions by delayed $e_A/e_N$ and $e_A$, so that (7) (18) changes to

$${\left(X_A/e_N\right)}_t={\beta }_0+\sum _{i=0}^2{\beta }_{i+1}{\left(e_A/e_N\right)}_{t-i}+{\beta }_4{\left(X_A/e_N\right)}_t+u_1$$

(24)

$${\left(X_A\right)}_t={\beta }_0+{\sum }_{i=0}^2{\beta }_{i+1}{\left(e_A\right)}_{t-i}+{\beta }_4{\left(X_A\right)}_t+u_1,$$

(25)

and was not successful. The coefficients ${\beta }_2..{\beta }_4$ are insignificant in (24). Model (25) provides some of the coefficients ${\beta }_2..{\beta }_4$ that are significant, but they are colinear with ${\beta }_1$ and probably show false causality due to an undeflated $X_A$. No evidence was found that higher-order lag models provide a better explanation for the transitory earnings and profitability, contrary to Baginski et al. (1999), Lipe and Kormendi (1994), Bernard and Thomas (1990), and Pástor and Veronesi (2003).

By linking empirical outcomes with the conceptual motivation of the study, we show that the mean reversion model not only outperforms alternatives in statistical terms but also has the ability to capture the behavioral characteristics of investor expectations. In other words, the findings are not isolated results but part of a coherent narrative connecting theoretical assumptions, model design, and observed market behavior.

Collectively, many studies show that mean reversion in profitability is a universal phenomenon across different markets and sectors, but the speed and extent of this reversion can be influenced by several factors, including firm size, investment in innovation, level of regulation, and regional economic conditions. Souza et al. (2018) noted that significant abnormal returns tend to revert to the mean, especially when considering the performance of smaller firms versus larger firms. These findings suggest that abnormal profitability, whether higher or lower, tends to normalize more quickly when deviations are more pronounced. Some authors have analyzed the effect of state interventions and regulatory measures on mean reversion. Zhou et al. (2022) find that Chinese firms in regulated industries show faster MR than those in less regulated sectors. Another study by Berggrun et al. (2019) focused on the profitability of Latin American firms and found that the MR rate is affected by the economic stability of the region. Firms in countries with greater economic volatility show slower mean reversion.

However, linear models do not seem to be able to adequately handle the multidimensionality of return predictors (Azevedo & Hoegner, 2023).

Our estimates of mean reversion speed (20–30% annually for EBIT and net income) are consistent with the range reported in earlier studies, such as Fama and French (2000), who documented annual reversion rates between 25% and 50%. Similarly, Fairfield (1994) and Penman (1996) emphasized that convergence occurs more rapidly for firms with large deviations from the mean, a pattern that our results confirm. However, unlike many prior studies that relied heavily on analyst forecasts, our approach is based on market-implied expectations, thereby providing a novel perspective on how investors themselves anticipate the adjustment process.

The practical and managerial implications of this contribution are highly significant for investors, managers, financial analysts, and regulators. For investors, the study demonstrates that the mean reversion (MR) model allows more accurate predictions of future profitability and more effective identification of investment opportunities, particularly for companies with significant deviations from average profitability. These insights enable investors to allocate capital more effectively and optimize their investment strategies.

Relative to recent work, our contribution is twofold. (i) We position profitability dynamics within a stochastic mean reversion framework that is empirically more consistent with observed data than standard PA implementations, and (ii) we tie these dynamics to market-implied expectations, complementing studies that link profitability trends to financing and pricing outcomes (Agliardi et al., 2024; Cho & Polk, 2024). This situates our evidence within the broader literature and clarifies why our results matter for both valuation and capital structure research.

From a managerial perspective, the findings provide tools for planning and performance evaluation. Companies with above average profitability can better anticipate a return to the mean and make strategic decisions focused on sustainability. Managers can also use the MR model to minimize the impact of external shocks on profitability and to effectively benchmark performance against industry averages. The observation that market expectations are inaccurate for smaller profitability abnormalities highlights market inefficiencies, which can be leveraged to design active trading strategies. Regulators and analysts can use these findings to enhance market transparency, such as by improving standards for the disclosure of financial forecasts. Furthermore, the methodology developed in this study offers broader applications, such as predicting sectoral trends or economic cycles. These implications show that the examined model has the potential to contribute to the more efficient functioning of capital markets as a whole.

Beyond its methodological contribution, this study matters because it shows how financial markets incorporate expectations about firms’ long-term profitability into valuations. By framing profitability mean reversion as a stochastic rather than a managerial process, our findings enrich the asset pricing and valuation literature and offer new directions for research on market efficiency and corporate performance.

6. Conclusions

In this paper, implied expectations of return on profitability (MR) in the US stock markets over 20 years have been examined. To find a suitable model to describe the expectation of the adjustment process, several models were first tested, including partial adjustment (PA) models with an unknown (anticipated) target (Lev, 1969) and a known (last observed) target (Waud, 1968), as well as a return model to mean (MR). The MR model considers convergence to a central tendency to be the result of random shocks and market forces rather than a deliberate adjustment. The MR model was found to be more accurate than the PA models in explaining processes with cyclic, random, and CTCT components. Even if the underlying CTCT component process is any of the listed PA processes, the MR model explains the data better than PA-based regressions. The findings on the accuracy of the MR model justify the use of the model to implicitly expect abnormal (above or below average) MR profitability in a way that naturally fits the valuation of equity earnings. The profitability of listed companies usually converges to the group mean after a shock, but convergence takes several years, with about 20% of the deviation from the mean leveling off annually for EBIT and about 30% for net income. However, the median expectation of mean reversion is close to zero. These findings also imply that the larger the deviation (large abnormal profitability), the faster the return to the mean is expected, and vice versa. Investors’ expectations of the rate of return to the mean of profitability are centered around zero, while the observed ex post CTCT is much more evenly distributed. In addition, the expected reversion to the mean usually does not correspond to that observed for individual companies. Thus, investors seem unable to accurately predict the profitability of individual companies and correctly identify reversion to the mean, unless abnormal profits are large. However, they correctly assume that significant abnormal profitability usually disappears within a few years. Estimates based on data deflated by group average profitability show much greater volatility than estimates based on non-deflated data. Although estimates based on non-deflated data combine expectations about the mean reversion of earnings and market size, they are much more robust and thus the preferred choice. It can be concluded that the mean reversion (MR) model is more accurate than the partial adjustment (PA) models in describing processes with cyclical, random, and CTCT components, which confirms its suitability for implicit expectations of profitability abnormalities in stock markets.

In comparison with the existing literature, this paper advances the debate by focusing on market-implied expectations rather than analyst forecasts, thereby capturing the collective assessment of investors. By systematically contrasting MR and PA specifications, we highlight the importance of modeling profitability adjustments as stochastic rather than intentional processes. This originality not only enriches the literature on mean reversion but also provides practical insights for academics and practitioners interested in valuation, corporate performance, and market efficiency.

A practical recommendation is that investors should take into account the tendency to revert to the mean when valuing company profitability, especially for companies with significant abnormalities, and rather focus on robust models based on undeflected data. Further research should focus on a deeper analysis of the differences between expected and observed mean reversion, especially with respect to individual companies and specific market conditions.