Wei-Qi, Liu; Jingxing, Zhang
Article
BM(book-to-market ratio) factor: Medium-term
momentum and long-term reversal
Financial Innovation
Provided in Cooperation with:
Springer Nature
Suggested Citation: Wei-Qi, Liu; Jingxing, Zhang (2018) : BM(book-to-market ratio) factor: Medium-
term momentum and long-term reversal, Financial Innovation, ISSN 2199-4730, Springer,
Heidelberg, Vol. 4, Iss. 1, pp. 1-29,
https://doi.org/10.1186/s40854-017-0085-6
This Version is available at:
https://hdl.handle.net/10419/237117
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RESEARCH Open Access
BM(book-to-market ratio) factor: medium-
term momentum and long-term reversal
Liu Wei-qi and Zhang Jingxing
*
* Correspondence:
[email protected]
Shanxi university, No.92, Wucheng
road, Xiaodian district, Taiyuan city,
Shanxi Province, China
Abstract
To explain medium-term momentum and long-term reversal, we use the difference
between the optional model and the CAPM model to construct a winner-loser portfolio.
According to the CAPM model’s zero explanatory ability with respect to stock market
anomalies, we obtain an anomaly interpretative model. This study shows that this
anomaly interpretative model can explain stock market perceptions and medium-term
momentum. Most importantly, BM is a critical factor in the model’s explanatory ability.
We present a robustness test, which includes selecting new sample data, adding new
auxiliary variables, changing sample years, and adding industry fixed effects. In general,
the BM effect does have considerable explanatory power in medium-term momentum
and long-term reversal.
Keywords: Stock market volatility, medium-term momentum, long-term reversal, holding
period, formation period, book-to-market ratio, return on equity
Introduction
In a completely effective market, stock market volatility is random. However, through
daily observations, we find that the stock market often shows regular fluctuations,
which is counter intuitive. These cyclical fluctuations often occur during a certain
period and are known as stock market visions. These phenomena have attracted
academic attention because they can no longer be interpreted by traditional methods.
The momentum and reversal effect, as two typical stock market visions, have caused
heated discussion in the academic community. Retail investors and institutional inves-
tors believe that grasping momentum or reversal means that a stock's future returns
can be predicted for profit. Some such cases do exist. Jegadeesh and Titman (1993)
first discovered and proposed that the US securities market momentum effect exists
for three to 12 months and provided a short-term profitability strategy. The authors
also found that this strategy can obtain 1% of the excess return rate within a certain
period. Wang Yonghong and Zhao Xuejun (2001) used one month for the smallest sort
period and discovered that China's stock market has a clear reversal effect. The authors
also proved that the inertial effect of China's stock market is not obvious. However,
scholars have not given a definitive answer as to how to grasp these visions. Research
is still advancing incrementally. This paper focuses on the interpretation of the above
two visions, constructs a model, and identifies a new visions model that can be applied
Financia
l
Innovation
© The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International
License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,
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indicate if changes were made.
Wei-qi and Jingxing Financial Innovation (2018) 4:1
DOI 10.1186/s40854-017-0085-6
to the explanation of the two anomalies at the same time. First, we explain several
important basic concepts and previous research results.
Momentum effect is also known as the inertial effect. It implies that the stock had a
higher return in the past and will have a higher return in the future; lower stock yields in
the past mean that the future return will be lower. Jegadeesh and Titman (1993) were the
first academics to discover that the momentum effect existed in the US securities market
in three to 12 months. Soon afterward, they offered a short-term profitability strategy
and found that the strategy, within a certain period, obtained 1% excess return.
Then, Kaul and Conrad (1993) used shares of the stock listed on the New York
Stock Exchange and the US Stock Exchange during the period 1926 to 1989 to
study the stock returns during eight different investment periods. The research
found that four of the investment strategies achieved significant profits, two strat-
egies belong to the momentum strategy, the other two are reverse strategy. Almost
at the same time, Rouwenhorst analyzed the momentum effects of 12 European
countries' stock markets and found that almost all of them showed short-term mo-
mentum effect and to a greater extent than the US market. While Rouwenhorst
confirmed that applying momentum strategies to emerging markets could acquire
significant profit, some scholars discovered that investors tended to adopt momen-
tum strategy in their decision making. For example, Chen found that US investors
have momentum tendency when they make medium-term investment decisions.
Given this, we find that the stock market and the investors are affected by mo-
mentum effect within three to 12 months, which indicates the existence of the
medium-term momentum effect. This effect brings tangible benefits for investors.
Griffin (2003) used 40 countries to explain the momentum effect of macroeco-
nomic cycle risk. He found that momentum gains were present and significant in
these countries, and regardless of whether the economic cycle was in an upswing
or a declining phase, the momentum gains were significantly positive. Thus, the
momentum effect and the macroeconomic cycle do not have a significant relation-
ship. Lee and Swaminathan (2000) first studied the relationship between the mo-
mentum effect and stock trading volume from the perspective of trading volume.
The author concluded that the trading volume of the stock can predict the income of
the momentum strategy and the duration of the momentum effect. When using
momentum strategies in high stock returns, the effect of the momentum effect will
become less and will continue for a shorter time. In addition to the long-term reversal
effects found in the US market, scholars found a reversal effect in other countries. This
finding indicates that the effect is not the reason for data mining. Chan et al. (1996)
and Chui et al. (2000) found short-term reversal effects in the Japanese market. Baytas
and Cakici (1999) note long-term reversal effects in the other seven countries.
After understanding the mid-momentum effect, we find that the reversal effect becomes
easier to interpret. In simple terms, the reversal effect is that the stock that performed
poorly in the past period will show a better result in the future. De bondt and Thaler
(1985) found that the portfolio of the worst performing 10% stocks paid 10% over the
stocks that were winners after the three-year formation; the loser portfolio showed a
higher than average market return at 19.6% while the winner portfolio was still lower than
5%. This study confirmed that the reversal effect exists over a long period, and reversal
strategies can be used to achieve substantial returns over a longer period.
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 2 of 29
Much controversy surrounds the existence of the momentum effect and the reversal
effect on developed countries’stock markets. Zhu (2003) is one of the representative
scholars and has proved that China does not have a reversal effect through the study of
the securities market. Chen Qiao and Wang Shi (2003) found that the inertial strategy
based on the industry portfolio showed significant excess returns. Hameed and Ting
(2000) found a significant price reversal effect in the Malaysian stock market indicating
that there is a reversal effect in developed country stock markets. Gaunt’s (2000) study
found that the Australian market also shows a significant reversal effect. His sample
interval was from 1974 to 1997, the formation period was five years and, after an 18-
month holding period, the reversal effect appeared. From the theoretical results of
these scholars, we conclude that there are divergent views on whether there are
momentum and reversal trends in the Chinese market, and the conclusions are totally
different. The focus of our study is to clarify medium-term momentum and long-term
reversal effects rather than prove their existence. Based on the above literature, we
know that the reversal effect is a ubiquitous phenomenon and is concentrated in the
medium and long term. Thus, our decision will be more inclined to choose the mature
US financial market so that the statement on the existence of medium-term momen-
tum and long-term reversal will be unified. The US stock market has become a better
option for our empirical analysis.
Typically, scholars are divided into two categories: behavioral finance and traditional
finance. Behavioral finance scientists often use the theory of overreaction and under-
reaction to explain the reversal and momentum effects. The view of long-term
overreaction was first introduced by De Bond and Thaler (1985, 1987). Additionally,
there are several interpretative models in behavioral finance. The BSV model (Barberis,
Shleifer, & Vishny, 1998) assumes that market investors have two deviations when they
make personal decisions. One is a representative bias; because investors so easily over-
look recent data, the result is over-reaction. The other is conservative bias; investors
are insensitive to new information causing inadequate response to the information. The
DHS model (Daniel, Hirshleifer, & Subramanyam, 1998) divides investors into
information-based investors and non-information investors and considers whether in-
vestors are sensitive to new information. The HS model (Hong & Stein, 1999), based
on assumptions concerning investors’sensitivity and type, interprets the under-reaction
from different perspectives. Fama and French (1996), who are representatives of
traditional finance schools, analyzed the reversal and momentum of stock returns and
confirmed that the benefits of long-term reversal strategies can be explained by their
three-factor model, but the model cannot explain the medium-term momentum effect.
Fama and French (1996) also believed that the CAPM vision is due to the CAPM
model, which lacks the consideration of other necessary risk factors. Considering the
problem with the CAPM model, our analysis is established based on the traditional
financial school, which uses a model similar to the factor model to explain the two
visions.
As mentioned, the BETA value in the CAPM model is not a reliable risk indicator.
Some scholars found that if the new risk factor is added into the factor models, some
excess returns can be explained. Some type of vision will disappear. But what variable
should be added to the factor model to explain the medium-term momentum and
long-term reversal? Scholars failed to reach a consensus on this point. Following the
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 3 of 29
traditional financial idea, we consider the type of impact that occurs from adding BM
and ROE into the model.
First, we validate the zero interpretation ability about the medium-term momentum
and long-term reversal of the CAPM model. Then, we use the benefit forecast model
obtained through the MC model (Lyle & Wang, 2015) to construct independent vari-
ables difference and obtain the vision interpretation model. This paper chooses the MC
model as the basic model. We need to prove the following: First, this model has better
ability than the CAPM model to forecast future stock earnings. Second, this model has
the same ability as the three-factor model and the four factors in returning a prediction.
The empirical test found that medium-term momentum was largely explained, and a
small part of the long-term reversal was explained.
Additionally, through a series of studies, we discovered the BM as the most critical
factor in explaining ability. Its value is totally different under the efficient market hy-
pothesis. The long-term reversal explaining ability is far more than that of the
medium-term momentum. We design a series of correlation experiments that are
deemed significant to prove that the BM factor gradually becomes noticeable over time,
which is also consistent with the traditional finance situation whereby the market tends
to become an effective market, and the vision will gradually disappear.
For the BM effect, Fama and French (1992) studied all the stocks listed on NYSE,
AMEX, and NASDAQ from 1963 to 1990 and found that the 1.53% combination of
BM’s highest value (the value combination) had the lowest monthly yield (the combin-
ation of charm). Xiao Jun and Xu Xinzhong (2004) used Shanghai and Shenzhen stock
market shares from June 1993 to June 2001 as a sample and calculated holding shares’
earnings in one year, two years, and three years to find that the BM effect does exist.
Finally, if we lack rigor in our research methods, we hope that other scholars in this
area provide more in-depth study and more proof will emerge in the future.
The remainder of this paper is organized as follows: section2 presents the MC model
and determines the parameters. Chapter 3 analyzes the BETA coefficients of the CAPM
model based on the empirical analysis. In Chapter 4, the existence of the stock market,
vision model, and empirical explanation are proposed. Chapter 5 analyzes the model’s
inherent explaining mechanism. Chapter 6 gives a robustness test, and Chapter 7 sum-
marizes the full text.
Model
The specific formula of the MC model is shown in (Lyle & Wang, 2015), and other
detailed derivation steps are given in the literature.
ri;tþ1¼μi1−ωi
α1
α2

|{z}
β0
þ 1−k1κi!
|{z}
β1
bmi;tþ ωi
1−k1κi
1−k1ωi!
|{z}
β2
roei;tþξi;tþ1ð1Þ
among β1¼1
α2
;β2¼ωi
α1
α2
;ξi;tþ1¼α1
α2
εi;t−ωiνi;t

þηi;tþ1
We obtain the industry parameters from the estimated coefficients of (Chen &
Wang, 2017):
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 4 of 29

κ¼1−β1

=k1ð2Þ
μ¼β0=1−β2
 ð3Þ
ω¼β2=β1
1þβ2=β1

k1ð4Þ
In developing the model, first, industry parameters must be determined. Over time,
any company tends to become more like its peers. Any abnormal expected ROE (or
excess expected return) will be gradually weakened due to the presence of industry
competition. Equation (1) is also the core model for the anomaly interpretation.
Sample selection and data sources
After obtaining the prediction model, the coefficients and other implied parameters of
this model must be confirmed. The sample is selected from the quarterly training data
(the sunken data), which is made up of 100 quarterly data points. The period was from
the first quarter of 1980 to the fourth quarter of 1994 for all the US financial markets
in DataStream. We implement the out-of-sample prediction from the first quarter of
2010 to the fourth quarter of 2015. At this time, we no longer use a sample of all
industries but select five, and these five industries will run through the anomaly inter-
pretation and robustness test. The main reason we gather data in this way is that col-
lecting whole industry involves extensive work and the probability of error is greater.
Empirical analysis
Sample regression analysis
First, the model is estimated through quarterly data by ordinary least squares (OLS).
The model coefficients and model implicit parameters are calculated using Equ. (1).
Panel B of Table 1 shows the time series mean for each industry. Panel A gives the
summary result. The mean values (median) of log BM and log ROE are 0.045638
(0.0461) and 0.339625 (0.265), respectively. The constant coefficients corresponding to
median and mean values are 0.032162 and 0.02985, respectively. By comparative
analysis, we find that the first-order autoregressive persistence parameters of mean
(median) values are 0.964003 and 0.962121, respectively, for a given industry; persis-
tence values are high. The standard deviation of the industry persistence parameter is
low, only 0.017613, while the ROE continuous parameter's standard deviation is
0.181218, which is larger than the log returns. Overall, the MC model has good persis-
tence for expected returns.
Table 1 shows that the coefficient and implicit parameters between industries differ,
which suggests that every industry’s response to external change varies because of its
unique characteristics. For most industries, the forecast returns are consistent with
actual earnings; that is, the k value is large enough. Some industries, such as gas, water,
and multiple utilities; life insurance; financial services; and equity investment instru-
ments have slightly larger coefficients. Therefore, their returns are vulnerable to outside
influences. Industries such as food and drug retail are different. ROE coefficient values
are close to one implying returns are easily influenced.
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 5 of 29
Table 1 Summary of model parameters
Regression coefficients Implied parameters
cons bm roe k w μ
Panel A: Regression results summary
5th percentile -0.0119 0.013 0.1106 0.93523 0.707112 0.02003
25th percentile 0.02114 0.0322 0.179 0.950202 0.817889 0.031925
Mean 0.032162 0.045638 0.339625 0.964003 0.878349 0.056226
Median 0.02985 0.0461 0.265 0.962121 0.893978 0.041286
75th percentile 0.03772 0.0584 0.461 0.976566 0.938257 0.05825
95th percentile 0.05589 0.07149 0.61 0.995 0.98653 0.12964
Standard deviation 0.013273 0.017437 0.198346 0.017613 0.084486 0.181218
Panel B: Industry Coefficient
Electricity 0.03705 0.0453 0.114 0.964343 0.731332 0.041817
Equity investment instrument 0.04606 0.0153 0.108 0.994646 0.89955 0.051637
Financial service 0.04952 0.0423 0.46 0.967374 0.941658 0.091704
Fixed line telecommunications 0.03535 0.0461 0.136 0.963535 0.763959 0.040914
Food and drug retail 0.03188 0.057 0.97 0.952525 0.972041 1.062667
food producers 0.05658 0.0288 0.51 0.98101 0.974212 0.115469
Forestry & paper 0.03507 0.0402 0.384 0.969495 0.930503 0.056932
Gas, water, & multiple utilities 0.06409 0.0225 0.59 0.987374 0.99193 0.156317
General industrial 0.03798 0.0436 0.262 0.966061 0.879962 0.051463
General retailers -0.01148 0.0622 0.53 0.947273 0.91966 0.024426
Health care equipment & services 0.02878 0.0475 0.188 0.962121 0.817889 0.035443
Household goods & home construction 0.03397 0.0512 0.117 0.958384 0.710426 0.038471
Industrial engineering 0.02985 0.0554 0.277 0.954141 0.854701 0.041286
industrial metals & mining 0.03686 0.0332 0.259 0.976566 0.910593 0.049744
Industrial transportation 0.02691 0.0499 0.26 0.959697 0.860642 0.036365
Leisure goods 0.01774 0.0632 0.354 0.946263 0.870677 0.027461
Life insurance 0.04108 0.0187 0.461 0.991212 0.989546 0.076215
Media 0.01825 0.0584 0.233 0.951111 0.81924 0.023794
Mining 0.02114 0.0594 0.62 0.950101 0.938257 0.055632
Mobile telecommunications 0.01277 0.0709 0.6 0.938485 0.918977 0.031925
No life insurance 0.03772 0.0322 0.265 0.977576 0.916162 0.05132
Oil equipment & services 0.02799 0.0433 0.192 0.966364 0.836456 0.034641
Oil & gas producers 0.03311 0.0266 0.541 0.983232 0.981192 0.072135
Personal goods 0.0288 0.0545 0.179 0.955051 0.78464 0.035079
Pharmaceuticals & biotechnology -0.01425 0.072 0.38 0.937374 0.86246 0.022984
Real estate investment & services 0.05535 0.0129 0.174 0.997071 0.957728 0.06701
Real estate investment trusts 0.04822 0.0132 0.269 0.996768 0.981286 0.065964
Software & computer services 0.02563 0.0593 0.56 0.950202 0.929461 0.05825
Sport services -0.01356 0.0769 0.222 0.932424 0.75965 0.017429
Technology hardware & equipment 0.02293 0.0613 0.138 0.948182 0.707112 0.026601
Tobacco 0.02047 0.0529 0.356 0.956667 0.893978 0.031786
Travel & leisure 0.02875 0.0442 0.159 0.965455 0.80129 0.034185
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 6 of 29
Expected returns analysis
After determining the coefficient, we calculate the out-of-sample return. First, we need to
predict in-sample return before interpreting vision. The reason is the following: first, the
anomaly excess return is also a part of the actual benefit; second, predicting ability in 3,
12, 24, and 36 months is mainly possible because long-term reversal typically occurs be-
tween three to five years of holdings, and the medium-term momentum typically occurs
between three to 12 months. Thus, if the model can predict returns within three years or
less, it is likely that the MC model can explain the above two anomalies better.
By estimating the coefficients and the implicit parameters of the model, the expected
returns of the holding period T are obtained:
XT
j¼1μtþj−1¼^
μTþ1−
^
κT
1−κ
^
β1bmtþ
^
β2roet−μðÞ
hi ð5Þ
For the construction of the holding period under 1, 4, 8, and 12 quarterly
periods according to (Chena et al., 2017) (see Table 2), we compare the average ex-
pected return and the average known earnings at different times. For example, the
mean (median) expected return is 0.7808 (0.82983), 0.56554 (0.5757), 1.4438
(1.6883), and 1.3871 (0.78332), respectively. When the lead time is 1, 4, 8, and 12
quarters, the known returns were 0.74912 (1.07651), 0.9445 (1.87121), 1.4047
(2.4969), and 1.6952 (2.00348), respectively. Table 2 shows that most short-term
gains can be predicted. This shows that the model reflects the actual value of
future returns, particularly in one and four quarters.
Predictive ability regression tests
By expecting cross-sectional property, we need to verify the MC model’s estimated
reliability. We focus on companies’cross-sectional property in the three-year period.
Table 2 Summary of expected returns
1Q
ahead
4Q
ahead
8Q
ahead
12Q
ahead
Long
term
LT-1Q
difference
12Q-1Q
difference
Panel A: Expected log returns
5th -0.4794 -0.5047 -0.83602 0.79245 0.3987 -0.80482 -0.5963
25th -0.2079 -0.3625 1.0216 0.9635 0.7758 -0.02842 -0.0692
Mean 0.7808 0.56554 1.4438 1.3871 1.3511 1.58205 0.721
Median 0.82983 0.5757 1.6883 0.78332 1.0587 1.4608 0.503
75th 0.89856 0.8399 1.7164 1.89945 2.0655 1.249 0.9803
95th 0.9563 1.56075 1.952 2.90988 2.7431 2.0729 1.5028
Standard deviation 0.633536 0.6896 1.0295 0.83071 0.868 0.7052 0.74976
Panel B: Realized log returns
5th -0.41356 -0.1694 -0.17981 -0.97937 0.3842
25th -0.17805 -0.0497 -1.044 0.20461 1.0284
Mean 0.74912 1.3646 1.1662 1.6952 1.2974
Median 1.07651 0.9445 1.4047 2.00348 1.3002
75th 1.5181 1.87121 2.3969 3.2014 1.5355
95th 1.8595 2.1495 3.86763 3.5383 2.4085
Standard deviation 0.910017 0.967892 1.675042 1.7374 0.662228
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 7 of 29
We use the regression test method to estimate the mixed holding period log return
within the limit time:
ri;tþT¼δ0þδ1Etri;tþT

þωi;tþTð6Þ
Under the condition of δ0=0,δ1 = 1, there will be an absolute true estimate of the
expected return for any holding period. Thus, by meeting such a benchmark as much
as possible, when δ1 is more important, the expected value is closer to the true value.
Table 3 lists the estimated results about (7) for the 3, 12, 24, and 36-ahead months.
In the presence of industry (not industry) fixed effect, the predicting coefficients are
0.717 (0.65284), 0.643 (0.5972), 0.5828 (0.50284), and 0.438 (0.4028), respectively, and
the coefficient is significantly not zero at the 1% level. Because the holding period re-
turn proxies conditionally change according to the change in T and the holding period
increases, the measurement error becomes large, and the coefficient is likely to grad-
ually decrease. However, the correspondence between the expected return and the
known return is better within two years, the predicted coefficient is greater than 0.5,
and the slope coefficient under the three-year lead period has been significantly less
than 0.5. This indicates that the MC model may lack the ability to explain long-term
reversal, but we cannot exclude the human factor that could be influential and lead to
such a result.
CAPM model parameters
After determining the basic coefficients of the MC model, we must determine the coef-
ficients of the CAPM model. To ensure the consistency of the data, before predicting
the CAPM model coefficients, we still use the historical quarterly data in DataStream
as sample data although the calculation method will slightly differ. The stock returns
Table 3 Revenue return
Panel A: Regression parameter summary
Data Regression coefficients Implied parameters
cons bm roe F k w μ
5th percentile 0.014 0.045 0.259 70.62 0.904 0.620 0.014
25th percentile 0.018 0.048 0.325 145.24 0.952 0.647 0.017
Mean 0.022 0.050 0.430 359.70 0.910 0.657 0.023
Median 0.025 0.058 0.539 879.02 0.948 0.800 0.041
75th percentile 0.029 0.065 0.872 1426.60 0.963 0.889 0.059
95th percentile 0.035 0.094 0.890 1552.22 0.976 0.913 0.064
standard deviation 0.027 0.054 0.205 819.93 0.015 0.090 0.036
3M
(1)
12M
(2)
24M
(3)
36M
(4)
3M
(5)
12M
(6)
24M
(7)
36M
(8)
E[r
(i,t+1)
] 0.717
***
0.643
***
0.5828
***
0.438
***
0.65284
***
0.5972
***
0.50284
***
0.4028
***
(0.073) (0.0615) (0.0295) (0.0103) (0.0624) (0.040) (0.0105) (0.004)
Cons -0.0343 0.0368 0.0329
***
0.0319
**
0.0286 0.3058 0.2528
**
0.2485
**
(0.063) (0.0132) (0.0202) (0.2840) (0.0468) (0.008) (0.0120) (0.025)
Number of observations 2851 2212 2105 1776 2851 2212 2105 1776
Fixed effects no no no no yes yes yes yes
Adj.R
2
0.032 0.031 0.025 0.030 0.0193 0.0284 0.5020 0.1598
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 8 of 29
Table 5 One hundred groups of wp, lp, wl, t value summary table (Continued)
t 5.5660 5.7970 4.0500 -6.3380 -6.9130
6 months wp 0.0690 0.0364 -0.0425 0.0217 0.0292
lp -0.0335 -0.0551 -0.0407 -0.1476 0.1022
wl 0.102505※0.091487※-0.001725※0.1693044※-0.0730656※
t 4.1339 8.3676 -3.3159 6.4333 2.4222
9 months wp 0.0597 0.0467 -0.0421 0.0054 -0.0460
lp -0.0442 -0.0427 0.0397 0.1088 -0.0912
wl 0.103965※0.089413※-0.0818328※-0.1034639※0.0452
t 6.1218 10.2500 -3.3015 -2.3915 -1.9165
12 months wp 0.0503 0.0450 -0.0387 -0.0693 -0.0594
lp -0.0404 0.0478 0.0383 0.0731 -0.0612
wl 0.090685※-0.0028797※-0.076932※-0.1423882※0.0018128※
t 5.1261 -6.2303 -3.9858 -4.9017 2.1925
18 months wp 0.0487 0.0393 0.0322 -0.0609 0.0515
lp -0.0437 -0.0439 -0.0396 -0.0672 0.0517
wl 0.092433※0.0832855※0.0719 0.0063171※-0.00022※
t 2.3860 4.2167 1.2674 3.3433 -2.0865
24 months wp 0.0477 0.0446 0.0385 -0.0723 -0.0552
lp -0.0200 -0.0398 -0.0326 0.0418 0.0564
wl 0.0676301※0.0844540※0.071153※-0.1141 -0.1116
t 3.0767 10.1525 7.2020 -1.2751 -1.9976
30 months wp 0.0291 0.0330 0.0318 0.0527 -0.0463
lp -0.0245 -0.0399 -0.0343 0.0604 -0.0495
wl 0.0536 0.0729 0.0661 -0.0077 0.0032618※
t 1.0736 1.0897 1.1173 0.8520 5.9153
36 months wp -0.0217 -0.0325 -0.0293 -0.0590 0.0477
lp 0.0241 0.0175 0.0158 0.0647 0.0484
wl -0.0457920※-0.0500914※-0.045144※-0.1237 -0.000688※
t -5.0737 -8.0590 -5.0440 -1.1230 -4.5070
48 months wp 0.0445 0.0504 0.0401 0.2460 0.1566
lp -0.0295 -0.0460 -0.0378 -0.0732 -0.0604
wl 0.0740122※0.0965 0.0778 0.3192 0.2169
t -3.0458 1.0312 0.9965 1.0826 -0.6063
60 months wp 0.0255 0.0267 0.0211 0.0041 0.0044
lp -0.0027 -0.0024 0.0264 0.0600 0.0465
wl 0.0282 0.0291 -0.0053 -0.0558800※-0.0421593※
t 1.0402 1.0100 -0.9474 -4.0311 -3.6525
Note: * is a significant result after the bilateral t-test with = “wp”corresponding to the winner portfolio. “lp”corresponds
to the average return of the strategy portfolio. We conduct the t-test about the “wl”value
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 15 of 29

Among them, the results were significant for 3, 6, and 24 months. The loser and winner
portfolios shown in Table 5 have obvious performance in the momentum effect.
Through observation, we find that the momentum effect has significant performance
through the three to nine months’holding period. During the same period, the wl port-
folio showed an increasing trend with an increase in the formation period. Therefore,
we conclude that when the combination period and holding period become longer, the
effect of the reversal and the momentum become less significant.
CAPM model zero interpretation ability
Chan (1988) argued that the risk of winners and losers is changing over time and, when
the risk factor is controlled, the reversal strategy can only produce minimal returns. In
accordance with the method of Chan (1988), the CAPM model can be used to analyze
whether the risk was controlled in the process of momentum or reversal of the strate-
gy's profitability:
rpt−rA¼αþβrmt−rA
ðÞþεtp∈ω;ιðÞ ð16Þ
rιt−rωt¼αcþβcrmt−rft
ðÞþεtð17Þ
The time interval t is one month, rmt is the return of the equal market index, and
the return of the loser and winner portfolio is checked by (16). (17) is used to test the
W-L portfolio (the same as L-W). The test results show that the BETA value can
explain most of the change in the winner and loser portfolio earnings (above 75%).
However, the W-L (or L-W) portfolio cannot be explained by this BETA. In the case of
strategy 3-36, which brings out momentum returns, there is a positive return on the
winner portfolio, a negative return on the loser portfolio, and this W-L portfolio’s
BETA value is not significant at all. Therefore, the BETA, as a risk measure value, has
no ability to explain the momentum and reverse profitability. This is theoretically
affirmed by the CAPM model having zero interpretation on the medium-term momen-
tum and long-term reversal.
Vision interpretation
We determined all the coefficients required for the MC model and the CAPM model,
and we also used the MC model to make a revenue forecast test for all data from
Fig. 3 Shows the different rates of W-L combinations in different holding periods
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 16 of 29
January 2010 to December 2015. Then, following the development of the paper, we
focus on the interpretation of medium-term momentum and long-term reversal.
Building an explanatory model
First, we compare the CAPM model with the MC model, see (1) and (7). To facilitate
understanding, we summarize two models that remove the constant term as follows:
MC model r
i, t + 1
=β
1
bm
i, t
+β
2
roe
i, t
+ξ
i, t + 1
(1)
CAPM model E(r
i, t
)−r
f, t
=β
i
(E(r
m
)−r
f, t
)+ε
i
(7)
E(ri,t) is the expected return, and ri,t+1is the expected log return. The coefficient of
ROE reflects the profitability of different firms in the same industry, and the βi in the
CAPM model is also an indicator of profitability.
Since the two models are not of the same magnitude, the expected log returns in the
MC model need to be transformed to the same magnitude as those in the CAPM
model:
nri;tþ1¼Aeβ0þβ1bmi;tþβ2roei;t−1þηnr
i;tþ1ð18Þ
The above formula relates the expected net return (ηi,t+1 nr), the expected log
returns (Aeβ0+β1bmi,t+β2roei,t −1), and known net returns (nri,t+1). The expected log
returns are transformed into the above function form so that we can obtain expected
net return. In this case, the expected return is multiplied by the expected logistic return
index (by the conditional logistic variance). Parameter A represents this multiplication.
Parameter A can be estimated using two non-linear irrelevant regression condition
equations, including (1) and (19):
bmi;t¼X∞
j¼1kj−1
1Etri;tþj

−Etroei;tþj
 ð19Þ
After obtaining the value of parameter A, the expected net return can be expressed
as:
r0
i¼Aeri−1ð20Þ
After that, we introduce a new parameter γ:
γ¼r
‘
i−Er
i
ðÞ ð21Þ
The meaning of γis easy to understand. It is the difference between the two models'
predictions of future returns. Logically, if the model has a certain ability to explain the
medium-term momentum and long-term reversal, this part of the interpretation is also
fully included in the γparameter. Then, we place the actual returns sample data
directly from the database into formula (22), which is:
R0¼R−γð22Þ
Finally, according to the series of R′data calculated by (22), we construct a new winner
and loser portfolio and use the same method to verify the existence of the anomalies in
the previous chapter to obtain the new returns value. Examining whether the mid-
momentum and the long-term reversal are weakened, if γcontains these two return
anomalies, then, through the new empirical analysis, the two anomalies will be well ad-
dressed. We offer an empirical test theory as follows. The sample data are expanded from
December 2015 to June 2016.
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 17 of 29
By comparing the sunken sample data in the DataStream database with the data
calculated using the MC model and the CAPM model into (20), (21), and (22), we
obtain the returns of Fig. 4 in Fig. 5, which corresponds to the change in Fig. 1.
Both the mid-momentum and the long-term reversal have weakened in the graphical
trend in Figs. 1, 2, and 4, and the change in the monthly return is more obvious than
the previous change. We show powerful graphical evidence in Figs. 5 and 6 and provide
a more convincing empirical test.
Empirical test
In addition to the processing of the sample, the interpretation process is consistent
with the anomalies existence test. That is, the winner and loser portfolios are con-
structed based on the new return data.
Table 6 shows that the winners’portfolio at different stages of formation is explained
to a large extent by the three to nine-month holding period’s mid-momentum and the
three to five-year holding period’s reversal effect (Fig. 6). Additionally, the different for-
mation periods of the loser portfolio in the three to nine-months holding period of the
momentum effect and the three to five-year holding period of the reversal effect are
also interpreted to a certain degree. For example, the (3, 3) winners (losers) portfolios
in Tables 5 and 6 are 0.0385 (-0.0255) and -0.1384 (0.0631), respectively. The (3, 48)
winners (losers) portfolios in Tables 5 and 6 are -0.0246 (0.0221) and 0.13942 (-0.0246),
respectively. The values in Table 6 satisfy the stochastic fluctuations in the effective
market returns. Thus, the model can explain the mid-momentum and long-term rever-
sal of the two anomalies to some extent, and the mid-momentum explaining capability
is greater than the long-term reversal. We present the three-dimensional line graph
corresponding to Table 6, which is more intuitive reflecting the change after the
change.
From the described empirical study, we draw two conclusions: first, the MC model
has a certain ability to predict income, which is excellent news for the future of the US
stock market. Second, based on the CAPM model, we obtain a new model with ex-
planatory ability for medium-term momentum and long-term reversal. According to
the empirical results, this model can explain the medium-term momentum and shows
weak ability, but not completely zero, for long-term reversal. In the existing study of
Fig. 4 Monthly change chart (adjusted)
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 18 of 29
the US market, there are few models that can explain both mid-term momentum and
long-term reversal.
Correlation analysis
Since we have verified that the MC model does exhibit some explanatory power for
mid-momentum and long-term reversal, an ensuing problem arises: whether the BM or
ROE is a factor in the interpretation of the model? Given this question, we conducted
the following tests.
First, we remove the BM factor and ROE factor. Second, we repeat the same process
as described above. The reconstructed model with only the BM factor or ROE factor
can explain the two stock market anomalies, but the ability of the ROE factor is weaker
than the model with only the BM factor.
As the two company basic indicators have some explanatory power concerning the
vision, we need to determine whether there is a correlation between them and between
ROE, BM and BETA.
Richard and Jeong (1997) proposed such a model:
Pt
Bt¼1þX∞
τ¼11þrðÞ
−τEr
ROEtþτ−rðÞBtþτ−1
Br
 ð23Þ
It has been proven that there is a correlation between the current book value (the
reciprocal of the book market ratio) and the current ROE, and the current book value
also contains more information on the future ROE compared to the current ROE,
which will cause change in the ROE. Additionally, Richard and Jeong (1997) tested the
correlation between the ROE and BETA values and found that the coefficients were
negative. This indicates that there is no correlation between the two values. Next, we
must verify the relevance of the BM factor and the BETA and the hybrid correlation
between the three.
We use the mixed regression method proposed by Newey and West (1987) to regress
Eqs. (24) and (25), in order to weaken heteroscedasticity and sequence dependency that
mixed regression may cause.
BM ¼r0
0þr0
1BETAtð24Þ
BE ¼r0þr1BETArþr2ROErð25Þ
Fig. 5 Annual average earnings graph (adjusted)
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 19 of 29
Table 6 Existential verification results
Hold period q Forming period p 3 months 6 months 9 months 12 months 18 months
3 months wp -0.1384 -0.0843 -0.2944 0.2095 -0.0283
lp 0.0631 0.0295 -0.0298 -0.3943 0.3943
wl -0.2015※-0.1138※-0.2646※0.6038※-0.4226※
t -7.5469 -9.5765 -2.3137 -6.5177 -2.1533
6 months wp -0.0103 -0.0384 0.0483 0.2048 0.1948
lp 0.0384 0.2940 -0.9387 0.2984 -0.0849
wl -0.0487※-0.3324※0.9870※-0.0936 0.2798※
t -4.3401 -2.4090 -4.4905 -1.0605 -1.1958
9 months wp -0.2934 -0.0206 0.0853 -0.4832 0.0519
lp 0.0238 -0.0394 0.3085 -0.5293 -0.2085
wl -0.3172※0.0188※-0.2232※0.0461※0.2604※
t -4.7922 -8.5792 -5.5912 -3.7257 -7.9083
12 months wp -0.7083 -0.3582 0.3849 0.2048 -0.3028
lp 0.2925 0.2490 -0.0294 0.2084 -0.0247
wl -1.0008※-0.6072※0.4143※-0.0036※-0.2781※
t -5.0284 -9.3962 -7.2975 -7.3374 -9.8217
18 months wp -0.9240 0.0108 0.1052 -0.4937 0.2806
lp 0.0284 -0.2420 0.1083 -0.0284 0.2874
wl -0.9524※0.25284※-0.0030※-0.4653※-0.0068※
t -8.06059 -7.7146 -0.0415 -2.7705 -9.0339
24 months wp -0.0183 -0.0083 0.0398 -0.0408 -0.2084
lp 0.0482 0.0029 0.0184 -0.1294 0.1083
wl -0.0665※-0.0112 0.0214※0.0886※-0.3167※
t -5.1487 -0.5971 -2.6853 -8.9878 -9.2583
30 months wp -0.2948 -0.1294 0.2948 0.0698 -0.2949
lp 0.0290 0.1084 0.0108 -0.0908 0.2844
wl -0.3238 -0.2378 0.2840 0.1606※-0.5793※
t -0.9548 -0.7403 -0.0126 -5.6271 -8.7080
36 months wp 0.2084 -0.2850 0.0203 -0.3985 0.5082
lp 0.0129 0.0940 0.2984 -0.2084 0.2952
wl 0.1955 -0.379※-0.2781※-0.1901※0.213※
t -1.8446 -3.6409 -2.0442 -3.9977 -6.2831
48 months wp -0.0024 0.2044 -0.2049 0.0385 -0.0108
lp 0.0597 0.1094 -0.2943 -0.0044 0.2044
wl -0.0621※0.095※0.0894※0.04285※-0.2152※
t -8.873 -7.2558 -8.5790 -5.5865 -6.3009
60 months wp -0.0070 0.1939 0.1033 -0.0188 0.4298
lp -0.2934 0.0139 -0.0210 -0.1944 0.0039
wl 0.2864※0.1800※0.1243※0.1756※0.4259※
t -2.8769 -5.2485 -8.0858 -6.7919 -8.9324
Hold period q Forming period p 24 months 30 months 36 months 48 months 60 months
3 months wp 0.0428 0.2490 0.1932 0.3942 0.1038
lp -0.0208 -0.0283 -0.0286 -0.0246 -0.0612
wl 0.06364※0.27734※0.22180※0.4188※0.1650※
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 20 of 29

Table 6 Existential verification results (Continued)
t -9.5661 -6.4948 -4.5017 -4.3875 -6.5235
6 months wp -0.0242 0.1084 -0.3942 0.0329 0.2494
lp 0.6092 0.0903 -0.3934 -0.2820 -0.3028
wl -0.6334※0.01812※-0.0087※0.31494※0.5522※
t -4.7564 -2.3861 -5.7409 -9.4873 -9.6719
9 months wp 0.2058 0.5038 0.4846 0.2045 0.5028
lp 0.4080 -0.0920 0.2058 0.0385 -0.3040
wl -0.2022※0.5958 0.2788※0.1660 0.8068※
t -7.3042 -1.2111 -7.6711 -1.7727 -2.5962
12 months wp 0.3856 -0.4941 0.3494 0.3842 -0.0794
lp -0.5927 0.3859 0.4928 -0.3842 -0.5938
wl 0.9783※-0.88※-0.1434※0.7684※0.5144※
t -7.2787 -8.0722 -2.9770 -7.2284 -5.0118
18 months wp 0.1958 -0.1858 -0.0184 0.1540 -0.1885
lp 0.0593 0.0284 -0.1084 -0.3749 0.0563
wl 0.13653※-0.2142※0.09※0.5289※-0.2448※
t -9.0714 -9.6582 -3.3249 -8.1674 -4.2962
24 months wp -0.0183 0.1098 0.1282 0.2853 0.0828
lp 0.2945 -0.2084 -0.0173 0.2943 -0.2084
wl -0.3128※0.3182※0.1455※-0.0090 0.2912※
t -3.5671 -5.7649 -3.3872 -1.3965 -7.8998
30 months wp 0.3930 -0.0109 -0.0014 0.0194 0.1854
lp -0.1050 0.0335 0.1042 -0.1988 -0.4851
wl 0.498※-0.0444 -0.1056※0.2182※0.6705※
t -1.2578 -1.7227 -3.6256 -5.0381 -6.5537
36 months wp 0.0189 -0.3940 -0.2840 -0.0018 -0.1030
lp -0.2848 0.0592 -0.2020 -0.0188 -0.0128
wl 0.3037※-0.4532※-0.0820※0.0170 -0.0902※
t -5.7673 -8.9516 -4.3568 -0.61444 -8.8585
48 months wp 0.3095 -0.0240 0.4085 0.2953 0.0044
lp -0.1084 0.3848 -0.0220 -0.0140 0.0199
wl 0.4179※-0.4088※0.4305※0.3093※-0.0155
t -1.7304 -5.2197 -2.3661 -3.6702 -1.1061
60 months wp 0.1088 -0.0053 -0.2420 -0.0110 -0.0399
lp 0.2842 -0.0910 0.2985 0.0503 0.0018
wl -0.1753※0.0857※-0.5405※-0.0613※-0.0417※
t -2.5214 -2.6269 -5.2042 -9.6926 -4.9969
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 21 of 29
Tables 7 and 8 show the results of the correlation regression. The results in Table 7
are almost negative. However, if BM can return the risk, it should be positive. There-
fore, the regression of (24) indicates that the BM value cannot be used to return the
risk but can be used to characterize corporate risk (Penman, 1991).
In the mixed regression in Table 8, the BETA coefficient is less significant, and the
fluctuation varies greatly compared to the regression results in Table 7. After adding
the industry dummy variable, we found that the coefficient of BETA has no signifi-
cance. This shows that company risk impacts the BM ratio. Thus, the industry's special
factors capture risk more effectively than BETA.
Overall, ROE and BM do not correlate with BETA, and BM is less relevant to BETA.
But BM and ROE are related, and BM will cause change in the ROE. This leads to the
following conclusion: the BM factor is the fundamental reason that models can explain
the medium-term momentum and long-term reversal. In other words, the BM effect
does have an impact on medium-term momentum and long-term reversal.
Test conclusions
Through the previous interpretation of the mid-momentum and the long-term reversal
of the two anomalies, we confirm that BM, as an influencing factor, does have some
explanatory power of the two visions in the US stock market. On the other hand, the
interpretation also explains that the BM effect does cause part of the mid-momentum
and long-term reversal to generate excess returns. The BM effect can only partially
Fig. 6 Sorting period W-L combinations or different holdings monthly rate of change (adjusted)
Table 7 Cross-sectional regression with BM as the dependent variable and BETA as the
independent variable year by year
Cons BETA Adj.R
2
2010 0.2554 -0.1546 0.0655
2011 0.4229 -0.2986 0.0055
2012 0.3500 -0.2014 0.0204
2013 0.4476 -0.1012 0.0253
2014 0.2759 -0.0281 0.0062
2015 0.3939 -0.2399 0.0772
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 22 of 29
explain the two visions. There is no in-depth study on this subject, and it is not the
focus of this paper. However, we believe that this issue will create a meaningful re-
search direction. Of course, if there is change in the future, we would consider con-
ducting more detailed research. Additionally, BM as a model impact factor is one for
which the sample data are easy to obtain and calculate, which is a rare advantage.
For the model capacity’s difference with respect to mid-momentum and long-term
reversal, we speculate that the reason may be that the book value and the market value
are infinitely close to the same mean when the time lengthens.
Overall, the research results are of great importance. First, the MC model has a good
ability to predict earnings, but company's basic data is not easy to obtain. The explain-
ing of stock mid-momentum and long-term reversal is convenient.
For the model’s ability to predict and explain, first, we identify whether the model's
capabilities are limited to return forecasts. Second, we identify whether this conclusion
can be applied to other financial indicator forecasts. In other words, we identify
whether the principles used in MC model construction can be exploited in different as-
pects of research in the future. Third, does this new model only apply to the above two
types of visions, or is it possible to have explanatory power for other stock market
anomalies? If the new model is applied to other stock market visions, what will be the
consequence? This study only presents ideas, and future practical application is
required. However, this new model will have deep significance for future research.
Research prospects
Our study uses a newly constructed model to prove the existence of medium-term mo-
mentum and long-term reversal, but it also obtains the intrinsic relationship between
the BM factor and these two visions. This is undoubtedly a great improvement in the
field of financial market vision research. Based on this research, the author suggests
that the following points should be discussed in depth.
First, the US market has significant mid-term momentum and long-term reversal.
China, as a representative developing country, may have the same characteristics, which
is contrary to the stated effective market hypothesis. Second, the MC model is used to
predict future earnings and to explain financial visions in this paper. Thus, we can infer
whether this model has good ability in other financial markets and even in other
Table 8 A comparison of the results of year-round mixed regression with account-to-market ratio
Panel A: No dummy variables
Independent variable 2010 2011 2012 2013 2014 2015
Cons 0.8302 1.2207 1.2830 0.7492 0.9298 1.0593
BETA -0.0824 -0.0240 -0.3023 -0.0427 -0.6222 -0.3812
ROE 1.1221 1.8324 0.8066 1.4899 1.0270 1.9774
Panel B: There are dummy variables
Independent variable 2010 2011 2012 2013 2014 2015
Intercept 0.7045 0.9152 0.6048 0.5362 0.5065 1.0709
BETA -0.0783 -0.0656 -0.1084 -0.0936 -0.0676 -0.0905
ROE 1.3985 1.7643 1.3274 1.5536 1.5359 1.7982
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 23 of 29
nonfinancial sectors. Future discussion will be based on existing research. Third, since
the BM factor does have a considerable internal relationship with medium-term mo-
mentum and long-term reversal, the BM factor also presents some explanatory power
for other visions in the financial market. This research question will become our future
main research direction.
Robustness test
Control the sample age
When the model is calculated for the model coefficients and results, the sample used is
the initial training sample from 1980 to 1994. When we use the complete model to test
the performance of quarterly log returns, we use a sunken sample from June 2010 to
June 2016. According to the empirical analysis conducted by Kelly and Pratt (2013), we
need to consider the estimated deviation of the expected return due to the different
sample. Therefore, to avoid this deviation, we choose the sample data from January
1996 to December 2009, which is a set of sample data at different time intervals to test
the sample sensitivity.
Figures 7 and 8 show the graphical representation of the slope coefficients after the
same regression as that shown in Table 3 under different samples. Figure 9 includes the
industry fixed effect. The two figures will have different test results considering the differ-
ent running time. However, in the mathematical sense, these results are still significant,
and the regression slope coefficient is stable with a certain economic significance.
Use roei,t−1 instead of roei,t
The next step concerns the choice of an auxiliary variable roei,t, which may lead to
unreliability in the test results. That is, in the process of using the log ROE(roei,t) to re-
place the expected log ROE(hi,t)of the next period, it is likely that the MC model will
not agree with the estimation of β2. If the final test result is moderate, this indicates
that the lag value of ROE is a useful auxiliary variable, and this auxiliary variable can be
used to weaken potential bias in the coefficient estimating process.
Panel A of Table 9 shows the summary statistics for the model estimate parameters
using the auxiliary variable regression method. Additionally, the fourth column of the
table records the statistical results of the F statistic.
The F statistic results for the fifth quartile were 93.1769, and the average quartile was
765.4844. The coefficient of roei,t increased from the least squares estimate 0.339625
(0.265) to the average score (median) of 0.494119 (0.3615). This result indicates that
the results of the least squares estimation may be affected by the measurement error,
resulting in an increase in the ROE coefficient estimation. In contrast, the coefficient
variance changes from 0.198346 to 1.176139, almost 10 times that of the previous re-
sult. Thus, as previously proposed, there is a trade-off between the estimates of bias
and efficiency. The constant coefficient estimates will increase mutations. The increase
in all estimated coefficients’anomalies will inevitably lead to anomalies in the implicit
model parameters. However, this can be reduced by adjusting the implicit persistence
parameter (0.9999). The mean (median) of the model estimation parameters indicates
that the long-term unconditional expectation decreases from the OLS estimates
0.056226 (0.041286) to 0.047367 (0.042206). The persistence parameter value
Wei-qi and Jingxing Financial Innovation (2018) 4:1 Page 24 of 29