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Electronic copy available at: http://ssrn.com/abstract=1763439Electronic copy available at: http://ssrn.com/abstract=1763439Electronic copy available at: http://ssrn.com/abstract=1763439
1
Latent Fundamentals Arbitrage with a Mixed Effects
Factor Model
Andrei Salem Gonçalves, andreisalem@hotmail.com (CEPEAD/UFMG)
Robert Aldo Iquiapaza, rbali@face.ufmg.br (CEPEAD/UFMG)
Aureliano Angel Bressan, bressan@face.ufmg.br (CEPEAD/UFMG)
Abstract: We propose a single factor mixed effects panel data model to compose an
arbitrage portfolio identifying differences in firm-level latent fundamentals. Furthermore,
we show that even though the characteristics that affect returns are unknown variables, it
is possible to identify the strength of the combination of these fundamentals for each
stock by following a simple approach using historical data. As a result, a trading strategy
that buys the stocks with the best fundamentals (strong fundamentals portfolio) and sells
the stocks with the worst ones (weak fundamentals portfolio) realized significant risk-
adjusted returns for the period between July 1986 and June 2008. In this case, this
arbitrage portfolio generated a significant monthly risk-adjusted return of 2.42%
considering the Carhart (1997) 4-factor model and presented a market sensibility (CAPM
Beta) of 0.14. For robustness we performed sub period and seasonal analyses as well as
an adjustment by trading costs. Finally, we find further empirical evidence to profit from
the usage of a simple investment rule identifying fundamentals from structure of pass
returns.
Keywords: Arbitrage, Latent Fundamentals, Factor Models, Mixed Effects.
JEL: C22, C53, G11
1. Introduction
In the investments area, securities selection is the attempt to distinguish prospective winners
from losers. The base for doing that is grounded in two main conjectures. First, even if one
believes the market is efficient in the intermediate or long run, there is no reason to believe
that instant market quotes reflect long-run value in the short run (Black 1986). Second,
market microstructure, liquidity, and behaviorist explanations suggest that at any given time
prices may be noisy and returns skewed.
The value and growth investing is one of the deviations that persists in the USA as well as in
international markets, attracting a great deal of academic empirical research (Chan and
Lakonishok 2004). There is vast empirical evidence (see Fama and French 1992, 1996, 1998;
Lakonishok, Shleifer, and Vishny 1994) suggesting that on average value stocks outperform
glamour (growth) stocks. Nevertheless, the debate about the explanations for this ―value
premium‖ anomaly in stock returns is going on with several proposals. The first branch is led
by Fama and French (1992, 1996), who argue that the market is efficient and that better
performance of value investing could be explained by value stocks being more risky. The
other area led by Lakonishok, Shleifer, and Vishny (1994) hold the view that investors’
cognitive biases and agency costs of professional investment, which lead both individuals
and institutional investors to prefer growth stocks and dislike value stocks, could explain the
value premium anomaly (Sharma, Hur and Lee 2008).
There are other significant aggregate factors associated with deviations from the equilibrium
model, such as the companies’ size, liquidity, cash flow, dividend yield, earnings yields,
volatility, accruals, etc. (Fama and French 1992, 1996; Acharya and Pedersen, 2005; Liu
Electronic copy available at: http://ssrn.com/abstract=1763439Electronic copy available at: http://ssrn.com/abstract=1763439Electronic copy available at: http://ssrn.com/abstract=1763439
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
2
2006; Ang et al., 2006; Ang and Bekaert 2007; Hirshleifer, Hou and Teoh, 2009; Kang, Liu
and Qi, 2010). However, Subrahmanyam (2010) stated that predictive variables emanate
from informal arguments, alternative tests of risk-return models, behavioural biases, and
frictions, such as limited arbitrage and leverage controls, especially for sophisticated
professional investors (Stein 2009). According to Subrahmanyam (2010) in the last twenty
five years ―more than fifty variables have been used to predict returns‖.
This paper does not aim to explain why those aggregate deviations persist, or what the
specific relevant variables are. Instead, the aim is to show that is it possible to profit the
deviations through the estimation of the combination of the unknown relevant variables.
The usual way to take advantage of persistent deviations, thus, to identify prospective
winners and losers, is to sort the stocks for a particular characteristic. For example, to identify
growth and value stocks it is necessary to estimate several indicators or fundamentals, such as
the price-to-earnings ratio and the price-to-book ratio. Growth stocks usually have high price-
to-earnings and price-to-book ratios, which means that these stocks are relatively high-priced
in comparison with the companies’ net asset values. On the other hand, value stocks have
relatively low price-to-earnings and price-to-book ratios. Other firm characteristics, such as
size, dividend yield, and liquidity, that contributes to stocks’ future performance, are
identified in a similar way. A combination of sorting variables is also possible.
There are, also, more complex procedures to identify prospective winners and losers. Some
researcher suggest the use of structural VAR and/or VECM models with the inclusion of
macroeconomic and/or fundamentals, such as growth rates of industrial production and stock
prices (Binswanger 2004; Louis and Eldomiaty 2010).
In this paper we take an alternative approach in order to identify prospective winners. Our
aim is to show that it is also possible to take advantage of any premium arbitrage opportunity
identifying firms’ strength based on the correlational structure of stocks past returns and a
single factor mixed effects panel data model. In this way, we also propose an alternative to
Subrahmanyam (2010: p27), when he said that ―more needs to be done to consider the
correlational structure amongst the variables, use a comprehensive set of controls, and
discern whether the results survive simple variations in methodology.‖
Furthermore, we found empirical evidence that a zero net investment portfolio (i.e., an
arbitrage portfolio), structured with a long position in a portfolio composed by the stocks that
presented the best combination of the N latent fundamental factors (the ―strong fundamentals
portfolio‖) and a short position in a portfolio composed by the stocks with the worst
combination of the N fundamental factors (the ―weak fundamentals portfolio‖), generated
positive and statistically significant risk-adjusted returns considering the Carhart (1997) 4-
factor model.
The remaining of the paper is organized as follows. Section 2 describes the theoretical
background in order to identify the stocks with the strongest and the weakest firm-level latent
fundamentals. Section 3 gives an overview of the data used, defines the period of the study
and presents the statistical model used to perform the analysis and the portfolios formation
approach. Section 4 reports the general results for the application of the strategy; a subsample
and seasonal analysis of the strategy; the results adjusted for trading costs. And, finally,
section 5 presents the conclusions of the study.
Electronic copy available at: http://ssrn.com/abstract=1763439
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
3
2. Theoretical Background
2.1. Preliminaries
The structure of stock returns was first studied by Markowitz (1952) through the mean-
variance framework, which represented the theoretical basis for the Factor Models and the
well-known Capital Asset Pricing Model (CAPM), proposed by Sharpe (1964), Lintner
(1965) and Mossin (1966).
According to Bodie et al. (2009) the single-factor model can be expressed as follow:
,,i t i i t i t
r E r M
(1)
where ri,t is the return on stock i at time t, E(ri) is the stock i expected return unconditioned
on time, Mt is the value of the market factor that affects all securities at time t, βi is the
sensibility of stock i to that factor and εi,t is the firm i characteristic that determines the
idiosyncratic risk at time t.
From Equation 1, it can be inferred that M captures the effects of unanticipated market
movements and ε expresses the firm-level specific latent fundamentals on stock i returns. The
expectation for micro innovations are equal to zero. Furthermore, Sharpe (1963) presented
the single-index model, which is a particular restriction of the single-factor model:
, , ,
()
i t i f i M t f i t
r r r r
(2)
where rf is the return on the economy risk-free asset and rM,t is the return on the market
portfolio at time t.
Observe that in Equation 2, (rM,t-rf) is the factor that expresses unanticipated market
surprises. As E(rM,t-rf) is not necessarily equal to zero, then the intercept of the model (αi+rf)
is not the stock’s i expected return unconditioned on time as in the single-factor model.
Besides, as βi is the market or macroeconomic risk (sensibility) of stock i and rf can be
obtained by a risk-free investment, αi can be viewed as the return above the risk or the risk-
adjusted return for the stock i.
In this context, under specific assumptions about the economy, the CAPM general
equilibrium predicts that the risk premium on individual assets, E(ri,t-rf), will be proportional
to the risk premium on the market portfolio, E(rM,t-rf) , and the securities’ market sensibility
(βi):
, , ,
,,
(i)
(ii)
i t f i M t f i t
i t f i M t f
r r r r
E r r E r r
According to the CAPM, the εi is the firm’s i idiosyncratic return and it is not related to any
risk source. In addition, E(εi)=0 and Var(εi)=σ2. Furthermore, comparing the equations 2 and
(3)
(3.i)
(3.ii)
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
4
3.i, it is possible to notice that the CAPM predicts that αi should be zero for all assets. As αi
is the risk-adjusted return for stock i, the CAPM implies that no asset can originate returns
above its exposition to the macroeconomic risk (market risk). As the CAPM is a prediction
about ex ante (expected) returns, it allows the stocks’ ex post (realized) returns to present
risk-adjusted returns different from zero; but it also states that the expected risk adjusted
returns of different stocks are random variables with mean equal to zero.
Although the CAPM is widely used in the context of investment analysis, the empirical
literature demonstrated that it suffers from serious problems that limit most of its applications
(see Friend and Blume , 1970; Jensen et al., 1972; Elton et al., 1993; Jagannathan and
McGrattan, 1995 and Fama and French, 2004). As an alternative, Ross (1976) proposed the
Arbitrage Pricing Theory (APT). The APT assures the same basic results presented in the
CAPM but implies that the risk-adjusted return must be equal to zero if and only if the factor
model used is an unrestricted multifactor model for the structure of stock returns.
In this context, some firm’s characteristics such as size and value play an important role in
determining the structure of stock returns. As a consequence, in order to account for the
effects of the variables that affect the returns the 3-factor model was proposed by Fama and
French (1993):
, , ,i t f i M t f i t i t i t
r r r r S SMB H HML
(4)
where SMBt is the return at time t of a portfolio composed by a long position in the biggest
companies and a short positions in the smallest ones (size factor), HMLt is the return at time t
of a portfolio composed by a long position in the highest value companies and a short
positions in the lowest value ones (value factor) and Si and Hi are the sensibility of stock i to
the size and value factors respectively.
According to tests performed by Fama and French (1996) the 3-factor model explained most
of the asset pricing anomalies described in the literature. However, Jegadeesh and Titman
(1993) demonstrated that a strategy that buys stocks with high past returns and sells stocks
with low past returns generated risk-adjusted returns robust to the 3-factor model. They
called this factor the ―Momentum Effect‖ and Carhart (1997) introduced the 4-factor model,
that is essentially the 3-factor model in addition to the momentum factor:
, , ,i t f i M t f i t i t i t i t
r r r r S SMB H HML U UMD
(5)
where UMDt is the return at time t of a portfolio composed by a long position in the winners
(stocks with the highest past returns) and a short positions in the losers (stocks with the
lowest past returns) and Ui is the sensibility of stock i to the momentum factor.
More recently, other anomalies that represent important factors for the structure of stock
returns were presented in the literature, such as the liquidity factor (Acharya and Pedersen,
2005 and Liu, 2006) and the exposure to the aggregated volatility (Ang et al., 2006). As can
be seen, the literature has provided several different asset pricing anomalies that must be
counted as factors in addition to the market portfolio excess return in order to create the
factor model that explains the structure of stock returns.
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
5
In this perspective we claim that a possible unrestricted model that explains the structure of
stock returns presents N+1 different factors1. One of them is a market portfolio factor that
affects all assets, but in different intensities (alternative assets have different sensibilities to
this factor) while the other N factors are firm-level latent fundamentals that compose the
―microeconomy‖ of firms which, in a panel data setting, correspond to the heterogeneity
between these firms. Still, we shall prove that although the N firm-level factors are unknown,
it is still possible to find what stocks present the best and the worst combinations of the N
latent fundamental factors by following a simple procedure.
2.2. The model
First, consider the following model:
,,i t f i t i i t
r r M
(6)
where Mt is the market factor at time t, γi is the influence of a specific firm-level fundamental
or microeconomic characteristic that exists but cannot be measured under the structure of its
stock’s ri,t returns and ωi represents the idiosyncratic return under the hypothesis that the
model proposed in 6 is the unrestricted model.
Let us assume that the microeconomic characteristic is the firms’ size. Therefore, the
influence of firm’s i size in the structure of its returns is equal to γi. Now, let the following
equation represent the real unrestricted factor model for the structure of stock returns:
, 1, 2, , ,
...
i t f i t i i N i i t
r r M
(7)
where Mt is the market factor at time t, γk,i is the influence of the k-th firm-level characteristic
under the structure of returns and ωi represents the real idiosyncratic return for the stock i
under the true unrestricted model for stock returns, such that
. and 0 2
ii VarE
Also, consider that each γk,i is related to a specific firm-level fundamental or microeconomic
characteristic Xk. The γk,i can be viewed as by which magnitude the microeconomic
characteristic Xk influence the structure of stock i returns. As the firms’ microeconomies are
stable, at least in the short run, we consider that the parameters γk,i are relatively stable trough
time and deterministic for a specific time t.
In this setting we maintain the original one risk factor model, or market model. Recently,
Cohen, Polk and Vuolteenaho (2009), using cross-sectional tests, found that there exist
specifications in which differences in relative price levels of individual stocks can be largely
explained by their fundamental betas. The author said, ―the CAPM fails to explain the one-
1 There is no difference in the reasoning line if we allow the number of factors vary along the firms, so Ni +1.
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
6
period expected returns on some dynamic trading strategies but, …, gets stock prices and
expected long-term returns approximately right‖ (p.2742). So, it is important not to relate the
microeconomic or firm-level characteristics Xk to aggregate risk factors, such as the size,
value and/or momentum factors, from models presented in the literature. They have different
concepts from those in Equation 7, where the fundamentals indicate influences of firm-level
characteristics that do not change trough time (in the short run).
Observe that as the model proposed in Equation 7 is the unrestricted model for the structure
of stock returns, its intercept is equal to zero. The expression βi∙Mt represents stock’s i return
generated by its exposition to the market factor and, thus, it is the return related to the stock’s
market risk. The expression
1, 2, ,
...
i i N i
is the stock return related to its own
microeconomy and, therefore, does not contribute as a risk factor for the structure of stock
returns.
Now, let the market factor M to be the market excess of return and
1, 2, ,
...
i i i N i
.
Rewriting Equation 7, we have:
, , ,i t f i M t f i i t
r r r r
(8)
As a consequence,
i
is the firm’s i microeconomy or firm-level fundamentals’ influence
under the structure of its returns. Comparing Equation 8 to the single index model in
Equation 2, we have that
ii
and, thus, the CAPM predicts that
0
i
E
. We do not
rule out this result, but now we have a different interpretation of it. As
i
represents the firms
i microeconomy influence under the structure of stock i returns and the firms’
microeconomies are relatively stable over time, we have that:
(i) 0
(ii) ( )
(iii) | 0
(iv) ( | ) 0
i
i
i
i
E
Var
Ei
Var i
Applying the Expectation and Variance operators to Equation 8, we have:
,,
2
, , ,
i t f i M t f
i t i M t i t
E r r E r r
VAR r VAR r v VAR
If we consider that εi,t =
i
+ ωi,t , then the results proposed in 10.i and 10.ii are the same basic
results of the CAPM and the idiosyncratic return for the CAPM model has two components:
the real idiosyncratic return and the return related to the firm’s own latent fundamentals.
As a consequence, now applying the conditioned Expectation and Variance operators to
Equation 8, we have:
(9)
(9.i)
(9.ii)
(9.iii)
(9.iv)
(10)
(10.i)
(10.ii)
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
7
,,
2
, , ,
||
||
i t f i M t f i
i t i M t i t
E r i r E r r E i
VAR r i VAR r VAR i
Observe that
i
contributes to the stock’s i expected return but does not contribute to its
volatility. Therefore, we have that as higher the value
|
i
Ei
, the stronger the firm’s i
microeconomy and higher its expected returns.
Most important, if our model prediction in 11 is valid, then a strategy that buys stocks from
firms with the best (strongest) fundamentals– expressed by higher values of
i
– and shorts
those with the worst (weakest) fundamentals should generate positive and statistically
significant risk-adjusted returns, since these microeconomies are not related to any market
risk source. In addition, as the Equation 8 represents an unrestricted model for the structure of
stocks’ returns, a significant risk-adjusted return must be observed regardless of the factor
model used to analyze the returns as long as it is not the unrestricted model itself.
One interesting question would be how to reconcile our firm-level implicit fundamentals with
the standard cash flow valuations models. The central issue in asset pricing is whether stock
prices move due to the revisions of expected future cash flows or/and revisions of expected
discount rates, and by how much of each. Chen and Zhao (2010), using direct cash flow
forecasts, showed that there is a significant component of cash flow news in stock returns,
whose importance relative to the discount rate news increases with investment horizons. For
horizons over two years, the importance of cash flow news far exceeds that of discount rate
news. So, the latent factor identified by our procedure would be more related to a
combination of those factors that change the discount rate and only partially to those
affecting the cash flows.
3. Data Collecting, Sampling and Treatments
3.1. Data Collecting and Sampling
This study considered the period between July 1985 and June 2008. All the measures
presented are monthly unless other time frequency is indicated. The statistical tests and
performance measures were performed for the period of July 1986 to Jun 2008, which is
defined as the sample period.
The sample study comprised the biggest 1,000 US companies2 of each year. The CRSP3
Value Weighted Index and the Monthly Treasury Bill rate were used as proxies for the
market portfolio and the risk free rate respectively.
2 The companies are ranked by their market cap (market cap = number of shares outstanding · price per share) at
the last trading day of May for each year.
3 Center of Research in Security Prices. <www.crsp.com>
(11)
(11.i)
(11.ii)
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
8
Moreover, the data set was obtained from three different databases. All the information of
stocks and indices was obtained from the CRSP monthly file database while the monthly
Treasury bill rate was collected from the Federal Reserve Bank interest rate database and the
data used for the 4-Factor model employed in the analysis was obtained from Kenneth
French’s website4. The statistical analysis was performed with Microsoft Excel 2007 and
with the R software, package nlme (http://www.R-project.org).
3.2. Estimation details
From the former derivations, we have that the model in Equation 8 is the unrestricted model
for the structure of stock returns. Also, we demonstrated that higher values of
i
imply in
stronger firm-level fundamentals for the i-th firm and, as a consequence, higher expected
returns. However, as in Equation 8 there are too many parameters to be estimated, there
would be a high loss in terms of statistical information. To manage that limitation, the
following longitudinal one factor model with random effects was used for the analysis and
estimated by the Restricted Maximum Likelihood5:
, , , , , , ,i t TB t CRSP t TB t i CRSP t TB t i i t
r r r r b r r
(12)
where rTB,t is the Treasury bill rate at month t, rCRSP,t is the CRSP Value Weighted Index
return at month t, β is the model’s fixed effect and bi and
i
are the model’s random effects.
In addition, we have that
2
( , ) ~ (0, )
ii
bN
, where
is the variance and covariance matrix
for the model’s random effects. The sensibility of stock i to the market is given by β+bi and
the firm-level latent fundamental’s influence on the structure of stock’s i returns is
i
.
At every July’s first trading day6 ranging from 1986 to 2007, we run the model proposed in
Equation 12 using the last 12 months information. As the 1,000 biggest companies were
evaluated every May, we verified which of them were still being traded in the market on the
first trading day of July on the same year, and those not traded were excluded from the
analysis7. Then, we ranked the stocks in an ascending order by their firm-level fundamentals
i
’s and constructed 10 equally weighted portfolios based on this ranking. The hundred
lowest
i
stocks formed the portfolio 1 (the ―weak fundamentals portfolio‖), the next
hundred formed the portfolio 2 and so on until the portfolio 10 (the ―strong fundamentals
portfolio‖) which would be formed by less than 100 stocks since some of the 1,000 stocks are
not being traded anymore at July first8.
4 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
5 For details about the longitudinal approach used, see Fitzmaurice et al. (2004).
6 By composing the portfolios at July first trading day we avoid possible effects of financial statements
disclosure.
7 We did that for each year to assure the applicability of the strategy.
8 The number of stocks in the strong portfolio ranged from 92 to 98 during the analysis.
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
9
In addition, a zero net investment portfolio (the ―arbitrage portfolio‖) is composed to take
advantage of the differences in firm-level latent fundamentals, thus is taking a long position
in the strong fundamentals portfolio and a short position in the weak fundamentals portfolio
every July first. To minimize trading costs, the portfolios were not rebalanced until the first
trading day of July of the next year, when the analysis was performed again and the new
portfolios were formed (i.e., we considered a one year investment horizon).
The arbitrage portfolio does not present real returns since there is no net investment. As a
consequence, to analyze its results we considered the returns per dollar long in this portfolio
as the arbitrage portfolio returns. In addition, there are two arbitrary adequacy rules for the
implementation of this strategy:
a) If a portfolio, at some point during the portfolio’s investment horizon, contains a company
that does not present information anymore, then it is considered that the stock is sold exactly
at the last price available for the last trading day of the previous month and the funds are not
reinvested until the next analysis date.
b) If any stock does not have any available information to perform the analysis for a specific
month, then the observation of this stock for this specific month is dropped.9
4. Results
4.1. Overall results
The portfolios’ results for the sample period are reported in Table 1. The average returns, risk
premium, tracking error and risk-adjusted returns seem to grow accordingly to the portfolio’s
firm-level fundamentals strength. In addition, the risk measures (CAPM Beta and Standard
Deviation) follow a similar pattern, but both the weak and the strong portfolios present high
risk levels in comparison to the other portfolios. The strong fundamentals portfolio had a
statistically superior variance than the weak portfolio (F= 2.31) in line with the explanation
and results of Fama and French (1992, 1993), and Petkova and Zhang (2005).
The strong portfolio presented average return and risk premium higher than the market and a
market sensibility of 0.71, meaning that its average return was above the market’s return and
its market sensibility was lower than it was for the market portfolio itself.
Furthermore, the strong fundamentals portfolio (weak fundamentals portfolio) risk-adjusted
return is positive (negative) and statistically significant while the arbitrage portfolio presents
average return, risk premium, tracking error and risk-adjusted returns that are positive and
statistically significant. These results are in accordance to our model’s theoretical background
and, therefore, represent empirical evidence in favor of its validity. This results are also in
line with the literature, Fama and French (1992, 1998, 2006), Lakonishok, Shleifer, and
Vishny (1994), Sharma, Hur and Lee (2008) among others. Moreover, the arbitrage
portfolio’s market sensibility was only 0.14, which indicates a low level of market risk, as it
9 This rule was applied for a very insignificant number of observations.
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
10
is expected given its composition (long and short positions in alternative portfolios). The
arbitrage portfolio had a statistically significant monthly 2.42% risk adjusted return (33.2%
annually). As a point of reference, this result is much more expressive than the expected
value premium of 6.1% per annum found by Chen, Petkova and Zhang (2008) for the period
from 1945 to 2005.
TABLE 1 - Portfolios’ Results: Sample Period Analysis, July 1986 to June 2008
The portfolios are formed every July first trading day according to the firms’ fundamentals strength in the
previous 12 months and the investment horizon is one year ahead. The results reported are monthly based and
refer to the period between July 1986 and June 2008 (n=264) and their t-statistics are reported in parenthesis.
The Risk Premium is the portfolio average excess of return considering the risk-free rate, the Tracking Error is
portfolio average excess of return considering the market portfolio return and the Risk-Adjusted returns are
related to the 4-factor model presented in Equation 5. The CRSP Value Weighted Index and the Monthly
Treasury Bill rate were used as proxies for the market portfolio and the risk-free rate respectively. The arbitrage
portfolio has value zero every July 1st and, therefore, we considered the returns per dollar long in this portfolio
as its returns.
Portfolio
Average
Return
Risk
Premium
Tracking
Error
Risk-
Adjusted
Return
CAPM
Beta
Standard
Deviation
Weak Fundamentals Portfolio
-0.0007
-0.0044
-0.0097
-0.0078
0.55
0.0542
(-0.20)
(-1.30)
(-3.00)
(-2.52)
Portfolio 2
0.0059
0.0022
-0.0031
-0.0017
0.48
0.0405
(2.36)
(0.88)
(-1.23)
(-0.79)
Portfolio 3
0.0087
0.0050
-0.0003
0.0003
0.47
0.0377
(3.74)
(2.14)
(-0.14)
(0.17)
Portfolio 4
0.0095
0.0058
0.0005
0.0011
0.49
0.0378
(4.08)
(2.49)
(0.20)
(0.56)
Portfolio 5
0.0113
0.0076
0.0023
0.0025
0.49
0.0385
(4.79)
(3.22)
(0.97)
(1.27)
Portfolio 6
0.0132
0.0095
0.0042
0.0039
0.54
0.0420
(5.10)
(3.66)
(1.69)
(1.82)
Portfolio 7
0.0148
0.0111
0.0058
0.0054
0.55
0.0449
(5.37)
(4.02)
(2.20)
(2.26)
Portfolio 8
0.0167
0.0130
0.0077
0.0074
0.58
0.0483
(5.62)
(4.37)
(2.76)
(2.85)
Portfolio 9
0.0216
0.0179
0.0126
0.0113
0.63
0.0550
(6.40)
(5.29)
(4.07)
(3.77)
Strong Fundamentals Portfolio
0.0312
0.0275
0.0222
0.0210
0.71
0.0823
(6.15)
(5.42)
(4.66)
(4.30)
Arbitrage Portfolio
0.0311
0.0274
0.0221
0.0242
0.14
0.0921
(5.48)
(4.83)
(3.61)
(4.10)
Market
0.0090
0.0053
-
-
1.00
0.0435
(3.36)
(1.99)
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
11
These results also lead us to conclude that our approach was able to measure the firm-level
latent fundamentals strengths of different stocks and that these fundamentals had a direct
influence on the stocks’ returns during the sample period. In addition, the main outcome is
that it was possible to obtain positive risk-adjusted returns by identifying the strongest and
weakest firm-level fundamentals for that period.
4.2. Subperiod and Seasonal Analyses Results
To assure that the results were not conditioned to the analyzed period, we divided the sample
period into five subperiods and performed the same analysis. Then, we had a total of 264
months (22 years) that were split into three subperiods of four years and two of five years, as
can be seen in Table 2.
The F-statistics indicated that the average returns were jointly equal during the five
subperiods for all portfolios considered, what leads us to conclude that the strategy proposed
is relatively stable through time. Moreover, the predictions generated by our model were
stable for all subperiods, given the positive risk-adjusted returns for the arbitrage portfolio
(although not always statistically significant). In addition, the average returns for the strong
fundamentals portfolio were higher than for the weak fundamentals one and market portfolios
in all subperiods analyzed as well as the risk-adjusted returns for the strong portfolio in
comparison to the weak portfolio.
TABLE 2 - Portfolios’ Results: Subperiod Analysis
The portfolios are formed every July first trading day according to the firms’ fundamentals strength in the
previous 12 months and the investment horizon is one year ahead. The results reported are monthly based and
refer to the specified subperiods and their t-statistics are reported in parenthesis. The Risk-Adjusted returns are
related to the 4-factor model presented in Equation 5. The CRSP Value Weighted Index and the Monthly
Treasury Bill rate were used as proxies for the market portfolio and the risk-free rate respectively. The
arbitrage portfolio has value zero every July 1st and, therefore, we considered the returns per dollar long in this
portfolio as its returns. The F-statistics refer to the null hypothesis that the average returns are jointly equal for
all subperiods.
Subperiod
Weak Fundamentals
Portfolio
Strong Fundamentals
Portfolio
Arbitrage Portfolio
Market
Average
Return
Risk-Adjusted
Return
Average
Return
Risk-Adjusted
Return
Average
Return
Risk-Adjusted
Return
Average
Return
1986 - 1990
-0.0019
-0.0094
0.0386
0.0356
0.0398
0.0391
0.0100
(-0.31)
(-1.88)
(3.03)
(3.58)
(3.76)
(4.03)
(1.27)
1990 - 1994
-0.0066
-0.0151
0.0349
0.0219
0.0402
0.0313
0.0080
(-0.99)
(-2.48)
(3.45)
(2.36)
(4.61)
(3.32)
(1.59)
1994 - 1998
0.0115
0.0039
0.0217
0.0105
0.0096
0.0013
0.0204
(1.39)
(0.37)
(3.69)
(1.42)
(1.01)
(0.10)
(4.50)
1998 - 2003
-0.0120
-0.0131
0.0330
0.0263
0.0431
0.0345
0.0007
(-1.32)
(-1.60)
(2.08)
(1.65)
(2.25)
(1.73)
(0.09)
2003 - 2008
0.0068
0.0014
0.0280
0.0247
0.0219
0.0217
0.0082
(1.18)
(0.23)
(3.68)
(3.02)
(2.57)
(2.26)
(2.21)
F-Statistics
(p-Value)
1.71
-
0.31
-
1.29
-
1.39
(0.15)
(0.87)
(0.27)
(0.24)
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
12
We also conducted an analysis by calendar months to verify whether seasonal effects exist in
the performance of our investment methodology. This seasonal analysis can be found in
Table 3. The F-statistics indicated that the average returns were not jointly equal for all
calendar months for any portfolio considered besides the market portfolio, indicating that our
strategy presented significant seasonal effects.
TABLE 3 - Portfolios’ Returns by Calendar Months
The portfolios are formed every July first trading day according to the firms’ fundamentals strength in the previous
12 months and the investment horizon is one year. The average monthly returns reported refer to the specified
calendar months and their t-statistics are reported in parenthesis. The CRSP Value Weighted Index was used as a
proxy for the market portfolio. The arbitrage portfolio has value zero every July 1st and, therefore, we considered the
returns per dollar long in this portfolio as its returns. The F-statistics refer to the null hypothesis that the average
returns are jointly equal for all calendar months.
Calendar
Month
Weak Fundamentals
Portfolio
Strong Fundamentals
Portfolio
Arbitrage
Portfolio
Market
January
-0.0015
0.0830
0.0844
0.0164
(-0.12)
(3.72)
(3.08)
(1.72)
February
-0.0011
0.0577
0.0630
0.0070
(-0.12)
(2.33)
(1.97)
(0.83)
March
-0.0150
0.0465
0.0590
0.0069
(-2.87)
(3.86)
(4.83)
(0.98)
April
-0.0123
0.0530
0.0617
0.0127
(-1.01)
(3.44)
(2.92)
(1.53)
May
0.0240
0.0500
0.0268
0.0188
(2.08)
(3.17)
(1.36)
(2.67)
June
-0.0263
0.0388
0.0596
0.0032
(-2.17)
(2.28)
(2.38)
(0.40)
July
-0.0105
-0.0038
0.0067
0.0023
(-0.81)
(-0.28)
(0.72)
(0.24)
August
0.0081
0.0126
0.0032
0.0007
(0.58)
(0.80)
(0.35)
(0.06)
September
-0.0049
-0.0149
-0.0093
-0.0048
(-0.42)
(-1.10)
(-0.83)
(-0.48)
October
-0.0020
0.0050
0.0063
0.0075
(-0.15)
(0.27)
(0.51)
(0.57)
November
0.0231
0.0100
-0.0129
0.0142
(2.05)
(0.60)
(-0.83)
(1.36)
December
0.0103
0.0363
0.0241
0.0233
(1.63)
(2.93)
(1.83)
(3.10)
F-Statistics
(p-Value)
1.75
3.00
3.08
0.78
(0.06)
(0.00)
(0.00)
(0.66)
The arbitrage portfolio generated negative average returns for two months of analysis
(September and November), but none of those are statistically significant, while for the other
months it presented positive and, mostly, statistically significant average returns. The strong
fundamentals portfolio presented similar pattern, but the two months of non significant
negative average returns were July and September.
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
13
On the other hand, the weak fundamentals portfolio generated, mostly, negative average
returns, but only two of them were statistically significant (March and June), while two of the
few positive average returns generated were statistically significant (May and November).
In addition, the market portfolio presented only one month of negative non significant
average return, but also only three months presented positive significant average returns
(January, May and December).
Finally, the subperiod and calendar month analysis indicated the stability of the strategy
throw the years, but the existence of seasonal effects. However, the main characteristics of
our strategy do not change significantly for different months since none of the negative
average returns for the arbitrage portfolio were statistically significant.
4.3. Trading Costs Adjusted Results
The former results provides evidence that our strategy (the arbitrage portfolio) can generate
positive risk-adjusted returns, but from a practical investment perspective it is important to
understand whether or not the abnormal returns can be sustained after accounting for trading
costs. In this context, our strategy was considered to have a turnover of 100%10. Berkowitz et
al. (1988) estimated a one way transaction cost of 23 basis points for institutional investors.
For the analysis of trading costs performed on our strategy, a two way transaction cost of
0.25% per trade was used with a degree of conservatism.
The results for the annual returns adjusted by trading costs can be found in Table 4. It can be
seen that annual returns for the strong fundamentals portfolio are positive for the whole
sample period while the weak fundamentals portfolio presented many negative annual
returns. The weak portfolio generated a higher return compared to the strong one only for 4
of the 22 years in the sample, while the strong fundamentals portfolio beats the market for 21
of 22 years of analysis. In addition, the arbitrage portfolio generated positive returns for 18 of
22 years of our analysis.
Furthermore, the average annual return for the strong fundamentals portfolio is 41.08% while
the market portfolio and the weak fundamentals portfolio generated respectively 10.95% and
-0.91% average annual returns. The strong fundamentals portfolio annual risk-adjusted return
was 36.67% while the weak fundamentals portfolio presented a negative risk-adjusted return
of -22.56%, both statistically significant. In addition, the arbitrage portfolio presented a
positive and statistically significant average return of 40.97% and a risk-adjusted annual
return of 52.86%.
10This is a conservative assumption since not all assets are being traded each year. In fact, some of the assets
remain in the portfolio for longer than one period, which would lead to a turnover lower than 100%.
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
14
The results suggest that our investment approach generated positive risk-adjusted returns for
the sample period even after adjusting by trading costs. That is, there is some empirical
evidence that supports our model, proposed in Equation 8, and they lead to the conclusion
that it is possible to profit from the usage of a simple investment approach based on it.
TABLE 4 - Portfolios’ Annual Returns Adjusted by Trading Costs
The portfolios are formed every July first trading day according to the firms’ fundamentals strength in the
previous 12 months and the investment horizon is one year. The results reported are annual based and refer to
the specified subperiods. The returns for the Weak, Strong and Arbitrage portfolios were adjusted by trading
costs considering a two way transaction costs of 0.25% and a turnover of 100%. The Risk-Adjusted returns are
related to the 4-factor model presented in Equation 5. The CRSP Value Weighted Index and the Monthly
Treasury Bill rate were used as proxies for the market portfolio and the risk-free rate respectively and are not
adjusted by trading costs. The arbitrage portfolio has value zero every July 1st and, therefore, we considered the
returns per dollar long in this portfolio as its returns. The t-statistics refer to the significance of the average
annual returns and were reported in parenthesis.
Subperiod
Weak
Fundamentals
Portfolio
Strong
Fundamentals
Portfolio
Arbitrage
Portfolio
Market
Risk-Free
Rate
1986 - 1987
-0.0896
0.4572
0.5369
0.1952
0.0557
1987 - 1988
-0.1069
0.4039
0.5010
-0.0625
0.0549
1988 - 1989
0.1237
0.4144
0.2797
0.1878
0.0756
1989 - 1990
-0.0677
0.7502
0.8078
0.1264
0.0822
1990 - 1991
-0.3897
0.2222
0.6035
0.0690
0.0708
1991 - 1992
0.0523
0.9571
0.8943
0.1395
0.0468
1992 - 1993
0.1278
0.1637
0.0251
0.1618
0.0309
1993 - 1994
-0.0641
0.6438
0.6980
0.0081
0.0306
1994 - 1995
-0.1814
0.2177
0.3898
0.2423
0.0486
1995 - 1996
0.1350
0.6097
0.4637
0.2568
0.0542
1996 - 1997
0.2130
0.1871
-0.0373
0.2875
0.0530
1997 - 1998
0.3958
0.1408
-0.2673
0.2808
0.0510
1998 - 1999
-0.2154
0.5395
0.7458
0.1881
0.0465
1999 - 2000
-0.1965
1.0017
1.1889
0.1105
0.0514
2000 - 2001
0.1259
0.0316
-0.1053
-0.1604
0.0568
2001 - 2002
-0.0528
0.3488
0.3917
-0.1672
0.0243
2002 - 2003
-0.3999
0.0564
0.4482
0.0236
0.0142
2003 - 2004
0.2723
0.8289
0.5452
0.2167
0.0090
2004 - 2005
0.2254
0.1685
-0.0681
0.0894
0.0181
2005 - 2006
-0.0672
0.3798
0.4371
0.1125
0.0388
2006 - 2007
0.0846
0.2701
0.1748
0.2131
0.0507
2007 - 2008
-0.1259
0.2448
0.3611
-0.1092
0.0337
Average Return
(1986-2008)
-0.0091
0.4108
0.4097
0.1095
0.0454
(-0.21)
(6.80)
(5.40)
(3.76)
(11.20)
Risk-Adjusted Return
(1986-2008)
-0.2256
0.3567
0.5286
-
-
(-5.79)
(4.23)
(6.25)
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
15
5. Concluding Remarks
It is possible to appraise firm’s latent fundamentals strength based on the structure of past
returns and a single factor mixed effects panel data model. We argue that the unrestricted
model for the structure of stock returns is given by the market variable and N firm-specific
latent fundamental characteristics, and that even though these characteristics are not directly
observed, it is possible to value the strength of the combination of these N microeconomic
characteristics for each stock.
In addition, a trading strategy that buys and sells stocks according to their latent fundamentals
strength realized significant risk-adjusted returns for the period between July 1986 and June
2008. This result was obtained with the examination of an investment strategy that buys the
stocks with the strongest latent fundamentals (the ―strong fundamentals portfolio‖) and sells
stocks with the weakest latent fundamentals (the ―weak fundamentals portfolio‖); hence, we
are able to explore the arbitrage opportunity originated from the differences in firm-level
latent fundamentals . The portfolio composed by this strategy (arbitrage portfolio) has a low
level of market risk and significant risk adjusted return due to its composition, generated by a
panel data mixed effects approach.
For instance, the arbitrage portfolio generated significant average monthly return of 3.11%
and risk-adjusted return of 2.42% considering the 4-factor model proposed by Carhart (1997)
and presented a market sensibility (CAPM Beta) of 0.14. Why this opportunity remain
through time, especially when sophisticated professional investors are dominant in the
market? This would be an interesting future research; in the meantime, the restrictions stated
by Stein (2009) would be a part of the answer.
We also provided subperiod and seasonal analyses in order to identify the strategies specific
characteristics. First, the sample period was divided into five subperiods and the analysis
suggested that the hypothesis that the arbitrage portfolio average returns for different
subperiods are jointly equal cannot be rejected. Then, we considered the average returns per
calendar month and found that the arbitrage portfolio returns differ for different months,
generating, mostly, positive and statistically significant returns, but two non significant
negative returns.
In sequence, to assure the strategy’s practical profitability we performed a trading cost
adjusted analysis and found that the arbitrage portfolio generated significant average annual
return of 40.97% and risk-adjusted return of 52.86% per annum. Furthermore, the strong
fundamentals portfolio presented positive annual returns adjusted by costs for all the 22 years
of analysis.
Finally, there is empirical evidence that our model is able to profit from the differences in
firm-level latent fundamentals, and that lead to the conclusion that it is possible to estimate
the firm-level fundamentals from the structure of past returns. These results confirm the
branch of literature related to value premium and other deviations, implying the existence of
some degree of predictability on stock returns. The paper also provides some evidence that it
is possible to generate profitable strategies in a panel data approach with mixed effects.
Therefore, the results provided evidence that contradicts the Efficient Market Hypothesis.
Latent Fundamentals Arbitrage with a Mixed Effects Factor Model
16
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