VaR and the cross-section of expected stock returns: an emerging market evidence.
Chen, Dar-Hsin ; Chen, Chun-Da ; Wu, Su-Chen 等
Introduction
The prominent capital asset pricing model (CAPM) of Sharpe (1964),
Lintner (1965), and Black (1972) has for decades been the major
framework for analyzing the cross sectional variation in expected asset
returns, but theory and practice might not always match. Fama and French
(1992) draw two different conclusions regarding CAPM--that is, when one
allows for variations in the CAPM market [beta] that are unrelated to
size, the univariate relationship between [beta] and the average return
for 1941-1990 is weak, and [beta] does not suffice to explain this
average return. They also find no cross-sectional return-beta
relationship while controlling for size and the ratio of book-to-market
equity (Chan, Chui 1996).
Several alternative risk factors have consequently been employed in
the literature, for example, the size effect of Banz (1981) and Nunes et
al. (2012). They finds the market value of equity (ME) and firm size
provide an explanation of the cross-section of average returns. Other
variables such as the book-to-market equity ratio (BE/ME) (Fama, French
1992, 1993, 1995, and 1996; Rosenberg et al. 1985), the price/earnings
ratio (Basu 1977), leverage (Bhandari 1988), Value-at-Risk (Bali, Cakici
2004), and profitability and investment patterns (Fama and French 2013)
also have significant explanatory power for making clear the average
expected returns. Hung et al. (2004), on the other hand, control for the
sign of realized market premia and use higher order asset pricing models
to test CAPM.
More noteworthy is that the conception and utilization of
Value-at-Risk (henceforth VaR) are designed to summarize the predicted
maximum loss (or worst loss) over a target horizon within a given
confidence interval (Jorion 2000). The movements of extreme price could
bring serious results to some corporations and cause disastrous
consequences for financial markets, although these cases are rare (1).
To a risk manager, a good measure of market risk is more than necessary.
As such, VaR was first used by major financial firms and has become the
most popular measurement for the risks of trading portfolios since the
late 1980s.
Modeling portfolio risk with a traditional standard deviation (a
good proxy of risk) measures implies in general that investors are
concerned only with the average variation in single stock returns.
Financial data, however, exhibit fat-tailed and asymmetric distributions
for market returns. For the last few decades, the popular and
traditional measure of risk has been volatility, yet the main problem
with volatility is that it does not involve the direction of an
investment's movement--a stock can be volatile, because it suddenly
jumps higher. Investors are not distressed by gains! By assuming that
investors care about the likelihood of a really big loss, VaR answers
the question, "What is my worst-case scenario?"
Traditional investment theory makes all possible uncertainty as
risk in spite of the direction. As investors have shown, there is a
problem if returns are not symmetrical. Investors worry about their
losses "to the left" of the average, but they are not
concerned with their gains "to the right" of the average. If
investors are risk-averse, then they request greater compensation to
hold stocks following higher downside risk. Many studies, including
Campbell et al. (2001), find that market volatility increases in bear
markets and recessions. Duffee (1995) also finds that idiosyncratic
volatility increases in down markets. Both of these effects generate
conditional beta that has little asymmetry across the downside and the
upside. In particular, this paper measures VaR in terms of a
company's market value at risk. The VaR is related to the
company's stock price and it reflects the shareholders'
perception of risk. The downside focus separates the loss from the
upside potential--namely, only the former truly constitutes risk and
only negative surprises to the stock market represent potential
litigation threats.
Only a few studies have surprisingly so far looked at VaR (a proxy
of risk, Bali, Cakici 2004). To the best of our knowledge, our attempt,
which incorporates VaR as a measure of the risk explaining the portfolio
expected returns in a less developed (emerging) stock market, is unique.
In particular, the high volatility of stock returns occurs frequently in
emerging stock markets. Under this circumstance, the dynamic risk
management of VaR could access the real risks accurately. The daily
price limit and short-sales constraint below last closing price are two
major features in the Taiwan stock market. According to the information
and overreaction hypothesis, the price limit does delay the process for
prices to reflect intrinsic value. The short-sales constraint could
prevent stock returns from any arbitrage momentum, but might tend to
cause stock overvaluation and the overvaluation effect is more dramatic
for individual stock reversely (Chang et al. 2007).
On the strength of the reasons mentioned above, we believe that the
VaR is relevant as a risk factor and an appropriate risk measurement,
and it could also provide a good explanatory power in stock return and
stock market variation (Jorion 1996). Another contribution of this paper
is we fulfill the literature gap to examine whether the maximum
potential loss measured by VaR plays a key role in explaining expected
cross-sectional stock returns in Taiwan. Furthermore, this study also
uses the Fama-French 3-factor model associated with VaR to investigate
the cross-sectional variation at the firm level. The empirical results
enrich our understanding of risk management and provide more evidence in
emerging financial markets.
The remainder of this paper is organized as follows. Section 1
discusses the previous studies. Section 2 introduces the data, the
variable definitions, and the models. The empirical results are
presented in Section 3. Concluding remarks are given in the final
section.
1. Literature review
Empirical research has provided several pieces of evidence that
reject the validity of the Sharpe-Linter capital asset pricing model
(CAPM). The existence of market frictions, the presence of irrational
investors, and inefficient markets may distort the cross-sectional
relationship between expected stock returns and market beta. This
research discusses the relevant evidence reported in empirical studies.
1.2 Fama-French three factor model
CAPM employs a single factor, beta, to compare a portfolio with the
market as a whole and was first introduced by Sharpe (1964) and Lintner
(1965), but it oversimplifies the complex financial market. Fama and
French (1992) start with the observation that two classes of stocks have
tended to do better than the market as a whole: (1) small caps and (2)
stocks with a high book-to-market equity ratio (BE/ME). They then add
these two factors to CAPM to reflect a portfolio's exposure--that
is, the Fama-French three factor model, which corresponds to the
following 3-factor regression:
R - RF = a + b x [RM - RF] + c x SMB + d x HML, (1)
where R is the portfolio's return rate, RF is the risk-free
return rate, and RM is the return of the aggregate stock market. The
"three factor" beta is analogous to the classical beta, but
not equal to it, since there are now two additional factors to do some
of the work. The SMB (Small Minus Big) portfolio represents a zero-
investment portfolio that is long in small-cap stocks and short in
big-cap stocks. The HML (High Minus Low) portfolio represents a
zero-investment portfolio that is long in high book-to-market stocks
(so-called "Value" stocks) and short in low book-to-market
stocks (so-called "Growth" stocks). The Fama-French model is
based on the observation that small cap stocks and "Value"
stocks historically tend to do better than the market as a whole. While
[bar.[R.sup.2]] for CAPM usually takes values of around 0.85, the
Fama-French model is capable of accounting for almost all variation in
individual assets (2).
1.2 Application of Value-at-Risk (VaR)
The VaR analysis originated with the variance-covariance model
introduced by J.P. Morgan's RiskMetrics in 1993. The
variance-covariance approach to calculating risk could be traced back to
the modern portfolio theory by Markowitz (1959), in which most of
today's risk managers are conversant. Engle and Manganelli (2004)
further extend the quantile regression to model the VaR directly instead
of modeling the underlying volatility generating process and also
introduce the conditional autoregressive Value-at-Risk (CAViaR) model.
Sequentially, Bali and Weinbaum (2005), Bali and Cakici (2004 and 2006),
Lim and Tan (2007), and Bail et al. (2009) also employ the concept of
conditional VaR to measure idiosyncratic risk and market returns. It is
worth noting that Bali and Cakici (2004) argue that the VaR captures the
cross-sectional differences in expected stock returns of firms on the
U.S. three major stock exchanges, whereas the market beta and total
volatility have no power in explaining the firm's average stock
returns. Therefore, this study is strongly motivated by their vigorous
findings and follows a similar methodology to apply Taiwan data that
compares the prediction ability (in terms of the [bar.[R.sup.2]] value)
of beta, size, BE/ME, and VaR and explains the cross-sectional variation
in portfolio returns (3).
2. Data, variable definitions and methodology
2.1. Data
According to Bali and Cakici (2004), this paper examines whether
the market factor (beta), firm size, BE/ME, and VaR provide different
explanatory powers to the average stock returns under diverse company
characteristics in an emerging market Taiwan. The stock returns covered
in this study include the all listed stocks of TSEC (Taiwan Stock
Exchange). Regarding the sample period, to avoid possible abnormal
trading activities during the 1997 Asia financial crisis period (1997 to
1998, when the total market value lost over 40%), the sample period for
this study covers seventeen years, from 1991 to 2009. The estimation
period spans from January 1991 through December 1995, while the test
period extends from January 1996 to December 2009 (4). The Taiwan
Economic Journal (TEJ) database provides these firms' stock
returns.
2.2. Variable definitions
Systematic risk (Beta)
Following Fama and French (1992), this study sorts all the stocks
by size (the market value of equity) to determine the stocks'
quintile breakpoints, because the size differences may be attributed to
a wide range of average returns and Ps (Chan and Chen, 1991). We then
subdivide each size quintile into five portfolios on the basis of
pre-ranking betas for all the stocks. The pre-ranking betas are
calculated by monthly returns of five years ending in December 1995 and
144 post-ranking monthly returns for each of 25 portfolios are estimated
from January 1996 to December 2009 (5). We follow Allen and Cleary
(1998) to evaluate beta incorporating Scholes and Williams (1997) as the
sum of the slopes in the regression of portfolio returns on portfolios
(6).
Size
Following the previous studies, we evaluate size, the market value
of a firm's outstanding shares, with the natural logarithm of the
market value of equity.
Book-to-market equity (BE/ME)
Book-to-market equity (BE/ME) is the natural logarithm of the ratio
of the book value of equity plus deferred taxes over the market value of
equity, which involves accounting- and market-based variables. This
paper uses a firm's market equity at the end of December of the
previous year to compute its BE/ME.
Value-at-Risk (VaR)
In this paper we follow the method of Bali and Cakici (2004) to
estimate VaR (7). After obtaining the VaR for each stock, we rank and
place VaR into 5 quintile portfolios. Portfolio 1 has the lowest VaR and
portfolio 5 has the highest VaR. The portfolio formation procedure is
very similar to Fama and French (1992), except that they update their
portfolios annually, whereas we update ours on a monthly basis. The
estimation period and test period of VaR respectively span from 1991 to
1995 and from 1996 through 2009. We calculate the one-month-ahead
portfolio returns from 1996 to 2009 with 144 time series for the 5
equally-weighted portfolios based on VaRs and find that the portfolios
with the higher VaR have greater rates of return.
2.3. Methodology
Cross-sectional regression
This paper utilizes Fama-MacBeth (1973) (hereafter FM) regressions
to examine whether the beta, size, BE/ME, 1% VaR, 5% VaR, and 10% VaR
can provide large and statistically significant cross-sectional
variations in expected stock returns. Monthly cross-sectional
regressions are run for the following econometric specifications:
[R.sub.j,t+1] = [[omega].sub.t] + [[gamma].sub.t][X.sub.j,t] +
[[epsilon].sub.j,t+1], (2)
X = BETA, ln(ME), ln(BE/ME), [VaR.sup.1%], [VaR.sup.5%], and
[VaR.sup.10%],
where [R.sub.j,t+1] is the realized average return on stock j in
month t + 1, and BETA and ln(ME) are the respective full-sample
pre-ranking beta and the natural logarithm of market equity.
VaR([alpha]) is--1 times the maximum likely loss (VaR) with the loss
probability level [alpha] = 1%, 5%, and 10%, and [[epsilon].sub.j,t+1]
is the residual series from the cross-sectional regressions.
Size-BE/ME portfolios and VaR-BE/ME portfolios
In the study we follow Fama and French (1993) to construct the
factor mimicking portfolio. Fama and French (1993, 1995, 1996) also
indicate the importance and calculations of RMRF, SMB, and HML. To test
the performance of VaR based on the 25 portfolios of Fama and French
(1993), this study devises a factor, HVARL (high VaR minus low VaR),
that is designed to mimic the risk factor in returns related to
Value-at-Risk and is defined as the difference between the simple
average returns on the high-VaR and low-VaR portfolios. The construction
of a 5% VaR portfolio is similar to the construction of Fama and
French's size portfolios. In December of each year t from 1996 to
2009, this study ranks all stocks according to a 5% VaR. The median 5%
VaR figure is used to divide the stocks into two groups--the high VaR
and low VaR groups.
Three-factor model and four-factor model
This study performs a four-step analysis of the various factors
(RMRF, HVARL, SMB, and HML) in explaining stock returns and then
examines the three-factor and four-factor models. The three-factor model
suggested by Fama and French (1993) provides an alternative to the CAPM
for the estimation of expected returns. This model includes two
additional factors to explain excess return: size and the book-to-market
ratio. The Fama and French model is of primary interest to us as one of
the objectives of the paper is to assess its implications for an
investor's investment decision. The explanatory variables in the
time series regressions include not only the returns on a market
portfolio of stocks, but also the mimicking portfolio returns for size
and book-to-market. The four-factor model adds another risk factor, i.e.
HVARL, into the three-factor model. The risk factor is a mimicking
portfolio that follows Bali and Cakici (2004) as follows:
R(t) - RF(t) = a + b x [RM(t) - RF(t)] + c x SMB + d x HML + e x
HVARL + u(t). (3)
3. Empirical results
3.1. VaR and cross-sectional regression
This section shows the results of the Fama and MacBeth regressions
(estimated by Ordinary Least Squares method) of excess returns on
characteristics that are best known to be associated with expected
returns--namely, the beta, firm size, BE/ME, and VaR (1%, 5%, and/or
10%) variables. Table 1 presents the time series average value of
[gamma]t, the t-statistics, and the time series averages of the
determination coefficient ([R.sup.2]) over the 144 months in the sample.
The t-statistics shown in the parentheses are the time series average
values of [[gamma].sub.t] divided by the corresponding time-series
standard errors. As can be seen from the estimated slopes of beta,
ln(ME) and ln(BE/ME) are all highly significant at the 1% level. The
average slopes provide standard Fama-MacBeth tests for determining which
explanatory variables had, on average, non-zero expected return premiums
during the January 1996 to December 2009 period. As expected, there is a
negative relationship between the realized stock returns and beta. The
empirical evidence shows that the lower the sensitivity is of the asset
return, the greater the realized return will be (Fama, MacBeth 1973;
Banz 1981). Meanwhile, the average slope for the monthly regressions of
the realized returns and size, ln(ME), is negative and about -0.49, with
a t-statistic of -33.81. We believe that size is related to
profitability. On average, the profitability of larger-cap stocks in the
Taiwan stock market is less than that of smaller-cap stocks. This result
also shows that, for a firm with larger capitalization, the performance
seems lower than for the small firm from the viewpoint of the
cross-sectional regressions.
The average slope based on the univariate regressions of the
monthly return on ln(BE/ ME) is about 0.94, with a t-statistic of 17.36,
implying that the risk captured by BE/ ME is the relatively distressed
factor of Chan and Chen (1991). They postulate that the earning
prospects of firms are associated with a risk factor in the returns.
Firms that the market judges to have poor prospects, signaled here by
low stock prices and high BE/ME ratios, have higher expected returns
(they are penalized with higher costs of capital) than firms with strong
prospects. It is also possible that BE/ME only captures the unraveling
of irrational market whims regarding the prospects of firms. This result
accords with the view put forward by Fama and French (1992) that BE/ME
has a stronger role in explaining average stock returns than size.
Furthermore, as we move to a lower significance level (higher confidence
level), the VaR estimation becomes more important in explaining the
cross-sectional average stock returns. We can see that the VaR([alpha])
is significant when a is 1% and 5%. The results of the positive
coefficients of VaR indicate that the greater a stock's potential
fall in value is, the higher the expected return should be. In addition,
we further report multivariate cross-sectional regression results as
Model 7 and Model 8. The result shows that the VaR does provide a higher
explanatory power to the stock monthly average returns. The [R.sup.2] of
Model 7 and Model 8 are higher than those of Model 1 to Model 6.
3.2. VaR and time-series variation of expected returns
In time-series regressions, the slopes and [R.sup.2] values are
direct evidence as to whether different risk factors capture a common
variation in stock returns. This study examines the explanatory power of
stock market factors. Table 2 of Panel A shows the simple statistics of
RMRF, SMB, HML and HVARL. The average value of the market risk premium
is 1.28% per month. This study also calculates the correlations between
RMRF, SMB, HML, and HVARL. Table 2 of Panel B presents the correlation
coefficients for the factors used. The last row shows that HVARL is
positively correlated with RMRF and SMB, whereas it is negatively
correlated with HML. A notable point is that the positive relationship
between HVARL and SMB is much stronger than the negative relationship
between HVARL and HML.
To compare the relative performances of HVARL, RMRF, SMB, and HML,
this study calculates the correlations between the returns for the 25
portfolios of Fama and French (1993) and the various factors. Table 3
shows that, not surprisingly, the excess return on the market portfolio
of stocks, RMRF, captures more common variations in stock returns, on
average, than HVARL, HML, and SMB. The average correlation between the
returns for the 25 portfolios and RMRF is 0.7673, whereas the average
correlation between HVARL and the monthly returns on the 25 portfolios
is 0.4824. Furthermore, the average correlation is 0.3011 for SMB and
-0.3809 for HML. Clearly, HVARL as a single factor is superior to SMB
and HML in explaining the time-series variation in stock returns.
3.3. Properties of portfolios formed based on size and pre-ranking
P
Fama and French (1992) find that after controlling for the size and
book-to-market effects, beta seems to have no power to explain the
average returns on a security. This finding is an important challenge to
the notion of a rational market, since it seems to imply that a factor
that should affect return--systematic risk--does not seem to matter.
Table 4 shows that forming portfolios on size and pre-ranking [beta]s,
rather than on size alone, magnifies the range of full-period
post-ranking [beta]s. The all column and row shows statistics for
equal-weighted size-decile portfolios and for equal-weighted portfolios
of the stocks in each [beta] group respectively.
The average row of Panel B of Table 4 shows that the portfolio beta
of each beta group averaged across the 5 different-sized portfolios
steadily increases from 0.81 to 1.21.
The average row in Panel C shows that the average portfolio size
within each beta group is almost identical, ranging from 8.63 to 8.81.
This allows us to interpret Panel A as a test of the net effect of beta
on average returns holding size fixed. Panel A of Table 4 clearly shows
that, for the period 1996-2009, average returns are not positively
related to beta. The highest-beta portfolios do not have the highest
returns, and this occurs in the fourth-beta portfolios. The results do
not support the central prediction of the CAPM, because average stock
returns are not positively related to the market beta at the portfolio
level. The CAPM insight is that volatility arising from specific events
(called specific or idiosyncratic risk) can be eliminated in a
diversified portfolio, and that investors will not be paid for bearing
these risks with extra returns. This result will support us as we
continue to further discuss the three- and four-factor models. We should
note that average monthly post-formation returns seem to be negatively
correlated with firm size. The smallest size quintile, on average, has
the highest average return (1.31% per month) and the biggest size
quintile has the lowest average return (0.22% per month).
3.4. Properties of portfolios formed on VaR and pre-ranking [beta]
Table 5 of Panel A reports that when common stock portfolios are
formed on 5% VaR, the average stock returns are positively related to
VaR. Going from the lowest 5% VaR quintile to the highest 5% VaR
quintile, the average stock returns from VaR portfolios increase from
0.55% per month to 1.27% per month monotonically. This result supports
our argument to the effect that if investors are more averse to the risk
of losses on the downside than of gains on the upside, i.e. a higher
VaR, then investors should demand greater compensation. Furthermore, we
see that the greatest average monthly post-formation return is about
1.92%, and not surprisingly it is apparent in the highest VaR-BE/ME
group. However, the average monthly post-formation returns are not
similar within the same [beta] quintile. For the smallest 5% VaR
quintile, the highest [beta] does not have the largest stock returns.
Beta seems to have much less power to explain the average stock returns
after controlling for the 5% VaR and book-to-market effects. These
results inform us that the more a stock can potentially fall in value,
the higher the expected return should be.
3.5. Main model results: three and four-factor models
Table 6 presents estimates from the three-factor model in which the
excess returns on 25 portfolios are regressed on RMRF, SMB, and HML.
Table 6 demonstrates that most of the coefficients for the three
Fama-French factors (coefficient (b) is for RMRF, coefficient (c) is for
SMB, and coefficient (d) is for HML) are highly significant. The lower
BE/ME quintile and bigger size quintile portfolios capture between 70%
and 90% of the variations in terms of the [bar.[R.sup.2]] values.
However, the higher BE/ME quintile and smaller size quintile seem to
leave 30-40% of variations that cannot be explained by Fama and
French's three-factor model. Furthermore, the results indicate
that, when controlling for the BE/ME effect, the SMB factor is highly
significant. What is initially surprising, however, is the fact that SMB
seems to work in a reverse manner than what would be expected, i.e.
small firms have on average higher returns than big firms do. This can
be seen by looking at the coefficients for SMB, which go from positive
to negative when moving from small stock portfolios to big stock
portfolios and after taking into account the fact that the size premium
is negative during our sample period. On the other hand, when
controlling for size, the HML factor clearly captures the higher returns
for the high BE/ME portfolios as compared to the low BE/ME stocks.
Subsequently, we will continue to see if another factor--the VaR--can
enhance and capture the variations.
Therefore, SMB, the mimicking return for the size factor, has more
power than HML. Not surprisingly, the slopes on HML are systematically
related to BE/ME. In every size quintile of stocks, the HML slopes
increase monotonically from lower to higher BE/ ME quintiles.
Table 7 presents the parameter estimates, t-statistics,
[bar.[R.sup.2]] values, and standard errors of estimate (S.E.E.) from
the time series regressions of excess stock returns on RMRF, SMB, HML,
and HVARL. As shown in Table 7, the slope coefficients for the market
factor, RMRF, are highly significant. Most of the slope coefficients for
SMB and HML factor are also significant. A notable point is that, for
the lowest size-quintile, none of the HVARL slopes are significant, but
the rest of 20 HVARL slopes are significant. The [bar.[R.sup.2]] values
of the four-factor model are greater than those of the three-factor
model. When viewed at the portfolio level, these empirical results show
that the VaR factor plays an important role in firms especially with
larger capitalization. This could be the reason why either the concept
of VaR is not very familiar to individual investors since they are the
major participants in the Taiwan stock market or else larger companies
always pay much attention to VaR in order to control for downside risk.
However, after the New Basle II Accord was implemented at the end of
2006, VaR is looking to play an increasingly important role to stock
returns in the future.
Conclusions
By focusing on downside risk as an alternative measure of risk
measured by VaR, this paper investigates whether the new VaR factor
plays an important role in explaining Taiwan's stock returns from
January 1996 to December 2009. The empirical results do not support the
central prediction of the CAPM, because average stock returns are not
positively related to the market beta at the portfolio level. From the
cross-sectional regressions in a Fama and French (1992) asset pricing
framework, we find that, in addition to market betas, idiosyncratic
factors (such as firm size, book value of equity to market value of
equity, 1% VaR, and 5% VaR) are related to the return at the individual
stock level. In particular, the BE/ME factor captures most of the
variations in average realized stock returns in terms of
[bar.[R.sup.2]]. From the time series regressions we investigate models
with factors ranging from one to four to test the empirical performance
at the portfolio level. From the results, which are based on 25
size/book-to-market portfolios of Fama and French (1993), and following
Bali and Cakici (2004), we find that the HVARL factor further captures
the variation in emerging/less developed stock markets, especially for
the larger companies in the Taiwan stock market.
doi:10.3846/16111699.2012.744343
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Received 13 September 2011; accepted 16 October 2012
Dar-Hsin Chen (1), Chun-Da Chen (2), Su-Chen Wu (3)
(1) Department of Business Administration, College of Business
National Taipei University, 151 University Rd., San Shia, Taipei, 237
Taiwan
(2) Department of Economics and Finance, College of Business,
Tennessee State University, AWC Campus, 330 10th Avenue North,
Nashville, TN, USA
(3) Graduate Institute of Finance, National Chiao Tung University,
Hsinchu, Taiwan
E-mails: (1) dhchen@mail.ntpu.edu.tw; (2) cchen2@tnstate.edu
(corresponding author)
(1) For instance, the New York stock market crashed in October
1987, and then, one decade later, the Asian stock market crashed in
1997. The Enron scandal also caused the Dow Jones Industrial Average
(DJIA) to drop sharply. These crises have harmed thousands of companies
and much of the value of their stocks has been wiped out within a short
period of time.
(2) For other important relevant studies following the Fama-French
three factor model, please refer to Heston et al. (1999) in beta, Berk
(1995), Chen and Zhang (1998), Chiu and Wei (1998), Rouwenhorst (1999),
Jarrett and Schilling (2008), and Bistrova et al. (2011) in size effect,
and Lakonishok et al. (1994), Fama and French (1993 and 1995), and Roll
(1995) in the BE/ME ratio effect.
(3) Please refer to the page 58 of Bali and Cakici (2004) for the
detailed VaR definitions and calculations.
(4) To be included in the sample, for a given month a stock has to
satisfy several criteria. First, the stock returns over the previous 60
months are available. Second, data are available from TEJ to calculate
the BE/ME ratio as of December of the previous year. Finally, we include
only securities defined by TEJ as ordinary common shares. This screening
process yields averages of 202 stocks per month.
(5) The choice of using portfolios instead of individual shares is
dictated by the evidence of Griffin and Lemmon (2002) which shows that
this is the way the sampling error is reduced. Additionally, using
portfolios also facilitates a comparison with past studies in the field,
as the majority of these studies use portfolios instead of individual
stocks. Further advantages of using portfolios instead of individual
firms in the regressions include the following: 1. A pooled sample of
individual firms used in CSR analysis allows us to eliminate the
potential threat posed by temporal and firm-specific effects in terms of
biasing the results. 2. There is significantly less computational effort
in using portfolios instead of individual stocks in the regression
analysis.
(6) Due to limited space, please refer to Allen and Cleary (1998)
for the detailed beta estimation.
(7) We further use EWMA (exponentially weighted moving average) to
check the performance of our model and the results are similar.
Dar-Hsin CHEN is a Professor of Finance at the Department of
Business Administration, National Taipei University and has received his
PhD from the University of Mississippi in 1998. His research interests
are corporate finance, international finance, and risk management.
Chun-Da CHEN is an Associate Professor of Finance at the Department
of Economics and Finance, Tennessee State University. He holds a PhD in
Finance from the Tamkang University in 2005 and has published in
journals such as Journal of Economic Behavior and Organization,
International Review of Financial Analysis, and International Review of
Economics and Finance.
Su-Chen WU receives a MBA in Finance from the Graduate Institute of
Finance, National Chiao Tung University in 2005.
Table 1. Cross-sectional regressions of stock returns on Beta,
size, BE/ME, and VaR
Monthly Constant BETA ln(Me) ln(BE/Me)
Regression
(N = 168)
Model 1 -0.7591 *** -1.2811 ***
(-10.94) (-16.20)
Model 2 -3.8170 *** -0.4888 ***
(-26.13) (-33.81)
Model 3 -0.7977 *** 0.9404 ***
(-9.82) (17.36)
Model 4 0.1972 ***
(4.3)
Model 5 0.1555 ***
(2.66)
Model 6 0.4240 ***
(7.24)
Model 7 -4.1359 *** -1.0331 *** -0.9747 *** 1.2487 ***
(-28.55) (-8.70) (-14.45) (9.34)
Model 8 -0.1359 *** -0.4569 ** -0.8964 *** 1.0013 ***
(-10.35) (-2.47) (-10.11) (6.55)
Monthly VaR 1% VaR 5% VaR 10% [R.sup.2]
Regression
(N = 168)
Model 1 0.1684
Model 2 0.3607
Model 3 0.4211
Model 4 0.0072 *** 0.0425
(3.08)
Model 5 0.0137 *** 0.0280
(3.01)
Model 6 -0.0034 0.0196
(0.35)
Model 7 0.4497
Model 8 0.0654 *** 0.4613
(3.35)
Notes: ***, **, and * mean significantly different from zero at
the 0.01, 0.05, and 0.10 levels, respectively. This table reports
the time-series average of the month-by-month regression slopes
from January 1996 to December 2009. The dependent variables are
the monthly average returns on individual stocks. The independent
variables include beta, firm size, the book-to-market ratio
(BE/ME), and [VaR.sup.[alpha]], where [alpha] = 1%, 5%, and 10%.
The betas that correspond to the portfolio they belong to are
assigned to individual stocks. The size is the natural log of the
market value. The BE/ME is the natural log of the book-to-market
value. The VaR is calculated using the historical simulation method
(the tails of the empirical return distribution). We use OLS
(ordinary least squares) method to estimate regressions of the
following form: [R.sub.j,t] = [[omega].sub.t] + [[gamma].sub.t] x
[X.sub.j,t] + [[epsilon].sub.j,t], where X contains 6 independent
variables. The t-statistic reported in the parentheses is the
average slope divided by its time-series standard error. For
robustness, we also try 1% VaR and 10% VaR in Model 8, but the
results have no significant change.
Table 2. Time series regressions--descriptive statistics
Panel A: Descriptive Statistics
Variable N Mean Std. Dev. minimum Maximum
RMRF 132 1.2753 9.0405 -23.3456 24.7122
SMB 132 -0.6008 4.6622 -18.1625 10.5123
HML 132 3.0023 6.2125 -19.4860 21.4324
HVARL 132 0.4437 5.6937 -17.0467 20.1614
Panel B: Pearson Correlation Coefficients, N = 144
Prob > [absolute value of r] under [H.sub.0]: [rho] = 0
RMRF SMB HML HVARL
RMRF --
SMB -0.0633 --
HML -0.0747 -0.5971 * --
HVARL 0.6754 ** 0.4769 ** -0.3760 * --
Notes: This table gives the correlation coefficients calculated
from the sample. An asterisk indicates that the correlation
coefficient is significant (i.e. the p-value is less than 0.05).
***, **, and * means significantly different from zero at the 0.01,
0.05, and 0.10 levels, respectively. This table presents simple
summary statistics for the stocks in the sample. The six size-PB
portfolios (S/L, S/M, S/H, B/L, B/M, and B/H) are formed in
December of each year t-1 and value-weighted monthly returns are
calculated from January to December of year t. Panel A presents the
basic statistics of the four factors. Panel B presents the Pearson
correlation coefficients that are calculated based on monthly
returns for each of the factors RMRF, SMB, HML, and HVARL. The
sample period extends from January 1996 to December 2009 exclusive
of 1997 and 1998--there being 144 monthly observations.
Table 3. Correlations of 25 portfolio returns with RMRF,
SMB, HML, and HVARL
Correlations RMRF SMB HML HVARL
S1B1 0.6737 0.5062 -0.5768 0.5339
S1B2 0.7325 0.4757 -0.5251 0.5462
S1B3 0.6938 0.5526 -0.5071 0.5679
S1B4 0.6502 0.4857 -0.5098 0.5022
S1B5 0.6966 0.4260 -0.3111 0.4999
S2B1 0.7463 0.4361 -0.5201 0.4929
S2B2 0.7591 0.4641 -0.4774 0.5686
S2B3 0.6941 0.4139 -0.4975 0.4497
S2B4 0.7451 0.3519 -0.1891 0.4630
S2B5 0.7502 0.4537 -0.6120 0.5643
S3B1 0.7338 0.3796 -0.4170 0.4759
S3B2 0.7667 0.3593 -0.3467 0.4469
S3B3 0.7718 0.2150 -0.1478 0.4606
S3B4 0.7651 0.3717 -0.6163 0.5862
S3B5 0.7728 0.3232 -0.5354 0.5427
S4B1 0.8096 0.2052 -0.3373 0.4011
S4B2 0.8132 0.0840 -0.0675 0.5476
S4B3 0.8299 0.1710 -0.4630 0.4536
S4B4 0.8543 0.0677 -0.2828 0.3222
S4B5 0.8305 0.0129 -0.2229 0.2653
S5B1 0.8879 -0.3264 0.2379 0.4791
S5B2 0.6737 0.5062 -0.5768 0.5339
S5B3 0.7325 0.4757 -0.5251 0.5462
S5B4 0.6938 0.5526 -0.5071 0.5679
S5B5 0.6502 0.4857 -0.5098 0.5022
Average 0.7673 0.3011 -0.3809 0.4824
Notes: S1B1 (S5B5) denotes a size-BE/ME portfolio that belongs
to the smallest (largest) size quintile and lowest (highest)
BE/ME quintile.
Table 4. Properties of portfolios formed on size and pre-ranking
[beta]: stocks sorted by size (down) then pre-ranking [beta]
(across), 1996-2009
Low- 2 3 4 High- Average
[beta] [beta]
Panel A: Average Monthly Post-formation Returns (in percent)
Small- 1.0293 1.3052 0.6207 1.0251 2.5847 1.3130
size
2 0.4303 0.7127 1.2314 0.6835 1.2962 0.8708
3 0.8353 0.2684 0.2441 1.2928 1.1026 0.7486
4 0.4469 0.6885 0.0076 1.2041 0.6368 0.5968
Big-size -0.1541 0.5779 1.0680 -0.2209 -0.1574 0.2227
Average 0.5175 0.7106 0.6344 0.7969 1.0926
Panel B: Post-ranking [beta]
Small- 0.7941 0.9160 0.9644 1.2843 1.4599 1.0837
size
2 0.8132 0.8710 1.0172 1.1230 1.1833 1.0015
3 0.8458 0.9948 1.0100 1.3498 1.3091 1.1019
4 0.8124 1.0384 1.0222 1.4002 1.2299 1.1006
Big-size 0.7715 0.8263 0.8643 1.0182 0.8916 0.8744
Average 0.8074 0.9293 0.9756 1.2351 1.2148
Panel C: Average Ln(Size)
Small- 7.1450 7.2802 7.2981 7.2590 7.4390 7.2842
size
2 7.9950 8.0069 8.0009 7.9556 8.0679 8.0053
3 8.5667 8.5161 8.5679 8.6229 8.5625 8.5672
4 9.1994 9.2539 9.2348 9.2685 9.2632 9.2440
Big-size 10.2412 10.7327 10.6171 10.4087 10.7113 10.5422
Average 8.6295 8.7579 8.7438 8.7030 8.8088
Notes: At the end of year t-1, the stocks obtained from the TEJ are
assigned to 5 size portfolios. Each size quintile is subdivided
into 5 p portfolios using the pre-ranking p of individual stocks
estimated with 60 monthly returns ending in December of year t-1.
The equal-weighted monthly returns on the resulting 25 portfolios
are then calculated for year t. The average returns are the time-
series average of the monthly returns, in percentage form. The
post-ranking Ps use the full 1996-2009 sample of post-ranking
returns for each portfolio. The pre- and post-ranking Ps are the sum
of the slopes from a regression of monthly returns for the current
and prior months' value-weighted market return. The average size of
a portfolio is the time-series average of each month's average value
of ln(Size) for a stock within the portfolio. Size is dominated in
millions of TWD. There are, on average, about 5 stocks in each
size-[beta] portfolio in each month.
Table 5. Properties of portfolios formed on VaR and pre-ranking
[beta]: stocks sorted by VaR (down) then pre-ranking P (across),
1996-2009
Low- 2 3 4 High- Average
[beta] [beta]
Panel A: Average Monthly Post-formation Returns (in percent)
Small-VaR -0.1722 0.8005 0.9174 0.4467 0.7515 0.5488
2 0.5899 0.6460 0.9079 0.8104 0.1219 0.6152
3 0.9873 0.5480 0.7447 1.0340 0.2751 0.7178
4 0.6689 0.3862 1.5197 0.0331 0.6422 0.6500
Big-VaR 0.7067 1.2596 1.2077 1.2487 1.9213 1.2688
Average 0.5561 0.7280 1.0595 0.7146 0.7424
Panel B: Post-ranking [beta]
Small-VaR 0.4900 0.7738 0.5932 0.8779 0.8600 0.7190
2 0.9007 0.9186 0.9306 0.8939 0.8637 0.9015
3 0.9327 0.9686 1.1463 1.1416 1.1855 1.0750
4 0.9866 0.9977 1.4168 1.0077 1.2284 1.1274
Big-VaR 1.1235 1.4283 1.3407 1.3202 1.5012 1.3428
Average 0.8867 1.0174 1.0855 1.0483 1.1278
Panel C: Average VaR
Small-VaR 13.6133 14.0500 14.0540 14.5902 14.9116 14.2438
2 16.7515 17.0488 16.7265 16.8485 16.7665 16.8284
3 18.9336 18.8091 18.7304 19.2716 19.5507 19.0591
4 20.9608 21.4593 22.0351 21.5672 21.8481 21.5741
Big-VaR 25.8988 25.6252 25.6738 25.8725 27.3475 26.0836
Average 19.2316 19.3985 19.4440 19.6300 20.0849
Notes: The formation of the VaR-beta portfolios is similar to that
of the size-[beta] portfolios. At the end of year t-1, stocks are
sorted by their 5% VaR and assigned to 5 portfolios. Each VaR
quintile is subdivided into 5 [beta] portfolios using the
pre-ranking [beta] ending in December of year t-1. The equal-
weighted monthly returns on the resulting 25 portfolios are then
calculated for year t. The average returns are the time-series
average of the monthly returns, in percentage form. The post-
ranking [beta]s use the full 1996-2009 sample of post-ranking
returns for each portfolio. The pre- and post-ranking [beta]s are
the sum of the slopes from a regression of monthly returns on the
current and prior months' value-weighted market return. The average
5% VaR of a portfolio is the time-series average of each month's
average value of 5% VaR for stock in the portfolio. There are, on
average, about 5 stocks in each VaR-[beta] portfolio each month.
Table 6. Three-factor model: regression of excess stock returns on
the excess stock-market return, SMB, and HML (Jan. 1996 to Dec.
2009, N = 144)
Panel A: R(t) - RF(t) = a + b x [RM (t) - RF(t)] + c x SMB +
d x HML + u(t)
BE/ME Quintile
Size Low 2 3 4 High
Quintiles
Slope Coefficient (a), Intercept
Small -1.24 -0.28 0.05 0.36 0.44
2 -0.43 0.56 0.54 1.58 1.86
3 0.47 0.57 0.92 0.69 2.18
4 0.58 1.43 2.18 0.71 1.43
Big 0.14 0.14 1.20 -0.43 -0.60
Slope Coefficient (b), Market Return
Small 1.10 0.97 0.89 0.79 0.84
2 1.03 0.96 0.90 0.80 0.98
3 1.10 1.09 1.02 1.00 1.06
4 1.03 0.97 1.05 0.92 1.20
Big 0.94 0.90 0.88 0.87 1.17
Slope Coefficient (c), SMB
Small 1.14 0.94 1.22 0.91 1.09
2 0.88 0.79 0.91 0.65 1.20
3 0.74 0.75 0.86 0.90 0.84
4 0.34 0.32 0.46 0.30 0.56
Big -0.08 -0.08 -0.12 -0.10 -0.37
Slope Coefficient (d), HML
Small -0.75 -0.48 -0.29 -0.41 0.05
2 -0.68 -0.53 -0.31 -0.46 0.30
3 -0.89 -0.61 -0.35 -0.14 0.20
4 -0.99 -0.77 -0.57 -0.33 0.24
Big -0.73 -0.39 -0.31 0.01 0.41
[[bar.R].sup.2]
Small 0.81 0.84 0.85 0.73 0.70
2 0.91 0.83 0.85 0.73 0.72
3 0.91 0.91 0.73 0.75 0.66
4 0.91 0.83 0.80 0.74 0.68
Big 0.85 0.77 0.71 0.74 0.88
Panel A: R(t) - RF(t) = a + b x [RM (t) - RF(t)] + c x SMB +
d x HML + u(t)
BE/ME Quintile
Size Low 2 3 4 High
Quintiles
t-statistic (a)
Small -1.74 -0.54 0.11 0.56 0.69
2 -1.06 1.06 1.21 2.64 2.83
3 1.06 1.38 1.29 1.08 2.80
4 1.37 2.68 3.64 1.21 1.73
Big 0.30 0.27 2.05 -0.82 -1.34
t-statistic (b)
Small 15.70 18.85 18.47 12.74 13.46
2 25.70 18.50 20.40 13.52 15.18
3 25.39 26.84 14.48 15.89 13.97
4 24.64 18.66 17.84 15.93 14.93
Big 20.70 17.76 15.32 17.05 26.57
t-statistic (c)
Small 6.76 7.64 10.48 6.05 7.28
2 9.14 6.30 8.58 4.60 7.72
3 7.11 7.65 5.06 5.92 4.57
4 3.36 2.53 3.27 2.16 2.90
Big -0.74 -0.68 -0.85 -0.83 -3.45
t-statistic (d)
Small -5.94 -5.20 -3.29 -3.65 0.48
2 -9.40 -5.60 -3.92 -4.33 2.53
3 -11.34 -8.32 -2.72 -1.23 1.45
4 -13.21 -8.16 -5.38 -3.19 1.62
Big -8.96 -4.24 -3.01 0.07 5.20
S.E.E.
Small 41.70 22.55 19.97 33.13 33.12
2 13.73 23.06 16.47 29.57 35.52
3 16.06 14.11 42.05 33.88 49.60
4 14.82 23.21 29.45 28.38 55.59
Big 17.46 21.86 27.93 22.21 16.56
Table 7. Four-factor model: regression of excess stock returns on
the excess stock-market return, SMB, HML, and HVARL (Jan. 1996 to
Dec. 2009, N = 144)
Panel A: R(t) - RF(t) = a + b x [RM (t) - RF(t)] + c x
SMB + d x HML + e x HVARL + u(t)
BE/ME Quintile
Size Low 2 3 4 High
Quintiles
Slope Coefficient (a), Intercept
Small -1.28 -0.15 0.03 0.42 0.57
2 -0.52 0.72 0.53 1.73 1.99
3 0.40 0.66 1.08 0.79 2.18
4 0.44 1.37 2.30 0.82 1.27
Big 0.21 0.40 1.59 -0.16 -0.92
Slope Coefficient (b), Market Return
Small 1.08 1.04 0.88 0.83 0.91
2 0.98 1.05 0.89 0.88 1.05
3 1.06 1.14 1.10 1.05 1.07
4 0.95 0.94 1.11 0.98 1.12
Big 0.97 1.05 1.09 1.02 0.99
Slope Coefficient (c), SMB
Small 1.10 1.07 1.20 0.96 1.21
2 0.80 0.93 0.89 0.79 1.32
3 0.68 0.83 1.00 0.99 0.84
4 0.20 0.26 0.57 0.40 0.42
Big -0.02 0.16 0.24 0.15 -0.67
Slope Coefficient (d), HML
Small -0.74 -0.52 -0.28 -0.42 0.02
2 -0.66 -0.56 -0.31 -0.50 0.26
3 -0.87 -0.63 -0.38 -0.16 0.20
4 -0.96 -0.75 -0.60 -0.36 0.27
Big -0.75 -0.45 -0.40 -0.06 0.49
Slope Coefficient (e), HVARL
Small 0.07 -0.25 0.05 -0.11 -0.24
2 0.27 -0.29 0.02 -0.27 -0.34
3 0.22 -0.16 -0.29 -0.18 -0.33
4 0.36 -0.14 -0.21 -0.35 0.38
Big -0.23 -0.49 -0.72 -0.50 0.60
[[bar.R].sup.2]
Small 0.81 0.85 0.85 0.73 0.70
2 0.92 0.84 0.85 0.74 0.73
3 0.91 0.91 0.74 0.75 0.66
4 0.91 0.83 0.81 0.74 0.68
Big 0.85 0.81 0.79 0.78 0.92
Panel A: R(t) - RF(t) = a + b x [RM (t) - RF(t)] + c x
SMB + d x HML + e x HVARL + u(t)
BE/ME Quintile
Size Low 2 3 4 High
Quintiles
t-statistic (a)
Small -1.78 -0.28 0.06 0.65 0.89
2 -1.28 1.38 1.17 2.90 3.03
3 0.90 1.58 1.52 1.23 2.77
4 1.06 2.55 3.83 1.39 1.55
Big 0.45 0.85 3.18 -0.33 -2.50
t-statistic (b)
Small 12.63 17.03 14.89 10.90 12.13
2 20.35 17.03 16.59 12.45 13.51
3 20.24 23.25 13.10 13.83 11.46
4 19.34 14.81 15.68 14.06 11.50
Big 17.79 18.61 18.54 18.03 22.64
t-statistic (c)
Small 5.89 7.96 9.23 5.77 7.35
2 7.54 6.90 7.60 5.09 7.71
3 5.89 7.69 5.42 5.90 4.12
4 1.90 1.87 3.65 2.62 1.98
Big -0.14 1.32 1.88 1.20 -6.94
t-statistic (d)
Small -5.58 -3.16 -3.72 0.21
2 -6.06 -3.81 -4.66 2.26
3 -8.56 -3.01 -1.42 1.42
4 -7.90 -5.60 -3.41 1.85
Big -5.31 -4.54 -0.67 7.44
t-statistic (e)
Small -2.19 0.43 -0.76 -1.68
2 -2.50 0.22 -2.03 -1.99
3 -1.69 -1.84 -1.27 -2.03
4 -1.95 -1.67 -1.73 1.66
Big -4.62 -6.45 -4.67 7.26
S.E.E.
Small 21.76 20.12 33.27 32.55
2 21.96 16.62 28.71 34.98
3 13.87 41.10 33.69 50.08
4 23.23 29.04 28.02 54.88
Big 18.28 20.08 18.50 11.06
