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A Valuation Study of Stock Market Seasonality and the Size Effect

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  • U.S. Securities and Exchange Commission, San Francisco, CA, United States

Abstract

Existing studies on market seasonality and the size effect are largely based on realized returns. This paper investigates seasonal variations and size-related differences in cross-stock valuation distribution. We use three stock valuation measures, two derived from structural models and one from book/market ratio. We find that the average valuation level is the highest in mid summer and the lowest in mid December. Furthermore, the valuation dispersion (or, kurtosis) across stocks increases towards year end and reverses direction after the turn of the year, suggesting increased movements in both the under-and over-valuation directions. Among size groups, small-cap stocks exhibit the sharpest decline in valuation from June to December and the highest rise from December to January. For most months, small-cap stocks have the lowest valuation among all size groups and show the widest cross-stock valuation dispersion, meaning they are also the hardest to value. Overall, large stocks enjoy the highest valuation uniformity and are the least subject to valuation seasonality.
A Valuation Study of Stock Market Seasonality and
the Size Effect
Zhiwu Chen and Jan Jindra
October 30, 2009
Abstract
Existing studies on market seasonality and the size effect are largely based on realized
returns. This paper investigates seasonal variations and size-related differences in cross-
stock valuation distribution. We use three stock valuation measures, two derived from
structural models and one from book/market ratio. We find that the average valuation
level is the highest in mid summer and the lowest in mid December. Furthermore, the
valuation dispersion (or, kurtosis) across stocks increases towards year end and reverses
direction after the turn of the year, suggesting increased movements in both the under-
and over-valuation directions. Among size groups, small-cap stocks exhibit the sharpest
decline in valuation from June to December and the highest rise from December to
January. For most months, small-cap stocks have the lowest valuation among all size
groups and show the widest cross-stock valuation dispersion, meaning they are also the
hardest to value. Overall, large stocks enjoy the highest valuation uniformity and are the
least subject to valuation seasonality.
One of the salient facts in finance is the documented seasonality in stock returns.
Specifically, recent losers tend to experience fortune reversals in January (hence, the
January effect), whereas recent winners tend to continue and expand their fortunes in
December (hence, the December effect).1Most studies on stock market seasonality have
relied on return differences across calendar times. In this paper, we analyze seasonality
from a different perspective: cross-sectional valuation movements from one calendar time
to another. Using alternative valuation measures, we investigate how the cross-stock
valuation distributions shift from month to month. Our goal is to study whether such
month-to-month variations exhibit any systematic patterns. If seasonal patterns exist,
we then want to see whether they can shed new light on the documented January and
December effects. This approach offers unique insights into the seasonality effects by
allowing us to gauge the relative and absolute valuation characteristics of stocks in various
months of the year. At the same time, given the documented seasonal return patterns,
our study allows us to evaluate alternative valuation measures, with the understanding
that a good stock valuation metric must be able to demonstrate in advance certain
seasonal dynamics and predict such return patterns. That is, which valuation measure
best anticipates the return seasonality?
The fact that beaten-down stocks, especially small firms, experience return reversals in
January has long been a puzzling phenomenon. There are several proposed explanations,
including ”window-dressing” by institutional investors (Lakonishok, Shleifer, Thaler, and
Vishny [1991]) and the tax-loss selling hypothesis (Grinblatt and Moskowitz [2004]).
The window-dressing hypothesis contends that institutional investors sell losers and buy
winners to prepare for year-end reporting. Such buying and selling create positive price
pressure on winners and downward pressure on losers before the turn of the year. As
the selling by institutions stops at year end, prices for beaten-down stocks rebound in
January, producing large returns for last year’s losers. The tax-loss selling hypothesis
asserts that it is the individual investors faced with capital gains taxation who sell poor-
performing stocks to reduce their taxes. To achieve such tax reduction, an investor must
sell the losing stocks before year end as capital losses can be used to offset gains only
upon realization.2Both hypotheses predict that losers as of late fall will likely continue
going lower in December, but will have high returns in January. The window-dressing
hypothesis also suggests that strong performers by late fall shall continue going strong
in December, which may be a key reason behind the December effect.
The purpose of this paper is to identify and understand what valuation picture is
behind the return seasonality. Such an exercise does not only help deepen our under-
standing of the seasonality phenomenon, but also shed new light on the development
of stock valuation models. Specifically, we implement three valuation measures. The
first measure is based on the dynamic stock valuation model developed by Bakshi and
Chen [2005] and extended by Dong [2000], (“BCD model”). We refer to the percentage
deviation by the market price from a stock’s model valuation as the BCD mispricing.
The second measure is the value/price (V/P) ratio, where “value” is determined using
the residual-income model as implemented in Lee, Myers and Swaminathan [1999]. The
last measure is the book/market (B/M) ratio, which is a traditional valuation metric.
Our results are briefly summarized below:
1
According to each valuation measure, stocks on average become less favorably val-
ued as the year end approaches. In December, stocks are the least favorably val-
ued (except that based on the V/P ratio, October/November instead of December
marks the lowest valuation month of the year), while stocks reach their highest
valuation in the May-June period. Towards the year end, the cross-stock valuation
distribution also becomes increasingly dispersed and flattened out, with either the
standard deviation or the kurtosis of the valuation distribution growing larger. This
means that undervalued stocks become more undervalued whereas overvalued ones
become more so, implying more differential and uneven treatments of stocks as the
year end approaches. After the turn of the year, all of these valuation trends are
reversed.
Across size groups, small-cap stocks are on average the least favorably valued,
followed by mid-cap and then by large-cap stocks. This is true for most calendar
months. According to each measure, the valuation spread between December and
June is the largest for small-cap and the second largest for mid-cap; Similarly,
the December-January valuation spread is also the largest for small-cap. Thus,
seasonal valuation patterns are the most severe for small-cap, in terms of changes
both around year end and from year end to mid-year.
For each given month, the cross-stock valuation distribution is the most widely
dispersed (i.e., with the highest standard deviation) for small-cap stocks and the
least dispersed for large-cap stocks. It is again true regardless of valuation measure.
When we estimate monthly Fama-McBeth cross-sectional return regressions, we
find that the BCD mispricing and size have the highest predictive power, whereas
the V/P and B/M ratios are at best marginal. In particular, the BCD mispric-
ing and size are substantially more significant in predicting the mid-December to
mid-January returns than in predicting any other monthly returns. The valuation
seasonality captured by the BCD model anticipates return seasonalities significantly
more closely than the V/P and B/M ratios.
The findings summarized above are consistent with most returns-based studies on
market seasonality. The fact that valuation is the lowest in December anticipates the well
known January return effect. The phenomenon that the valuation distribution becomes
more dispersed towards year end possibly captures two simultaneous, but different trends:
tax-loss and window-dressing selling of the losers (making the losers more undervalued)
and window-dressing buying of winners (making the winners more overvalued).
We also add a few items to the growing list of conclusions regarding small-cap versus
large-cap stocks: that most of the year small-cap stocks are the least favorably valued,
that their valuation dispersion is the highest, that their June-December valuation pat-
terns are the strongest, and that their December-to-January valuation increase is the
highest. These valuation-based findings together suggest that (i) small-cap stocks may
simply be much harder to value and (ii) their mispricing is more difficult to be arbi-
traged away. Forces that have been proposed as factors behind these difficulties include
informational asymmetry, price-impact and trading costs (e.g., Hasbrouk [1991], and
2
Stoll [2000]), liquidity constraints, high information-acquisition costs but low economic
benefits for institutional portfolio managers, and high arbitrage risk (e.g., Wurgler and
Zhuravskaya [2002]). All these factors favor large-cap and work against small-cap stocks.
The rest of the paper proceeds as follows. Next section discusses the three valuation
measures and describes the data sample used. The second section documents valua-
tion seasonality for the overall sample. In the third section, we divide the sample into
three size groups and address their differences. Section four focuses on Fama-McBeth
cross-sectional return-forecasting regressions, studying a formal linkage between valua-
tion seasonality and return seasonality. Concluding remarks are given in section five.
Appendices A and B provide overview of valuation models and their implementation.
VALUATION MEASURES
To make our results independent of a particular valuation model, we use three value
measures for each stock: a mispricing measure based on the valuation model in Bakshi
and Chen [2005] and Dong [2000] (BCD mispricing), a value/price (V/P) ratio based on
the residual-income model in Lee, Myers and Swaminathan [1999], and book-to-market
(B/M) ratio. These value metrics have been applied in the literature and shown to be
significant in predicting cross-sectional stock returns. See, for example, Chen and Dong
[2001], Fama and French [1992, 1993], and Lee, Myers and Swaminathan [1999].
The BCD Mispricing
ValuEngine, Inc. provided us with data necessary to calculate the BCD mispricing in this
paper: predicted monthly model prices as well as the concurrent market prices of individ-
ual stocks. ValuEngine’s estimation technique is based on the implementation method
used in Bakshi and Chen [2005] and Chen and Dong [2001].3The BCD model assumes
a parameterized stochastic process for each of the firm’s earnings-per-share (EPS), ex-
pected future EPS and economy-wide interest rates and leads to a closed-form valuation
formula. Appendix A outlines the main points of the BCD model.
It is important to note that the estimation process is independently and separately
applied to every stock for each month. For this reason, all model prices used here are
determined out of sample. For each month and each stock, the BCD mispricing is defined
to be the difference between the market and model price, divided by the model price.
The V/P Ratio
The second measure is based on the residual income model as outlined in Lee, Myers, and
Swaminathan [1999]. Specifically, we calculate the V/P ratio, where value is determined
by the residual income model and price is taken from the market as of each mid-month.
The residual-income model valuation is based on a multi-stage discounting procedure
that involves estimating both the firm’s future EPS in excess of the required return on
equity and a terminal value of the stock at the end of valuation horizon. Appendix B
details our implementation.
3
B/M Ratio
Market ratios have long been used as indirect measures of value, including book/market
(B/M), earnings/price (E/P), cashflow/price, and sales/price. These ratios have been
shown to be significant predictors of future returns.4Since E/P, cashflow/price and
sales/price are generally highly correlated with B/M, we focus on B/M, where the book
value of equity is measured once a year at the firm’s fiscal year end, whereas the stock
price is taken at each mid-month. Therefore, there is a monthly B/M value for each
stock.
Data Sample
Four databases are used to estimate the three valuation measures for each stock and
in every month. First, we identify all companies with stock price and return data in
the CRSP database. Second, the BCD mispricing data starting in 1977 is provided by
ValuEngine, Inc. Third, the actual book value data is from COMPUSTAT. Finally, for
determining each stocks V/P ratio, we use analyst consensus earnings forecasts from
I/B/E/S. Since the I/B/E/S database starts in 1976, our original sample does not start
until 1976. When it started, I/B/E/S did not cover more than a few hundred large firms.
To maintain a robust sample size, we restrict our attention to the 27-year period from
January 1980 to December 2007.
The temporal distribution of the three measures is shown in Exhibit 1. The BCD
mispricing sample has 71,392 stock-month observations over the period; The B/M sample
has 105,697 observations, where any observation with either (i) a B/M less than 0.05 or
greater than 30 or (ii) a book value less than $0.1 million is excluded; there are 81,687
observations for the V/P ratio sample. The varying sample sizes across the three datasets
reflect the availability of data required for each measure. The sample size for each measure
has steadily increased from 1980 through late 1990’s, due to both the increasing coverage
of firms by I/B/E/S and the increasing number of publicly traded firms. Following the
market decline and ensuing consolidation after 2000, all samples decrease correspondingly.
Exhibit 1 also provides summary statistics for each valuation measure, both over
the entire period and year-by-year. Over the entire period, the average estimate for
all observations with available data is 1.63% for the BCD mispricing, 84.85% for V/P,
and 79.60% for B/M. From 1980 to 2007, both the average valuation and the cross-
sectional standard deviation have varied significantly, regardless of the valuation metric.
For example, the average BCD mispricing was 5.96% (overvalued) in 1997, the last highest
valuation until the end of the sample; The average V/P continued its downward trend
from 1994 to 1997, whereas B/M started its down trend in 1990. All measures indicate
that U.S. stocks had become increasingly more overvalued in 2006 and 2007.
SEASONALITY IN VALUATION MEASURES
To examine how the valuation distribution changes from month to month, for each given
month of the year and a given valuation metric, we compute the mean, median, standard
deviation, skewness and kurtosis of the metric across all stocks in that calendar month
4
and for all years from 1980 to 2007. That is, we pool together the same-calendar-month
observations from each year, resulting in 12 calendar-month pools. In total, we obtain
12 calendar-month valuation distributions across individual firms.
Exhibit 2 reports valuation statistics for each calendar month. Both the mean and
median BCD mispricing estimates are the highest in May, implying stocks are on average
the highest valued during mid-year, with a mean and median mispricing of 4.02% and
2.79%, respectively. From May to December, the mean and median BCD mispricing levels
decrease steadily; Then, from December onward the BCD mispricing reverses direction
and rises to approach the highest in May. For a given stock, the average BCD mispricing
difference between December and June is 4.2%. Therefore, stocks are on average the most
underpriced before year end, with a mean mispricing of -0.72% in December. January
marks the beginning of the “correction” process.
The kurtosis of the BCD mispricing also shows a seasonality, increasing each month
from October to December, reaching its local high in December. In January, the kurto-
sis begins to decrease, reaching its minimum in April. Thus, the cross-stock mispricing
distribution becomes more fat-tailed towards year end than at the beginning of the year.
This suggests that as the year end approaches, stocks that are beaten down and under-
valued will become even more undervalued, while stocks that are already overvalued will
grow more overvalued. This phenomenon of extremely mispriced stocks becoming even
more so (on both the under and the over-valued side) is likely due to the working of
two different factors: one due to performance-chasing at year end by portfolio managers
and the other due to tax-loss selling. It anticipates both the December and the January
effect.
Exhibit 3 displays how the fractions of undervalued, fair-valued, and overvalued stocks
based on the BCD mispricing, change from month to month. The undervalued fraction
has a hump-shaped pattern from June to December to May, reaching its peak in Decem-
ber, whereas the seasonal variation for the overvalued fraction is U-shaped. On the other
hand, the fair-valued fraction is relatively stable over the year: with around 35% of the
stocks fair-valued throughout the year according to the BCD model.5
B/M exhibits similar seasonality as the BCD mispricing. In Exhibit 2, the mean and
standard deviation for B/M start increasing in the summer and reach their peak in De-
cember. Thus, again, stocks on average become more undervalued, and the cross-stock
valuation distribution grows more dispersed, as December approaches. For a typical
stock, its average change in B/M is -6.4% from December to June and -4.3% from De-
cember to January. The skewness for B/M is the second lowest in December, increasing
slightly from November. For V/P, its mean and standard deviation each gradually in-
crease from the summer and reach their highest values in October (instead of December),
and they start gradually decreasing from October to January.
Based on Exhibits 2 and 3, our conclusions can be summarized as follows. First,
regardless of the valuation metric, stocks on average start from year-end’s undervaluation
to mid-year’s overvaluation and then back to year-end’s undervaluation again, a seasonal
valuation pattern that anticipates, and is consistent with, the well documented January
stock-return effect. Second, as the year end approaches, the dispersion of the cross-
stock valuation distribution increases, or both tails of the distribution become fatter: the
overvalued stocks grow more overvalued and the undervalued become more undervalued.
5
This finding is consistent with both the January effect (which is mostly about movements
in the undervalued tail of the distribution around the turn of the year) and the December
effect as documented in Grinblatt and Moskowitz [2004]. The December effect is about
the fact that stocks that are winners by the end of November tend to continue doing well
before the end of December. Given the relative high correlation between recent return
performance and overvaluation (the correlation coefficients of BCD mispricing, VP ratio,
and B/M with past 12-month momentum is 0.37, -0.15, and -0.16, respectively), these
winners are also likely to be overvalued in November. Therefore, the December effect is
about how the stocks in the right (overvalued) tail of the mispricing distribution move
from November to the end of December.
VALUATION SEASONALITY BY SIZE GROUPS
The size effect is another well-documented phenomenon in which small-cap stocks out-
perform large-cap stocks (e.g., Banz [1981], Blume and Stambaugh [1983], Fama and
French [1992, 1993], Keim [1983], and Roll [1983]). Furthermore, there is substantial
evidence suggesting that the January effect is largely due to small-cap firms. To help us
understand the seasonal patterns in valuation, we divide each June’s sample into three
size groups: small-cap, mid-cap and large-cap, each group consisting of 30%, 40%, and
30% of that June’s stock universe, respectively. Once the size groups are formed, they are
fixed for the next 12 months until the following June when the size groups are re-formed.
Within each group, we determine its cross-stock valuation distribution for each month
of the year and according to a given valuation measure. Exhibit 4 shows the resulting
seasonal distributions, one panel for each size group.
First, note that in Exhibit 4 the annual re-sorting in June of the stock universe into
three size groups creates noticeable discontinuity from June to July, especially for the
small-cap group. When stocks are re-grouped according to their market cap in June, the
small-cap group collects a disproportionately large number of losers. Since those losers
tend to be beaten down and hence likely to be undervalued, the average mispricing for
the small-cap group suddenly drops from its pre-sorting level of 3.54% to its post-sorting
level of 0.17% in July (see Panel A). In contrast, for the mid-cap and large-cap groups,
their changes in mispricing are not as dramatic from June to July (Panels B and C).
Panel A of Exhibit 4 shows the seasonal variation in valuation distribution for small-
cap stocks. Seasonal changes in average BCD mispricing are much stronger and more
pronounced for small-cap than for the overall sample (Exhibit 3). The average BCD
mispricing for small-cap is 3.54% in June, but is -3.24% in December, resulting in a
December-June difference of 6.8%. In contrast, the December-June spread in average
BCD mispricing is 4.2% for the overall sample. By mid-January, the average BCD
mispricing jumps to by 3.1% to -0.12% from its December level of -3.24% for small-cap,
implying a sharp reversal in valuation from December to January.
Small-cap stocks have the lowest valuation in December according to the B/M ratio,
and in October based on the V/P ratio. For these ratios, the change from June to
December (or, October for the V/P) is gradual, monotonic and highly significant. The
average net change in B/M from June to December is 14.2%, while the decline in B/M
6
from December to January is 8.8%, for small-cap, suggesting that most of the year’s
correction in small-cap’s misvaluation occurs in January.
For the mid-cap group in Panel B of Exhibit 4, these stocks also become gradually
less favorably valued going from June towards year end. But, regardless of the valuation
measure, the change in valuation is not as dramatic from December to June. As noted
above, the average December-June difference in BCD mispricing is 6.8% for small-cap
firms, but it is only 3.1% for mid-cap firms. The average December-June difference in
B/M is -14.2% for small-cap, but only -5.5% for mid-cap stocks.
According to the BCD mispricing, large-cap stocks still exhibit a seasonal pattern in
valuation. In Panel C of Exhibit 4, the average mispricing is 4.50% in July and then
gradually goes down to 1.22% by December. The average December-June difference for
large-cap is only 2.3% in BCD mispricing. Neither V/P nor B/M exhibit a clear seasonal
pattern for large-cap stocks.
In summary, small-cap firms show the strongest valuation variations both from mid-
year to year-end and around the turn of the year, while large-cap stocks show the least.
In fact, according to both B/M and V/P, large-cap stocks have only slight seasonal
variations and these variations even have the wrong sign.
The fact that among all size groups, small-cap stocks have the lowest valuation in
December is consistent with the extensive evidence that these stocks show the strongest
January return effect. As large-cap stocks do not show much seasonal variations in
valuation, they should not exhibit much return seasonality either and they do not.
Besides the afore-mentioned difference in valuation seasonality, the valuation distri-
bution behaves quite differently among the size groups. We can examine this for each
given calendar month, in terms of both the median level and standard deviation of val-
uation. First, for each calendar month, the median valuation is the lowest for small-cap
and the highest for large-cap. For example, in June, the median B/M ratio is 69.54%
for small-cap, 61.46% for mid-cap and 57.26% for large-cap. In December, the median
B/M is 79.63% for small-cap, 64.57% for mid-cap and 58.22% for large-cap. The same
observations can be made based on V/P and, to a lesser degree, on BCD mispricing.
The fact that small-cap stocks are consistently less favorably valued than large-caps may
also explain the persistence of the famous size premium, that is, small stocks on average
outperform large stocks (e.g., Banz [1981], Grinblatt and Moskowitz [2004], Keim [1983],
Loughran and Ritter [2000], and Roll [1983]).
Finally, the cross-stock valuation dispersion (or standard deviation) is the highest for
small-cap and the lowest for large-cap. For instance, in August, the standard deviation
for BCD mispricing is 28.54% for small-cap, 24.05% for mid-cap and 19.29% for large-cap
stocks; In November, the standard deviation for B/M is 126.74% for small-cap, 119.10%
for mid-cap and 108.66% for large-cap stocks.
This result suggests that small firms are perhaps harder to value and that mispricing
for small-cap is more difficult to be arbitraged away. One explanation may be that
generally less information is available about small firms. For example, typically, the larger
a firm, the more security analysts and portfolio managers following the firm, and hence
the more monitoring and scrutiny of the firm’s activities and news. For large institutional
portfolio managers, it is economically not meaningful to invest in small firms and hence
they may not attempt to gather and process information on these firms. The relative
7
lack of information and the higher information-production costs can be enough to make
small stocks bounce more easily between different valuation levels. Wider bid-ask spreads
and higher price-impact costs for small-cap firms are also reasons for arbitraging to be
difficult (e.g., Hasbrouk [1991] and Stoll [2000]).
Another explanation is that it is more difficult to find close substitutes for small stocks
(Wurgler and Zhuravskaya [2002]). Presumably, when valuation is dispersed across small
stocks, one would want to construct a long-short arbitrage portfolio between the two tails
of the mispricing distribution. Such arbitraging may be the most effective to correct the
mispricing dispersion. But, since it is difficult to make the long and short sides match
in risk for small-cap stocks, such arbitraging is more risky to do among small-cap than
among large-cap stocks.
VALUATION AND RETURN SEASONALITY
Using the three valuation measures, the preceding sections have shown that stocks are on
average the most favorably priced before year end and the least favorably priced in mid-
year. Broadly speaking, this valuation seasonality is consistent with the known January
effect, that is, favorable valuations of stocks in December are followed by abnormally high
returns in January. However, we have established this timing consistency between the
two types of seasonality only at the average stock level and across calendar months. What
remains to be shown is whether cross-sectionally those stocks that contribute the most to
the January effect are also the most favorably priced in the preceding December. Such an
exercise is important because the association between the valuation seasonality and the
return seasonality could be spurious: even though the average valuation level is the lowest
in December, it could happen that stocks that are the most overvalued in December go
up in the following January, whereas the undervalued ones stay unchanged or even go
down further in January. If that would occur, the average valuation in December would
be the lowest and the average January return could still be abnormally high, but the low
December valuation would not be the cause behind the high average January return. To
rule out such a possibility, we estimate Fama-McBeth return-forecasting regressions in
this section.
Exhibit 5 serves to provide a general picture of the predictive power by different
valuation measures and size. For this part, we include all months as well as December
only cross-sectional regressions. In the univariate regressions (not shown), each of the
BCD mispricing and V/P ratio is a statistically significant predictor of future stock
returns, implying that the more favorably priced a stock, the higher its future one-month
return. When all the valuation measures and size are included in a joint forecasting
Regression 1, only BCD mispricing, V/P and size are significant and their respective
coefficient estimates are of the correct sign, whereas the B/M ratio’s coefficient has the
wrong sign.
To establish that the valuation seasonality anticipates the return seasonality, we are
interested in the December results as the January effect is the major contributor to stock
return seasonality. Recall that because of the mid-month sampling practice by I/B/E/S,
most empirical results of this paper are based on the mid-month value for each measure
8
and mid-month to mid-month returns. Therefore, the December regression results are
from using the mid-December to mid-January returns as the dependent variable, hence
covering a crucial time period for the January effect. Thus, the December regression
results should be the most informative of which ex ante variables are the most predictive
of high January returns.
Based on the December results in Regression 2 in Exhibit 5, BCD mispricing and firm
size are the most significant in predicting a stocks up-coming January return: the more
underpriced a stock in mid December and/or the smaller the firm, the higher its return
over the next month. In unreported regressions for each calendar month, we note that
both the magnitude and t-statistic for these two variables’ coefficient estimates are the
highest for December than for any other calendar month. Thus, the return-based size
effect is mostly due to the month of January and the low December valuation (a result
consistent with the findings in Blume and Stambaugh [1983] and Keim [1983]).
The results in Exhibit 5 suggest that tax-loss-selling may not explain all of the January
effect or the size effect. Note that if the year-end tax-loss-selling were the exclusive
reason behind the January effect, then the high January returns must be exclusively due
to “valuation corrections” and one would expect the valuation factors to be the only
significant predictors of the January effect. In this sense, we can think of the valuation
factors in Exhibit 5 as capturing the tax-loss-selling effect.6For the regressions, we have
included all of the BCD mispricing, V/P and B/M, as these are the known valuation
measures in the literature. The fact that both BCD mispricing and size are significant in
jointly explaining the January effect implies that while tax-loss-selling (i.e., the valuation
factor) is a major reason, it is not the only reason behind the high January returns. Size
appears to capture something beyond the correction of the mis-valuation caused by tax-
loss-selling.7
It is worth noting that the BCD mispricing reflects more of a stock’s current valuation
relative to its own past valuation levels, and this “mispricing” assessment is completely
independent of how other stocks are and have been valued. On the other hand, the size
factor, when defined by market capitalization, is a cross-sectional variable and serves as a
proxy for factors that set firms of different sizes apart and that are not yet known. That
is, the BCD mispricing captures the stock’s “temporal” variation in valuation, whereas
size is a cross-sectional measure. This may explain why both valuation and size are
significant predictors of the January return effect.
In Exhibit 5, we also divide the sample into three size groups and run the same Fama-
McBeth regressions separately for each group. The size-based results re-confirm the above
finding about BCD mispricing and size. Within each size group and among all calendar
months, the BCD mispricing, V/P and B/M are statistically significant predictors of
future one-month returns (Regressions 3, 5 and 7). In predicting the January effect
within the size groups, BCD mispricing and size are significant (Regressions 4, 6 and 8).
The fact that BCD mispricing predicts future returns, and explains the January effect,
better than V/P and B/M supports the BCD model valuation as a better measure of a
stock’s value. The persistence of stock return seasonality is a phenomenon indicative of
regularly recurring mis-valuation by the market, and any empirically acceptable valuation
model must produce a mispricing pattern that is consistent with, and predicts, the return
seasonality. Among the three valuation measures implemented in this paper, the BCD
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model has performed the best in this regard.
CONCLUDING REMARKS
The focus of this paper has been on documenting and understanding seasonal valuation
patterns for stocks. This is in contrast with most existing studies on stock market season-
ality where the focus has been on observed return patterns. When researchers first started
investigating stock market seasonality, the most natural and direct approach was clearly
to use realized returns across calendar times as the basis. A return-focused approach is
in some sense free of valuation models, hence not subject to model misspecifications.
Given the abundant evidence for return seasonality, it has been a challenge to find
a fundamental economic explanation. Window-dressing and tax-loss selling are among
the front runners in this direction. While the true cause for return seasonality may be
window-dressing and/or tax-loss selling and possibly others, for asset valuation theory
itself the challenge still remains: how can valuation theory capture and reconcile such
return seasonality from a modeling perspective?
In this paper, we have relied on two recent stock valuation models, the BCD model
and residual income model, and one indirect valuation metric, B/M, to study market
seasonality. Our finding indicates that regardless of the valuation measure, stocks are
on average the least favorably priced towards year end and the most overvalued in mid-
summer. The correction process of December’s low valuation starts in mid-December,
accelerates in early January, and ends by March, after which point stocks tend to begin
an overvaluation season of the year. Another important finding concerns the differences
across size groups. For small-cap, the valuation seasonality is by far the strongest and
the January valuation correction is also the sharpest. For most of the year, small-cap
stocks are the least favorably valued with the widest valuation dispersion, while large-
cap stocks are the most favorably priced with the lowest dispersion. Overall, our study
suggests that the BCD model captures a stock’s true value better than V/P and B/M.
APPENDIX
A The BCD Model
For detailed derivations and discussions of this stock valuation model, see Bakshi and
Chen [2005] and Dong [2000].
To describe the BCD model, assume that a share of a firm’s stock entitles its holder to
an infinite dividend stream {D(t) : t0}. Our goal is to determine the time-tper-share
value, S(t), for each t0. Bakshi and Chen [2005] make the following assumptions:
The firm’s dividend policy is such that at each time t
10
D(t) = δ Y (t) + ²(t) (A-1)
where δis the target dividend payout ratio, Y(t) the current EPS (net of all ex-
penses, interest and taxes), and ²(t) a mean-zero random deviation (uncorrelated
with any other stochastic variable in the economy) from the target dividend policy.
The instantaneous interest rate, R(t), follows an Ornstein-Uhlenbeck mean-reverting
process:
d R(t) = κrhµ0
rR(t)idt +σrr(t) (A-2)
for constants κr, measuring the speed of adjustment to the long-run mean µ0
r, and
σr, reflecting interest-rate volatility. This is adopted from the well-known single-
factor Vasicek [1977] model on the term structure of interest rates.
In Bakshi and Chen [2005], the assumed process for Y(t) does not allow for negative
earnings to occur. To resolve this modeling issue, Dong [2000] extends the original
Bakshi-Chen earnings process by adding a constant y0to Y(t):
X(t)Y(t) + y0(A-3)
X(t) can be referred to as the displaced EPS or adjusted EPS. Next, Dong [2000]
assumes that X(t) and the expected adjusted-EPS growth, G(t) follow
dX(t)
X(t)=G(t)dt +σxx(t) (A-4)
d G(t) = κghµ0
gG(t)idt +σgg(t) (A-5)
for constants σx,κg,µ0
gand σg, where G(t) is the conditionally expected rate of
growth in adjusted EPS X(t). The long-run mean for G(t) is µ0
g, and the speed at
which G(t) adjusts to µ0
gis reflected by κg. Further, 1
κgmeasures the duration of the
firm’s business growth cycle. Volatility for both the adjusted-EPS growth and changes
in expected adjusted-EPS growth is time-invariant. The correlations of ωx(t) with ωg(t)
and ωr(t) are respectively denoted by ρg,x and ρr,x.
Under the given model assumptions, the equilibrium stock price is
S(t) = δZ
0{X(t)exp [ϕ(τ)%(τ)R(t) + ϑ(τ)G(t)] y0exp[φ0(τ)%(τ)R(t)]}
(A-6)
where
ϕ(τ) = λxτ+1
2
σ2
r
κ2
r"τ+1e2κrτ
2κr
2(1 eκrτ)
κr#κrµr+σxσrρr,x
κr"τ1eκrτ
κr#
11
+1
2
σ2
g
κ2
g"τ+1e2κgτ
2κg
2
κg
(1 eκgτ)#+κgµg+σxσgρg,x
κg"τ1eκgτ
κg#
σrσgρg,r
κrκg(τ1
κr
(1 eκrτ)1
κg
(1 eκgτ) + 1e(κr+κg)τ
κr+κg)(A-7)
%(τ) = 1eκrτ
κr
(A-8)
ϑ(τ) = 1eκgτ
κg
(A-9)
φ0(τ) = 1
2
σ2
r
κ2
r"τ+1e2κrτ
2κr
2(1 eκrτ)
κr#,(A-10)
subject to the transversality conditions that
µr>1
2
σ2
r
κ2
r
(A-11)
µrµg>σ2
r
2κ2
r
σrσxρr,x
κr
+σ2
g
2κ2
g
+σgσyρg,x
κg
σgσrρg,r
κgκr
λx(A-12)
where λxis the risk premium for the systematic risk of earnings shocks, µgand µrare
the respective risk-neutralized long-run means of G(t) and R(t). Formula (A-6) represents
a closed-form solution to the equity valuation problem, except that its implementation
requires numerically integrating the inside exponential function. Therefore, the equilib-
rium stock price is a function of interest rate, current EPS, expected future EPS, the
firm’s required risk premium, and the structural parameters governing the EPS and in-
terest rate processes. We refer to this stock-pricing formula as the Bakshi-Chen-Dong
(BCD) model.
In the original research by Bakshi and Chen [2005], the structural parameters needed
to be estimated. To reduce the number of parameters to be estimated, the following
parameters were preset ρg,x = 1 and ρg,r =ρr,x ρ, that is, actual and expected
adjusted-EPS growth rates are subject to the same random shocks. In addition, for each
individual stock estimation, the three interest-rate parameters are preset at µr= 0.0794,
κr= 0.109, σr= 0.0118. These parameter values are based on a maximum-likelihood
procedure using a 30-year yield time-series (Bakshi and Chen [2005]). A justification for
this treatment is that the interest-rate parameters are common to all stocks and equity
indices. Note that these estimates are comparable to those reported in Chan, Karolyi,
Longstaff, and Sanders [1992]. Then, there are 8 firm-specific parameters remaining to
be estimated: Φ = {y0, µg, κg, σg, σx, λx, ρ, δ}. At each time point of valuation, the most
recent 24 monthly observations on a stock (and interest rates) are used as the basis to
estimate Φ (see Chen and Dong [2001] for details on this choice). Specifically, for each
stock and for every month in the sample, Φ is chosen so as to solve
MinΦ
24
X
t=1
[ˆ
S(t)S(t) ]2(A-13)
12
where S(t) is as given in formula (A-6) and ˆ
S(t) the observed market price in month
t.
Once the parameters are estimated for a given stock and in a given month (using
data from the prior years), the parameter estimates, plus the current R(t), Y(t) and G(t)
values, are substituted into formula (A-6) to determine the current model price for the
stock in that month. After the model price is calculated for the stock in that month, the
parameter estimation steps and the calculation are repeated for the same stock, but for
the following month and so on. This process is independently and separately applied to
every stock in the sample and for each month. For this reason, all the model prices are
determined out of sample.
B The Residual Income Model
A stock’s intrinsic value is defined as the present value of the expected future cash flows
to shareholders:
V(t) =
X
i=1
Et(D(t+i))
(1+re)i(B-1)
where Et(D(t+i)) is the expected future dividend for period t+iconditional on all
available information at time t, and reis the cost of equity. Ohlson [1990, 1995] demon-
strates that if the firm’s earnings and book value are forecasted in a manner consistent
with clean-surplus accounting (i.e. a dollar of earnings increases either dividends paid
out or book value by a dollar), the intrinsic value may be rewritten as:
V(t) = B(t) +
X
i=1
Et[NI (t+i)reB(t+i1)]
(1+re)i(B-2)
=B(t) +
X
i=1
Et[(ROE (t+i)re)B(t+i1)
(1+re)i(B-3)
where B(t) is the book value at time t,Et[NI(t+i)] and Et[ROE (t+i)] are the
conditional expectations of both net income and after-tax return on book value of equity
for period t+i.
The above equation expresses a stock’s intrinsic value in terms of an infinite sum.
However, for practical purposes, only limited future earnings forecasts are available. This
limitation introduces a need for an estimate of a terminal value. That is, we measure the
intrinsic value in the following way:
V(t) = B(t) +
2
X
i=1
EPS (t+i)reB(t+i1)
(1+re)i+T V (B-4)
where EP S (t) is the consensus earnings forecast for period t, and T V is the terminal
value estimate based on the average of the last two years of data (in order to smooth
cases of unusual performance in the last year, D’Mello and Shroff [2000]):
13
T V =B(t) + 1
2re(1+re)2
3
X
i=2
EPS (t+i)reB(t+i1)
(1+re)2(B-5)
When implementing the model, we estimate the current book value per share, B(t),
from the most recent financial statement. Book value for any future period t+i,B(t+i),
is given by the beginning-of-period book value, B(t+i1), plus the forecasted EPS,
EP S (t+i), minus the forecasted dividend per share for year t+i. The forecasted
dividend per share is estimated using the current dividend payout ratio. The forecasted
earnings per share, EP S (t+ 1), are given by the analyst consensus forecast for the
relevant year and as reported in I/B/E/S. The cost of equity capital, re, is estimated
using the CAPM and following Fama and French [1997]: 60 monthly observations prior
to the month of estimation are used to estimate the stock’s beta, and then the cost of
equity capital is determined by the market T-bill rate of the month plus the beta times
the market risk premium, where the market risk premium is the average excess return
on the NYSE/AMEX/Nasdaq portfolio from January 1945 to month t-1.
ENDNOTES
Zhiwu Chen is a Professor of Finance at Yale School of Management, 135 Prospect
Street, New Haven, CT 06520, Zhiwu.Chen@yale.edu, Tel. (203) 432-5948; Jan Jindra
is an Assistant Professor at Menlo College, 1000 El Camino Real, Atherton, CA 94025,
JJindra@menlo.edu and telephone: (650) 804-6807.
The authors would like thank ValuEngine, Inc. for graciously providing data used to
estimate BCD mispricing. Any errors are our responsibility alone.
1The January effect has been documented by Rozeff and Kinney [1976], Dyl [1977],
Roll [1983], Keim [1983,1989], Reinganum [1983] and others. Grinblatt and Moskowitz
[2004] find such a December effect.
2Sikes [2008] finds calendar year end stock return patterns for stocks with negative
year-to-date returns are related to year end changes in ownership of institutions with
incentives to sell such stocks for tax purposes.
3See http://www.valuengine.com/pub/main?p=0.
4For example, Daniel and Titman [1997], Davis, Fama, and French [2000], Fama and
French [1992,1993,1995,1996,1997], Frankel and Lee [1998], Jegadeesh and Titman [1993],
Lakonishok, Shleifer, and Vishny [1994], Grinblatt and Moskowitz [1999].
5As a robustness check, we also examine the distribution of the undervalued, fair-
valued, and overvalued stocks for a period prior to 1998. For this earlier period, the
patterns in the distribution of undervalued and overvalued stocks are even more pro-
nounced.
14
6This reasoning can be best seen in Roll [1983, p. 20]: “There is downside price
pressure on stocks that have already declined during the year, because investors sell
them to realize capital losses. After the year’s end this price pressure is relieved and the
returns during the next few days are large as those same stocks jump back up to their
equilibrium values.”
7Reinganum [1983] constructs a measure of a stock’s tax-loss-selling potential, to study
the extent to which the January effect is due to tax-loss-selling. He concludes that after
controlling for tax-loss-selling, firms still exhibit a January seasonal effect that seems to
be related to market capitalization. In our case, we use mid-December’s valuation as a
proxy for the extent of tax-loss-selling potential.
15
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18
Distribution of Valuation Measures by Year
BCD Mispricing V/P B/M
Year N Mean Std. Dev. N Mean Std. Dev. N Mean Std. Dev.
1980 1,001 14.85% 23.66% 1,562 104.43% 61.61% 3,333 112.59% 104.58%
1981 1,279 20.04% 26.15% 1,630 94.40% 53.93% 3,358 99.87% 91.72%
1982 1,508 8.15% 28.27% 1,598 109.25% 60.72% 3,366 111.99% 96.47%
1983 1,540 26.09% 31.19% 1,684 78.90% 42.49% 3,179 76.16% 89.08%
1984 1,731 8.50% 28.49% 2,244 86.00% 50.48% 3,429 83.74% 93.72%
1985 1,734 1.08% 24.63% 2,197 78.02% 46.29% 3,424 81.30% 105.89%
1986 1,812 -6.99% 22.30% 2,173 72.79% 46.15% 3,260 73.28% 111.62%
1987 2,068 4.16% 27.55% 2,375 79.01% 51.31% 3,444 73.22% 107.21%
1988 1,929 -6.47% 24.78% 2,347 87.52% 52.69% 3,451 84.79% 114.46%
1989 2,004 -0.41% 25.15% 2,419 74.85% 44.11% 3,437 82.32% 111.75%
1990 2,186 -6.94% 26.72% 2,453 88.42% 58.37% 3,399 100.44% 129.34%
1991 2,274 2.20% 27.69% 2,422 81.14% 59.43% 3,381 92.61% 128.81%
1992 2,360 1.87% 25.31% 2,679 79.02% 52.01% 3,497 80.58% 120.09%
1993 2,526 -2.91% 23.04% 2,958 80.07% 49.52% 3,716 71.71% 111.57%
1994 2,673 2.95% 25.05% 3,502 85.52% 53.42% 4,534 75.97% 112.46%
1995 2,924 -1.20% 25.00% 3,669 76.64% 48.85% 4,840 75.93% 115.92%
1996 3,113 3.51% 26.89% 3,928 72.37% 44.95% 4,958 72.17% 119.31%
1997 3,360 5.96% 27.39% 4,231 68.98% 46.54% 5,072 71.12% 128.74%
1998 3,456 -6.84% 28.71% 4,060 77.70% 67.53% 4,926 78.44% 141.01%
1999 3,570 -4.21% 33.49% 3,885 89.64% 71.77% 4,217 73.91% 79.44%
2000 3,435 -7.19% 36.43% 3,517 99.41% 82.11% 4,151 86.18% 108.52%
2001 3,176 -5.09% 33.45% 3,367 90.87% 79.67% 3,782 96.20% 125.99%
2002 2,978 -6.97% 33.43% 3,262 103.34% 84.27% 3,362 93.53% 118.68%
2003 3,152 -5.38% 32.12% 3,297 103.28% 84.27% 3,391 74.14% 82.24%
2004 3,341 2.02% 31.41% 3,439 85.63% 65.67% 3,620 52.67% 47.43%
2005 3,397 -1.94% 27.65% 3,571 82.08% 68.41% 3,819 52.10% 42.30%
2006 3,450 3.18% 26.79% 3,644 76.45% 66.09% 3,737 49.36% 41.14%
2007 3,415 3.69% 27.71% 3,574 69.99% 58.88% 3,614 52.46% 45.58%
All Years 71,392 1.63% 27.87% 81,687 84.85% 58.98% 105,697 79.60% 100.89%
EXHIBIT 1
Difference in means is calculated only for stocks present during both relevant months. S.E. denotes standard error, and p reports the p-value for the hypothesis that the valuation level is
the same between the two calendar months.
p
0.000
0.000
0.42%
0.43%
S.E.
B/M Ratio
V/P Ratio
BCD Mispricing
0.16%
0.22%
p
0.000
0.000
Test of Differences in Valuation Measure Between Calendar Months
S.E.
0.12%
0.06%
p
0.000
0.000
S.E.
Fair Valued group contains observations with BCD mispricing greater than -10% and less than or equal to 10%.
EXHIBIT 3
Seasonal Distribution of All Firms Based on BCD Mispricing
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
40.0%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
40.0%
EXHIBIT 4
Seasonal Distributions of BCD Mispricing, V/P and B/M Ratios by Size
Month Mean Med Std. D. Skew. Kurt. Mean Med Std. D. Skew. Kurt. Mean Med Std. D. Skew. Kurt.
Jul 0.17% -0.97% 29.14% 0.70 1.96 107.95% 89.15% 82.08% 4.07 25.52 99.63% 74.77% 110.20% 4.86 40.84
Aug -0.65% -2.08% 28.54% 0.66 1.85 110.96% 91.43% 83.03% 4.13 26.67 102.24% 75.64% 115.32% 4.73 38.78
Sep -0.91% -2.50% 28.48% 0.72 2.12 111.80% 92.22% 81.93% 3.86 23.61 102.65% 75.74% 116.85% 4.85 39.96
Oct -1.73% -3.25% 28.51% 0.69 1.89 117.30% 96.44% 86.35% 3.67 20.31 103.76% 76.03% 119.45% 4.90 41.17
Nov -2.61% -3.96% 28.30% 0.68 1.92 112.08% 92.89% 82.91% 3.90 24.17 106.69% 77.02% 126.74% 4.86 39.98
Dec -3.24% -4.46% 29.04% 0.75 2.24 103.01% 84.74% 80.73% 4.44 32.28 109.95% 79.63% 127.41% 4.78 37.16
Jan -0.12% -1.81% 28.91% 0.73 2.10 99.98% 83.82% 73.00% 3.78 23.58 101.14% 75.48% 113.86% 4.92 42.01
Feb 0.99% -0.63% 28.88% 0.66 1.86 103.58% 86.18% 75.27% 3.87 25.71 99.46% 74.81% 108.72% 4.55 35.63
Mar 0.96% -0.49% 29.08% 0.67 1.91 106.51% 88.61% 75.41% 3.72 24.06 99.03% 73.21% 114.13% 4.90 40.27
Apr 2.25% 0.56% 29.62% 0.58 1.51 107.95% 89.81% 77.30% 3.69 23.21 98.10% 72.16% 115.63% 5.26 48.50
May 3.58% 2.01% 30.07% 0.57 1.39 106.45% 88.07% 75.14% 3.47 20.22 96.84% 70.28% 113.00% 4.99 42.60
Jun 3.54% 1.56% 30.53% 0.65 1.72 106.16% 87.64% 78.80% 3.83 23.29 95.78% 69.54% 112.66% 5.45 50.98
All 0.19% -1.33% 29.09% 0.67 1.87 107.81% 89.25% 79.33% 3.87 24.39 101.27% 74.53% 116.16% 4.92 41.49
Diff. Diff. Diff.
Jun v. Dec 6.8% 3.1% -14.2%
Jan v. Dec 3.1% -3.0% -8.8%
Panel A of EXHIBIT 4: Small Cap
BCD Mispricing
V/P Ratio
B/M Ratio
Test of Differences in Valuation Measure Between Calendar Months
0.19%
0.000
0.79%
0.000
0.69%
0.000
0.35%
0.000
1.03%
0.000
0.79%
0.000
p
S.E.
p
S.E.
p
S.E.
Month Mean Med Std. D. Skew. Kurt. Mean Med Std. D. Skew. Kurt. Mean Med Std. D. Skew. Kurt.
Jul 4.39% 2.80% 24.43% 0.76 2.79 83.72% 73.04% 56.33% 4.27 34.40 81.03% 60.56% 103.69% 7.23 84.03
Aug 3.10% 1.72% 24.05% 0.78 3.14 86.97% 75.37% 60.71% 4.89 44.68 83.39% 61.17% 109.17% 7.00 78.76
Sep 2.38% 1.02% 23.94% 0.86 3.62 89.57% 77.06% 61.88% 4.44 36.77 84.68% 61.31% 114.87% 7.49 88.02
Oct 1.02% -0.15% 23.83% 0.65 2.47 93.77% 79.55% 66.82% 4.65 40.31 86.84% 62.57% 113.44% 6.88 75.86
Nov 0.53% -0.70% 23.57% 0.67 2.78 89.88% 76.19% 68.04% 5.37 50.46 87.14% 62.07% 119.10% 7.23 81.35
Dec 0.27% -0.86% 23.83% 0.66 2.74 81.74% 69.28% 63.17% 5.73 61.31 89.28% 64.57% 117.62% 6.92 74.86
Jan 2.26% 0.72% 24.03% 0.71 2.70 80.09% 69.26% 55.78% 5.06 54.80 85.66% 64.56% 110.69% 7.45 86.68
Feb 2.54% 0.96% 24.18% 0.77 3.02 85.47% 73.68% 59.35% 4.99 51.57 83.62% 62.83% 107.62% 7.49 86.53
Mar 1.74% 0.35% 23.90% 0.63 2.55 87.78% 75.70% 59.40% 4.41 37.56 83.93% 63.49% 104.65% 7.12 82.03
Apr 3.52% 2.41% 24.29% 0.62 2.47 87.43% 74.96% 60.20% 4.55 40.51 83.99% 62.95% 105.44% 7.15 82.35
May 4.37% 3.13% 23.98% 0.64 2.69 86.15% 74.18% 59.53% 4.46 37.80 83.45% 61.84% 107.72% 7.32 83.75
Jun 3.36% 2.15% 24.20% 0.65 2.76 86.36% 74.70% 59.03% 4.13 31.44 83.75% 61.46% 111.49% 7.45 83.60
All 2.46% 1.13% 24.02% 0.70 2.81 86.58% 74.41% 60.85% 4.75 43.47 84.73% 62.45% 110.46% 7.23 82.32
Diff. Diff. Diff.
Jun v. Dec 3.1% 4.6% -5.5%
Jan v. Dec 2.0% -1.7% -3.6%
0.000
0.23%
0.000
0.66%
0.000
0.69%
0.000
0.12%
0.000
0.51%
0.001
0.68%
S.E.
p
S.E.
p
S.E.
p
Panel B of EXHIBIT 4: Mid Cap
BCD Mispricing
V/P Ratio
B/M Ratio
Test of Differences in Valuation Measure Between Calendar Months
Month Mean Med Std. D. Skew. Kurt. Mean Med Std. D. Skew. Kurt. Mean Med Std. D. Skew. Kurt.
Jul 4.50% 2.94% 20.03% 0.80 3.49 77.34% 67.80% 51.20% 4.55 39.76 76.74% 57.81% 103.79% 7.65 91.01
Aug 3.40% 1.96% 19.29% 0.70 3.06 79.15% 69.13% 53.37% 5.13 49.86 77.62% 57.66% 106.37% 7.70 91.61
Sep 2.21% 0.88% 19.52% 0.79 3.39 81.17% 71.01% 55.19% 5.47 59.33 76.92% 56.99% 107.11% 7.80 93.56
Oct 1.69% 0.31% 20.01% 0.75 3.31 83.45% 72.50% 57.60% 4.89 44.73 78.41% 57.55% 106.34% 7.46 85.34
Nov 1.51% 0.14% 19.51% 0.63 2.76 80.20% 69.66% 54.24% 4.65 42.98 78.39% 57.86% 108.66% 7.69 90.41
Dec 1.22% -0.25% 19.67% 0.65 3.27 76.01% 63.32% 66.70% 6.54 62.10 78.16% 58.22% 106.83% 7.49 85.43
Jan 2.18% 0.90% 19.24% 0.66 3.19 73.84% 64.19% 54.43% 6.15 68.26 78.32% 59.99% 104.05% 7.52 87.97
Feb 2.42% 1.02% 19.52% 0.59 2.87 78.68% 68.44% 53.26% 5.12 50.39 76.50% 58.49% 100.82% 7.66 93.32
Mar 1.66% 0.43% 19.71% 0.62 3.35 79.33% 69.46% 52.47% 4.85 45.65 76.14% 58.09% 100.65% 7.70 93.43
Apr 3.15% 2.16% 19.76% 0.55 3.07 79.65% 69.21% 57.28% 5.67 56.37 77.16% 58.00% 105.95% 8.21 101.89
May 4.25% 2.88% 19.63% 0.74 3.74 79.46% 68.58% 57.68% 5.70 56.66 75.66% 57.36% 102.48% 8.15 102.18
Jun 3.55% 2.70% 19.82% 0.61 3.23 80.42% 69.14% 60.88% 6.33 69.84 75.06% 57.26% 103.33% 8.04 98.59
All 2.65% 1.34% 19.64% 0.68 3.23 79.06% 68.54% 56.19% 5.42 53.83 77.09% 57.94% 104.70% 7.76 92.90
Diff. Diff. Diff.
Jun v. Dec 2.3% 4.4% -3.1%
Jan v. Dec 1.0% -2.2% 0.2%
Small-, Mid-, and Large-cap groups are determined based on the market cap of each firm in June of year t and remain fixed for July-December of year t and for January to June of
year t+1. Difference in means is calculated only for stocks present during both relevant months. S.E. denotes standard error, and p reports the p-value for the hypothesis that the
valuation level is the same between the two calendar months.
0.11%
0.000
0.67%
0.001
0.75%
0.790
p
0.22%
0.000
0.80%
0.000
0.76%
0.000
S.E.
p
S.E.
p
S.E.
Panel C of EXHIBIT 4: Large Cap
BCD Mispricing
V/P Ratio
B/M Ratio
Test of Differences in Valuation Measure Between Calendar Months
EXHIBIT 5
Regression of Stock Returns on Valuation Measures
No. Specification Intercept BCD Mispr. V/P B/M Ln(Size) Adj- R2No. Obs.
1 All Months 2.171 -0.013 0.448 -0.075 -0.101 0.028 324
Whole Sample (3.22) (5.02) (3.68) (2.61) (2.60)
2 December 9.956 -0.043 0.330 0.069
-0.561 0.046 27
Whole Sample (5.35) (4.28) (0.72) (0.67) (6.64)
3 All Months 1.977 -0.012 0.637 -0.115 -0.082 0.016 324
Small Cap (1.78) (4.46) (4.18) (1.72) (0.95)
4 December 15.512 -0.048 -0.164 0.194
-0.988 0.035 27
Small Cap (4.41) (5.01) (0.31) (1.02) (3.37)
5 All Months 1.634 -0.016 0.432 -0.120 -0.059 0.026 324
Mid Cap (1.33) (5.09) (2.53) (2.00) (0.70)
6 December 8.504 -0.052 0.381 0.135
-0.499 0.048 27
Mid Cap (2.28) (4.29) (0.49) (0.41) (2.02)
7 All Months 1.314 -0.013 0.323 -0.193 -0.038 0.046 324
Large Cap (1.54) (3.07) (1.65) (1.76) (0.79)
8 December 8.394 -0.035 -0.372 0.744
-0.450 0.059 27
Large Cap (3.24) (2.23) (0.44) (1.49) (3.41)
The dependent variable is the future 1-month holding return. For each given month during the sample period, a cross-
sectional regression of future returns is run on BCD mispricing, V/P and B/M ratios, and natural log of company's
market capitalization. The table reports the time-series average of coefficient estimates from cross-sectional regressions
with a corresponding t-statistic (given in parentheses). Adj-R 2is the time-series average of the Adjusted R 2for the cross-
sectional regressions. The number of observations reported in the last column is the total number of monthly cross-
sectional regressions.
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