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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

9

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) : t≥0}. Our goal is to determine the time-tper-share

value, S(t), for each t≥0. 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

r−R(t)idt +σrdωr(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 +σxdωx(t) (A-4)

d G(t) = κghµ0

g−G(t)idt +σgdωg(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)]}dτ

(A-6)

where

ϕ(τ) = −λxτ+1

2

σ2

r

κ2

r"τ+1−e−2κrτ

2κr

−2(1 −e−κrτ)

κr#−κrµr+σxσrρr,x

κr"τ−1−e−κrτ

κr#

11

+1

2

σ2

g

κ2

g"τ+1−e−2κgτ

2κg

−2

κg

(1 −e−κgτ)#+κgµg+σxσgρg,x

κg"τ−1−e−κgτ

κg#

−σrσgρg,r

κrκg(τ−1

κr

(1 −e−κrτ)−1

κg

(1 −e−κgτ) + 1−e−(κr+κg)τ

κr+κg)(A-7)

%(τ) = 1−e−κrτ

κr

(A-8)

ϑ(τ) = 1−e−κgτ

κg

(A-9)

φ0(τ) = 1

2

σ2

r

κ2

r"τ+1−e−2κrτ

2κr

−2(1 −e−κrτ)

κr#,(A-10)

subject to the transversality conditions that

µr>1

2

σ2

r

κ2

r

(A-11)

µr−µg>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+i−1)]

(1+re)i(B-2)

=B(t) +

∞

X

i=1

Et[(ROE (t+i)−re)B(t+i−1)

(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+i−1)

(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+i−1)

(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+i−1), 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

EXHIBIT 2

Calendar-Month Distributions of BCD Mispricing, V/P and B/M Ratios

Month Mean Med Std. D. Skew. Kurt. Mean Med Std. D. Skew. Kurt. Mean Med Std. D. Skew. Kurt.

Jul 2.97% 1.83% 25.41% 0.69 2.91 83.42% 71.95% 56.44% 4.48 39.92 88.77% 62.26% 123.43% 6.94 75.93

Aug 1.94% 1.01% 24.96% 0.65 2.78 84.86% 73.25% 57.60% 4.65 42.82 90.47% 62.93% 125.85% 6.78 72.12

Sep 1.22% 0.08% 24.84% 0.71 2.95 85.69% 73.99% 58.30% 4.57 40.72 90.84% 62.91% 126.71% 6.79 72.43

Oct 0.23% -0.71% 24.97% 0.63 2.53 87.96% 75.27% 61.57% 4.49 37.80 92.88% 63.73% 127.71% 6.45 65.86

Nov -0.35% -1.28% 24.89% 0.61 2.62 87.61% 74.93% 61.87% 4.75 41.93 93.75% 63.79% 131.02% 6.41 63.88

Dec -0.72% -1.60% 25.26% 0.63 2.84 80.40% 68.85% 57.98% 5.32 52.43 94.67% 65.01% 132.45% 6.43 63.09

Jan 1.27% 0.15% 25.15% 0.65 2.76 78.89% 68.22% 54.07% 4.91 48.37 90.32% 64.55% 123.57% 6.96 74.68

Feb 1.89% 0.67% 25.33% 0.64 2.70 80.98% 69.86% 55.16% 4.72 44.67 88.81% 63.60% 121.37% 7.04 76.90

Mar 1.34% 0.19% 25.19% 0.61 2.70 82.01% 70.73% 55.29% 4.49 40.10 88.72% 63.53% 121.51% 7.05 76.57

Apr 3.02% 1.98% 25.64% 0.57 2.39 82.66% 70.99% 56.86% 4.64 41.53 88.87% 63.13% 122.47% 7.21 80.73

May 4.02% 2.79% 25.77% 0.58 2.49 82.53% 71.07% 56.26% 4.55 40.35 87.91% 62.16% 121.98% 7.19 79.87

Jun 3.50% 2.41% 26.25% 0.61 2.67 83.00% 71.01% 58.08% 4.77 43.68 88.28% 61.70% 126.02% 7.33 80.95

Diff. Diff. Diff.

Jun v. Dec 4.2% 2.6% -6.4%

Jan v. Dec 2.0% -1.5% -4.3%

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.