Decomposing Value∗

Joseph Gerakos Juhani T. Linnainmaa

September 19, 2012

Abstract

We show that all pricing-relevant variation in book-to-market ratio-sorted portfolios is due to

changes in the market value of equity. Although the remaining variation across these portfolios

explains comovement among stocks, it does not spread returns. The HML factor therefore consists

of a priced and unpriced risk component. Because every factor has just one price of risk, the three-

factor model gives the appearance of high risk-adjusted returns for strategies that covary negatively

with the unpriced component. This finding explains why the three-factor model appears to price

anomalies associated with earnings-to-price and cashflow-to-price ratios. When we adjust these

ratios to not overlap with the unpriced part of the HML factor, these anomalies resurface.

∗Joseph Gerakos and Juhani Linnainmaa are at University of Chicago Booth School of Business. We thank JohnCochrane, Peter Easton, Gene Fama, Mark Grinblatt, John Heaton, Matti Keloharju, Ralph Koijen, Martin Lettau, TobyMoskowitz, Lubos Pastor, and the seminar participants at the University of Chicago Booth School of Business and theChicago Quantitative Alliance fall 2012 conference for helpful discussions and Jan Schneemeier for research assistance.

1 Introduction

Value stocks are stocks with high BE/ME (book-to-market) ratios; those with low ratios are growth

stocks.1 The average return on value stocks exceeds that on growth stocks. The HML factor, for

example, averaged 4.6% per year (t-value = 3.4) from 1926 through 2011. This return differential is

central to empirical finance because of its role in the three-factor model, which is used to evaluate

performance and anomalies. Stocks are sorted into portfolios by their BE/ME ratios to measure the

value premium and construct the HML factor. We empirically investigate what information univariate

sorts on BE/ME ratios pick up in the data.

Following Daniel and Titman (2006) and Fama and French (2008), we start with two identities

that govern the time series evolution of book-to-market ratios: one describes changes in the book

value of equity and the other changes in the market value of equity. These identities can be further

decomposed into returns on book and market values of equity, and into terms describing (book and

market effects of) dividend and issuance policies. We depart from the prior research by observing that,

unless every component of the decomposition has the same variance and they are all uncorrelated, the

value premium is not the sum of these individual components. Because the value premium is based on

how the book-to-market ratio sorts stocks into portfolios, what matters is the extent to which these

components drive variation in book-to-market ratios.

When firms’ current book-to-market ratios are decomposed into their book-to-market ratios five

years earlier, and the changes in the book and market values of equity, these components contribute

52%, −15%, and 63% to the cross-sectional variation in today’s book-to-market ratios. The weights

1Value and growth can also be defined in terms of earnings-to-price, dividend-to-price, and cashflow-to-price ratios,or as a combination of multiple measures. See, for example, Lettau and Wachter (2007). We discuss the relation betweenBE/ME ratios and other price-scaled variables later in this paper.

1

on the annual changes in the market value of equity are initially high and decrease rapidly because

changes in book and market values of equity covary. The negative contribution of the book value of

equity indicates that when book values of equity increase then, on average, market values of equity

increase even faster, thereby resulting in lower book-to-market ratios in the cross-section.

Changes in the market value of equity also give book-to-market ratios predictive power over future

returns. In Fama-MacBeth regressions that examine individual components’ contributions, the changes

in the market value of equity are negative and highly significant. The changes in the book value of

equity are not. A further decomposition shows that the changes in the market value of equity predict

future returns because they pick up both past stock returns and recent net issuances. This finding

explains why five-year reversal is a good proxy for value, yet the return spread on BE/ME-sorted

portfolios is higher than that based on five-year reversals. Our results can be interpreted as suggesting

that book-to-market ratio-sorted portfolios spread returns because these portfolios keep track of the

accumulation of discount rate shocks2—sidestepping these portfolios’ “growth” and “value” labels, a

high BE/ME portfolio then has a high expected return simply because it is riskier than a low BE/ME

portfolio.

It is easy to demonstrate that the recent changes in the market value of equity subsume the power

of book-to-market ratios to explain variation in average returns. Regressions (1), (4), and (7) in

Table 1 report Fama-MacBeth regressions of monthly returns against size and book-to-market ratios

for all stocks, All but Micro, and Microcaps. These regressions are updated versions of those in Fama

and French (1992). Book-to-market ratio is statistically significant among both large and small stocks.

2If a firm’s discount rate (expected return) increases, its market value of equity decreases. The argument here is thata sort of stocks into portfolios by BE/ME picks up portfolios that have recently experienced a sequence of large discountrate shocks. Berk (1995) proposes that this same mechanism explains why firm size explains variation in average returns.

2

Regressions (2), (5), and (8) show that when the regressions also control for the five-year change in the

market value of equity, book-to-market ratios lose their significance. A disaggregation of the changes

in the market value of equity (regressions (3), (6), and (9)) pushes the slopes on the book-to-market

ratios even closer to zero. This paper explains (1) why the changes in the market value of equity

subsume the power of book-to-market ratios and (2) what consequences this result has for the HML

factor.

The variation in book-to-market ratios can be broken into two parts: a priced component, due

to the recent changes in the market value of equity, and an unpriced component, largely due to the

variation inherited from the distant past. The importance of this finding for asset pricing can be

illustrated in terms of the arbitrage pricing theory (Ross 1976). If the true model describing asset

returns is

rit − rft = β1,iF1t + β2,iF2t + · · · + εit, (1)

in which E(F1t) = λ1 > 0 and E(F2t) ≈ 0, then our results suggest that the book-to-market ratio as a

risk factor (such as HML) is akin to F1t+ F2t. This is a problem for the three-factor model—HML has

just one price of risk, λhml, but a strategy can covary with HML not only because β1,i in equation (1)

is high, but also because β2,i is high. The three-factor model can assign an incorrect expected return

to a portfolio or strategy because it does not distinguish with which component a strategy covaries.

We show this unpriced component poses significant problems. First, if portfolios are sorted by the

(approximate) unpriced part of the book-to-market ratio, then the average return and CAPM alpha

on the high-minus-low strategy are close to zero. But, the three-factor model alpha on this hedge

portfolio is −47 basis points per month with a t-value of −3.5, and the regression’s R2 is 34%. This

decomposition of HML into its priced and unpriced components uses information only from the current

3

cross-section. Hence, one can circumvent the three-factor model by combining any strategy with a

strategy that exploits the unpriced component. The unpriced component also presents a problem

for using the 25 size- and BE/ME-sorted portfolios as test assets to evaluate asset pricing models.

A model can price these portfolios not only by covarying with their priced components but also by

covarying with their unpriced components.

The unpriced part of the HML factor is also important in practice. The data, for example,

marginally reject the three-factor model for its ability to price BE/ME-sorted portfolios because these

portfolios themselves covary with HML’s unpriced component. When the unpriced component of book-

to-market ratios is removed and HML is reconstituted (so that it has a uniform price of risk), the new

model prices these portfolios perfectly. Our alternative versions of value here are based directly on the

changes in the market value of equity. They are attractive alternatives to the BE/ME-based factor

because they share its source of returns (and nothing more), but do not require accounting-based

variables.

Nevertheless, the original three-factor model is marginally better than our alternatives at pricing

portfolios sorted by earnings-to-price and cashflow-to-price ratios. We find an intuitive resolution

to this puzzling result. Similar to BE/ME, other commonly considered price-scaled variables inherit

significant variation from distant changes in the market values of equity. The unpriced part of the HML

factor therefore covaries with portfolios sorted by such variables. That is, in terms of equation (1),

β2,i increases uniformly when portfolios are formed by sorting on any price-scaled variable. The HML

factor appears to explain the E/P and C/P anomalies because it assigns its one price of risk to this

covariation. But, the model misses their actual source of returns. When we adjust the E/P and

C/P ratios so that they only reflect information accrued into these ratios over the past five years,

4

the anomalies reappear with sizable three-factor model alphas: 50 basis points per month on an E/P

strategy (t-value = 3.06) and 32 basis points per month on a C/P strategy (t-value = 2.06). These

estimates represent returns on strategies that only use information from the cross-section at the time

stocks are sorted into portfolios.

All of our tests are designed to study the mechanics of the univariate sorts that measure the

value premium and underlie the HML factor. Hence, our tests do not directly address the overall

information content of BE/ME ratios. An example illustrates this distinction. Fama and French

(2008) show that the changes in the book value of equity are informative about future returns in a

multivariate regression that also includes the lagged book-to-market ratio, the changes in the market

value of equity, and net issuances. Their result suggests that, for the book value of equity to spread

returns, one has to sort stocks into portfolios simultaneously in four dimensions: (1) the change in the

book value of equity minus the change in the market value of equity (the slopes on these two variables

are of the same magnitude but with opposite signs in Fama and French’s regression), (2) the lagged

book-to-market ratio, (3) net issuances, and (4) market capitalization. The fourth dimension (size)

is necessary because the Fama and French result only holds for a sample that excludes the smallest

two deciles of stocks. Such a multivariate sort is relevant neither for the value premium nor the HML,

which are based on univariate BE/ME sorts.

Our results do not imply that the changes in fundamentals have no information about expected

returns. In fact, Fama and French (2008) show the opposite. But, when the BE/ME ratio is used

by itself in univariate sorts, all such additional information either remains hidden out of sight or

generates unnecessarily noisy sorts, or leads to an asset pricing factor with a non-uniform price of

risk. In contrast, a sort on the changes in the market value of equity spreads future returns more than

5

BE/ME-sorted portfolios, and a factor based on these same changes performs better than HML in asset

pricing regressions. Moreover, we show that even if one conditions on the other components of BE/ME,

the trading strategy based on the change in the book value of equity (which is profitable in isolation), is

spanned by the market-value-of-equity and net-issuance trading strategies. Our results do not preclude

that fundamentals other than the change in the book value of equity have information about expected

returns. For example, Novy-Marx (2012b) shows that a factor based purely on fundamentals, gross

profitability, spreads future returns even after controlling for BE/ME.

The other components of book-to-market ratios could contain information about long-run expected

returns. Such a mechanism is, however, not relevant for our conclusions, because portfolios are sorted

annually when measuring the value premium or constructing the HML factor. Therefore, any dif-

ferences in long-run expected returns have only limited time to materialize. Moreover, Cohen, Polk,

and Vuolteenaho (2003) estimate that differences in expected 15-year stock returns account for only

a quarter of the cross-sectional variation in today’s book-to-market ratios.

We find that the changes in the market value of equity provide a sufficient statistic for the predictive

power of BE/ME-sorted portfolios. This result has several important implications. First, it provides a

powerful testable restriction on theories of value. Whatever the source of the value premium is within

a model, this premium should disappear after controlling for the changes in the market value of equity.

For example, if as in Zhang (2005) the argument is that the value premium is due to irreversibility of

investment, this attribute must be subsumed, within the model, by past changes in the market value

of equity. If “value” within a model is something that exists independent of these changes, such a

model is inconsistent with the data. Moreover, if a theory models the behavior of book-to-market

ratios, these ratios should decompose into priced and unpriced components.

6

Second, our results suggest that, because of the flaw in the HML factor, the three- and four-factor

model alphas do not represent risk-adjusted returns. This finding is important for understanding

anomalies. Which of the anomalies that are known to “beat” the three-factor model would remain

anomalies if we replace the value factor with one that has a uniform price of risk? And, are there other

anomalies, similar to the E/P and C/P anomalies, that we thought were subsumed by the three-factor

model but that are actually distinct from BE/ME ratios? Third, the problem with the HML factor is

relevant for performance evaluation. Studies that search for skill among money managers use the three-

and four-factor models extensively. The concern then is that if managers employ strategies that covary

with the unpriced component of HML, their estimated alphas are biased away from their true alphas.

In particular, if the average covariation between mutual fund returns and this unpriced component

is positive, then the distribution of estimated alphas over all managers does not display enough skill.

Fourth, the three-factor model is important for corporations and thus for overall economic activity

when it is used to estimate costs of equity (Fama and French 1997). The non-uniform price of risk of

the HML factor can cause firms to mismeasure their cost of equity, which in turn would lead them to

undertake either too many or too few projects.

The rest of the paper is organized as follows. Section 2 introduces the data. Section 3 decomposes

cross-sectional variation in book-to-market ratios. Section 4 uses Fama-MacBeth regressions to evalu-

ate the extent to which different components of BE/ME ratios help or hurt their ability to spread out

future returns. Section 5 demonstrates that strategies can circumvent the three-factor model and give

the appearance of high risk-adjusted returns. Section 6 demonstrates that alternative versions of the

HML factor have the same good pricing properties as the actual HML (while not inheriting its flaw),

and that E/P and C/P can be resurrected as anomalies by not letting HML’s unpriced part covary

7

with them. Section 7 examines trading strategies and the information content of the other BE/ME

components. Section 8 concludes.

2 Data

We take stock returns from CRSP and accounting data from Compustat. Our sample starts with all

firms traded on NYSE, Amex, and Nasdaq. For these firms, we calculate the book value of equity

(shareholder equity, plus balance sheet deferred taxes, plus balance sheet investment tax credits, minus

preferred stock). We set missing values of balance sheet deferred taxes and investment tax credit equal

to zero. To calculate the value of preferred stock, we set it equal to the redemption value if available,

or else the liquidation value, or the carrying value. If shareholders equity is missing, we set it equal

to value of common equity if available, or total assets minus total liabilities. We then use the Davis,

Fama, and French (2000) book values of equity from Ken French’s website to fill in missing values of

the book value of equity. Because we require the return on equity for many of our tests, we start our

sample in January 1963. We end it in December 2011.

For BE/ME, we use the market value of equity as per the fiscal year end and calculate it as the

CRSP month end share price times the Compustat shares outstanding if available, or else the CRSP

shares outstanding. We follow prior research and lag book-to-market ratios by at least six months so

that companies have released their annual financial statements. For example, if a firm’s fiscal year

ends in December, this December information will become available in the asset pricing tests at the

end of June. When we calculate the BE/ME, E/P, C/P, and D/P ratios, we align the numerator and

denominator, and all the components of the decomposition at the same point in time. For example,

8

the five-year change in the market value of equity is the five-year change up to the date when the

book-to-market ratio is computed. Thus, our decompositions are exact.3

3 Variance Decomposition of Book-to-Market Ratios

The book-to-market ratio can be decomposed using the following algebraic identity:

bmt ≡ bmt−k +

t∑

τ=t−k+1

dbeτ −

t∑

τ=t−k+1

dmeτ , (2)

where bmt is the log-book-to-market ratio at time t, dbet = ln(BEt/BEt−1) is the change in the book

value of equity, and dmet = ln(MEt/MEt−1) is the change in the market value of equity. This identity

implies that in a regression of bmt against the components on the right-hand side of equation (2), the

slopes on bmt−k and dbeτ ’s would equal (identically) “1” and those on dmeτ ’s would equal “−1.”

Components of bmt can, however, differ significantly from each other in their contribution to the

variation of current book-to-market ratios. This variation is what ultimately matters when we sort

stocks by book-to-market ratios to measure value and growth or to construct the HML factor. Two

time-series features of dbeτ and dmeτ are important. First, because stock returns are more volatile

than accounting variables, the dmeτ s drive more of the cross-sectional variation in book-to-market

ratios. Second, the market and book values of equity are not independent of each other: an increase

in the book value of equity, generally through higher earnings, usually reflects in market valuations.

Moreover, changes in one component can be reflected in the others either at a lead or lag. The

3We trim BE/ME ratios at 0.005 and 0.995 levels each month to discard outliers. The literature alternatively winsorizesor trims outlying BE/ME ratios. We trim because winsorization breaks the exact decomposition identities. See Famaand French (2008, footnote 1).

9

analogy here to modern portfolio choice theory is clear. Today’s book-to-market ratio is a “portfolio”

of the changes in the book and market values of equity. The question is how the covariance structure

between the change terms plays out in the formation of the book-to-market ratios, and how much of

the component-specific variation is diversified away.

3.1 High-level Decomposition

We use a variance decomposition to explore the significance of the book-to-market ratio components.

Our cross-sectional decomposition starts from the identity that a variable’s covariance with itself equals

its variance:

var(bmt) = cov(bmt, bmt−k +

t∑

τ=t−k+1

dbeτ −

t∑

τ=t−k+1

dmeτ )

= cov(bmt, bmt−k) +

t∑

τ=t−k+1

cov(bmt, dbeτ ) +

t∑

τ=t−k+1

cov(bmt,−dmeτ ). (3)

Dividing both sides of this equation through by var(bmt) gives each term’s percentage contribution to

the variance of today’s book-to-market ratios.

Our use of the term “variance decomposition” is consistent with its usage in prior research.4

Our decompositions, however, measure the covariation between today’s book-to-market ratios and

their components, and can therefore be negative. These covariances have the same interpretation as

the analysis of Fama and French (1995), who, for example, show that value firms experienced low

profitability for the prior five years. Our estimates can be inverted to ask what the typical value or

growth firm looks like. Below we find that the covariance between today’s book-to-market ratios and

4See, for example, Cochrane (1992).

10

the changes in the book values of equity is negative. This result restates Fama and French’s finding—a

firm with a decrease in the book value of equity is typically a value firm. The connection to the value

premium and the HML factor is that these covariances tell us what type of firms end up in different

portfolios when sorted by their book-to-market ratios. In particular, if a component’s covariance with

today’s book-to-market ratios is zero, then any information contained in this component is lost in the

univariate portfolio sorts, because this component does not vary across portfolios.5

Table 2 Panel A presents the variance-decomposition estimates. We estimate the covariances of

equation (3) with year fixed effects. After computing the variance decomposition, we bootstrap the

data (in annual blocks) to estimate standard errors. A firm is used to estimate the covariances between

years t and t− k if necessary information for the firm is available for both years. Thus, as the lag of

the decomposition grows, data from fewer firms are available. Hence, the amount of variation due to

the cumulative changes in the book and market values of equity (shown in the last two columns) does

not exactly match the sums of the one-year estimates (shown in the “one-year changes” columns).

However, the differences are empirically small.

The main result here is that most of the variation in book-to-market ratios arises from lagged book-

to-market ratios and changes in the market value of equity. In the one-year decomposition (the first

row), 81.84% of the variation is due to the prior year’s book-to-market ratio, 23.1% is due to (minus)

the change in the market value of equity, and the rest, −4.93%, is due to the change in the book value

of equity. The change in the book value of equity enters with the “wrong” sign. The negative sign

indicates that when the book value of equity increases, the market value of equity generally increases

5This argument is related to Lewellen, Nagel, and Shanken’s (2010) critique of using 25 size- and BE/ME-sortedportfolios to test asset pricing models. Their argument is that a sort of stocks in these two dimensions imposes a rigidfactor structure—whatever asset pricing factors there were in stock returns prior to sorting stocks into these portfoliosare now largely gone.

11

even more thereby resulting in lower book-to-market ratios in the cross-section.6

The importance of one-year changes in the book and market values of equity decreases from year

to year. This result may seem unexpected. If a random variable is decomposed into k independent

components with equal variances, then each component contributes 1/kth of the overall variance. The

implication of this observation for the results in Table 2 is that the importance of one-year changes

decreases because the changes in the book and market values of equity are significantly autocorrelated

or cross-serially correlated. If the market value of equity increases in year t− k, Table 2 suggests that

this increase is often offset by increases in the book value of equity in years t − k, t − k + 1, . . . , or,

alternatively, by decreases in the market value of equity in years t− k + 1 ≤ τ ≤ t− 1.

The cross- and serial-correlation structure between the dbeτ and dmeτ terms is such that changes

in market values of equity older than five years contribute negligible amounts of variation to today’s

book-to-market ratios. At this horizon, over half of the variation is due to the old book-to-market

ratios, 63% is due to the cumulative changes in market value of equity, and the difference (−15%) is

due to cumulative changes in the book value of equity.

Although the role of past book-to-market ratios initially decreases very fast, this decrease slows

down significantly after just a few years. Even at the ten-year mark, 41.7% of the variation in today’s

book-to-market ratios is attributable to bmt−10. At the 15-year mark (not shown), the contribution of

the past book-to-market ratio is 36%. These estimates suggest that a part of book-to-market ratios is

equivalent to a fixed firm characteristic—at least a part of whether a firm is more like a “value” firm

6The covariance term cov(bmt, dbeτ ) can be written as the variance of dbeτ and its covariances with all other terms ofthe decomposition. The results here suggest that the sum of these other covariances is large enough to more than offsetdbeτ ’s own variance. These results are similar to that observed in the price-dividend ratio decompositions where futurereturns appear to account for more than 100% of the variation in the price-dividend ratios (Cochrane 2005, p. 400).

12

or a “growth” firm appears to go back to the firm’s beginning.7

These decomposition results address the question of what variation book-to-market ratio-sorted

portfolios pick up from the data. The algebraic decomposition could lead one to believe that, because

bmt is the cumulation of past changes in the book and market values, every part of the history plays

as prominent a role. Table 2 rejects this view. The book-to-market ratios observed in the cross-section

today are almost entirely due to what these ratios were in the past plus an adjustment for the changes

in the market value of equity during the intermittent years. Thus, a sort on today’s book-to-market

ratios mostly sorts stocks by their old book-to-market ratios and the changes in the market value of

equity.

3.2 Low-level decomposition

Changes in the book and market value of equity can further be decomposed. The change in the book

value of equity arises from (1) the log-return on equity (roe) with tax adjustments, (2) dividends,

which we measure by the difference in the with- and without-dividends log-returns on equity (rbdiv,τ ),

and (3) the remainder term, which represents net issuance (rbiss,τ ) and is the difference between the

log-change in the book value of equity and the without-dividends log-return on equity. Similarly, the

change in the market value of equity is traceable to (1) total stock return, rτ , (2) the dividend yield,

rdiv,τ , measured as the difference in with- and without-dividends stock returns, and (3) the remainder

term, which represents net issuance (riss,τ ) and is the difference between the log-change in the market

7Lemmon, Roberts, and Zender (2008) find a similar effect for capital structure. They show that the majority ofcross-sectional variation in leverage is explained by firm fixed effects.

13

value of equity and the without-dividends stock return. The identities are

dbeτ ≡ roeτ − rbdiv,τ + rbiss,τ , (4)

−dmeτ ≡ −rτ + rdiv,τ − riss,τ , (5)

in which the signs keep track of each term’s algebraic contribution to the book-to-market ratio. These

identities indicate that

dbmτ = dbeτ − dmeτ = (roeτ − rτ )−(rbdiv,τ − rdiv,τ

)+(rbiss,τ − riss,τ

),

which gives the appearance that the change in the book-to-market ratio is approximately equal to the

difference in returns on book and market equity, roeτ and rτ . This approximation, however, is only

reasonable if a firm’s pre-issuance (or pre-dividend) BE/ME ratio is close to one. Otherwise equity

issuances (ISS), for example, pull BE/ME ratios towards one.

Table 2 Panel B shows the variance-decomposition estimates in which the current book-to-market

ratio is decomposed into the old book-to-market ratio, the cumulative sums of the three book-value-

of-equity terms, and the cumulative sums of the three market-value-of-equity terms. Each row in the

table sums to 100% because it accounts for every component of today’s book-to-market ratio. The

main result in Panel B is that stock returns drive in large part the covariance between the change in

the market value of equity and book-to-market ratios. The dividend and net issuance terms are small

in comparison.

If dividends and issuances were to wash out, then the covariances of the respective terms with

today’s book-to-market ratios in Panel B would sum to zero. For the dividend term, the sum of the

14

ten-year covariances, however, is 12.0% + 5.1% = 17.1%. That is, higher book and market dividend

yields in the past covary positively with today’s book-to-market ratios. Issuances do not wash out

either. On the book side, the net issuance term is the component responsible for Panel A’s result that

the one-year change in the book value of equity covaries negatively with bmt—the behavior of this one

component more than overshadows the positive contributions of the return-on-equity and dividend

terms.

Table 2 Panel C shows the correlations between bmt, bmt−5, and the five-year cumulative changes

in the components of bmt. Consistent with Daniel and Titman (2006), the changes in the book and

market value of equity are highly correlated (0.636) over five years. With respect to returns, the

five-year change in the market value of equity is more highly correlated with the five-year total stock

return than the five-year change in the book value of equity is with the five-year return on equity

(0.926 and 0.608).

4 Book-to-Market Ratio in Fama-MacBeth Regressions

The book-to-market ratio correlates with future returns because some or all of its components correlate

with those returns. A regression of stock returns against a firm’s log-market capitalization (met) and

log-book-to-market ratio (bmt) constrains the regression slopes on the BE/ME components to equal

15

each other:

rj,t+1 = b0 + b1mej,t + b2bmj,t + ej,t+1 (6)

= b0 + b1mej,t + b2bmj,t−k + b2

(t∑

τ=t−k+1

dbeτ

)+ b2

(−

t∑

τ=t−k+1

dmeτ

)+ ej,t+1

= b0 + b1mej,t + b2bmj,t−k + b2

(t∑

τ=t−k+1

roeτ

)+ b2

(−

t∑

τ=t−k+1

rbdiv,τ

)+ b2

(t∑

τ=t−k+1

rbiss,τ

)

+ b2

(−

t∑

τ=t−k+1

rτ

)+ b2

(t∑

τ=t−k+1

rdiv,τ

)+ b2

(−

t∑

τ=t−k+1

riss,τ

)+ ej,t+1,

where the second row replaces today’s book-to-market ratio with its high-level decomposition, and the

third row uses the low-level decomposition.

Regression (6) can be used to assess which components of the book-to-market ratio contribute to

its ability to predict future returns in the absence of all other conditioning information but size. We

do so by first estimating a baseline specification, which includes only the log-size and today’s book-

to-market ratio. We then estimate univariate regressions of these first-stage residuals against each

component. This second-stage regression measures how close each component’s optimal slope (bj) is

to the common slope (b∗) when all other components’ slopes remain fixed at b∗. As an illustration,

suppose the slope on the book-to-market ratio is 0.25 in the baseline specification and that the slope

from regressing the residuals against component j is −0.2. This estimate suggests that, if all other

components’ slopes are kept at 0.25, a slope of bj = 0.25 − 0.2 = 0.05 on component j maximizes the

amount of variance the model explains.

We first test whether the component’s optimal slope differs from zero, H0 : bj = 0. In the example

above, the test is whether bj = 0.05 is statistically different from zero. Second, we test whether

16

the component’s optimal slope differs from the common slope, H0 : bj = b∗. In the example above,

the test now is whether −0.2 differs from zero. These two sets of t-values measure whether, at

a first approximation, one would be better off excluding a variable from the book-to-market ratio

when we do not condition on any additional information. Component j’s optimal slope maximizes

the regression’s predictive power. Therefore, any deviation from this value hurts the model. If for

a specific component the zero benchmark is statistically closer than the common-slope benchmark,

then the regression suggests that the book-to-market ratio performs worse than it would with the

component removed from the ratio.

Table 3 shows the baseline regressions (Panel A) as well as various decompositions (Panels B

through D). Panel B decomposes the book-to-market ratio into three parts: the book-to-market ratio

five years ago, and the cumulative log-changes in the book and market values of equity. The three

univariate regressions summarized here show that the optimal slopes on these three components are

very different: 0.133 (bmt−k), 0.087 (∑

dbeτ ), and 0.439 (−∑

dmeτ ). The first column of t-values

shows that while the change in the market value of equity is very significant (t-value of 4.98), the

other two are not. The second column of t-values indicates that while the slope on∑

dmeτ exceeds

the common slope of 0.261 from Panel A, the other components’ optimal slopes are closer to zero.

The significant t-values here (−2.80 and −4.04) suggest that the book-to-market ratio would perform

better in Fama-MacBeth regressions after excluding one (or possibly both) of these components from

the ratio.8

8Asness and Frazzini (2011) find that adjusting book-to-market ratios to use more timely price data increases thevalue premium. Their results are consistent with those in Table 3. Our Fama-MacBeth regressions indicate that theBE/ME ratio would perform better without the book value of equity because its optimal slope is closer to zero than tothe common slope. Asness and Frazzini (2011) increase the value premium by using more recent market values of equity,thereby reducing the role that recent changes in the book value of equity play in the BE/ME ratios. In fact, one obtainsa similar result not only by using more timely records of market values of equity but also by using older records of bookvalue of equity.

17

Panel C disaggregates log-changes in the book and market values of equity by year. The left half

of the table uses a five-year decomposition and the right half a ten-year decomposition. The message

here is similar to that in Panel B. None of the one-year changes in the book value of equity are

significantly different from zero and, except in one case, the estimated slope is statistically closer to

zero than to the common slope. These computations suggest that today’s book-to-market ratio, as

an explanatory variable in Fama-MacBeth regressions only with log-size, does not derive its power

from either old book-to-market ratios or the changes in the book value of equity. The slopes on

the disaggregated changes in the market value of equity are significantly above both the zero and

common-slope benchmarks.

The ten-year decomposition uses the same sample of firms as the five-year decomposition. For a

firm less than 10 years old, bmt−k is the firm’s first book-to-market observation, and those changes in

the book and market values of equity that extend beyond the firm’s first appearance in the data are set

to zero. The decomposition identity thus holds firm by firm. The results here show that the additional

changes in the book value of equity, except in one case, are not statistically significant but some of

the market value changes are. Hence, these regressions do not rule out the possibility that there is

information about future returns in old changes in the market value of equity. This conclusion, however,

mostly pertains to the Fama-MacBeth regressions. Table 2 Panel A shows that each additional lag

after the fifth lag contributes relatively little to the cross-sectional variation in book-to-market ratios,

and so this older information is not picked up to a significant extent by portfolio sorts.

Table 3 Panel D presents a low-level decomposition in which the changes in the book and market

values of equity are further separated into three components each using equations (4) and (5). The

results on the book side are easy to summarize. The optimal slopes on the return on equity are positive

18

but insignificant and the slopes on book dividend yields, while large in magnitude, are noisy and not

significant either way. The first two optimal slopes on the book issuance variables are significant but in

the wrong direction, and therefore work against the book-to-market ratio in predicting future returns.

These net issuance variables are thus also responsible for the slightly negative slopes on the first two

changes in the book value of equity in Panel B.

With respect to the market-side variables, stock market returns and net issuances contribute power

to the book-to-market ratio. In fact, the slopes on all net-issuance variables are significantly above

the common slope. The market-side dividend yield variables, similar to their book-side counterparts,

are thoroughly insignificant.

5 Repackaging Value

5.1 Fama-MacBeth Regressions and Portfolio Sorts

The variance decompositions and Fama-MacBeth regressions raise the question of how well the sub-

components of value would perform as value “factors.” In the decompositions, half of the variation

in the book-to-market ratio is due to the changes in the market value of equity over the previous

five years and the rest is inherited from old book-to-market ratios. The Fama-MacBeth regressions

suggest that only the changes in the market value of equity help this ratio predict returns. The old

book-to-market ratios do not.

Table 4 Panel A reports Fama-MacBeth regressions that use alternative repackagings of value.

Panel B sorts stocks into deciles based on the NYSE breakpoints for the distributions of these alter-

19

native variables. We use five subcomponents of bmt as our alternative repackagings of value: bmt−5

is the book-to-market ratio from five years ago; −∑

dmeτ is minus the change in the market value

of equity over the past five years;∑

dbeτ is the change in the book value of equity over the past five

years; −∑

rτ is minus the five-year stock returns; and bmt is the part of bmt that is due to the past

five years of market value of equity changes.

We construct bmt by first computing for each monthly cross-section the covariances between cur-

rent book-to-market ratios and Xt = [dmet · · · dmet−5]. We denote the column vector of these five

covariances by Ct. We then compute the covariance matrix of Xt, denoted by ΣXt , and construct a

vector of weights wt = (Σxt )

−1Ct. The alternative repackaging bmt is then bmt = Xtwt. By construc-

tion, the covariances of this variable with the changes in the market value of equity are identical to

what they are for the book-to-market ratio itself, that is, cov(bmt, dmet−k) ≡ cov(bmt, dmet−k) for

0 ≤ k ≤ 4.9 This construction therefore discards any information not contained in the changes in the

market value of equity, and only uses these five variables in the same proportions as they are present

in bmt.10 We only use the information from each cross-section to construct this variable, and so it

uses no past or future information.

The construction bmt is not the same as the projection of today’s book-to-market ratios against

the changes in the market value of equity. Such a projection maximizes the amount of variation in

bmts these variables can explain. By contrast, the method used here “explains” variation in bmts

only to the same extent as they do when they occur naturally within the book-to-market ratios. This

9We created a similar variable based on the five-year stock returns and two-year net issuances because the low-leveldecomposition suggests that these variables may be ultimately responsible for the significance of the changes in themarket value of equity. Results using this alternative measure are very similar to bmt across all tests.

10Because bmt is constructed to replicate the variance due to the changes in the market values of equity, it is notthe same as summing the past five years of changes in the market value of equity. For such a sum, in general,cov(bmt, dmet−k) 6= cov(

∑0

τ=t−4dmeτ , dmet−k).

20

matching implies that when we sort on bmt, we sort on the changes in the market value of equity to

the same extent as we do when we sort on bmt. The difference is that bmt throws out variation we

suspect does not spread returns.11

The first row of Table 4 Panel A reports the baseline regression that only includes the market

capitalization and the current book-to-market ratio as the explanatory variables. We report two

specifications for all repackaged value variables, one with and the other without the current book-to-

market ratio.

The old book-to-market ratio is not very informative. It is insignificant by itself and significantly

negative when used in conjunction with the current book-to-market ratio. This result suggests that the

old BE/ME ratio teases out fresher information from the current book-to-market ratio. The remaining

rows of Table 4 Panel A examine what this fresh information might be. The cumulative log-change

in the market value of equity is very significant by itself with a t-value of 5.7 and it retains most

of its significance when the regression also controls for the current book-to-market ratio. Whatever

information there is in bmt, most of it is embedded into these log-changes. The next row shows that

the same is not true for the changes in the book value of equity. The sign on this variable is negative

because, as Table 2 shows, a firm is more likely to be a “growth” firm after experiencing positive

changes in the book value of equity. When the regression includes both these changes and the current

book-to-market ratio, they are of approximately equal importance.

The last two rows examine what drives the importance of the market value of equity changes.

11One significant difference between the priced component, bmt, and the residual component, bmt − bmt, is that theformer does not vary significantly by industry. The R2 from regressing bmt on indicator variables for the 49 Fama andFrench industries (with year fixed effects) is 1.23%. By contrast, the R2 from regressing the residual component on thesesame indicator variables is 11.96%. Thus, industries explain a significant amount of the variation in book-to-marketratios that does not appear to be priced, but that explains comovement in returns.

21

Total stock returns, dividend yield, and issuance activity comprise these changes. A comparison of

−∑

dmeτ and −∑

rτ shows that past stock returns by themselves are not as powerful predictors

of future returns as are changes in the market value of equity over the same period. The last row

suggests, consistent with Table 3, that the additional information in the changes in market value is

due to net issuances. When the regression contains both the current book-to-market ratio and the

same ratio stripped out of all the variation that is not driven by the five-year changes in the market

value of equity, bmt is statistically and economically insignificant.12 This result is in contrast to Fama

and French (1996) who show that BE/ME ratios subsume the five-year reversal in returns found by

De Bondt and Thaler (1985). We show the opposite: changes in the market value of equity subsume

BE/ME ratios.

Table 4 Panel B shows average excess returns, CAPM alphas, and Fama and French (1993) three-

factor model alphas for portfolios sorted by alternative value factors. The results here are similar to

the Fama-MacBeth regressions. The portfolios sorted on the five-year changes in the market value

of equity spread out returns more than the current book-to-market ratio. The excess-return spread

between the top and bottom decile is 50 basis points (t-value = 2.48) per month for bmt but 56 basis

points (t-value = 2.85) for the market value of equity changes. A CAPM adjustment increases the

spreads to 52 and 65 basis points. The stripped-down version of bmt, which uses dmeτ s in the same

proportions they are present in bmt, generates similar return spreads.

The average monthly return (35 basis points) and CAPM alpha (39 basis points) are lower for the

12Table 1 shows that, in a multivariate Fama-MacBeth regression of returns on the year-by-year changes in marketvalue of equity, the slopes are all statistically significant but slightly different. This finding explains why the currentbook-to-market ratio has a higher t-value (although it is statistically insignificant) in the −

∑dme regression than in

the bmt regression. Because bmt and −∑

dme load differently on the changes in the market values of equity, bmt helps

the first of these regressions to span the optimal slopes. Because bmt, in contrast, loads in exactly the same way on thechanges in the market value of equity as bmt, bmt is not useful even for this purpose in Table 4’s last regression.

22

high-minus-low portfolio based on −∑

rτ than they are for that based on −∑

dmeτ . The largest

return differences are in the tails of the distribution. Although the returns on deciles 2 and 9 are

fairly similar for these two sorts, those on deciles 1 and 10 are not. This behavior is expected because

the difference between the two sorts is (mostly) about net issuances. The unconditional covariation

between net issuances and today’s book-to-market ratios is modest (see Table 2 Panel B) because

issuances are relatively rare. Conditional on issuing equity, however, a firm’s movement in the book-

to-market ratio can be substantial. Table 4 Panel B shows indirectly that firms with recent net

issuances are mostly in deciles 1 or 10 when firms are sorted by their BE/ME ratios. The results here

explain why five-year reversal (De Bondt and Thaler 1985) is a good proxy for value13, yet the return

spread on BE/ME-sorted portfolios is higher than that based on five-year reversals. The difference is

that a sort on BE/ME ratios gets an additional return boost from net issuances.

An important result in Panel B is that the explanatory power of the three-factor model is signifi-

cantly higher for the current book-to-market ratio-sorted hedge portfolio (73.8%) than what it is for

alternative value factors such as bmt (42.1%). Moreover, the explanatory power for the bmt−5 -sorted

portfolio is 36.2%. The implication here, together with the negative alpha on the high-minus-low

hedge portfolio for bmt, is that HML captures some systematic variation in returns that is not priced,

or that at least has a lower price than the rest of HML.

5.2 Cheating the Value Factor

Table 5 tests the hypothesis that there is unpriced, but systematic, variation in HML. We begin by

creating a variable that picks up only the variation in bmt that is not already part of bmt. This

13See, for example, Fama and French (1996) and Asness, Moskowitz, and Pedersen (2012).

23

variable, defined as the difference bmet = bmt − bmt, does not covary with the past five-year change

in the market value of equity. For example, cov(bmet , dmet) = cov(bmt, dmet) − cov(bmt, dmet), but

this difference is zero by the definition of how we constructed bmt. We sort stocks into deciles based

on this residual component of bmt and estimate both CAPM and three-factor model regressions for

returns on these deciles as well as the return earned by the high-minus-low portfolio.

Excess returns earned by the different deciles are similar. The return spread between the highest

and the lowest decile is only −7 basis points per month and insignificant with a t-value of −0.39. The

residual component of bmt also does not covary significantly with the market. The market betas on

different deciles range from 0.89 to 1.04, but the beta on the high-minus-low strategy is only 0.13

(t-value = 3.64). The CAPM alpha is thus −12 basis points per month with a t-value of −0.70.

For the three-factor model, the deciles differ markedly. They correlate significantly with SMB and,

in particular, HML. The loadings increase almost monotonically from the lowest to the highest decile

and the high-minus-low portfolio has loadings of 0.30 (t-value = 6.41) and 0.74 (t-value = 15.22) on

SMB and HML. The three-factor model alpha on a strategy that purchases “value” stocks and sells

“growth” stocks, as classified according to bmet , earns a monthly risk-adjusted alpha of −49 basis

points. This estimate has a t-value of −3.48. The high R2 in the three-factor model regression for the

high-minus-low portfolio (33.6%) implies that the unpriced component is an economically important

part of the overall variation in HML.

The results in Table 5 suggest that a value factor constructed from the book-to-market ratios is

a problematic variable for risk adjustment. The problem is that bmt has two types of systematic

variation, but only the systematic variation related to the changes in the market value of equity is

priced. If a strategy, such as the one highlighted here, correlates only with the unpriced part, HML

24

still assigns the λhml price of risk to this strategy thereby giving it a low risk-adjusted alpha. Hence,

the three-factor model alpha is not a fair representation of a strategy’s risk-adjusted performance.

Here the seemingly profitable strategy is to purchase “growth” stocks and to short “value” stocks,

where growth and value are defined using the variation in bmts not related to five-year change in the

market value of equity.

5.3 BE/ME-Sorted Portfolios and Asset Pricing Tests

The 25 Fama-French portfolios are used extensively to test asset pricing models.14 If the unpriced

component varies across BE/ME-sorted portfolios, then the 25 Fama-French portfolios can generate

misleading inferences when used as test assets. Table 6 examines this issue and measures how prevalent

the priced and unpriced components are within each of the 25 portfolios by regressing their returns

on the market factor, SMB factor, and the priced and unpriced components of the HML factor. We

construct the (priced) bmt-based HML factor in the same way as the standard HML factor except that

we replace bmt with the approximate priced component, bmt. We construct the (unpriced) bmet -based

HML factor by replacing bmt with the approximate unpriced component, bmet .

The estimates in Table 6 show that although the loadings on the priced component increase steadily

as we move from low BE/ME portfolios to high BE/ME portfolios, so do the loadings on the unpriced

component. For example, in the highest size quintile the loading on the priced component increases

from BE/ME quintile 1’s −0.12 (t-value = −6.0) to quintile 5’s 0.44 (t-value = 10.05), and the loading

on the unpriced component increases monotonically from −0.50 (t-value = −20.5) to 0.68 (t-value =

13.10).

14See, for example, Lewellen, Nagel, and Shanken (2010, p. 175) for a list of studies.

25

The unpriced-component loadings are problematic. This issue is the flip side of the results presented

in Section 5.2 on how the unpriced component within the HML factor generates non-zero alphas for

portfolios that load on this component. Table 6 suggests that an asset pricing model can price the 25

Fama-French portfolios not only because it covaries correctly with these portfolios’ priced components

but also because it covaries with their unpriced components. Hence, BE/ME-sorted portfolios, such

as the 25 Fama-French portfolios, can be ill-suited for testing asset pricing models for a reason distinct

from that detailed in Lewellen, Nagel, and Shanken (2010). A better set of test assets would be those

sorted only by the priced component of BE/ME ratios or by the five-year change in the market value

of equity.

6 Asset Pricing Implications

6.1 Explaining Anomalies

Fama and French (1996) show that the three-factor model provides a good description of average

returns on portfolios sorted by, among other things, E/P, C/P, and D/P. Our decomposition results

suggest that HML prices these anomalies because BE/ME and these other price-scaled variables pick

up the same variation in past changes in the market value of equity. If so, alternative HML variables,

based either on the cumulative change in the market value of equity over the past five years (∑

dme)

or on the approximate priced component of BE/ME (bmt), should price these anomalies just as well.15

15One benefit of using dmeτs to reconstitute the HML factor is that it does not require using any accounting infor-mation, which is sometimes either missing or unreliable. Because Compustat, for example, does not contain reliablebook-value-of-equity records for the pre-1963 period, a project that wants to use the longest possible sample periodhas to fill in the missing values using the hand-collected data of Davis, Fama, and French (2000). Moreover, differentapproximations have to be used depending on the detail of information available (see, for example, Novy-Marx (2012a,footnote 1) for a hierarchy of the common approximations). If we use dmeτs, we also do not have to take a stance onthe interpretation of negative book values of equity (that are usually trimmed), or the meaning of extreme outliers that

26

The other possibility, which we discuss in detail below, is that there is something special about these

other price-scaled variables (that is, they are separate anomalies), but that the three-factor model

nevertheless matches their expected returns.

Table 7 examines the proposition that all price-scaled variables get their return spreads from the

past changes in market prices of equity. We follow Fama and French (1996) and sort stocks into

portfolios based on size, BE/ME, E/P, C/P, D/P, and jointly on size and BE/ME (the 25 Fama-

French portfolios). The first row of each block reports the average excess returns (and t-values in

parentheses) on each of the ten portfolios based on the underlying sort variable. (For the 25 Fama-

French portfolios, we show the estimates for the stocks in the highest and smallest size quintiles.) The

row labeled bmt uses the three-factor model’s standard HML factor (based on bmt), the row labeled

∑dme sorts stocks based on the changes in the market value of equity to create an alternative HML

factor, and the last row uses the approximate priced component of bmt, bmt, for this purpose. The

rightmost column reports the Gibbons, Ross, and Shanken (1989) test statistic and its associated

p-value.

A comparison of the models with the standard and alternative HML factors in Table 7 supports

the view that HML gets its pricing ability by picking up variation in the changes in the market value of

equity. The results for the ten BE/ME-sorted portfolios offer the strongest evidence. The three-factor

model does not price these portfolios as well as the two alternative models. The joint pricing errors

are insignificant for models with∑

dme- and bmt-based HML factors, but closer to significance with

a p-value of 0.052 for the model that uses the standard HML factor. The differences in test statistics

do not arise from noise: the alpha estimates are approximately as precise across the three models.

are either trimmed or winsorized.

27

Instead, the standard three-factor model fails because it predicts that the extreme value (growth)

stocks should have even higher (lower) returns than they actually have in the data.

The problem with the three-factor model is that a significant fraction of the variation in the book-

to-market ratios comes from what these ratios were five years ago but, as shown in Table 4 Panel B,

this variation does not generate any return differences. Because the standard HML factor does not

distinguish whether something covaries with its priced or unpriced component, it assigns too high an

expected return to the extreme value stocks and too low an expected return to the extreme growth

stocks. The alternative HML factors are free of this old component, and so the covariances with these

factors are proper measures of the quantity of risk in each portfolio.

The three-factor model cannot price the size-sorted portfolios. The issue is that the very smallest

stocks do not earn high enough average returns and the biggest stocks earn too high average returns

compared to what SMB predicts. In contrast, the other two models based on alternative HML factors

have p-values associated with the GRS statistics that are above the 0.05 threshold (0.103 for∑

dme

and 0.123 for bmt). In the 25 Fama-French portfolios, the largest pricing error belongs to the small-

growth corner of these portfolios. The three models predict that the returns on these portfolios should

be significantly higher than what they are in the data, resulting in estimated alphas between −0.66

and −0.57 percent per month.

The results on the E/P-, C/P-, and D/P-sorted portfolios also support the view that BE/ME

prices portfolios because it captures variation in the changes in the market value of equity. All three

models price these portfolios according to the GRS test statistics. However, the puzzling finding here

is that the standard three-factor model does a slightly better job in pushing the estimated alphas

towards zero. The two highest C/P deciles, for example, have positive alphas (significant for the ninth

28

decile and marginally significant for the highest decile) when the asset pricing model is based on the

two alternative HML factors, but these alphas are insignificant in the standard three-factor model.

Although the overall differences between the models here are economically rather small, this finding

points to an interesting puzzle. How can it be that the original three-factor model can price E/P and

C/P portfolios even better than the alternative model, even though the alternative model is a better

description of the average returns earned by the BE/ME-sorted portfolios? We now turn to variance

decompositions of E/P and C/P ratios and show that the three-factor model better prices these

portfolios because they correlate with the unpriced part of the HML factor.

6.2 Decomposing Other Price-Scaled Variables

Daniel and Titman (2006) note that any price-scaled variable can be decomposed in the same way as

the book-to-market ratio. For example, the log-earnings-to-price ratio can be decomposed as

ept = ept−k +t∑

τ=t−k+1

deτ −t∑

τ=t−k+1

dmeτ (7)

where deτ is the log-change in earnings in year τ . This decomposition requires that earnings are

positive over the span of the decomposition. A similar decomposition can be written for other price-

scaled variables, such as cash flow-to-price and dividend-to-price ratios. Size can also be decomposed

in the same way:

met = met−k +t∑

τ=t−k+1

dmeτ . (8)

This decomposition states that a firm’s current size is the firm’s size k years ago, plus the cumulative

change in the market value of equity. Equation (8) implies that size reflects (very) long-run reversals.

29

Hence, size has a role identical to that played by book-to-market ratios: it picks up reversals in market

capitalizations. The difference between size and value is that size, after controlling for value, is about

even longer-term reversals.16

Table 8 Panel A shows five-year variance decompositions for log-size and the BE/ME, E/P, C/P,

and D/P ratios. In the size decomposition, the loadings on the changes in the market value of equity are

close to each other because annual returns exhibit only weak negative serial correlation. The pattern

here is thus very different from the one observed for the book-to-market ratio, in which the loadings

on these market value changes decrease rapidly because of the covariances between the changes in

fundamentals and market values.

The E/P, C/P, and D/P decompositions indicate that changes in the market value of equity play a

smaller role than for the book-to-market ratio. Both E/P and C/P, for example, reflect to a significant

extent the most recent changes in earnings and cash flows. The reason is that while the book value

of equity keeps track of the accumulation of earnings and therefore usually changes slowly, both E/P

and C/P can change abruptly in response to earnings or cash-flow shocks.

Panel B use the same method as Table 3 to evaluate the extent to which the different components

of price-scaled variables explain cross-sectional variation in returns in absence of other conditioning

information. The baseline specification regresses returns against log-size and the current price-scaled

variable. The bottom rows estimate univariate regressions of the first-stage residuals against each

component. The first column reports each component’s optimal slope (bj) when all other components’

slopes are held at the common slope b∗. The second column is the deviation between the optimal and

16A Fama-MacBeth regression that also controls for short-term reversal, momentum, and value forces size to be aboutvariation in the changes in the market value of equity that is largely orthogonal to the changes in the market value ofequity over the past five or six years.

30

common slope. The E/P and C/P regressions show that both variables are significant in explaining

variation in returns. The D/P ratio enters with a positive sign but is insignificant. The “fundamentals”

of E/P and C/P behave differently than those of BE/ME in the analysis of deviations. The optimal

slopes on the changes in earnings and cash flows are always significant and also statistically closer to

the common-slope benchmark than to zero. This analysis suggests that E/P and C/P ratios derive

some of their predictive power from the fundamentals, which is consistent with Ball (1978) who argues

that in a misspecified asset pricing model earnings can proxy for dividends and therefore correlate

with differences in expected returns.17

It is perhaps surprising that the three-factor model prices E/P- and C/P-sorted portfolios as

well as it does in Table 7. The issue is that both of these ratios appear to contain information

about expected returns that does not appear (at least directly) in the book-to-market ratios. This

seeming contradiction could be resolved if E/P and C/P correlate positively with the unpriced part

of the BE/ME ratios, thereby increasing the covariance between HML and the anomalies. Because

regressions assign the same price of risk to every unit of covariation, the three-factor model would

therefore predict that the highest E/P- and C/P-sorted portfolios should earn very high returns. Put

differently, it could be that the three-factor model gets the numbers right but the story wrong: it gets

the alphas to zero, not because it understands what the risks in E/P and C/P are, but because it

thinks the risks have something to do with the variation inherited mostly from the distant past.

Table 8 Panel C supports this view. It shows that the E/P, C/P, and D/P ratios covary not only

with the (approximate) priced part of book-to-market ratios, but also with the unpriced part. E/P

and C/P, for example, covary significantly more with the unpriced part than with the priced part.

17Berk (1995) makes the same argument for size. Namely, if the asset pricing model is misspecified, differences inmarket values of equity will correlate with differences in expected returns.

31

The fact that these price-scaled variables covary positively with the (approximate) unpriced part of

book-to-market ratio is not surprising. The decompositions in Panel A suggest that if we were to

extend the decompositions backwards in time, we would find significant loadings on the changes in

the market value of equity in years t− 5, t− 6, and so forth. This pattern is common across all price-

scaled variables, including BE/ME, and this commonality shows up in Panel C as positive covariances

between E/P, C/P, D/P, and the unpriced part of bmt.

6.3 Resurrecting Anomalies

If E/P and C/P correlate significantly with the unpriced part of HML, then the three-factor model

does not explain these anomalies for what they truly are. The anomalies might be resurrected by

modifying the price-scaled variables so that they no longer covary as much with the (approximate)

unpriced part of the HML factor. The three-factor should then no longer be able to price them.

Table 9 evaluates the extent to which the three-factor model prices the E/P, C/P, and D/P anoma-

lies through the unpriced-risk channel. The first three columns show average excess returns, CAPM

alphas, and three-factor model alphas for portfolios sorted using each ratio. These results mirror those

shown in Table 7.18 The returns on the extreme portfolios sorted by these variables are well described

by the three-factor model, and so the high-minus-low portfolio earns excess returns close to zero: 7

basis points per month for E/P-, 6 basis points per month for C/P-, and −15 basis points per month

for D/P-sorted high-minus-low portfolios. Not one of these high-minus-low portfolio returns is close

to being statistically significant.

18We use the E/P, C/P, and D/P return series provided by Ken French in Table 7. In Table 9, because we want tocompare portfolios that are based on unadjusted and adjusted price-scaled variables, we compute both sorts ourselvesusing the CRSP/Compustat data. The numbers in the first three columns of Table 9 are therefore not identical to thecorresponding numbers in Table 7.

32

The second set of columns sorts stocks into portfolios based on the information in E/P, C/P, and

D/P ratios, but does so based on the variation accrued into these variables over the past five years. At

the end of each June when stocks are assigned to portfolios, we first estimate the covariances between

each price-scaled variable and the five changes in fundamentals (such as det, . . . , det−4) and the five

changes in the market value of equity. These are the same covariances that underlie the variance

decomposition results shown in Table 8 Panel A except that we here carry out this decomposition

separately for each cross-section without using any lagged or forward-looking information. We use

these covariances to create a linear combination of the ten variables (the five changes in fundamentals

and the five changes in market values) such that its covariances with the ten variables are the same

as what they are for the original price-scaled variable. For the earnings-to-price ratio, for example, we

solve for weights such that cov(ept, dft−k) = cov(ept, dft−k) and cov(ept, dmet−k) = cov(ept, dmet−k)

for 0 ≤ k ≤ 4. These variables, which we call the adjusted price-scaled ratios, capture the information

accrued into the original ratios over the past five years but are free of the “old” information that

covaries significantly with the unpriced part of HML.19

The results for the portfolios sorted by adjusted variables are striking. The three-factor model

no longer prices portfolios sorted by E/P and C/P. Consider the portfolios sorted by E/P. Whereas

the three-factor model alphas for the lowest and highest unadjusted-E/P deciles were −1 basis point

(t-value = −0.13) and 6 basis points (t-value = 0.58) per month, the alphas on these two portfolios

diverge significantly when stocks are sorted into portfolios based on the adjusted E/P. The three-factor

model alphas for the low and high portfolios are now −22 basis points (t-value = −2.29) and 28 basis

19This computation results in a variable that differs from the one that would be obtained by subtracting a five-year-oldprice-scaled variable from the current price-scaled variable. The results in Table 8 show that such a differencing wouldcreate an oddly behaving variable. In the case of E/P and C/P, for example, such differencing would create a variablethat loads significantly on both year t and t−5 changes in fundamentals. The covariance-matching method, by contrast,properly purges variation that is associated with the “old” price-scaled variable.

33

points (t-value = 2.70) per month. Hence, a high-minus-low portfolio based on the adjusted E/P ratio

earns a per-month three-factor model alpha of 50 basis points with a t-value of 3.06. The return on

the high-minus-low portfolio based on C/P also increases significantly, from 3 basis points (t-value =

0.25) to 32 basis points (t-value = 2.06) after adjustment.20

Although the return on the D/P strategy is affected as well, the return on the high-minus-low

portfolio turns from being moderately negative (−15 basis points) to moderately positive (18 basis

points) but remains statistically insignificant. The reason is that, as Table 8 shows, D/P behaves very

similarly to BE/ME in terms of not only the sources of variation (mostly the changes in the market

value of equity) but also the source of profits (these same changes). The three-factor model prices

these portfolios because their returns arise from the same source as those of the priced part of the

HML factor.

Table 9’s results are important because they reflect returns on two sets of strategies that are equally

implementable. We do not use any forward-looking information to construct the adjusted versions of

E/P, C/P, and D/P, and even though we construct the portfolios by matching covariances, we do

so only to impose some structure and to facilitate fair comparisons. The key observation is that the

three-factor model cannot explain the “true” returns of E/P and C/P anomalies. The three-factor

model appears to explain these anomalies because BE/ME, E/P, and C/P all reflect some variation in

the very old changes in the market value of equity. Because of this shared variation, the HML factor

covaries positively with the high unadjusted-E/P portfolio, and it takes credit for this portfolio’s high

return. However, when E/P (or C/P) is adjusted so that it is only about recent changes in earnings

20Novy-Marx (2012b) finds that profitability, as measured by the ratio of revenues minus cost of goods sold to totalassets, complements BE/ME in predicting the cross-section of returns. Profitability is highly correlated with earningsand cash flows, suggesting that profitability and adjusted E/P and C/P ratios share some commonality in terms of thevariation they pick up in the data.

34

(or cashflows) and market values of equity, the three-factor model fails.

We did not adjust the price-scaled variables so that they should be as difficult as possible for the

three-factor model to price. That is, we did not try to construct a component that is unique to these

ratios. In fact, the adjusted price-scaled variables load in the exact same way on the changes in the

market value of equity as the unadjusted ratios load. As expected, the three-factor model properly

prices this component. Hence, the three-factor model alphas are significantly lower than the CAPM

alphas in Table 9.

7 Trading Strategies and the Information Content of Book-to-Market

Ratio Components

Up to this point we examine what happens if one unconditionally sorts stocks based on their book-

to-market ratios. For these univariate sorts, we show that pricing relevant information comes from

the change in the market value of equity. Following Fama and French (2008) we now consider the

information content in book-to-market ratios when conditioning on the other components. Using

multivariate Fama-MacBeth regressions, Fama and French (2008) find that for one-, three-, and five-

year decompositions, the log-changes in the book value of equity and market value of equity are of

approximately equal importance in their ABM sample. In their sample of Microcaps, the changes in

the market value of equity are more informative than the changes in the book value of equity.

Table 10 Panel A replicates the main results of Fama and French (2008). Columns (1) and (3)

correspond to the high-level decomposition results presented in Table 3, estimated here as multivariate

regressions. In this specification, the changes in the market value of equity contain more information

35

about future returns than the changes in the book value of equity. Columns (2) and (4) add the five-

year net issuance as an additional regressor and they represent Fama and French’s (2008) specification.

The inclusion of net issuances alters the slopes on both the changes in the book and market values

of equity because net issuances are part of both variables. For the ABM sample, the slopes on the

changes in the book and market value of equity are now close to each other, while for the Microcaps

sample the changes in market value of equity contain more information than the changes in the book

value of equity.

We next consider the profitability of trading strategies implied by the Fama and French (2008)

specification. Specifically, we consider how well one can do by simultaneously trading on the compo-

nents of book-to-market ratios. As per Fama (1976, ch. 9), the slope coefficients from a multivariate

regression can be interpreted as the returns on zero-investment portfolios. If Xt is an N ×K matrix

of the intercept and the explanatory variables then the rows of a K × N matrix W ′

t = (X ′

tXt)−1X ′

t

represent the weights on K different strategies. The self-financing strategy on row k > 1 has a unit

loading on the kth variable and zero loadings on all other variables. The time series of the Fama-

MacBeth coefficient estimates (summarized in columns (2) and (4) of Panel A) are returns on these

zero-investment strategies. For example, the period t + 1 slope estimate on the change in the book

value of equity is, in column (2)’s regression, the fourth element of vector (X ′

tXt)−1X ′

trt+1 = W ′

trt+1.

These are implementable strategies—only the characteristics are required to compute the portfolio

weights Wt. In these tests we focus on three strategies that reflect different components of the book-

to-market ratio: the changes in the book value of equity, changes in the market value of equity, and

net issuances. We denote the returns on these strategies by Rbook value, Rmarket value, and Rnet issuance.

Table 10 Panel B examines the profitability of these trading strategies for the ABM and Microcaps

36

samples. Columns (1), (3), and (5) present the annualized percentage returns of the three zero-

investment strategies. The returns on all three strategies are positive and significant.21 Columns (2),

(4), and (6) test whether each strategy is inside the span of the other strategies. We regress the returns

of each strategy on the returns of the other two strategies. A statistically insignificant intercept in

these regressions indicates that the other two strategies span the strategy and the strategy does not

therefore add to the investment opportunity set.22 A significant intercept suggests that the strategy

adds to the investment opportunity set beyond what could be attained using the other two strategies.

Column (2) presents the spanning tests for the change in the book value of equity. For the ABM

sample, the intercept is negative and insignificant, implying that the market-value-of-equity and net-

issuance trading strategies span the trading strategy based on the change in book value. Hence, an

investor who has access to all three strategies would ignore the strategy based on the changes in book

value. For the Microcaps sample, the intercept in the Rbook value regression is negative and significant.

This negative coefficient implies that, if an investor has access to all three strategies, it would be

optimal to trade in the opposite direction of the change in the book value of equity. Columns (4)

and (6) present similar spanning regressions for strategies based on the change in the market value

of equity and net issuances. In contrast with the change in the book value, the intercepts for both

strategies are positive and significant in both the ABM and Microcaps samples.

Why is the change in book value of equity important in the Fama-MacBeth regressions (Panel A)

but unimportant (or has the “wrong” sign) in the spanning regressions (Panel B)? The Fama-MacBeth

regressions are multivariate regressions and therefore provide marginal effects. The positive and sig-

nificant slope on the change in the book value of equity shows that if the other regressors are held

21These returns can be directly calculated by multiplying the coefficients in columns (2) and (4) of Panel A by 12.22See, for example, Pastor and Stambaugh (2003) and Novy-Marx (2012a) for discussions on spanning tests.

37

constant, stocks that experience an increase in the book value of equity on average have higher returns

in the cross-section than those that experience a decline in the book value of equity.

The portfolio interpretation of the marginal effects is that the trading strategies in Panel B are

strategies that trade on the independent variation in each variable. The Rbook value strategy, for

example, trades on that part of the change in the book value of equity that is orthogonal to all other

regressors. But, even though this strategy trades on variation that is unique to the change in the book

value of equity, the returns on this strategy can be correlated with those earned by the other strategies.

The returns on the strategies are correlated if the slope estimates in the multivariate Fama-MacBeth

regressions are correlated. Indeed, the spanning regressions presented in Panel B demonstrate that

the “everything else” is rarely held fixed when the book value of equity changes.

The returns on the strategy that trades on the independent variation in the change in the book

value of equity are correlated with the returns on the other two strategies. An investor who trades

on this variation usually earns positive returns in those states of the world in which he would also

have earned positive returns by trading on the other two strategies. The insignificant intercept for the

ABM sample and the significant and negative intercept in the Microcaps sample demonstrate that the

correlations between these strategies are so high, and that the other strategies are so profitable, that

they wipe out (from the investor’s perspective) the gains that would be unique to the independent

variation in the change in the book value of equity. In fact, the result in the Microcaps sample indicates

that, if anything, an investor would improve his portfolio’s overall Sharpe ratio if he were to combine

the reverse of the change-in-the-book-value-of-equity strategy with the other two strategies.23

23Another way to reconcile the results presented in Panel B with those presented in Panel A is to examine the changein the adjusted R2 when the change in book value and the change in market value are removed from the regressionpresented in column (2) of Panel A. If the change in book market value is removed, the adjusted R2 drops from 3.88%to 2.91%. By contrast, the adjusted R2 only drops from 3.88% to 3.56% if the change in the book value of equity isremoved. That is, even though the coefficient on the change in book value is statistically significant it does not explain

38

8 Conclusions

We decompose the variance in book-to-market ratios to investigate the origins of the value premium.

We find that the priced variation in BE/ME-sorted portfolios is mostly about recent changes in

the market values of equity. A further decomposition tracks the relevant information down to total

stock returns and net issuances. The remaining variation in book-to-market ratios, while picking up

comovement between stocks, does not spread out returns. This variation is in large part about how

the market values of equity changed in the distant past, and this pricing-irrelevant variation is thus

shared by many price-scaled variables.

Our finding about the priced and unpriced components of the HML factor has two consequences.

First, we show that strategies can circumvent the three-factor model. The model gives the impression

of high risk-adjusted returns because it assigns the same price of risk to any strategy, independent

of whether the strategy covaries with the priced or unpriced component of HML. Second, we show

that the three-factor model inadvertently prices the E/P and C/P anomalies. The returns on these

strategies covary positively with the unpriced component of the HML factor and the three-factor model

therefore pushes their expected returns in the right direction. Indeed, when these ratios are adjusted

in real-time to not depend (as much) on how the market values of equity changed in the distant past,

the three-factor model no longer explains the high average returns of E/P- and C/P-based strategies.

The unpriced component of HML is a problem to the three-factor model because it distorts infer-

ences in unpredictable ways. An intuitive modification to this model would be to replace HML with

a factor that has all of its pricing-relevant information—but nothing more. A natural candidate is to

sort stocks based on the cumulative change in the market value of equity over a five-year period from

much variation in returns.

39

year t− 6 to t− 1. This factor is attractive because it does not require any accounting information (so

absent, noisy, limited, or delayed data are not issues) and it has a uniform price of risk.24 The unpriced

component also presents a problem in using BE/ME-sorted portfolios (such as the 25 Fama-French

portfolios) to test asset pricing models. Because these portfolios’ loadings on both the priced and

unpriced component increase as BE/ME increases, an asset pricing model can price these portfolios

for the “wrong” reason.

A replacement of the BE/ME-based HML factor with an alternative factor would prompt the re-

evaluation of some of the key questions in empirical asset pricing. First, which of the anomalies that

are known to “beat” the three-factor model25 remain anomalies if we modify the value factor? If some

supposed anomalies correlate negatively with the unpriced component of HML, the model gives the

mirage of high risk-adjusted returns. Second, similar to our results on E/P- and C/P-based anomalies,

there may be other anomalies that the three-factor model explains through the unpriced-risk channel.

Third, the performance evaluation literature extensively uses the three- and four-factor models

(Carhart 1997) to search for skill among money managers.26 It is likely that, in the large cross-section

of managers, some follow strategies that covary with the unpriced component of HML. If we adjust

the value factor in the three-factor model, estimated alphas will change. Do these changes wash out,

or would the results lead us to think differently about skill? The three-factor model is also used to

estimate firms’ costs of equity (Fama and French 1997). To what extent does covariance with the

24The proposal here is that if we want to adjust for differences in “risk” that are captured by differences in book-to-market ratios, an HML factor based on the changes in the market value of equity is an attractive alternative. However,because the changes in the market value of equity explain differences in future returns through two different channels—return reversals and the net-issuance effect—it is not obvious that these two channels should be kept bundled in an assetpricing model. In fact, the spanning tests in Table 10 Panel B suggest that these two channels are distinct. This bundlingis thus attractive only as far as the goal is to maximize consistency with prior research.

25See, for example, Stambaugh, Yu, and Yuan (2012) for a list of 11 such anomalies.26Kosowski, Timmermann, Wermers, and White (2006), Barras, Scaillet, and Wermers (2010), Fama and French

(2010), and Linnainmaa (2012), among others, measure the prevalence of skill among mutual fund managers.

40

unpriced component of HML lead to estimates of the cost of capital that distort investment decisions?

Our results on the origins of the value premium are of separate interest. In addition to providing

a clear testable prediction that challenges all theories of value—do the changes in the market value

of equity subsume the value attribute within a model?27—they also point to other challenges. For

example, if we have a model of value, is the size of the value premium predictable in the same way

it is in the data? Moreover, to match the data, a theory that models the behavior of book-to-market

ratios should produce ratios that decompose into priced and unpriced components. These empirical

facts about value impose structure and restrictions, thereby guiding the debate about whether value

is about mispricing or risk.

27We are not aware of a model that conforms to this prediction. The class of models based on differences in durationbetween growth and value stocks, explored in papers such as Lettau and Wachter (2007, 2011), and Croce, Lettau, andLudvigson (2010), may hold promise. However, when we simulate data from the Lettau and Wachter (2007) modelwith stochastic share processes using the paper’s parametrization, neither the changes in the market value of equity norpast returns correlate with the value premium. In the deterministic-share process case, the changes in the market valueof equity are connected to the value premium, but only because these changes keep track of where a firm lies in itsshare-process cycle.

41

REFERENCES

Asness, C. and A. Frazzini (2011). The devil in HML’s details. Working paper, AQR Capital Man-

agement.

Asness, C., T. J. Moskowitz, and L. H. Pedersen (2012). Value and momentum “everywhere”.

Working paper, University of Chicago.

Ball, R. (1978). Anomalies in relationships between securities yields and yield-surrogates. Journal

of Financial Economics 6, 103–126.

Barras, L., O. Scaillet, and R. Wermers (2010). False discoveries in mutual fund performance:

Measuring luck in estimated alphas. Journal of Finance 65 (1), 179–216.

Berk, J. B. (1995). A critique of size-related anomalies. Review of Financial Studies 8 (2), 275–286.

Carhart, M. M. (1997). On persistence in mutual fund performance. Journal of Finance 52 (1),

57–82.

Cochrane, J. H. (1992). Explaining the variance of price-dividend ratios. Review of Financial Stud-

ies 5 (2), 243–280.

Cochrane, J. H. (2005). Asset Pricing, Revised Edition. Princeton, New Jersey: Princeton University

Press.

Cohen, R. B., C. Polk, and T. Vuolteenaho (2003). The value spread. Journal of Finance 58 (2),

609–641.

Croce, M., M. Lettau, and S. Ludvigson (2010). Investor information, long-run risk, and the duration

of risky cash flows. UC Berkeley working paper.

Daniel, K. and S. Titman (2006). Market reactions to tangible and intangible information. Journal

42

of Finance 61 (4), 1605–1643.

Davis, J. L., E. F. Fama, and K. French (2000). Characteristics, covariances, and average returns:

1929 to 1997. Journal of Finance 55 (1), 389–406.

De Bondt, W. F. M. and R. H. Thaler (1985). Does the stock market overreact? Journal of Fi-

nance 40 (3), 379–395.

Fama, E. F. (1976). Foundations of Finance. New York, NY: Basic Books.

Fama, E. F. and K. French (1992). The cross-section of expected stock returns. Journal of Fi-

nance 47 (2), 427–465.

Fama, E. F. and K. French (1993). Common risk factors in the returns of stocks and bonds. Journal

of Financial Economics 33 (1), 3–56.

Fama, E. F. and K. French (1995). Size and book-to-market factors in earnings and returns. Journal

of Finance 50 (1), 131–155.

Fama, E. F. and K. French (1996). Multifactor explanations of asset pricing anomalies. Journal of

Finance 51 (1), 55–84.

Fama, E. F. and K. French (1997). Industry costs of equity. Journal of Financial Economics 43 (2),

153–193.

Fama, E. F. and K. French (2008). Average returns, B/M, and share issues. Journal of Fi-

nance 63 (6), 2971–2995.

Fama, E. F. and K. French (2010). Luck versus skill in the cross section of mutual fund returns.

Journal of Finance 65 (5), 1915–1947.

Gibbons, M., S. A. Ross, and J. Shanken (1989). A test of the efficiency of a given portfolio.

43

Econometrica 57 (5), 1121–1152.

Kosowski, R., A. Timmermann, R. Wermers, and H. White (2006). Can mutual fund “stars” really

pick stocks? New evidence from a bootstrap analysis. Journal of Finance 61 (6), 2551–2595.

Lemmon, M. L., M. R. Roberts, and J. F. Zender (2008). Back to the beginning: Persistence and

the cross-section of corporate capital structure. Journal of Finance 63 (4), 1575–1608.

Lettau, M. and J. A. Wachter (2007). Why is long-horizon equity less risky? A duration-based

explanation of the value premium. Journal of Finance 62 (1), 55–92.

Lettau, M. and J. A. Wachter (2011). The term structures of equity and interest rates. Journal of

Financial Economics 101 (1), 90–113.

Lewellen, J., S. Nagel, and J. Shanken (2010). A skeptical appraisal of asset pricing tests. Journal

of Financial Economics 96 (2), 175–194.

Linnainmaa, J. (2012). Reverse survivorship bias. Journal of Finance. Forthcoming.

Novy-Marx, R. (2012a). Is momentum really momentum? Journal of Financial Economics 103 (3),

429–453.

Novy-Marx, R. (2012b). The other side of value: The gross profitability premium. University of

Rochester working paper.

Pastor, L. and R. F. Stambaugh (2003). Liquidity risk and expected stock returns. Journal of

Political Economy 111, 642–685.

Ross, S. A. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory 13 (3),

341–360.

Stambaugh, R. F., J. Yu, and Y. Yuan (2012). The short of it: Investor sentiment and anomalies.

44

Journal of Financial Economics 104 (2), 288–320.

Zhang, L. (2005). The value premium. Journal of Finance 60 (1), 67–103.

45

Table 1: Average returns, B/M, and changes in the market value of equity

This table shows average Fama-MacBeth regression slopes and their t-values from cross-section regressions topredict monthly returns for all individual stocks, ABM stocks (above the 20th percentile of market capitalizationfor NYSE stocks), and Microcaps (below the 20th NYSE market capitalization percentile). The regressions,estimated monthly, use variables that are updated six months after the firm’s fiscal year end. The regressionsare estimated using data from July 1968 through December 2011. In these regressions met is the natural log ofmarket capitalization, bmt is the natural logarithm of the book-to-market ratio, and dmet is the change in themarket value of equity over fiscal year t, ln(MEt/MEt−1).

Explanatory All stocks All but Micro (ABM) stocks Microcapsvariable (1) (2) (3) (4) (5) (6) (7) (8) (9)met −0.074 −0.053 −0.060 −0.050 −0.058 −0.063 −0.261 −0.183 −0.182

(−1.74) (−1.33) (−1.57) (−1.29) (−1.50) (−1.72) (−3.53) (−2.66) (−2.71)bmt 0.261 0.104 0.051 0.172 0.059 0.005 0.323 0.165 0.113

(3.51) (1.40) (0.71) (2.08) (0.80) (0.07) (4.02) (1.84) (1.22)t∑

τ=t−4

dmeτ −0.272 −0.173 −0.311

(−5.17) (−3.20) (−4.81)dmet −0.339 −0.084 −0.429

(−2.91) (−0.69) (−3.23)dmet−1 −0.299 −0.202 −0.369

(−3.37) (−2.09) (−3.40)dmet−2 −0.280 −0.162 −0.313

(−3.78) (−1.80) (−3.55)dmet−3 −0.214 −0.126 −0.262

(−3.38) (−1.60) (−3.63)dmet−4 −0.207 −0.213 −0.217

(−3.55) (−2.71) (−3.11)

Average R2 2.15% 2.56% 3.42% 2.38% 3.09% 4.96% 1.28% 1.62% 2.25%

46

Table 2: Variance Decompositions of BE/ME Ratios

This table decomposes the book-to-market ratio in year t into the book-to-market ratio in year t − k plus thechanges in the book (dbeτ ) and market (−dmeτ ) values of equity from year t− k to t. The first row presents aone-year decomposition, the second row a two-year decomposition, and so forth. Each estimate is the percentageof variation in today’s book-to-market ratios explained by the component indicated by the column. In Panel A,columns dbeτ and −dmeτ indicate the amount of variation explained by the change in the book or market valueof equity τ years ago; columns

∑dbeτ and −

∑dmeτ indicate the amount explained by the cumulative change

in the book or market value of equity from year t−k+1 to t. Panel B decomposes the cumulative changes in thebook and market values of equity using identities dbeτ ≡ roeτ − rbdiv,τ + rbiss,τ and −dmeτ ≡ −rτ + rdiv,τ − riss,τ ,

where roeτ is the log-return on book equity, rbdiv,τ is the difference in with- and without-dividends log-returns on

book equity (book dividend yield), rbiss,τ is the remainder term (book net issuance), rτ is the total stock returnwith dividends, rdiv,τ is the difference in with- and without-dividends stock returns (market dividend yield),and riss,τ is the remainder term (market net issuance). A firm is used for estimating year t−k covariances if thenecessary information for it is available in years t− k and t. The sample is from 1963 through 2011 for horizon= 1, from 1964 through 2011 for horizon = 2, and so forth. The covariances used to measure these percentages(see equation (3)) are estimated with year fixed effects. Standard errors in Panels A and B, reported in squarebrackets, are block bootstrapped by year. Panel C reports correlations between today’s book-to-market ratios,the ratios from five years earlier, and the five-year cumulative changes in the high- and low-level components ofbmt.

Panel A: High-level decomposition of BE/ME ratiosOne-year changes Cumulative changes

Horizon bmt−k dbeτ −dmeτ∑

dbeτ −∑

dmeτ1 81.84 −4.93 23.09 −4.93 23.09

[1.41] [0.60] [1.54] [0.60] [1.54]

2 70.36 −4.23 14.80 −9.09 38.72[1.89] [0.46] [1.11] [0.77] [1.98]

3 62.61 −2.74 9.80 −11.49 48.88[1.92] [0.41] [1.09] [0.93] [2.08]

4 56.74 −2.27 7.58 −13.43 56.69[1.96] [0.34] [0.77] [1.07] [2.06]

5 51.73 −1.90 6.60 −14.85 63.13[1.48] [0.31] [0.91] [1.15] [1.68]

6 48.61 −1.38 4.38 −15.81 67.20[1.41] [0.30] [0.86] [1.35] [1.78]

7 46.33 −1.45 3.89 −16.69 70.36[1.33] [0.34] [0.86] [1.53] [1.92]

8 44.51 −1.52 3.37 −17.31 72.81[1.20] [0.28] [0.85] [1.41] [1.76]

9 42.70 −1.50 3.17 −17.89 75.19[1.19] [0.24] [0.74] [1.44] [1.96]

10 41.74 −1.49 2.59 −18.58 76.85[1.45] [0.23] [0.68] [1.43] [2.14]

47

Panel B: Low-level decomposition of BE/ME ratiosCumulative dbeτ components Cumulative dmeτ components

Horizon bmt−k

∑roeτ −

∑rbdiv,τ

∑rbiss,τ −

∑rτ

∑rdiv,τ −

∑riss,τ

1 81.84 1.96 0.67 −7.56 20.82 0.61 1.66[1.41] [0.78] [0.08] [0.50] [1.55] [0.07] [0.19]

2 70.36 0.37 1.58 −11.04 35.05 1.17 2.50[1.89] [1.07] [0.14] [0.73] [2.01] [0.14] [0.31]

3 62.61 −2.04 2.69 −12.14 44.62 1.66 2.60[1.92] [1.46] [0.20] [1.08] [2.08] [0.20] [0.44]

4 56.74 −4.82 3.90 −12.51 52.16 2.14 2.39[1.96] [1.47] [0.23] [1.19] [2.08] [0.29] [0.55]

5 51.73 −7.97 5.13 −12.01 58.59 2.61 1.93[1.48] [1.54] [0.28] [1.21] [1.62] [0.37] [0.60]

6 48.61 −11.25 6.43 −10.99 62.88 3.05 1.26[1.41] [1.57] [0.39] [1.23] [1.62] [0.45] [0.67]

7 46.33 −14.44 7.78 −10.03 66.41 3.53 0.42[1.33] [1.85] [0.42] [1.30] [1.61] [0.52] [0.74]

8 44.51 −17.83 9.22 −8.70 69.27 3.98 −0.45[1.20] [1.62] [0.48] [1.21] [1.34] [0.64] [0.73]

9 42.70 −20.98 10.60 −7.50 71.86 4.52 −1.19[1.19] [1.56] [0.52] [1.36] [1.33] [0.68] [0.84]

10 41.74 −24.38 12.00 −6.20 73.77 5.10 −2.02[1.45] [1.73] [0.57] [1.24] [1.60] [0.75] [0.87]

Panel C: Correlations between bmt, bmt−5, and five-year cumulative changes in the components of bmt

bmt bmt−5

∑dbeτ −

∑dmeτ

∑roeτ −

∑rbdiv,τ

∑rbiss,τ −

∑rτ

∑rdiv,τ

bmt−5 0.481∑dbeτ −0.139 −0.326

−∑

dmeτ 0.469 −0.184 −0.636∑roeτ −0.067 −0.145 0.608 −0.416

−∑

rbdiv,τ 0.195 0.112 0.053 0.013 −0.280∑

rbiss,τ −0.128 −0.218 0.348 −0.197 −0.503 0.136

−∑

rτ 0.458 −0.239 −0.498 0.926 −0.497 0.127 0.031∑rdiv,τ 0.158 0.211 −0.128 0.050 0.151 −0.629 −0.162 −0.098

−∑

riss,τ 0.041 0.052 −0.419 0.321 0.109 −0.087 −0.591 −0.038 0.057

48

Table 3: Fama-MacBeth Regressions with Decomposed BE/ME Ratios

This table reports Fama-MacBeth regressions that assess which components of the book-to-market ratio con-tribute to its ability to explain cross-sectional variation in monthly returns. Panel A presents the baselineregression. Panels B, C, and D report estimates from univariate regressions of first-stage residuals against eachcomponent. These regressions measure how close each component’s optimal slope (bj) is to the common slope(b∗) when all other components’ slopes remain fixed at b∗. The first t-value tests whether each component’soptimal slope is different from zero. The second t-value tests whether the optimal component slope is signifi-cantly different from the common slope reported in Panel A. The regressions presented in Panel B decomposethe book-to-market ratio into the book-to-market ratio five years ago and the cumulative log-changes in thebook and market values of equity. Panel C presents a more detailed decomposition that disaggregates thelog-changes by year. The left half uses a five-year decomposition and the right half a ten-year decomposition.Panel D presents a five-year low-level decomposition in which the annual changes in the book and market valuesof equity are further disaggregated into total return, dividend yield, and net issuance components.

Panel A: Baseline regressionRegressors EST t-valuemet −0.074 −1.86bmt 0.261 3.63

R2 2.15%

Panel B: Five-year changes in the book and market value of equityt-values

Regressors EST H0 : bj = 0 H0 : bj = b∗

bmt−k 0.133 1.40 −2.80∑t

τ=t−k+1dbeτ 0.087 1.32 −4.04

−∑t

τ=t−k+1dmeτ 0.439 4.98 4.98

49

Panel C: Five-year and ten-year high-level decompositions of BE/ME ratiosFive-year decomposition Ten-year decomposition

t-values t-values

Regressors EST H0 : bj = 0 H0 : bj = b∗ EST H0 : bj = 0 H0 : bj = b∗

bmt−k 0.133 1.40 −2.80 0.114 1.24 −3.87dbet −0.026 −0.24 −2.83 −0.026 −0.24 −2.83dbet−1 −0.054 −0.58 −3.94 −0.054 −0.58 −3.94dbet−2 0.131 1.40 −1.71 0.131 1.40 −1.71dbet−3 0.133 1.48 −1.67 0.133 1.48 −1.67dbet−4 0.161 1.71 −1.26 0.161 1.71 −1.26dbet−5 0.240 2.12 −0.22dbet−6 0.137 0.98 −0.95dbet−7 −0.563 −0.87 −1.28dbet−8 −0.655 −0.83 −1.17dbet−9 0.113 0.31 −0.41−dmet 0.447 3.68 2.04 0.447 3.68 2.04−dmet−1 0.442 4.16 2.51 0.442 4.16 2.51−dmet−2 0.415 4.22 2.29 0.415 4.22 2.29−dmet−3 0.387 3.69 2.03 0.387 3.69 2.03−dmet−4 0.421 4.09 2.71 0.421 4.09 2.71−dmet−5 0.213 2.16 −0.81−dmet−6 0.322 3.04 0.79−dmet−7 0.210 1.53 −0.43−dmet−8 0.095 0.60 −1.17−dmet−9 0.215 1.97 −0.53

Panel D: Five-year low-level decomposition of BE/ME ratios

t-values t-values

Regressors EST H0 : bj = 0 H0 : bj = b∗ Regressors EST H0 : bj = 0 H0 : bj = b∗

bmt−k 0.133 1.40 −2.80Book side: Market side:

roet 0.247 1.60 −0.11 −rt 0.348 2.79 0.86roet−1 0.202 1.50 −0.63 −rt−1 0.382 3.57 1.46roet−2 0.213 1.66 −0.57 −rt−2 0.387 3.97 1.66roet−3 0.204 1.49 −0.56 −rt−3 0.363 3.60 1.55roet−4 0.203 1.66 −0.72 −rt−4 0.412 4.23 2.52−rb

div,t −0.509 −0.52 −0.75 rdiv,t −1.337 −0.75 −0.92

−rbdiv,t−1 −0.377 −0.39 −0.63 rdiv,t−1 −0.193 −0.10 −0.23

−rbdiv,t−2

−0.100 −0.10 −0.36 rdiv,t−2 −0.128 −0.06 −0.19

−rbdiv,t−3

0.355 0.39 0.10 rdiv,t−3 −0.186 −0.10 −0.24

−rbdiv,t−4 −0.179 −0.19 −0.45 rdiv,t−4 −0.755 −0.39 −0.54

rbiss,t −0.265 −2.20 −4.18 −riss,t 1.617 7.54 7.65

rbiss,t−1 −0.256 −2.09 −4.09 −riss,t−1 1.402 6.23 6.11

rbiss,t−2 0.083 0.63 −1.34 −riss,t−2 0.949 4.01 3.31

rbiss,t−3 −0.008 −0.06 −1.98 −riss,t−3 0.816 3.52 2.84

rbiss,t−4 0.110 1.05 −1.41 −riss,t−4 0.664 2.91 2.07

50

Table 4: Alternative Repackagings of the Book-to-Market Ratio

This table reports Fama-MacBeth regressions and portfolio sorts that compare the information content of the

current book-to-market ratio (bmt) to alternative book-to-market constructions. Panel A reports the average

coefficients, the t-values associated with these averages, and average adjusted R2s from monthly Fama-MacBeth

regressions. All coefficients are multiplied by 100. Panel B sorts stocks into ten portfolios based on NYSE

breakpoints on the alternative book-to-market ratio construct at the end of each June and holds the portfolio

for the next year. The first row of each block reports the average monthly excess returns over the one-month

Treasury bill rate, the second row reports the CAPM alphas, and the third row reports the three-factor model

alphas. The R2 in the last column is the adjusted R2 from a time-series regression of high-minus-low (10− 1)

hedge portfolio returns against the market factor or the three-factor model factors. The five alternative book-

to-market constructs are (1) the book-to-market ratio from five years earlier, (2) minus the sum of log-changes

in the market value of equity over the past five years, (3) the sum of log-changes in the book value of equity

over the past five years, (4) the negative of the five-year stock returns, and (5) a modified version of the current

book-to-market ratio that only includes variation due to the five-year change in the market value of equity.

Panel A: Fama-MacBeth regressionsAlternative Regressorsrepackaging met bmt Alternative bmt

of bmt EST t-value EST t-value EST t-value R2

Baseline −0.074 −1.74 0.261 3.51 2.15%

bmt−5 −0.106 −2.55 −0.018 −0.28 1.97%−0.075 −1.80 0.344 4.51 −0.176 −2.86 2.51%

−∑

dmeτ −0.059 −1.60 0.311 5.70 2.11%−0.053 −1.33 0.104 1.40 0.272 5.17 2.56%

∑dbeτ −0.089 −2.25 −0.219 −4.36 1.82%

−0.062 −1.46 0.244 3.33 −0.186 −3.95 2.40%

−∑

rτ −0.060 −1.73 0.254 4.14 2.09%−0.051 −1.34 0.159 1.97 0.197 3.02 2.66%

bmt −0.055 −1.48 0.730 5.17 2.25%−0.053 −1.32 0.056 0.76 0.679 4.74 2.63%

51

Panel B: Portfolio sortsSorting Decilesvariable α 1 2 9 10 10-1 10-1 R2

Current re 0.283 0.452 0.705 0.780 0.496book-to-market: (1.23) (2.07) (3.30) (3.16) (2.48)

bmt CAPM −0.144 0.035 0.334 0.381 0.524 0.31%(−1.74) (0.58) (3.19) (2.63) (2.62)

FF 0.088 0.103 0.021 −0.059 −0.147 73.79%(1.50) (1.77) (0.31) (−0.64) (−1.41)

Old re 0.390 0.463 0.548 0.578 0.188book-to-market: (1.81) (2.13) (2.54) (2.47) (1.05)

bmt−5 CAPM 0.001 0.049 0.165 0.180 0.179 −0.12%(0.01) (0.76) (1.67) (1.48) (0.99)

FF 0.181 0.083 −0.016 −0.059 −0.240 36.20%(2.36) (1.28) (−0.19) (−0.56) (−1.64)

Five-year changes re 0.251 0.460 0.650 0.813 0.563in ME: (0.91) (1.99) (2.95) (3.21) (2.85)

−∑

dmeτ CAPM −0.257 0.022 0.274 0.390 0.647 4.40%(−2.49) (0.31) (2.40) (2.81) (3.34)

FF −0.054 0.093 0.000 0.182 0.235 33.07%(−0.60) (1.35) (0.00) (1.43) (1.43)

Five-year changes re 0.703 0.688 0.331 0.387 −0.316in BE: (3.03) (3.34) (1.39) (1.44) (−1.84)∑

dbeτ CAPM 0.298 0.322 −0.124 −0.114 −0.411 7.65%(2.66) (3.47) (−1.83) (−1.21) (−2.48)

FF 0.110 0.180 −0.052 0.091 −0.018 41.44%(1.08) (2.08) (−0.79) (1.18) (−0.14)

Five-year re 0.386 0.419 0.647 0.737 0.351stock returns: (1.42) (1.97) (2.74) (2.63) (1.68)

−∑

rτ CAPM −0.115 0.020 0.241 0.273 0.388 0.58%(−1.13) (0.28) (2.01) (1.74) (1.85)

FF 0.091 0.067 −0.023 0.060 −0.030 35.11%(1.03) (0.93) (−0.24) (0.46) (−0.18)

Variation due to five-year re 0.244 0.383 0.760 0.856 0.612changes in ME: (0.90) (1.67) (3.35) (3.18) (2.87)

bmt CAPM −0.259 −0.043 0.375 0.414 0.673 1.86%(−2.57) (−0.52) (3.16) (2.72) (3.17)

FF −0.054 0.033 0.074 0.095 0.149 42.10%(−0.62) (0.41) (0.81) (0.75) (0.90)

52

Table 5: Cheating the Value Factor

This table sorts stocks into ten portfolios at the end of every June based on an alternative version of the book-to-market ratio. This alternative

ratio, bmet , picks up the variation in bmt that is not part of bmt. It is defined as the difference bme

t = bmt − bmt and it therefore does notcovary with the past five-year change in the market value of equity, but its remaining covariances are unaffected. This variable is reconstructedat the end of every June by estimating the required covariances and linear combinations from the cross-section. This table shows the averagemonthly excess returns, CAPM betas (m) and alphas (a), and three-factor model loadings (m for the market factor; s for the size factor; andh for the value factor) and alphas (a) for value-weighted deciles based on this variable. The t-values are reported in parentheses.

Portfolio decile High − LowModel Low 2 3 4 5 6 7 8 9 High α R2

Excess return 0.40 0.45 0.49 0.44 0.49 0.46 0.46 0.57 0.47 0.34 −0.07(1.98) (2.00) (2.26) (2.02) (2.21) (2.18) (2.24) (2.79) (2.21) (1.40) (−0.39)

CAPM, m 0.91 1.04 1.01 0.98 1.01 0.94 0.91 0.89 0.92 1.03 0.13 2.21%(51.00) (71.25) (67.15) (56.90) (56.91) (53.51) (50.75) (45.87) (41.93) (41.94) (3.58)

a 0.03 0.02 0.08 0.04 0.08 0.07 0.09 0.21 0.10 −0.09 −0.12(0.39) (0.35) (1.18) (0.47) (0.96) (0.90) (1.03) (2.26) (0.97) (−0.73) (−0.70)

FF3 model, m 0.90 1.05 1.00 1.03 1.07 1.01 0.97 0.98 1.01 1.12 0.22 33.56%(54.57) (68.54) (61.71) (58.41) (60.45) (59.56) (55.71) (58.56) (62.36) (47.86) (6.73)

s −0.24 −0.13 −0.03 −0.05 −0.11 −0.07 −0.04 −0.03 0.14 0.05 0.30(−10.13) (−5.74) (−1.27) (−1.80) (−4.15) (−3.05) (−1.45) (−1.14) (5.80) (1.62) (6.41)

h −0.28 −0.07 −0.05 0.20 0.21 0.28 0.29 0.43 0.60 0.46 0.74(−11.12) (−3.16) (−2.20) (7.60) (7.94) (10.76) (10.77) (16.96) (24.46) (13.00) (15.22)

a 0.18 0.07 0.11 −0.05 −0.01 −0.05 −0.04 0.01 −0.19 −0.31 −0.49(2.53) (1.02) (1.55) (−0.70) (−0.15) (−0.67) (−0.58) (0.08) (−2.73) (−2.99) (−3.48)

53

Table 6: BE/ME- and Size-Sorted Portfolios and Their Loadings on Priced and Unpriced Componentsof HML

This table measures how prevalent the priced and unpriced components of BE/ME ratios are in the 25 portfoliossorted by size and BE/ME ratios (“the 25 Fama-French portfolios”). We estimate factor loadings on priced andunpriced components by regressing each portfolio’s monthly returns on the market factor, the SMB factor, and

the priced and unpriced components of the HML factor. We construct the (priced) bmt-based HML factor inthe same way as the standard HML factor except that we replace bmt with the approximate priced component,

bmt. The (unpriced) bmet -based HML factor is constructed by replacing bmt with the approximate unpriced

component, bmet .

Priced long-short factor based on bmt

Factor Loadings t-valuesBE/ME quintile BE/ME quintile

Size Low 2 3 4 High Low 2 3 4 HighSmall −0.26 0.06 0.22 0.32 0.52 −6.05 1.83 8.33 11.73 16.662 −0.31 0.03 0.19 0.31 0.56 −9.82 1.17 7.04 10.58 18.263 −0.32 0.06 0.15 0.31 0.50 −10.78 1.73 4.83 9.34 13.294 −0.26 0.03 0.21 0.29 0.50 −8.93 0.91 6.13 8.67 12.76Big −0.12 0.09 0.05 0.33 0.44 −6.03 3.24 1.77 11.09 10.05

Unpriced long-short factor based on bmet

Factor Loadings t-valuesBE/ME quintile BE/ME quintile

Size Low 2 3 4 High Low 2 3 4 HighSmall −0.13 0.00 0.16 0.28 0.43 −2.46 −0.10 5.15 8.55 11.642 −0.21 0.14 0.37 0.45 0.57 −5.58 4.19 11.42 12.86 15.453 −0.27 0.22 0.49 0.55 0.60 −7.51 5.82 12.94 13.69 13.294 −0.32 0.27 0.43 0.49 0.66 −9.10 6.98 10.30 12.52 14.31Big −0.50 0.11 0.41 0.50 0.68 −20.52 3.32 11.16 13.89 13.10

54

Table 7: Asset Pricing with Alternative Value Variables

This table sorts stocks into deciles based on size, book-to-market ratios (BE/ME), earnings-to-price ratios (E/P),cashflow-to-price ratios (C/P), and dividend-to-price ratios (D/P), and into 25 portfolios based jointly on fivesize and five book-to-market quintiles (the 25 Fama-French portfolios). For the 25 Fama-French portfolios,the estimates are for the stocks in the smallest (first five estimates) and largest (remaining five estimates) sizequintiles. Stocks are resorted into portfolios at the end of each June. The data are from 1963 through 2011.The first row of each block reports the average excess returns (and t-values in parentheses) on each of theten portfolios based on the underlying sorting variable. The row labeled bmt uses the three-factor model’sstandard HML, the row labeled

∑dme reconstructs HML factor by using the five-year change in the market

value of equity as the measure of value, and the last row uses the approximate priced component of bmt, bmt, toconstruct the HML factor. The rightmost column reports the Gibbons, Ross, and Shanken (1989) test statisticand its associated p-value.

55

HML Decile GRSvariable Low 2 3 4 5 6 7 8 9 High (p-value)

Test assets: Size deciles

re 0.52 0.54 0.61 0.58 0.62 0.56 0.59 0.55 0.50 0.37(1.82) (1.86) (2.21) (2.18) (2.42) (2.34) (2.46) (2.37) (2.32) (1.88)

bmt −0.13 −0.12 −0.03 −0.03 0.03 0.01 0.05 0.03 0.03 0.05 2.52(−1.63) (−2.40) (−0.78) (−0.74) (0.67) (0.25) (1.05) (0.61) (0.69) (1.97) (0.006)∑

dme −0.11 −0.12 −0.03 −0.02 0.04 0.05 0.08 0.04 0.06 0.03 1.60(−1.06) (−1.74) (−0.56) (−0.34) (0.80) (0.93) (1.58) (0.83) (1.38) (1.34) (0.103)

bmt −0.14 −0.14 −0.04 −0.02 0.03 0.04 0.08 0.03 0.06 0.03 1.54(−1.30) (−1.95) (−0.71) (−0.42) (0.73) (0.88) (1.54) (0.71) (1.19) (1.37) (0.123)

Test assets: BE/ME deciles

re 0.30 0.42 0.52 0.53 0.48 0.52 0.57 0.58 0.70 0.78(1.28) (1.95) (2.44) (2.40) (2.34) (2.48) (2.82) (2.84) (3.22) (2.94)

bmt 0.12 0.07 0.11 0.00 −0.03 −0.03 −0.02 −0.09 −0.01 −0.12 1.84(2.08) (1.19) (1.56) (0.03) (−0.43) (−0.49) (−0.23) (−1.57) (−0.10) (−1.27) (0.052)∑

dme 0.04 0.05 0.10 0.07 0.04 0.04 0.08 0.02 0.10 0.01 0.52(0.51) (0.82) (1.38) (0.90) (0.48) (0.45) (0.93) (0.28) (1.08) (0.07) (0.874)

bmt 0.06 0.05 0.10 0.06 0.04 0.03 0.06 −0.01 0.07 −0.04 0.52(0.73) (0.80) (1.36) (0.79) (0.43) (0.41) (0.70) (−0.11) (0.77) (−0.37) (0.874)

Test assets: E/P deciles

re 0.29 0.34 0.47 0.42 0.47 0.55 0.66 0.63 0.67 0.78(1.11) (1.56) (2.25) (2.06) (2.27) (2.75) (3.32) (3.08) (3.05) (3.26)

bmt 0.06 0.02 0.10 0.03 0.02 0.03 0.14 0.06 0.01 0.05 0.80(0.68) (0.22) (1.29) (0.33) (0.21) (0.38) (1.96) (0.74) (0.11) (0.59) (0.629)∑

dme −0.04 0.00 0.11 0.06 0.05 0.08 0.23 0.13 0.12 0.19 1.12(−0.46) (0.00) (1.34) (0.78) (0.63) (0.99) (2.62) (1.40) (1.19) (1.64) (0.348)

bmt −0.02 0.01 0.11 0.06 0.05 0.07 0.20 0.11 0.09 0.17 1.00(−0.24) (0.19) (1.39) (0.75) (0.64) (0.86) (2.41) (1.16) (0.90) (1.47) (0.442)

56

HML Decile GRSvariable Low 2 3 4 5 6 7 8 9 High (p-value)

Test assets: C/P deciles

re 0.28 0.39 0.45 0.49 0.52 0.44 0.51 0.59 0.70 0.78(1.11) (1.77) (2.16) (2.33) (2.49) (2.15) (2.59) (2.92) (3.43) (3.31)

bmt 0.05 0.07 0.08 0.03 0.02 −0.05 0.01 0.04 0.10 0.08 0.56(0.76) (1.03) (1.02) (0.35) (0.22) (−0.60) (0.14) (0.50) (1.27) (0.83) (0.849)∑

dme −0.04 0.04 0.08 0.08 0.06 0.03 0.08 0.12 0.20 0.21 0.75(−0.48) (0.58) (0.99) (1.07) (0.76) (0.27) (0.83) (1.23) (2.03) (1.78) (0.680)

bmt −0.03 0.05 0.08 0.07 0.04 0.01 0.08 0.11 0.19 0.19 0.72(−0.29) (0.66) (0.97) (0.96) (0.56) (0.05) (0.80) (1.18) (1.90) (1.68) (0.709)

Test assets: D/P deciles

re 0.31 0.35 0.48 0.51 0.40 0.53 0.51 0.65 0.59 0.51(1.19) (1.49) (2.09) (2.38) (1.84) (2.61) (2.51) (3.33) (3.20) (2.47)

bmt −0.03 −0.01 0.10 0.07 −0.09 0.07 0.01 0.13 0.06 −0.14 0.98(−0.34) (−0.18) (1.17) (0.87) (−0.95) (0.85) (0.11) (1.55) (0.70) (−1.19) (0.459)∑

dme −0.04 −0.01 0.13 0.10 −0.02 0.10 0.07 0.19 0.15 0.01 0.97(−0.51) (−0.08) (1.54) (1.14) (−0.18) (1.17) (0.81) (2.04) (1.44) (0.06) (0.469)

bmt −0.03 0.00 0.15 0.11 −0.04 0.10 0.06 0.18 0.12 −0.06 1.05(−0.34) (0.01) (1.69) (1.22) (−0.37) (1.15) (0.68) (1.88) (1.13) (−0.41) (0.398)

Test assets: 25 Size and B/M sorted portfolios (size quintiles 1 and 5 shown)

re −0.04 0.61 0.67 0.84 0.94 0.36 0.51 0.43 0.47 0.52(−0.10) (1.96) (2.50) (3.33) (3.43) (1.66) (2.48) (2.13) (2.38) (2.29)

bmt −0.57 0.00 0.02 0.15 0.10 0.16 0.10 −0.05 −0.12 −0.18 3.37(−5.55) (−0.02) (0.29) (2.42) (1.46) (2.91) (1.48) (−0.60) (−1.75) (−1.75) (0.000)∑

dme −0.66 −0.04 0.04 0.21 0.18 0.09 0.12 0.03 0.00 −0.05 3.69(−4.83) (−0.37) (0.57) (2.72) (2.05) (1.42) (1.67) (0.29) (−0.03) (−0.38) (0.000)

bmt −0.65 −0.05 0.01 0.17 0.14 0.10 0.11 0.03 −0.03 −0.08 3.65(−4.65) (−0.51) (0.18) (2.28) (1.65) (1.51) (1.58) (0.29) (−0.31) (−0.67) (0.000)

57

Table 8: Other Price-Scaled Variables

This table decomposes size, book-to-market ratio (BE/ME), earnings-to-price ratio (E/P), cashflow-to-priceratio (C/P), and dividend-to-price ratio (D/P) into what these variables were five years prior and how thefundamentals and market values of equity have changed between the two points in time. Panel A shows variancedecomposition estimates. The first row (Xt−5) shows the amount of variation in today’s variable that is due tothe variation in the old variable. The rows labeled df show how much the changes in fundamentals (book valueof equity, earnings, cashflows, or dividends) contribute to this variance, and the rows labeled −dme show howimportant the changes in the market values of equity are. The covariances used to measure these percentages areestimated with year fixed effects. Standard errors, reported in square brackets, are clustered by year. Only firmsthat have existed for five years and for which the required fundamentals are positive are included in the sample.Panel B reports average coefficients (t-values associated with these averages) from monthly Fama-MacBethregressions. The baseline specification explains returns using size and the current price-scaled variable. Theanalysis of deviations reports univariate regressions of the first-stage residuals against each component. Thefirst column reports each component’s optimal slope (bj) when all other components’ slopes remain fixed at thecommon slope (b∗) and the second column reports the deviation between the component’s optimal slope andthe common slope. Panel C reports the pairwise cross-sectional covariances between book-to-market ratio, itstwo components, and ME, E/P, C/P, and D/P. All variables here are in logs. The two BE/ME components arethe approximate priced and unpriced component of BE/ME (see text for details). All variables are demeanedcross-sectionally before estimating the covariances from the panel.

Panel A: Variance decompositions of size and other price-scaled VariablesPrice-scaled variable:

Component ME BE/ME E/P C/P D/PXt−5 83.86 52.22 28.85 34.05 68.44

[1.13] [1.63] [1.40] [1.30] [1.75]dft −4.05 46.66 39.79 1.94

[0.58] [1.89] [1.70] [0.76]dft−1 −2.52 9.22 5.71 0.09

[0.53] [0.99] [0.93] [0.70]dft−2 −1.35 2.10 0.32 −1.07

[0.47] [0.95] [0.80] [0.25]dft−3 −1.35 −1.61 −3.29 −1.24

[0.43] [0.74] [0.69] [0.30]dft−4 −1.26 −2.74 −2.79 −1.90

[0.40] [1.05] [0.63] [0.47]−dmet −4.07 22.27 5.24 7.85 12.05

[0.46] [1.51] [1.52] [1.54] [0.99]−dmet−1 −3.45 14.00 0.74 2.71 7.02

[0.44] [1.11] [1.20] [1.15] [0.84]−dmet−2 −2.93 9.21 2.63 4.59 5.26

[0.44] [1.02] [1.01] [0.95] [0.78]−dmet−3 −2.80 6.92 4.00 5.14 4.88

[0.40] [0.82] [0.92] [0.86] [0.61]−dmet−4 −2.90 5.90 4.91 5.92 4.52

[0.37] [0.94] [0.85] [0.83] [0.64]

58

Panel B: Fama-MacBeth regressions with other price-scaled variablesPrice-scaled variable:

BE/ME E/P C/P D/P

Regressors bj bj − b∗ bj bj − b∗ bj bj − b∗ bj bj − b∗

Baseline me −0.073 −0.067 −0.073 −0.045regression (−1.71) (−2.05) (−2.22) (−1.46)

Xt 0.255 0.180 0.280 0.061(3.41) (2.94) (4.57) (0.93)

Adj. R2 2.20% 2.11% 2.03% 2.69%

Deviations Xt−5 0.130 −0.125 0.098 −0.082 0.219 −0.060 0.031 −0.030(1.36) (−2.75) (1.18) (−2.12) (2.68) (−1.42) (0.43) (−1.04)

dft −0.008 −0.263 0.136 −0.045 0.188 −0.092 −0.009 −0.070(−0.07) (−2.51) (2.42) (−0.93) (3.15) (−1.69) (−0.09) (−0.93)

dft−1 −0.069 −0.324 0.121 −0.059 0.193 −0.087 −0.102 −0.163(−0.72) (−3.86) (1.96) (−1.63) (2.97) (−2.17) (−1.16) (−2.59)

dft−2 0.113 −0.142 0.152 −0.028 0.227 −0.052 −0.078 −0.139(1.17) (−1.83) (2.48) (−0.92) (3.62) (−1.43) (−0.94) (−2.16)

dft−3 0.109 −0.146 0.121 −0.059 0.222 −0.058 −0.077 −0.138(1.17) (−1.84) (1.98) (−1.57) (3.42) (−1.41) (−0.90) (−2.14)

dft−4 0.140 −0.115 0.170 −0.011 0.241 −0.039 0.009 −0.051(1.44) (−1.38) (2.84) (−0.31) (3.77) (−0.94) (0.12) (−0.83)

−dmet 0.434 0.179 0.206 0.026 0.304 0.024 −0.060 −0.121(3.56) (1.94) (1.55) (0.25) (2.34) (0.24) (−0.46) (−1.06)

−dmet−1 0.434 0.178 0.380 0.199 0.479 0.199 0.108 0.047(4.04) (2.45) (3.28) (2.38) (4.32) (2.43) (0.94) (0.52)

−dmet−2 0.410 0.154 0.302 0.122 0.415 0.135 0.218 0.158(4.14) (2.27) (2.65) (1.49) (3.83) (1.68) (2.12) (1.95)

−dmet−3 0.378 0.123 0.311 0.130 0.393 0.113 0.287 0.226(3.61) (1.98) (2.78) (1.76) (3.66) (1.58) (2.82) (2.99)

−dmet−4 0.426 0.171 0.343 0.163 0.441 0.161 0.113 0.052(4.12) (2.91) (3.17) (2.35) (4.20) (2.37) (1.07) (0.66)

Panel C: Covariances between bmt, bmt, bmet , met, and other price-scaled variables

bmt bmt bmet met ept cpt dpt

bmt 0.497 0.145 0.351 −0.490 0.150 0.171 0.202

bmt 0.152 −0.007 −0.266 0.034 0.045 0.112bme

t 0.359 −0.225 0.115 0.126 0.090

59

Table 9: Resurrecting Anomalies

This table estimates monthly average excess returns, CAPM alphas, and three-factor model alphas for portfoliossorted on E/P, C/P, and D/P ratios. The columns labeled “Sorts on unadjusted variables” use each ratio as suchdirectly to compute the deciles. The columns labeled “Sorts on adjusted variables” replace original price-scaledvariables with repackaged versions that are based on five-year changes in fundamentals (E, C, or D) and marketvalues of equity. This repackaging is done every June at the same time as the portfolios are formed by firstcomputing (from the cross-section) the covariance of the price-scaled variable with the five fundamental andfive market value variables. The adjusted version of a price-scaled variable is a linear combination of these tencomponents such that its covariances with these components are the same as what they are for the originalprice-scaled variable. This table shows the average excess returns, CAPM alphas, and three-factor model alphasfor deciles 1 and 10 and for a high-minus-low portfolio. t-values are reported in parentheses.

Sorts on unadjusted variables Sorts on adjusted variablesAnomaly Deciles High − Low Deciles High − Lowvariable Model 1 10 α R2 1 10 α R2

E/P re 0.25 0.78 0.53 0.17 0.86 0.69(0.98) (3.27) (2.76) (0.70) (3.76) (3.95)

CAPM −0.22 0.38 0.60 2.42% −0.28 0.46 0.74 2.13%(−2.22) (3.01) (3.11) (−2.85) (4.12) (4.28)

FF3 −0.01 0.06 0.07 52.30% −0.22 0.28 0.50 13.70%(−0.13) (0.58) (0.49) (−2.29) (2.70) (3.06)

C/P re 0.29 0.76 0.47 0.23 0.77 0.54(1.15) (3.28) (2.55) (0.90) (3.38) (3.15)

CAPM −0.18 0.36 0.54 3.16% −0.24 0.37 0.61 3.86%(−1.95) (3.03) (2.95) (−2.52) (3.35) (3.60)

FF3 0.04 0.07 0.03 51.85% −0.14 0.18 0.32 22.16%(0.59) (0.78) (0.25) (−1.52) (1.76) (2.06)

D/P re 0.29 0.52 0.23 0.17 0.64 0.47(1.14) (2.48) (0.96) (0.66) (2.78) (2.41)

CAPM −0.19 0.24 0.43 19.47% −0.30 0.27 0.56 5.80%(−2.11) (1.55) (2.05) (−2.72) (1.99) (2.97)

FF3 −0.04 −0.18 −0.15 62.94% −0.21 −0.03 0.18 31.95%(−0.44) (−1.66) (−1.00) (−1.94) (−0.31) (1.07)

60

Table 10: Trading Strategies and the Information Content of Book-to-Market Ratio Components

This table presents multivariate Fama-MacBeth regressions of monthly stock returns against BE/ME compo-nents (Panel A) and reports spanning tests of the implied portfolio strategies (Panel B). Regressions (1) and (3)in Panel A correspond to the high-level decomposition regressions of Table 3, presented separately for All butMicro (above the 20th percentile of market capitalization for NYSE stocks) and Microcap (below the 20th NYSEmarket capitalization percentile) samples. Regressions (2) and (4) add the five-year net issuance as an additionalregressor. Panel B takes the time series of Fama-MacBeth coefficient estimates from Panel A’s specifications (2)and (4) and estimates spanning tests for the strategies implied by these coefficient estimates. Rbook value is thestrategy that purchases stocks with high changes in the book value of equity and sells stocks with low changesin the book value of equity, and that has zero loadings on the other characteristics. Rmarket value is a strategythat purchases stocks with low changes in the market value of equity and sells stocks with high changes in themarket value of equity. Rnet issuance is a strategy that purchases stocks with low net issuance and sells stockswith high net issuance. The intercept in regressions (2), (4), and (6) is the annual percentage return on eachstrategy after controlling for the returns on the other two strategies.

Panel A: Multivariate Fama-MacBeth regressions of returns on five-year changes in BE/ME componentsABM Microcaps

Regressors (1) (2) (3) (4)

met −0.058 −0.055 −0.185 −0.187(−1.52) (−1.48) (−2.66) (−2.73)

bmt−k 0.071 0.138 0.153 0.121(0.88) (1.73) (1.50) (1.31)∑t

τ=t−k+1dbeτ 0.047 0.237 0.176 0.205

(0.58) (2.87) (2.02) (2.20)

−∑t

τ=t−k+1dmeτ 0.216 0.225 0.476 0.435

(2.39) (2.50) (5.84) (5.40)

−∑t

τ=t−k+1niτ 0.567 0.391

(4.87) (2.65)

R2 3.55% 3.88% 1.79% 2.06%

61

Panel B: Spanning tests for net-issuance, book-value, and market-value strategiesIndependent Rbook value Rmarket value Rnet issuance

variable (1) (2) (3) (4) (5) (6)

Sample: All but Micro (ABM) stocks

Intercept 2.843 −0.801 2.694 2.276 6.810 6.174(2.87) (−1.20) (2.50) (3.05) (4.87) (5.14)

Rbook value 0.821 0.940(24.42) (13.53)

Rmarket value 0.651 −0.756(24.42) (−11.84)

Rnet issuance 0.278 −0.281(13.53) (−11.84)

Adjusted R2 57.1% 54.3% 27.5%

Sample: Microcaps

Intercept 2.457 −3.334 5.218 4.768 4.697 7.804(2.20) (−4.90) (5.40) (7.62) (2.65) (5.86)

Rbook value 0.721 1.335(26.71) (20.85)

Rmarket value 0.802 −1.224(26.71) (−16.52)

Rnet issuance 0.341 −0.282(20.85) (−16.52)

Adjusted R2 65.7% 58.7% 46.6%

62