The Effect of Portfolio Weighting on

Anomaly Returns

Steven CrawfordRice University

James HansenUniversity of New Mexico

Richard Price∗

Rice University

January 5, 2011

∗Corresponding author (email: richardp@rice.edu, office tel: (713)348-6303). We thank KarenNelson, Barb Ostdiek, Mike Pinegar, Brian Rountree, Darren Roulstone, and workshop participantsat Rice University, the University of Illinois at Chicago, the BYU Accounting Research Symposium,and the European Accounting Association annual meeting for helpful comments.

The Effect of Portfolio Weighting on Abnormal Returns

Abstract

In this paper, we document how rebalancing assumptions can significantly

impact evidence of market efficiency. Specifically, many academic studies com-

pute equal-weighted portfolio returns as the simple average of monthly security

returns which assumes portfolios are rebalanced monthly. This practice is un-

likely to reflect actual investment returns, and we show that it alters returns

to several pricing anomalies. We also identify problematic assumptions under-

lying the benchmark return methodology used by the Center for Research in

Security Prices (CRSP), which researchers would be unlikely to know or mimic.

In particular, CRSP includes non-common stock securities that are excluded

by most researchers, does not follow a typical buy-and-hold methodology, and

excludes delistings. We discuss these issues and show how they can affect re-

sults in a number of research settings. The commonly used value-weighted,

size-based benchmark returns, and all equal-weighted daily benchmark returns

are particularly problematic.

Keywords: abnormal returns; anomalies; portfolio weights; CRSP

JEL Classifications: G14; M41; G33; G34

1 Introduction

In the research investigating market efficiency, many researchers make a seemingly

benign assumption related to how portfolios are rebalanced. The common practice is

to compute equal-weighted portfolio returns as the simple average of monthly security

returns, hereafter simple portfolio returns. This is presumably done because it is easier

than computing buy-and-hold portofolio returns.

The implicit assumption with this methodology is that every month following

formation, the portfolio is rebalanced to restore equal weights to each security in the

portfolio. This means selling shares of securities with returns above the portfolio

mean and purchasing additional shares of securities with returns below the portfolio

mean, i.e., consistently selling winners and buying losers.

Although hedge funds typically rebalance frequently, even on a daily or intra-day

basis, their rebalancing is very different from the rebalancing implied by using simple

portfolio returns. When hedge funds rebalance, they remove some or all securities

from the portfolio and replace them with other securities. Hedge funds choose the

rebalancing period to improve performance. The rebalancing academics assume is a

shift of weights among stocks already in the portfolio.

Imposing this rebalancing assumption on every trading strategy is likely to be

problematic. The alternative to simple portfolio returns is buy-and-hold portfolio

returns, which are more reflective of actual investment returns. Equal-weighted buy-

and-hold returns are initially equally weighted, but in subsequent months cumulative

prior raw returns are the weights.

The primary contribution of this paper is documenting the effects of rebalancing

assumptions, i.e., weighting methodology, on evidence of market inefficiency. Typical

asset pricing tests are conducted by regressing monthly excess portfolio returns on

excess market returns and other factors. We show that average raw portfolio returns

and consequently abnormal returns (alphas) are significantly affected by rebalancing

1

assumptions. We show the effects for the following strategies: momentum, the book-

to-market ratio, cash flow, accruals, and earnings.

We find that raw simple portfolio returns are generally larger than raw buy-and-

hold portfolio returns, especially in the lowest decile of these variables. Because the

lowest decile is generally the short side of the strategy, this generally leads to lower

trading strategy returns. For example, in the lowest momentum decile over an ex-

tended time period, the average monthly buy-and-hold return is 0.3% compared with

0.64%, a difference of 0.34% per month. In the 1990s, this difference was 0.66% per

month. The effects with cash flow and earnings are similarly striking, with average

monthly differences of 0.44% and 0.55% respectively. However, with book-to-market

and accruals strategies, the returns to buy-and-hold strategies are slightly lower be-

cause the long portfolio returns are reduced by more than the short portfolio returns.

These problems are unique to equal-weighted returns; value-weighted returns in-

herently compound prior returns and avoid monthly rebalancing of simple equal-

weighted portfolio returns. However, equal-weighted returns are often chosen as the

primary return measure (e.g. Jegadeesh and Titman (1993)) and have intuitive ap-

peal for investing. They will remain a frequently chosen weighting methodology by

academics and practitioners.

Academics are not the only ones subject to problematic assumptions in the com-

putation of portfolio returns. We also identify several problematic issues with bench-

mark1 returns provided by the Center for Research in Security Prices (CRSP). These

returns are used in many research settings to adjust security and portfolio returns

for the effects of risk and the market. Because these issues are subtle methodological

choices that are not clearly identified in CRSP documentation, most researchers are

unaware of them. Other carefully constructed benchmark returns (e.g., the Fama and

French factors) would not employ the methods used by CRSP.

1Benchmark returns include broad-based market returns and returns based on firm characteristicsincluding size, return volatility, and beta.

2

Academic researchers generally consider CRSP to be the gold standard in security

price data and rely heavily upon CRSP for accurate data. CRSP’s methods have

occasionally been questioned. For example Shumway (1997) raises concerns with

CRSP’s treatment of delistings (which has since been corrected). Canina et al. (1998)

raise concerns with CRSP’s equal-weighted daily market return. Our findings related

to CRSP returns extend Canina et al. (1998); we document additional problems with

all CRSP benchmark returns, including value-weighted returns.

Because CRSP benchmark returns are most often used as control variables, they

will have a second order effect in most research settings (although they are arguably

the most important control variable); consequently, bias or noise in CRSP benchmark

returns is unlikely to significantly affect inferences in most research settings. But in

some settings the effects can be large. We demonstrate in a few targeted research

settings how CRSP methodology can affect results and inferences.

The degree to which the issues we identify impact results depends on the bench-

mark measure and on the research design. The most affected measures include all

equal-weighted daily benchmark returns, which are severely biased upward when com-

pounded over extended periods, and the commonly used value-weighted, size-based

returns. The least affected of CRSP’s benchmark returns are the daily and monthly

value-weighted market returns.

Through the process of replication, we identify three problematic issues that re-

searchers should be aware of before they use the CRSP benchmark returns. First,

CRSP includes non common stock securities, such as exchange-traded funds, in bench-

mark returns. The percentage of non common stocks in CRSP has increased from

2% prior to 1970 to 28% in 2007. The increase is attributable to increasing numbers

of small and volatile closed-end funds (CEFs) and exchange-traded funds (ETFs),

many of which track commodities or bonds or take long or short leveraged positions.

Second, the weighting methodology CRSP uses differs from a typical buy-and-hold

3

methodology, which can result in large differences for equal-weighted and size-based

portfolio returns. Third, benchmark returns provided by CRSP generally exclude

delisting returns. This can cause problems for researchers who follow the recommen-

dations of Shumway (1997), Shumway and Warther (1999) and Beaver et al. (2007)

by including delistings in the sample but who fail to adjust the benchmark portfolio

for delistings.

We demonstrate the potential effects in several settings. First, we document the

effect on excess returns to trading strategies based on cash flows, accruals, and the

book-to-market ratio. Using a value-weighted size adjustment following Lakonishok

et al. (1994) and Sloan (1996), we document a shift of returns in the direction of

the short side. Second, we show that betas calculated using CRSP market returns

are generally larger than betas calculated using our corrected market returns. Third,

we document that the correlation between the returns of securities in the highest

and lowest CRSP size-deciles has increased substantially over the last decade (from

roughly 0.5 to 0.75) due to the disproportionate representation of ETFs in the lowest

size decile; this correlation is significantly lower for common stocks in the lowest

size decile. Fourth, we show that equal-weighted dividend yields are overstated by

nearly 100% when ETFs and other non common stock securities are included in the

market portfolio. Value-weighted yields are overstated by roughly 10%, which is large

considering that CRSP value-weighted market returns are virtually unaffected by the

inclusion of ETFs.

Overall, we document that problematic rebalancing assumptions related to equal-

weighted returns can directly affect results through the computation of portfolio re-

turns, or indirectly through the computation of benchmark returns. In the first case,

problematic implicit assumptions are made by academics, and in the second case

are made by CRSP in the creation of benchmark returns that are widely used in

academic research. Over time, these problems inherent in CRSP benchmark returns

4

have become more significant.

In section 2, we discuss how benchmark returns are computed and used in the

literature. In section 3, we discuss the three problematic issues with benchmark

returns in CRSP. In section 4, we present the effect of CRSP’s assumptions in various

research settings. In section 5, we conclude.

2 Computing Portfolio Returns

Academic research in accounting and finance often requires the computation or use

of portfolio returns. Once constructed, these portfolio returns are either directly

used in asset pricing tests, or are used to adjust the returns of individual securities

or portfolios for the effects of risk or the market. When creating portfolio returns

researchers make a number of research design choices, which should be driven by the

specific context of the research question. After determining which securities to include

in the portfolio, the weighting methodology must be chosen. The most common

methodologies are equal- and value-weighting.

The holding period and frequency of rebalancing are also important choices. The

holding period is the length of time securities remain in the portfolio before being

replaced by a new set of securities. With momentum strategies, holding periods of

three or six months are common. With accounting-based strategies, twelve-month

holding periods are also common. Rebalancing occurs when the weights (holdings)

of securities in the portfolio are set or reset. If researchers use value-weighted or

equal-weighted buy-and-hold returns there is no rebalancing between portfolio for-

mation dates. However, with simple equal-weighted portfolio returns, portfolios are

rebalanced monthly.

In the appendix, we present the mathematical framework used to compute buy-

and-hold portfolio returns. To make clear which methodology is being used through-

5

out our analysis, we use the following notation for portfolio returns R:

XRWTp,b (1)

X is an optional indicator for whether non common stocks are included in the portfolio

(N). The superscript, WT , indicates the weighting methodology: equal-weighted

(EW ) or value-weighted (V W ). The first subscript, p, indicates whether the index is

constructed with daily (d) or monthly (m) returns. The second subscript, b, indicates

the rebalancing period: daily (d), monthly (m), quarterly (q), or yearly (y).

Using this notation, simple portfolio returns of common stocks are expressed as

REWm,m; the portfolio is equal-weighted EW , constructed with monthly returns m, and

rebalanced monthly m. Buy-and-hold equal-weighted portfolio returns are expressed

as REWm,y ; the only difference is the second subscript, which indicates yearly rebalanc-

ing.

We also use this notation to discuss benchmark returns created comparing CRSP

methodology with other methodologies. For example, REWm,y (NREW

m,y ) is the equal-

weighted, yearly rebalanced return for a portfolio of common stocks (all securities)

that is constructed using monthly returns.

3 Rebalancing and Market Inefficiency

We begin by examining the effect of rebalancing assumptions on evidence of market

efficiency. We examine the effect of using simple portfolio returns versus buy-and-

hold portfolio returns with the following strategies: momentum, the book-to-market

ratio, cash flow, accruals, and earnings.

6

3.1 Data and Variable Definitions

We obtain security price data from CRSP and financial statement data from Compu-

stat. The sample consists of all common stocks in CRSP from 1973, which is when

CRSP adds Nasdaq firms, to 2007. We incorporate delistings following Beaver et al.

(2007).

The momentum strategy ranks firms based on returns accumulated over a five

month period. Portfolios are formed one month after the ranking period and held for

three months. Each month, new portfolios are formed, and the total portfolio return

is computed as the average of the three concurrent portfolios.

The book-to-market ratio (BM) is computed as common equity (ceq) divided by

CRSP market value at the beginning of the return accumulation period. Earnings

(EARN) is operating income after depreciation (oiadp) deflated by average assets

(at). Cash Flows (CF ) and accruals (AC) are estimated using balance sheet data

following Sloan (1996).

Portfolios based on accounting variables are formed at the end of each April using

the most recently reported annual financial statement data. Portfolios are held for

twelve months. We present average raw monthly returns and alphas with the two

weighting methodologies discussed previously: simple portfolio returns with weights

reset to one each month, and buy-and-hold portfolio returns with securities held

without rebalancing for the holding period (three months for momentum, twelve

months for accounting-based strategies).

3.2 Results

3.2.1 Momentum

Table 1 presents the results for the momentum strategy. Panel A shows the results

over the extended time period, 1973-2007. With both buy-and-hold and simple port-

7

folio returns, the expected pattern in returns is observed. In particular the lowest

decile exhibits the lowest average monthly returns (0.3% buy-and-hold, 0.64% sim-

ple), and the highest decile exhibits the highest returns (1.64% buy-and-hold, 1.66%

simple).

The absolute difference between the buy-and-hold and simple portfolio returns

is the largest in the smallest momentum decile (i.e., recent losers). The difference

is −0.34%, compared to −0.02% in the highest decile. In fact, the lowest momen-

tum decile is the only decile with a noticeable difference between the two weighting

methodologies.

We also present alphas using the Fama and French (1993) three-factor model. In

general, the results with the alphas are consistent with the raw returns. In particular,

the buy-and-hold alpha in the lowest momentum decile is −1.08% compared with the

simple alpha of −0.66%, a difference of −0.42%.

Table 1, Panel B shows the results by time period for the extreme momentum

deciles. Consistent with the results in Panel A, the differences between the returns

using the two weighting methodologies in the lowest momentum decile are large; the

minimum is −0.18% in the 1970s, reaching a maximum of −0.66% in the 1990s.

Similarly, the minimum difference between alphas is −0.28% in the 1970s, reaching a

maximum of −0.63% in the 1990s.

These findings are striking in that simple portfolio returns produce higher average

returns. These findings are consistent with short-term price reversal (i.e., firms with

large drops in price subsequently recover). However, since the momentum strategy

involves taking short positions in the securities in this decile, the higher returns reduce

the returns to the momentum strategy. Note that this analysis ignores any potential

trading costs involved in the monthly rebalancing typically assumed by academics,

which would further reduce strategy returns.

8

3.2.2 Accounting-based Anomalies

Tables 2 through 5 present the results for the following accounting-based anomalies:

the book-to-market ratio, cash flow, accruals, and earnings respectively. The findings

with these anomalies are generally similar to those with momentum.

The results in Table 2 present the results for the book-to-market strategy. The

largest differences in returns are in the extreme deciles. In the lowest decile (growth

firms), the average monthly difference is relatively small (−0.11% in Panel A for the

extended time period). In the highest decile (value firms), the difference is larger,

−0.23%, but somewhat smaller than seen with momentum. Because both the long

and short portfolios are affected, there is a relatively small effect on the overall returns

to the strategy; they are lower by −0.12% per month with buy-and-hold returns. In

contrast, the momentum returns are much lower with simple portfolio returns vs.

buy-and-hold portfolio returns.

The results in Table 3 present the results for the cash flow strategy. The largest

differences in returns are in the lowest decile, and the differences are monotonically

decreasing across the deciles. In the lowest decile, the average monthly difference is

small (−0.44% in Panel A for the extended time period). In the highest decile (value

firms), the difference is very small, −0.04%. Similar to the momentum strategy,

the overall effect is a reduction in average monthly returns (−0.4%) as a result of

rebalancing monthly as opposed to employing a buy-and-hold strategy. The results

in Panel B do not reveal significant trends over time.

The results in Table 4 present the results for the accrual strategy. Similar to the

results with the book-to-market strategy, the raw returns in both the long (decile 1)

and short (decile 10) portfolios are lower with buy-and-hold returns. In the lowest

decile, buy-and-hold returns are lower by −0.29% vs. −0.1% in the highest decile. The

returns to the strategy would be lower by −0.19% per month if buy-and-hold returns

are used. The results in Panel B reveal that the overall strategy is only profitable in

9

the 1970s and 1990s. Again, there is not a notable trend in the differences over time.

The results in Table 5 present the results for the earnings strategy. The results

are similar to the momentum and cash flow strategy results, but the differences are

larger in magnitude. In the lowest decile, the average monthly difference between buy-

and-hold and simple portfolio returns is −0.55% vs. −0.03% in the highest decile.

The overall effect on profitability is a striking 0.52% per month. By rebalancing

monthly, the trading strategy is unprofitable (raw difference between deciles is 0.16%

per month, with an alpha of 0.32%). By following a buy-and-hold strategy, the

strategy is profitable (raw difference between deciles is 0.68% per month, with an

alpha of 0.84%).

The results show that the choice of weighting methodology can significantly alter

strategy profitability. In general, the extreme deciles of the anomaly variables are the

most likely to be affected. Depending on whether the short portfolio is affected more

than the long portfolio (e.g., momentum, cash flow, earnings) or whether the long

portfolio is affected more (e.g., book-to-market, accruals), strategy returns are either

higher or lower with a buy-and-hold strategy.

4 Problematic Issues with CRSP Portfolio Returns

CRSP provides several benchmark portfolio return measures along with index levels

based on these portfolios. These returns are commonly used due to their convenience.

However, the methodologies used by CRSP to generate these returns are often mis-

understood or simply unknown. For this reason, we provide an in-depth discussion

of the methods CRSP uses and contrast them with the more common buy-and-hold

portfolio methodology.

In this section, we discuss the inclusion of non common stock securities and other

important issues that researchers should consider before using CRSP-provided data to

10

adjust returns. We compare portfolio returns we create using CRSP methodology with

various buy-and-hold portfolio returns to demonstrate how these issues affect CRSP-

provided returns. In reconstructing the CRSP returns, we use the CRSP portfolio

assignments rather than creating our own portfolios to ensure that differences are not

attributable to differences in portfolio cutoff points. We are able to replicate CRSP

portfolio returns exactly in most periods. The few existing differences are extremely

small and are presumably due either to minor programming differences or to changes

in CRSP data over time.

4.1 ETFs and Other Non Common Stock Securities Included

in CRSP

In calculating benchmark portfolio returns, CRSP includes most2 of the securities

listed on the NYSE, AMEX, and Nasdaq exchanges. These securities include com-

mon stocks, ETFs, CEFs, American Depositary Receipts (ADRs), and real estate

investment trusts (REITs). The inclusion of non common stocks in benchmark re-

turns can be motivated by the theoretical definition of the market portfolio under the

capital asset pricing model (CAPM). Under the CAPM, the market portfolio is mean-

variance efficient and it includes all assets in the economy, not just stocks, weighted

by their market values. Thus including ADRs (foreign firms), REITs (real estate),

and CEFs and ETFs that track commodities or bonds is justifiable because the un-

derlying assets increase diversification. However, unless the weights are appropriately

set, their inclusion will not result in a more theoretically correct market portfolio.3

Two strong reasons exist for excluding non common stocks, like CEFs and ETFs,

from the market portfolio. First, the majority of these securities are portfolios of

2Some securities, such as Exchange-Traded Notes (ETNs), are not included in the benchmarkreturns, and are not included in the stock database. ETNs are included in CRSP’s mutual funddatabase.

3See Roll (1977) for a discussion of the market portfolio and the use of proxies for the marketportfolio in tests of the CAPM.

11

publicly traded common stocks. Including these securities in the market portfolio re-

sults in the over-weighting of the securities included in the ETFs and CEFs.4 Second,

a large number of ETFs are constructed to amplify short and long position returns

in certain asset classes through the use of derivative instruments. These ETFs are

among the smallest and most volatile of securities, causing significant problems for

portfolios based on size and volatility, and otherwise inducing bias or noise in all

benchmark returns.

4.1.1 Composition of CRSP Database over Time

Table 6, Panel A shows the total number of securities tracked in CRSP over time in

the NYSE, AMEX, and Nasdaq markets and the fraction of total securities that are

CEFs, ETFs, REITs, ADRs, common stock (STOCK), and other types of securities

(OTH). The fraction of the sample represented by common stocks falls from 0.986

in 1925 to 0.722 in 2007. This dramatic decrease, which largely began in the 1980s,

is primarily due to the increase in the number of CEFs and ETFs. The fraction of

CEFs (ETFs) increases from 0.011 (0.000) in 1980 to 0.096 (0.092) in 2007.

Table 6, Panel B shows the median size and the fraction of the lowest size-decile

represented by security type for CEFs, ETFs, and common stocks. The median size

of CEFs has increased from $11 million in 1925 to $199 million in 2007. The median

size of common stocks has increased from $15 million in 1925 to $356 million in 2007.

In contrast, the median size of ETFs was $45 million in 2000, $255 in 2004 and $68

million in 2007. The decrease in 2007 is due to the large number of small ETFs

created in 2006 and 2007.

While the decline in the median size of ETFs is intriguing, the most striking finding

in Panel B is the large fraction of size-decile 1 represented by ETFs. The fraction

4For example, the SPDR S&P 500 ETF, SPY, is designed to track the performance of the S&P500. It is included in all of CRSP’s portfolios along with the individual S&P 500 stocks. Thus, allelse equal, the returns of the S&P 500 stocks receive more weight in the computation of benchmarkreturns than non-S&P 500 stocks.

12

increases from 0.03 in 2000 to 0.43 in 2007. In other words, by 2007, nearly half of

all the securities in the smallest CRSP size-decile are ETFs. Given the heavy use

of these size-adjusted returns in the accounting literature, this finding is especially

relevant. In the appendix we discuss the history of ETFs and provide descriptive

statistics related to their growth and diversity in Table 14.

4.1.2 Non Common Stock Securities

In Table 7, we examine the differences between market returns calculated using com-

mon stocks and using CRSP’s method of including all security types. Monthly se-

curity returns are used to create all indices. Panel A presents the average annual

difference for monthly-rebalanced, equal-weighted market returns including all secu-

rity types (CRSP Returns, NREWm,m) vs. only common stocks (STOCK Returns, REW

m,m)

by decade. Two patterns emerge. First, the variance of the difference increases over

time, i.e., there is an increase in the absolute values of the minimum, maximum, first

quartile, and third quartile of the difference. Second, the average difference, although

relatively small, is increasingly negative over time (for annual returns it goes from

−0.003 to −0.011).

Table 7, Panel B shows that for value-weighted market returns (CRSP Returns,

NRV Wm,m vs. STOCK Returns, RV W

m,m) the effect is significantly muted, with a differ-

ence that is essentially zero. Given that CEFs and ETFs are smaller than common

stocks and are composed of portfolios of other stocks, this difference is not surprising.

However, the minimum and maximum differences are −0.8% and 1.9%, respectively,

illustrating that returns can differ even for value-weighted market returns.

Table 7, Panel C shows the difference for value-weighted, size-based returns (CRSP

Returns, NRV Wm,m vs. STOCK Returns, RV W

m,m). CRSP provides size-based returns for

ten portfolios formed annually based on the size-decile assignment at the end of

the previous year. Results are shown by decade for the smallest size-decile, where

13

increasing differences between the two measures are observed over time. For example,

for 2000-2007, the minimum annual difference is −0.313. This difference is for the year

2003 and is attributable to a closed-end fund, Evergreen Income Advantage (EAD),

that was listed with a very small number of shares in CRSP and was therefore included

in the smallest size-decile. Shortly after being listed the number of shares in CRSP

increased dramatically, and EAD comprised nearly half of the market capitalization

of the entire size-decile for the rest of the year.5 The CRSP methodology, which

includes all securities in benchmark returns, produces severely mis-stated returns in

this instance.

In sum, the inclusion of non common stocks, most notably ETF and CEF securities

affects benchmark returns. In some cases there tends to be a slight average negative

bias. Also, a significant concern is the noise introduced by including these securities

in the market return. This issue is particularly problematic with size-based returns.

4.2 Weighting Methodology

Selecting value- or equal-weighted portfolio returns is an important research design

choice. In the following paragraphs, we discuss how the weighting methodology affects

CRSP-provided benchmark returns and compare index levels and returns for CRSP

vs. buy-and-hold methodology. For this comparison we include all securities to isolate

the effect of the difference in the weighting methodologies.

4.2.1 Equal-Weighted Benchmark Returns

CRSP provides daily and monthly equal-weighted market returns as well as indices

based on these returns. In addition, CRSP provides daily risk-based, equal-weighted

portfolio returns using beta and the standard deviation of returns as risk measures.

The risk-based portfolios are rebalanced yearly based on the previous year’s risk

5In 2004, EAD was assigned to size-decile 8.

14

measure.

Compounding CRSP equal-weighted daily benchmark returns over long periods of

time (i.e., a year or longer) can lead to large differences when compared to benchmark

returns calculated using a buy-and-hold methodology. This is because CRSP equal-

weighted returns are the simple average of returns every period. For daily returns,

this is equivalent to daily rebalancing. CRSP documentation formally states the

weighting methodology on page 121 of the CRSP Data Description Guide,6 namely

that every period each stock receives a weight of 1. However other language suggests

yearly rebalancing.7 We confirm CRSP’s weighting methodology through the process

of replication. It can be described as follows: annual formation with daily rebalancing

(or monthly rebalancing for monthly returns).

We are not the first to find problems with the CRSP equal-weighted return

methodology. Canina et al. (1998) document that CRSP’s equal-weighted methodol-

ogy results in an upward bias in the CRSP equal-weighted daily market returns. They

compare CRSP-provided, compounded daily returns with the average of monthly se-

curity returns (i.e., the monthly return provided by CRSP) and find that compounded

daily returns are significantly higher than the average of monthly returns. They at-

tribute this to the effects of market microstructure, primarily bid-ask bounce and

nonsynchronous trading. They caution researchers against using equal-weighted daily

market returns to compute long-run excess returns. Canina et al. (1998) also com-

pare value-weighted compounded daily returns with the average of value-weighted

monthly security returns, but they compare the two return series only to show that

value-weighted returns are not plagued by market microstructure effects. In the fol-

lowing subsection, we examine value-weighted returns in detail and demonstrate that

6CRSP documentation states on page 121 “In an equal-weighted portfolio, wn(I) = 1 for everystock. Such a portfolio would consist of n stocks with the same dollar amount invested in eachstock.”

7The CRSP Data Description Guide states on page 29 “CRSP Stock File Risk-Based DecileIndices are rebalanced each year.”

15

they are subject to their own set of problems.

Earlier studies discuss the issue of compounding daily returns without referring

explicitly to CRSP. Blume and Stambaugh (1983) and Roll (1983) address how alter-

native strategies to forming portfolios can bias returns and lead to an exaggeration

of the size effect. They argue the bias is mainly a result of the “bid-ask effect” in

closing price and to a lesser extent nonsynchronous trading. They show analytically

that returns to a buy-and-hold strategy are free from the bias. Furthermore, they

document that the bias is more prevalent in stocks with low prices where the bid-ask

spread is a significant portion of the price.

Table 8, Panel A presents the results for equal-weighted market returns: daily

rebalanced CRSP returns, NREWd,d vs. monthly rebalanced, buy-and-hold benchmark

returns (PORT), NREWd,m .8 The 2007 index level9 associated with the CRSP method-

ology is 151,525, which is significantly higher than the buy-and-hold portfolio index

level of 9,191. We also present the distribution of the difference between the two re-

turns. In general, CRSP returns are significantly higher than buy-and-hold returns.

These results are essentially a replication of Canina et al. (1998), who show that the

differences are explained by the bid-ask bounce. The problem is especially severe in

the 1990s, when the minimum difference between the return measures is 0.144, and

the maximum difference is 0.382. In untabulated results, we find that this difference is

eliminated using CRSP methodology if mid-point prices are used to compute returns

rather than quoted prices, which is consistent with bid-ask bounce biasing returns

upward.

Table 8, Panel B shows the results for portfolios formed on the standard devia-

tion of returns for Nasdaq10 securities (daily rebalanced CRSP Returns, NREWd,d vs.

8Results are very similar if PORT returns are rebalanced yearly (NREWd,y ).

9CRSP sets all indices to 100 at the end of 1972. We do the same.10CRSP provides risk-based daily portfolio returns for the sample of Nasdaq firms and for the

sample of NYSE and AMEX firms, but not for the combined sample of all securities (CRSP marketand size-based portfolios are provided for the combined sample). Results are similar with a sampleof NYSE and AMEX firms.

16

yearly rebalanced PORT Returns, NREWd,y ). Decile 1 includes firms with the highest

standard deviation of returns in the preceding year. The index level in decile 1 is

a staggering 2.52 trillion compared to a buy-and-hold index level of 2,046. The dif-

ference is the most extreme in decile 1, and it decreases nearly monotonically across

deciles. Untabulated results examining the CRSP beta portfolios are similar to those

presented for the standard deviation of returns. These benchmark returns are not

commonly used by researchers, but these results are useful in highlighting the severity

of the problems with CRSP weighting methodology.

4.2.2 Value-Weighted Benchmark Returns

CRSP value-weighted benchmark returns use lagged market value as the portfolio

weight and are largely free of the bid-ask effect that biases CRSP equal-weighted

daily benchmark returns. However, two distinct issues are raised when compounding

CRSP value-weighted returns.

First, the CRSP method assumes that dividends are not reinvested in the same

firm, but are reinvested among all stocks in the portfolio. This assumption is not

inherently problematic, but relative to a buy-and-hold methodology that reinvests

dividends in the same stock, portfolio weights are different and, consequently, bench-

mark returns are different. Second, and more likely to be problematic, the CRSP

method of weighting by lagged market value of equity assumes that portfolios are

rebalanced whenever there is a change in the number of shares outstanding due to

events such as stock repurchases and seasoned equity offerings, which change the

market value of the firm, but do not affect the weight of a buy-and-hold portfolio.

Depending on the research objectives, the CRSP method of weighting returns

by lagged market value may be appropriate. By using lagged market value as the

weight, CRSP ensures that changes in market value due to changes in capital structure

are incorporated in the weights. However, with respect to benchmark portfolios,

17

such as size-based portfolios, this methodology can result in firms being significantly

over- or under-weighted. For example, if a firm issues additional shares through a

seasoned equity offering and doubles in market capitalization, the CRSP portfolio

weight doubles whereas the buy-and-hold weight remains the same. Another concern

is that changes in shares outstanding are often incorporated with a delay in CRSP

(Ince and Porter 2006).

Table 9 presents the comparison of value-weighted index levels and returns using

CRSP methodology vs. buy-and-hold methodology. For this comparison we include

all securities except ADRs (which are excluded by CRSP in the computation of value-

weighted returns), to isolate the effect of the difference in the weighting methodologies.

Table 9, Panel A shows that the difference in index levels and returns is extremely

small for value-weighted market returns (daily rebalanced CRSP Returns, NRV Wd,d vs.

monthly rebalanced PORT Returns, NRV Wd,m ), consistent with Canina et al. (1998).

Table 9, Panel B shows the results by size-decile (daily rebalanced CRSP Returns,

NRV Wd,d vs. yearly rebalanced PORT Returns, NRV W

d,y ), where decile 1 contains the

smallest securities. In contrast to the absence of an effect with market returns in Panel

A, the returns for the smallest size-decile show increasing variance and an increasingly

negative average difference over time. The 2007 CRSP index level is 44,028 vs. a buy-

and-hold index level of 68,147. This shows that CRSP methodology results in lower

returns, on average. As an example, from 2000 to 2007 the average difference was

−0.039, with a minimum (maximum) annual difference of −0.356 (0.159). In the

1970s, the difference was essentially zero.

As discussed previously, the extremely large difference of −0.356 is attributable

to the Evergreen Income Advantage (EAD) closed-end fund in 2003. When buy-

and-hold returns are used, changes in market capitalization that are not attributable

to past returns are excluded from the portfolio weight. In this instance, the use of

buy-and-hold weights prevents EAD from dominating the lowest decile of size-based

18

returns in 2003.

4.3 Delisting Returns

The final issue we address is that CRSP benchmark returns generally do not incor-

porate delisting returns.11 CRSP daily benchmark returns include all standard daily

returns, but the delisting return itself is excluded. CRSP monthly portfolio returns

include all standard monthly returns prior to the month of delisting, but exclude

partial-month returns and the delisting return. This presents obvious problems with

benchmark returns and index levels. If researchers take steps to include delisting re-

turns in the sample following the recommendations of Shumway (1997), Shumway and

Warther (1999) and Beaver et al. (2007), but do not adjust the benchmark portfolio,

the potential for bias exists.

In untabulated analysis we find that the effect of delistings on benchmark returns

is relatively small, and therefore we do not discuss them in detail. The delisting

effect is most significant in the smallest size decile. In this setting, the exclusion of

delistings results in higher returns than if they are included. However, the variation

in returns is primarily attributable weighting methodology.

5 Application in Research Settings

With an understanding of the issues related to CRSP benchmark returns, we select

four targeted research settings to illustrate how results and inferences can be affected.

We choose the following settings: excess annual size-adjusted returns to anomalies,

the calculation betas, the size effect, and dividend yields.

11The cap-based portfolios provided by CRSP do include delistings, but they are not commonlyused. The most commonly used benchmark returns exclude delistings. These measures include stockfile capitalization decile indices (decret), equal-weighted market returns (ewretd), and value-weightedmarket returns (vwretd).

19

5.1 Excess Returns to Anomalies

First, we examine how returns to cash flow, accruals and the book-to-market anoma-

lies (Lakonishok et al. 1994; Sloan 1996) are affected by the CRSP weighting method-

ology. We present the raw strategy returns including delistings for the extreme deciles

of cash flows (CF) and accruals (AC) deflated by average assets and for the book-

to-market ratio (BM). We compute size-adjusted excess returns similar to Lakon-

ishok et al. (1994) and Sloan (1996). We report the size adjustment using the CRSP

methodology (CRSP Returns, NRV Wm,m, which includes non common stocks, rebalances

monthly, and excludes delistings) with our corrected measure (STOCK returns, RV Wm,y ,

which excludes non common stocks, rebalances yearly, and includes delistings).

The sample period is fiscal years 1973−2006 with firms ranked yearly based on

the end-of-year value of CF, AC, and BM, with return measurement starting four

months after fiscal year-end, continuing for twelve months.12 Pooled average returns

are presented.

Table 10, Panel A shows a large difference between the raw returns in the lowest

CF decile (0.052) and highest CF decile (0.190). The size adjustment and, con-

sequently, excess returns vary depending on the methodology. For the lowest CF

decile, the size adjustment is 0.144 using CRSP methodology (excluding delistings

and rebalancing monthly) vs. 0.149 using a yearly rebalanced size adjustment com-

puted with common stocks, including delistings. A similar pattern is observed in the

highest CF decile, and in Panels B and C with accruals and the book-to-market ratio.

We also present the results for several time periods in the latter part of the sample.

The results show that over time the difference between size adjustments varies. In

Panel A, for the lowest decile of cash flows, there is a difference of −0.024 in the 2001-

2003 time period. In contrast, the difference is 0.009 in the 2004-2007 time period.

12We begin the sample period in 1973 because CRSP adds Nasdaq firms in this year and sets allindex levels as of the end of 1972. Also, measured returns extend to the beginning of 2008.

20

Similar trends are seen in the highest decile of cash flows and in Panels B and C with

accruals and the book-to-market ratio.

The results in Table 10 show that excess returns to the short positions generally

increase in absolute magnitude for all strategies when using our corrected buy-and-

hold benchmark returns based on common stocks. In contrast, the excess returns to

the long position decrease. Overall there is a shift of trading strategy returns in the

direction of the short side. Given the difficulty of taking and holding short positions

for extended periods of time, the increased dependency of these strategies on the

short side for their profitability suggests their implementation may be difficult.

Figure 1 plots the annual difference between the size adjustment for the lowest

deciles of CF, AC, and BM. The minimum difference is −0.121 for CF in 2003. The

maximum difference is 0.044 for CF in 2007. The results in Table 10 document that

the difference between the CRSP size adjustment and our corrected size adjustment

tends to be negative. However, there is significant variation from year to year.

5.2 Estimation of Betas

Second, we examine the effect of using CRSP market returns to estimate betas. We

estimate betas following Scholes and Williams (1977), which is also the method used

by CRSP. We estimate betas for each common stock for each calendar year from 1973-

2008 using daily data. Betas are computed using equal-weighted and value-weighted

market returns.

We compare average betas calculated using CRSP market returns (which include

all non common stocks) vs. our proposed market returns (composed only of common

stocks). In general, an average beta of one is expected for the universe of stocks.

Average betas may not equal one for if the market return includes other securities

(such as ETFs and CEFs) or if value-weighted market returns are used to compute

beta, but the simple average of beta is computed (which is typically the case).

21

In Table 11, Panel A, we compare the betas estimated using CRSP market returns

with the betas estimated using our proposed market returns, which exclude non

common stocks. On the left side of the panel, we present the comparison of betas

computed with equal-weighted returns (equal-weighted betas) using CRSP (βallEW ) vs.

our corrected market returns (βcsEW ). On the right side of Panel A, we present the

comparison of betas computed with value-weighted returns (value-weighted betas)

using CRSP (βallV W ) vs. our corrected market returns (βcs

V W ).

For equal-weighted betas in the early time period, 1973-1989, the average βCEW of

1.011 is slightly higher than the average βPEW 0.992 using our corrected returns. In

the later time periods βCEW increases steadily, and in the 2000-2008 time period the

average beta is 1.071. This is noticeably higher than the average βPEW of 1.013 using

our corrected returns. It is striking that our corrected betas are much closer to the

theoretical value of one for the entire time period. Also presented is the ninety-fifth

percentile of the difference between βCEW and βP

EW . In 1973-1989, the ninety-fifth

percentile of the difference is 0.085. This difference increases over time and in 2000-

2008 is 0.207.

For value-weighted betas in the early time period, 1973-1989, the average βallV W of

0.753 is slightly higher than the average βcsEW of 0.746 using our corrected returns.

In the latest time period, 2000-2008, βallV W increases steadily to 0.943, compared with

0.934 for βcsV W . The ninety-fifth percentile of the difference between βC

V W and βPV W is

under 0.05 for all time periods.

We also present average betas for the lowest and highest size-deciles of common

stocks for 2000-2008. In general, the difference between CRSP betas, βall, and our

corrected betas, βcs, does not appear to be driven by small firms. It is interesting to

note that the average beta for small stocks (decile 1) is significantly lower than the

rest of the sample. In addition, equal-weighted betas, βcsEW , are significantly higher

than value-weighted betas, βcsV W for small firms: βcs

EW is 0.796, but βcsV W is 0.528.

22

These results demonstrate that estimated betas differ when using CRSP market

returns vs. our corrected market returns. The difference is more pronounced when

using equal-weighted market returns to compute betas. We show that betas are

generally higher when estimated using CRSP market returns.

5.3 The Size Effect

Third, we examine how including non common stock securities may affect inferences

on the size effect. The size effect is a well-known asset pricing anomaly first reported

by Banz (1981); he showed that small firms earn higher returns than large firms. A

vast literature has explored the size effect since Banz’s paper (Blume and Stambaugh

1983; Roll 1983), including many studies that argue the size effect disappeared in

the 1980s (Gompers and Metrick 2001), and studies that conclude the size effect is

a product of research design choices (Shumway and Warther 1999). However, other

papers provide evidence the size effect still exists in security returns (Hou and VanDijk

2008).

Non common stock securities, particularly ETFs and CEFs, may significantly

influence the size effect because they consist of other securities of varying size. To

examine this, we report average returns and correlations for securities in the smallest

size-decile (decile 1) and stocks in the highest size-decile (decile 10). In the largest

size-decile, we include only common stocks. In the smallest size-decile, we examine

the partition of only ETF and CEF securities, the partition of only common stocks,

and report the correlation between large stocks and all securities in the lowest size-

decile. We use monthly returns, assuming monthly rebalancing to avoid weighting

issues. We report results with both equal-weighted and value-weighted returns.

Table 12, Panel A reports average returns for large stocks (D10cs), small stocks

(D1cs), and small ETFs and CEFs (D1etf). The results show that, in general,

there is more variation in the returns of small stocks, and that the returns to small

23

ETFs/CEFs are more similar to large stocks than small stocks. Consistent with this,

Panel B shows that the correlation between large stocks and small ETFs/CEFs is

significantly higher than the correlation between small stocks and large stocks. The

correlation after 2000 (when the number of ETFs in smallest size-decile began to

increase) is generally in excess of 0.70, with a maximum in 2007 (correlation is 0.979

for equal-weighted returns and 0.967 for value-weighted returns).

By including increasing numbers of ETF securities in the lowest size-decile, the

returns of this decile start to resemble the returns to large stocks, with the potential

to mute the size effect.13 This highlights the problems with the size-based benchmark

returns frequently used in the accounting literature. The CRSP size adjustment for

the smallest size-decile is artificially similar to the highest size-decile in recent years

and thus it is an inadequate and inappropriate size adjustment for small firms (which

are disproportionately represented in extreme deciles of the anomaly variables).

5.4 Dividend Yields

Finally, we examine how the inclusion of non common stock securities affects market-

wide dividend yields (Fama and French 1988; Cochrane 2008). We compute annual

dividend yields for the market portfolio using equal- and value-weighted returns and

index levels - a method that is commonly employed in the finance literature. We

compare yields computed using CRSP methodology, with yields that exclude non

common stocks.14 We use monthly returns, assuming monthly rebalancing to avoid

weighting issues.

The results in Table 13 show that in early years, the inclusion of non common stock

securities does not significantly affect yields. Starting in the early 1990s, when CEF

13Most papers which examine the size effect are careful to include only common stocks (Hou andVanDijk 2008) so their inferences will not change as a result of the increase in the number of ETFs.Furthermore, two securities can be perfectly positively correlated, but have different average returns.The results in Table 12, Panel A suggest that the average returns of the sample of ETF and CEFsecurities in the lowest size-decile are more similar to large stocks than to small stocks.

14Delistings are included. Inferences are unaffected if delistings are excluded.

24

and ETF securities became more common, the difference between dividend yields

computed with all securities begins to be significantly higher than yields computed

with common shares. By 2007, CRSP equal-weighted dividend yields are nearly

double (0.018 vs. 0.010). Because CEF and ETF securities are composed of many

(often large) stocks, the dividend yield is significantly higher for these securities.

Table 13 also shows that the difference in value-weighted dividend yields is not

as dramatic as with equal-weighted dividend yields; CRSP value-weighted dividend

yields are roughly 10% higher after 2000 (in 2007 the yield is 0.020 vs. 0.018).

However, given the extremely minor differences observed in Table 7, Panel B between

value-weighted market returns for common stocks vs. all securities, this difference is

striking.

6 Conclusion

We show that the common practice in academic research of taking the simple average

of monthly security returns to compute equal-weighted portfolio returns can impact

returns to strategies that exploit pricing anomalies. By computing these simple port-

folio returns, academics assume portfolios are rebalanced monthly, but monthly re-

balancing of the same set of securities effectively leads to consistently selling winners

and buying losers. It is unlikely that managers of hedge funds and other practioners

rebalance portfolios this way. We show that simple returns are typically larger than

buy-and-hold portfolio returns, especially in the lowest decile of these variables. Our

results are important for academics who design tests of market efficiency so that their

returns are more representative of returns that can be achieved in practice.

We also examine the methodology used by CRSP to compute benchmark returns

and identify three problematic issues that can have a significant effect on computed

returns and the inferences researchers draw from their results. First, CRSP bench-

25

mark returns include closed-end funds, exchange-traded funds, and other non com-

mon stock securities that are generally excluded by researchers. Second, the weighting

methodologies used for CRSP equal- and value-weighted returns differ from the typ-

ical buy-and-hold methodology. Third, CRSP benchmark returns generally exclude

delisting returns.

We show that including non common stock securities such as ETFs significantly

influences CRSP-provided benchmark returns. We also show that CRSP’s weighting

methodology and exclusion of delistings can cause large differences between CRSP

benchmark returns and those calculated using a more traditional buy-and-hold method-

ology that includes delistings.

In four applied research settings we document the effects of CRSP methodology

on inferences. First, we document changes in the excess returns to trading strategies

based on cash flows, accruals, and the book-to-market ratio. Long strategy returns

are reduced, and short strategy returns are increased resulting in an overall shift in

strategy returns to the short side, suggesting the strategies may be more difficult to

implement. Second, we show that betas calculated using CRSP market returns are

generally larger than betas calculated using our corrected market returns. This di-

rectly affects beta-adjusted market returns and has the potential to change inferences

in an event study. Third, we show that because of the increasingly large number

of ETFs in the smallest size-decile, the correlation between small stocks and large

stocks is significantly overstated. Finally, we show that dividend yields are signif-

icantly higher when non common stock securities are included in the portfolio of

securities used to compute returns.

The potential number of areas where our research is relevant is strikingly large

because so many researchers use CRSP benchmark in their research. Our paper serves

to inform researchers of assumptions underlying CRSP market returns so they can

determine whether the returns provided by CRSP are appropriate for their specific

26

research design, and to help ensure that more accurate assessments of market returns

are used to study market-related phenomena.

27

A Appendix: Computing Portfolio Returns

We describe in general terms how traditional buy-and-hold portfolio returns are cal-

culated and present a basic mathematical framework for deriving portfolio returns.

We use returns calculated using buy-and-hold methodology as our benchmark be-

cause they are equivalent to investment returns. Blume and Stambaugh (1983) and

Roll (1983) show analytically that buy-and-hold returns are free from the weighting

issues we discuss in the body of the paper.

Portfolio returns for period t, Rt, are determined by the return for each security

i during period t, ri,t, and by the weight of each security, ωi,t.

Rt =∑

i

ωi,tri,t (2)

The weight of each security, ωi,t, is determined by the initial unscaled weight15 of

the security, ui,0, and the cumulative returns through period t − 1. Let ui,t be the

unscaled weight of security i on day t.

ui,t = ui,0

t−1∏

τ=1

(1 + ri,τ ) if t > 1 (3)

= ui,0 if t = 1 (4)

For an equal-weighted portfolio, the initial unscaled weight is equal for all securities

(i.e., ui,0 = 1) . For a value-weighted portfolio, the initial unscaled weight is the

beginning market value of equity. If a portfolio is equal-weighted, ωi,0 = 1/n, where

n is the number of firms in the portfolio. The weight of security i on date t, ωi,t, can

15The initial unscaled weight can be thought of as the dollar amount initially invested in thesecurity.

28

be expressed as follows:

ωi,t =ui,t∑j uj,t

(5)

After a period of time, portfolios may be rebalanced. This may involve altering

portfolio assignments or resetting portfolio weights. The following notation expands

the definitions above to incorporate rebalancing intervals. Let b be the rebalancing

interval (e.g. daily, monthly, quarterly or yearly). Portfolio returns with rebalancing

interval b are defined as follows:

Rb,t =∑

i

ωb,i,tri,t (6)

Let b(t) be the most recent rebalancing date preceding t. The unscaled portfolio

weight based on rebalancing interval b is defined as follows:

ub,i,t = ub,i,b(t)

t−1∏

τ=b(t)

(1 + ri,τ ) if t > b(t) + 1 (7)

= ub,i,b(t) if t = b(t) + 1 (8)

The weight of security i on date t with rebalancing period b, ωb,i,t, can be expressed

as follows:

ωb,i,t =ub,i,t∑j ub,j,t

(9)

B Appendix: ETF Descriptive Statistics

Due to the relative novelty of ETFs, and the lack of academic research in the area,

we provide some descriptive statistics about the growth of ETFs. Although ETFs are

similar to mutual funds and closed-end funds, specific features of ETFs give them an

29

advantage over either of these alternatives. The fact that ETFs are exchange-traded

facilitates trading and has had the effect of reducing fees. ETFs are structured to

allow the creation and redemption of additional shares. This open-ended feature of

ETFs is an arbitrage mechanism that helps to ensure that shares of ETFs closely track

the value of the underlying securities, thus avoiding the premiums and/or discounts

that often plague CEFs (Lee et al. 1991; Dimson and Minio-Kozerski 1999).

Table 14, Panel A provides details about the number and types of ETFs (e.g.,

index, commodity, equity) over time based on ETF descriptions. The first ETF,

SPY, was listed in 1993 and was designed to track the S&P 500. In 2000 the number

of ETFs jumped significantly from 31 to 92. Growth again slowed until 2005-2008.

The year with the largest growth was 2007, when the number increased from 374 to

634. At the end of 2008, the number of ETFs was 717.

In addition to the number of listed ETFs, Panel A also shows the proportion of

ETFs designed to track equities, bonds, or commodities. Prior to 1996, all ETFs were

designed to track U.S. equities. In 1996, 17 ETFs were created to track various indices

around the world. For example, the iShares MSCI United Kingdom Index Fund was

created to provide returns that correspond to the performance of the British equity

market. Non-equity ETFs began to appear in 2001 with the creation of commodity

ETFs. Bond ETFs followed in 2002. At the end of 2008, 54% of all ETFs track U.S.

equities, 30% track global equities, 9% track commodities (including currencies), and

7% track bonds. Untabulated analysis reveals that as of the end of 2008, 82 ETFs

had been delisted since their inception in 1993. Of these, 71 ETFs were delisted in

2008 with many of these ETFs being delisted within one year of listing. We examined

press releases for many of the delisted ETFs and discovered that the primary reason

for delisting was the lack of investor demand for highly specialized ETFs.16

Table 14, Panel B shows descriptive statistics of the investment strategies of ETFs.

16The primary analysis ends in 2007 because data for 2008 were unavailable to us when the analysiswas conducted.

30

Prior to 2006, most ETFs were designed to track an index, an industry, or followed a

size or value/growth (BM) strategy. In 2006, ETFs with short or leveraged strategies

appear in the sample, and by 2008, 21% of all ETFs are leveraged (with most of these

tracking equities).17 Together, Tables 6 and 14 demonstrate the remarkable growth,

in size and diversity, experienced by ETFs in the last ten years.

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33

-0.15

-0.10

-0.05

0.00

0.05

0.10

1990 1992 1994 1996 1998 2000 2002 2004 2006

Difference

Year

Annual Difference between CRSP Size adjustmentand Stock Portfolio Size Adjustment

CF

♦ ♦ ♦♦ ♦ ♦

♦♦

♦

♦

♦

♦

♦

♦

♦

♦

♦

♦

♦AC

? ??

? ? ??

?

?

? ?

?

?

?

?

?

?

?

?BM

rr r

rr

r

r

r

r

r

r

r

r

r

r

r

r

r

r

Figure 1: This figure presents the difference between the size adjustment computed us-ing CRSP methodology (which includes non common-stocks, rebalances monthly andexcludes delistings) and our proposed size adjustment (which excludes non common-stocks, rebalances yearly, and includes delistings). The difference is presented byyear for the lowest decile of operating cash flow (CF), accruals (AC), and the book-to-market ratio (BM).

34

Table 1

Portfolio Rebalancing and Momentum Returns

Raw Returns AlphasDecile Buy-Hold Simple Diff Buy-Hold Simple Diff

Panel A: Pooled Momentum Returns, 1973-20071 0.0030 0.0064 −0.0034 −0.0108 −0.0066 −0.00422 0.0072 0.0080 −0.0008 −0.0061 −0.0046 −0.00153 0.0100 0.0104 −0.0004 −0.0028 −0.0021 −0.00074 0.0113 0.0115 −0.0002 −0.0014 −0.0010 −0.00045 0.0124 0.0126 −0.0002 −0.0002 0.0001 −0.00036 0.0130 0.0131 −0.0001 0.0007 0.0008 −0.00017 0.0128 0.0129 −0.0001 0.0006 0.0007 −0.00018 0.0136 0.0137 −0.0001 0.0015 0.0015 0.00009 0.0149 0.0150 −0.0001 0.0031 0.0031 −0.000110 0.0164 0.0166 −0.0002 0.0049 0.0052 −0.0003

10−1 0.0134 0.0103 0.0032 0.0157 0.0117 0.0040

Panel B: Momentum Returns By DecadeDecade Decile 11970 0.0098 0.0116 −0.0018 −0.0084 −0.0056 −0.00281980 −0.0043 −0.0002 −0.0042 −0.0192 −0.0153 −0.00391990 0.0079 0.0145 −0.0066 −0.0035 0.0028 −0.00632000 −0.0001 0.0035 −0.0035 −0.0031 0.0010 −0.0040

Decile 101970 0.0144 0.0145 −0.0001 0.0034 0.0042 −0.00081980 0.0139 0.0144 −0.0005 0.0000 0.0005 −0.00051990 0.0192 0.0193 −0.0001 0.0057 0.0056 0.00022000 0.0179 0.0182 −0.0003 0.0078 0.0079 0.0000

Decile 10-11970 0.0046 0.0029 0.0017 0.0118 0.0098 0.00191980 0.0182 0.0145 0.0037 0.0192 0.0158 0.00341990 0.0113 0.0048 0.0065 0.0092 0.0028 0.00642000 0.0180 0.0148 0.0032 0.0109 0.0069 0.0040

This table presents the raw returns and alphas for momentum deciles. Themomentum strategy ranks firms based on returns accumulated over a fivemonth period. Portfolios are formed one month after the ranking period andheld for three months. Each month, new portfolios are formed, and the totalportfolio return is computed as the average of the three concurrent portfolios.We present average returns assuming portfolios are formed and held withoutrebalancing for three months (Buy-Hold), and also as the simple average ofsecurity returns, assuming monthly rebalancing (Simple). The sample consistsof all common stocks in CRSP from 1973-2007.

35

Table 2

Portfolio Rebalancing and Book-to-Market Returns

Raw Returns AlphasDecile Buy-Hold Simple Diff Buy-Hold Simple Diff

Panel A: Pooled Book-to-Market Returns, 1973-20071 0.0045 0.0056 −0.0011 −0.0065 −0.0058 −0.00072 0.0083 0.0091 −0.0008 −0.0028 −0.0023 −0.00053 0.0107 0.0116 −0.0009 −0.0011 −0.0006 −0.00044 0.0115 0.0119 −0.0004 −0.0007 −0.0008 0.00015 0.0124 0.0133 −0.0008 −0.0001 0.0003 −0.00046 0.0135 0.0141 −0.0006 0.0007 0.0008 0.00007 0.0137 0.0146 −0.0009 0.0005 0.0011 −0.00068 0.0156 0.0164 −0.0008 0.0019 0.0022 −0.00039 0.0162 0.0172 −0.0010 0.0018 0.0023 −0.000610 0.0162 0.0185 −0.0023 0.0012 0.0029 −0.0017

10−1 0.0117 0.0129 −0.0012 0.0077 0.0088 −0.0010

Panel B: Book-to-Market Returns By DecadeDecade Decile 11970 0.0117 0.0137 −0.0021 0.0015 0.0026 −0.00111980 −0.0004 0.0000 −0.0004 −0.0149 −0.0133 −0.00161990 0.0086 0.0099 −0.0013 −0.0043 −0.0030 −0.00132000 −0.0006 −0.0001 −0.0005 −0.0043 −0.0017 −0.0027

Decile 101970 0.0184 0.0221 −0.0037 −0.0015 0.0006 −0.00211980 0.0154 0.0164 −0.0010 −0.0008 0.0007 −0.00151990 0.0169 0.0211 −0.0042 0.0036 0.0057 −0.00212000 0.0144 0.0150 −0.0006 0.0025 0.0041 −0.0016

Decile 10-11970 0.0068 0.0084 −0.0016 −0.0030 −0.0020 −0.00101980 0.0158 0.0164 −0.0006 0.0141 0.0140 0.00011990 0.0083 0.0112 −0.0029 0.0079 0.0087 −0.00082000 0.0151 0.0152 −0.0001 0.0068 0.0057 0.0011

This table presents the raw returns and alphas for deciles of the book-to-marketratio (BM) over different time periods. The strategy ranks firms based on BMfrom the most recently reported fiscal year. We present average monthly re-turns and alphas assuming portfolios are formed and held without rebalancingfor twelve months (Buy-Hold) and as the simple average of security returnsassuming monthly rebalancing (Simple). The sample consists of all commonstocks in CRSP with data in CRSP for fiscal years 1969-2006; the correspond-ing return period is 1970-2007. Portfolios are assigned at the end of eachApril.

36

Table 3

Portfolio Rebalancing and Cash Flow Returns

Raw Returns AlphasDecile Buy-Hold Simple Diff Buy-Hold Simple Diff

Panel A: Pooled Cash Flow Returns, 1973-20071 0.0056 0.0100 −0.0044 −0.0076 −0.0035 −0.00412 0.0103 0.0123 −0.0020 −0.0027 −0.0011 −0.00163 0.0120 0.0133 −0.0013 −0.0013 −0.0004 −0.00094 0.0133 0.0140 −0.0007 0.0002 0.0005 −0.00035 0.0131 0.0137 −0.0006 0.0000 0.0002 −0.00026 0.0139 0.0144 −0.0005 0.0009 0.0011 −0.00027 0.0147 0.0153 −0.0007 0.0019 0.0022 −0.00038 0.0155 0.0159 −0.0004 0.0030 0.0030 0.00009 0.0149 0.0152 −0.0004 0.0025 0.0026 0.000010 0.0152 0.0155 −0.0004 0.0032 0.0033 0.0000

10−1 0.0095 0.0055 0.0040 0.0108 0.0068 0.0041

Panel B: Cash Flow Returns By DecadeDecade Decile 11970 0.0139 0.0171 −0.0032 −0.0057 −0.0040 −0.00171980 0.0005 0.0048 −0.0044 −0.0137 −0.0078 −0.00591990 0.0150 0.0196 −0.0046 0.0022 0.0067 −0.00452000 −0.0062 −0.0014 −0.0047 −0.0124 −0.0055 −0.0068

Decile 101970 0.0157 0.0168 −0.0010 0.0038 0.0036 0.00031980 0.0165 0.0165 0.0000 0.0036 0.0037 0.00001990 0.0161 0.0164 −0.0004 0.0034 0.0036 −0.00022000 0.0118 0.0122 −0.0004 0.0037 0.0047 −0.0011

Decile 10-11970 0.0018 −0.0003 0.0021 0.0095 0.0075 0.00201980 0.0160 0.0116 0.0044 0.0174 0.0115 0.00591990 0.0011 −0.0032 0.0043 0.0012 −0.0031 0.00432000 0.0180 0.0136 0.0044 0.0160 0.0103 0.0058

This table presents the raw returns and alphas for deciles of operating cash flowdeflated by average assets (CF) over different time periods. The strategy ranksfirms based on CF from the most recently reported fiscal year. We presentaverage monthly returns and alphas assuming portfolios are formed and heldwithout rebalancing for twelve months (Buy-Hold) and as the simple average ofsecurity returns assuming monthly rebalancing (Simple). The sample consistsof all common stocks in CRSP with data in CRSP for fiscal years 1969-2006;the corresponding return period is 1970-2007. Portfolios are assigned at theend of each April.

37

Table 4

Portfolio Rebalancing and Accrual Returns

Raw Returns AlphasDecile Buy-Hold Simple Diff Buy-Hold Simple Diff

Panel A: Pooled Accrual Returns, 1973-20071 0.0122 0.0151 −0.0029 −0.0013 0.0013 −0.00262 0.0142 0.0155 −0.0013 0.0013 0.0021 −0.00083 0.0149 0.0161 −0.0012 0.0023 0.0032 −0.00094 0.0147 0.0152 −0.0005 0.0021 0.0021 0.00005 0.0144 0.0148 −0.0004 0.0019 0.0018 0.00016 0.0131 0.0140 −0.0009 0.0004 0.0009 −0.00057 0.0131 0.0138 −0.0006 0.0007 0.0008 −0.00018 0.0124 0.0135 −0.0011 −0.0002 0.0005 −0.00079 0.0126 0.0133 −0.0008 0.0000 0.0003 −0.000310 0.0077 0.0087 −0.0010 −0.0056 −0.0052 −0.0004

10−1 0.0045 0.0064 −0.0019 0.0043 0.0065 −0.0022

Panel B: Accrual Returns By DecadeDecade Decile 11970 0.0192 0.0220 −0.0028 0.0034 0.0041 −0.00081980 0.0075 0.0100 −0.0025 −0.0054 −0.0026 −0.00271990 0.0172 0.0211 −0.0039 0.0034 0.0073 −0.00392000 0.0055 0.0076 −0.0021 −0.0035 0.0011 −0.0045

Decile 101970 0.0117 0.0136 −0.0019 −0.0045 −0.0039 −0.00071980 0.0062 0.0064 −0.0001 −0.0077 −0.0067 −0.00101990 0.0081 0.0094 −0.0013 −0.0056 −0.0062 0.00072000 0.0060 0.0067 −0.0007 −0.0016 0.0005 −0.0021

Decile 10-11970 0.0076 0.0084 −0.0009 0.0079 0.0080 −0.00011980 0.0013 0.0037 −0.0024 0.0024 0.0041 −0.00171990 0.0091 0.0117 −0.0026 0.0090 0.0135 −0.00452000 −0.0005 0.0009 −0.0014 −0.0019 0.0006 −0.0025

This table presents the raw returns and alphas for deciles of operating accrualsdeflated by average assets (AC) over different time periods. The strategy ranksfirms based on AC from the most recently reported fiscal year. We presentaverage monthly returns and alphas assuming portfolios are formed and heldwithout rebalancing for twelve months (Buy-Hold) and as the simple average ofsecurity returns assuming monthly rebalancing (Simple). The sample consistsof all common stocks in CRSP with data in CRSP for fiscal years 1969-2006;the corresponding return period is 1970-2007. Portfolios are assigned at theend of each April.

38

Table 5

Portfolio Rebalancing and Earnings Returns

Raw Returns AlphasDecile Buy-Hold Simple Diff Buy-Hold Simple Diff

Panel A: Pooled Earnings Returns, 1973-20071 0.0063 0.0118 −0.0055 −0.0069 −0.0018 −0.00512 0.0105 0.0132 −0.0027 −0.0025 −0.0002 −0.00233 0.0133 0.0142 −0.0009 −0.0005 −0.0001 −0.00044 0.0140 0.0146 −0.0006 0.0005 0.0006 −0.00015 0.0137 0.0139 −0.0003 0.0009 0.0007 0.00026 0.0144 0.0145 −0.0001 0.0013 0.0010 0.00037 0.0143 0.0145 −0.0002 0.0014 0.0012 0.00018 0.0138 0.0140 −0.0002 0.0011 0.0009 0.00029 0.0138 0.0139 −0.0001 0.0011 0.0010 0.000110 0.0131 0.0134 −0.0003 0.0015 0.0014 0.0001

10−1 0.0068 0.0016 0.0052 0.0084 0.0032 0.0052

Panel B: Earnings Returns By DecadeDecade Decile 11970 0.0138 0.0184 −0.0045 −0.0062 −0.0036 −0.00261980 −0.0009 0.0049 −0.0058 −0.0150 −0.0079 −0.00711990 0.0165 0.0231 −0.0066 0.0017 0.0091 −0.00742000 −0.0036 0.0009 −0.0045 −0.0097 −0.0027 −0.0069

Decile 101970 0.0122 0.0132 −0.0010 0.0015 0.0017 −0.00021980 0.0152 0.0148 0.0004 0.0021 0.0018 0.00031990 0.0142 0.0141 0.0001 0.0013 0.0011 0.00032000 0.0100 0.0107 −0.0007 0.0029 0.0043 −0.0014

Decile 10-11970 −0.0016 −0.0052 0.0035 0.0077 0.0053 0.00241980 0.0161 0.0099 0.0062 0.0171 0.0097 0.00741990 −0.0023 −0.0090 0.0066 −0.0003 −0.0080 0.00772000 0.0136 0.0098 0.0038 0.0126 0.0070 0.0056

This table presents the raw returns and alphas for deciles of income before ex-traordinary items deflated by average assets (EARN) over different time peri-ods. The strategy ranks firms based on EARN from the most recently reportedfiscal year. We present average monthly returns and alphas assuming portfo-lios are formed and held without rebalancing for twelve months (Buy-Hold)and as the simple average of security returns assuming monthly rebalancing(Simple). The sample consists of all common stocks in CRSP with financialstatement data in Compustat for fiscal years 1969-2006; the correspondingreturn period is 1970-2007. Portfolios are assigned at the end of each April.

39

Table 6

Types of Securities in CRSP

Panel A: Security Types in CRSP Over TimeYear CEF ETF REIT STOCK ADR OTH n1925 0.002 . . 0.986 0.006 0.006 5031940 0.009 . . 0.985 0.004 0.003 7921960 0.008 . . 0.984 0.006 0.002 1,1161970 0.009 . 0.012 0.967 0.007 0.004 2,4191980 0.011 . 0.017 0.951 0.015 0.007 5,0061990 0.040 . 0.019 0.892 0.022 0.028 6,7052000 0.058 0.012 0.025 0.839 0.058 0.008 8,1342001 0.066 0.016 0.026 0.819 0.064 0.009 7,4392002 0.077 0.019 0.027 0.802 0.064 0.010 7,0202003 0.087 0.020 0.028 0.790 0.064 0.011 6,6902004 0.093 0.025 0.030 0.779 0.062 0.011 6,7332005 0.095 0.033 0.031 0.768 0.060 0.012 6,7482006 0.094 0.055 0.028 0.750 0.058 0.015 6,8522007 0.096 0.092 0.023 0.722 0.050 0.017 7,001

Panel B: Size by Security Type Over TimeCEF ETF STOCK

Median Fraction of Median Fraction of Median Fraction ofSize Size Decile 1 Size Size Decile 1 Size Size Decile 1

1925 11 0.00 . . 15 1.001940 7 0.00 . . 9 1.001960 54 0.00 . . 62 1.001970 61 0.00 . . 42 0.991980 56 0.00 . . 36 0.971990 101 0.00 . . 24 0.982000 124 0.01 45 0.03 108 0.942001 137 0.00 74 0.02 144 0.962002 128 0.01 105 0.02 132 0.942003 158 0.02 229 0.01 278 0.952004 182 0.03 255 0.07 335 0.872005 194 0.05 218 0.11 344 0.822006 221 0.03 125 0.25 403 0.702007 199 0.03 68 0.43 356 0.52

Panel A shows the fraction of the CRSP database represented by the followingtypes of securities: closed-end funds (CEF), exchange-traded funds (ETF), realestate investment trusts (REIT), common stocks (STOCK), American Depos-itory Receipts (ADR), and other types (OTH). Panel B shows the median sizedecile assignment made by CRSP on the universe of non-ADR NYSE, AMEX,and Nasdaq securities where 1 is the lowest size decile, the median end-of-yearmarket value of equity in millions, and the fraction of the lowest size-decilerepresented by that type of security.

40

Table 7

Effect of Security Type on Returns

Panel A: Difference for Equal-Weighted Market Returns2007 Index Level CRSP STOCK

9,933 10,856

Mean CRSP Return (NREWm,m) − STOCK Return (REW

m,m)Decade NREW

m,m Mean Min. Q1 Med. Q3 Max.1970 0.207 −0.003 −0.018 −0.013 0.000 0.006 0.0091980 0.149 0.002 −0.018 −0.007 0.001 0.007 0.0281990 0.177 −0.007 −0.029 −0.017 −0.003 0.002 0.0102000 0.144 −0.011 −0.084 −0.025 0.001 0.014 0.020

Panel B: Difference for Value-Weighted Market Returns2007 Index Level CRSP STOCK

3,802 3,860

Mean CRSP Return (NRV Wm,m) − STOCK Return (RV W

m,m)Period NRV W

m,m Mean Min. Q1 Med. Q3 Max.ALL 0.124 −0.001 −0.008 −0.003 −0.001 0.002 0.019

Panel C: Difference for Value-Weighted Returns by Size-Decile2007 Index Level CRSP STOCK

Size Decile 1 54,258 56,976Size Decile 10 3,438 3,484

Size Mean CRSP Return (NRV Wm,m) − STOCK Return (RV W

m,m)Period Decile NRV W

m,m Mean Min. Q1 Med. Q3 Max.1970 1 0.283 −0.003 −0.026 −0.022 0.005 0.010 0.0101980 1 0.171 0.006 −0.017 −0.002 0.000 0.006 0.0681990 1 0.278 −0.003 −0.022 −0.012 −0.003 0.006 0.0212000 1 0.262 −0.030 −0.313 −0.020 0.005 0.011 0.083ALL 2 0.191 −0.003 −0.109 −0.013 −0.004 0.005 0.099ALL 3 0.165 0.001 −0.094 −0.013 −0.001 0.008 0.131ALL 4 0.160 −0.006 −0.135 −0.011 −0.002 0.008 0.053ALL 5 0.154 −0.007 −0.102 −0.016 0.001 0.009 0.030ALL 6 0.151 −0.004 −0.091 −0.011 −0.002 0.009 0.054ALL 7 0.143 0.001 −0.045 −0.006 0.001 0.006 0.042ALL 8 0.146 0.001 −0.026 −0.006 −0.001 0.008 0.062ALL 9 0.141 −0.001 −0.028 −0.009 −0.002 0.003 0.074ALL 10 0.120 0.000 −0.006 −0.002 −0.001 0.000 0.013

This table shows the difference between annual market returns that includeall securities (CRSP) and those with only common shares (STOCK). PanelA presents (for equal-weighted, monthly rebalanced returns) the differencebetween CRSP returns (NREW

m,m) and STOCK returns (REWm,m). Panel B

presents (for value-weighted, monthly rebalanced returns) the difference be-tween CRSP returns (NRV W

m,m) and STOCK returns (RV Wm,m). Panel C presents

value-weighted results by size decile. Monthly security returns are used tocreate indices.

41

Table 8

Effect of Weighting Methodology on Equal-Weighted Returns

Panel A: Equal-Weighted Market Returns2007 Index Level CRSP PORT

All Securities 151,525 9,191

Mean CRSP Return (NREWd,d ) − PORT Return (NREW

d,m )Decade NREW

d,d Mean Min. Q1 Med. Q3 Max.1970 0.243 0.038 0.021 0.024 0.027 0.051 0.0711980 0.191 0.043 0.006 0.017 0.026 0.084 0.1111990 0.398 0.222 0.144 0.164 0.187 0.297 0.3822000 0.206 0.066 0.018 0.020 0.059 0.108 0.135

Panel B: Standard Deviation of Returns (SDEV ) Deciles2007 Index Level CRSP PORT

Decile 1 2,515,718,218,164 2,046Decile 2 1,934,402 917Decile 5 68,143 13,849Decile 10 23,478 16,986

Mean CRSP Return (NREWd,d ) − PORT Return (NREW

d,yr )Decade Dec. NREW

d,d Mean Min. Q1 Med. Q3 Max.1970 1 0.479 0.216 0.073 0.087 0.204 0.360 0.4011980 1 0.374 0.369 0.089 0.133 0.329 0.396 0.8271990 1 4.771 4.415 1.051 2.239 3.756 6.208 10.1282000 1 0.864 0.681 0.016 0.141 0.226 1.270 2.155ALL 2 0.469 0.324 −0.002 0.057 0.165 0.534 1.418ALL 3 0.343 0.165 −0.034 0.022 0.103 0.279 0.695ALL 4 0.260 0.091 −0.052 0.006 0.064 0.156 0.579ALL 5 0.243 0.055 −0.070 −0.008 0.043 0.102 0.412ALL 6 0.232 0.045 −0.062 −0.010 0.028 0.086 0.336ALL 7 0.205 0.025 −0.045 −0.011 0.010 0.051 0.236ALL 8 0.212 0.020 −0.049 −0.007 0.016 0.040 0.196ALL 9 0.200 0.015 −0.031 −0.010 0.010 0.037 0.160ALL 10 0.183 0.012 −0.038 −0.009 −0.001 0.022 0.236

This table presents (for equal-weighted, yearly returns) the difference betweendaily rebalanced market returns (CRSP returns, NREW

d,d ) and buy-and-holdreturns (PORT returns, NREW

d,m , NREWd,yr ). In Panel A, equal-weighted market

PORT returns are rebalanced monthly (NREWd,m ). In Panel B, equal-weighted

risk-decile (standard deviation of returns) PORT returns are rebalanced yearly(NREW

d,yr ) for the sample of Nasdaq securities. Daily returns are used to createindices.

42

Table 9

Effect of Weighting Methodology on Value-Weighted Returns

Panel A: Value-Weighted Market Returns2007 Index Level CRSP PORT

All Securities 3,810 3,793

Mean CRSP Return (NRV Wd,d ) − PORT Return (NRV W

d,m )Decade NRV W

d,d Mean Min. Q1 Med. Q3 Max.1970 0.067 0.0002 0.0001 0.0001 0.0002 0.0003 0.00041980 0.170 0.0001 −0.0001 −0.0001 0.0001 0.0003 0.00031990 0.183 0.0001 −0.0004 −0.0001 0.0002 0.0002 0.00062000 0.042 0.0001 −0.0003 0.0001 0.0002 0.0003 0.0004

Panel B: Value-Weighted Returns by Size Decile2007 Index Level CRSP PORT

Decile 1 44,028 68,147Decile 2 11,924 14,012Decile 5 5,930 6,460Decile 10 3,448 3,521

Mean CRSP Return (NRV Wd,d ) − PORT Return (NRV W

d,yr )Decade Dec. NRV W

d,d Mean Min. Q1 Med. Q3 Max.1970 1 0.284 0.001 −0.003 −0.003 0.000 0.005 0.0051980 1 0.169 −0.009 −0.031 −0.024 −0.015 0.004 0.0291990 1 0.274 −0.027 −0.058 −0.035 −0.029 −0.018 0.0042000 1 0.243 −0.039 −0.356 −0.046 −0.027 0.016 0.159ALL 2 0.190 −0.005 −0.034 −0.016 −0.004 0.000 0.036ALL 3 0.163 −0.002 −0.050 −0.007 −0.002 0.002 0.056ALL 4 0.162 −0.002 −0.044 −0.006 −0.002 0.001 0.039ALL 5 0.153 −0.002 −0.026 −0.005 −0.001 0.001 0.013ALL 6 0.151 −0.001 −0.043 −0.003 −0.001 0.001 0.019ALL 7 0.144 −0.001 −0.026 −0.004 −0.001 0.002 0.009ALL 8 0.144 −0.002 −0.031 −0.003 −0.001 0.001 0.008ALL 9 0.142 −0.001 −0.038 −0.003 0.000 0.001 0.015ALL 10 0.120 −0.001 −0.014 −0.001 0.000 0.001 0.004

This table presents (for value-weighted, yearly returns) the difference betweendaily rebalanced market returns (CRSP returns, NRV W

d,d ) and buy-and-holdreturns (PORT returns, NRV W

d,m , NRV Wd,yr ). In Panel A, value-weighted market

PORT returns are rebalanced monthly (NRV Wd,m ). In Panel B, value-weighted

size-decile PORT returns are rebalanced yearly (NRV Wd,yr ). Daily returns are

used to create indices.

43

Table 10

Effect on Anomaly Returns

Panel A: Size-Adjusted Returns to a Cash Flow StrategyCRSP STOCK Size-Adjusted

Raw No DL DL ReturnsDecile Period Returns NRV W

m RV Wyr CRSP STOCK DIFF

a b c a − b a − c b − c1 1973 − 2007 0.052 0.144 0.149 −0.091 −0.097 −0.006

2 − 9 1973 − 2007 0.143 0.138 0.141 0.005 0.002 −0.00410 1973 − 2007 0.190 0.131 0.135 0.060 0.056 −0.004

1 1993 − 1996 0.007 0.150 0.160 −0.144 −0.154 −0.0101 1997 − 2000 0.128 0.133 0.145 −0.005 −0.017 −0.0121 2001 − 2003 0.199 0.276 0.300 −0.077 −0.101 −0.0241 2004 − 2007 −0.094 0.068 0.059 −0.162 −0.153 0.009

10 1993 − 1996 0.225 0.156 0.164 0.069 0.061 −0.00810 1997 − 2000 0.230 0.138 0.145 0.092 0.084 −0.00810 2001 − 2003 0.201 0.156 0.169 0.045 0.032 −0.01310 2004 − 2007 0.125 0.091 0.087 0.034 0.038 0.005

Panel B: Size-Adjusted Returns to an Accrual Strategy1 1973 − 2007 0.123 0.143 0.147 −0.019 −0.024 −0.004

2 − 9 1973 − 2007 0.148 0.136 0.140 0.012 0.008 −0.00410 1973 − 2007 0.077 0.143 0.148 −0.066 −0.071 −0.005

1 1993 − 1996 0.101 0.151 0.161 −0.049 −0.060 −0.0111 1997 − 2000 0.136 0.130 0.138 0.006 −0.003 −0.0081 2001 − 2003 0.267 0.258 0.278 0.009 −0.011 −0.0201 2004 − 2007 −0.005 0.071 0.063 −0.076 −0.068 0.007

10 1993 − 1996 0.067 0.156 0.165 −0.089 −0.098 −0.00910 1997 − 2000 0.019 0.141 0.151 −0.121 −0.131 −0.01010 2001 − 2003 0.237 0.233 0.251 0.004 −0.014 −0.01810 2004 − 2007 0.064 0.081 0.075 −0.016 −0.010 0.006

44

Panel C: Size-Adjusted Returns to a Book-to-Market StrategyCRSP STOCK Size-Adjusted

Raw No DL DL ReturnsDecile Period Returns NRV W

m RV Wyr CRSP STOCK DIFF

a b c a − b a − c b − c1 1973 − 2007 0.044 0.130 0.132 −0.086 −0.088 −0.002

2 − 9 1973 − 2007 0.155 0.152 0.155 0.003 0.000 −0.00310 1973 − 2007 0.218 0.190 0.192 0.028 0.026 −0.002

1 1993 − 1996 0.034 0.143 0.151 −0.109 −0.117 −0.0081 1997 − 2000 0.088 0.119 0.127 −0.030 −0.038 −0.0081 2001 − 2003 0.086 0.169 0.185 −0.083 −0.099 −0.0161 2004 − 2007 0.014 0.076 0.069 −0.061 −0.054 0.007

1 1993 − 1996 0.238 0.168 0.180 0.070 0.058 −0.0121 1997 − 2000 0.138 0.165 0.175 −0.027 −0.037 −0.0101 2001 − 2003 0.445 0.325 0.346 0.120 0.099 −0.0211 2004 − 2007 0.098 0.079 0.073 0.019 0.025 0.006

This table shows the effect of weighting methodology on returns to tradingstrategies based on cash flows (CF), accruals (AC) and the book-to-marketratio (BM). Raw strategy returns are presented along with the CRSP size ad-justment returns (which include non common-stocks, rebalance monthly, andexclude delistings) and our proposed size adjustment returns (which excludenon common-stocks, rebalance yearly, and include delistings). The sampleperiod is fiscal years 1973−2006, with firms ranked yearly based on the prioryear’s data. Returns begin four months after fiscal year-end and continue fortwelve months. Pooled average returns are presented. Raw strategy returns,and all index returns are computed using monthly returns for consistency.

45

Table 11

Effect on Betas, Beta-Adjusted Returns, and Inferences

Panel A: Comparison of Betas, CRSP (βall) vs. Portfolio Methodology (βcs)Equal-Weighted Value-Weighted

Size Mean Mean 95% Mean Mean 95%Period Decile βall

EW βcsEW Diff βall

V W βcsV W Diff

1973-1989 ALL 1.011 0.992 0.085 0.753 0.746 0.0251990-1999 ALL 1.049 1.006 0.194 0.721 0.706 0.0492000-2008 ALL 1.071 1.013 0.207 0.943 0.934 0.0392000-2008 1 0.796 0.754 0.227 0.528 0.519 0.0432000-2008 2−9 1.101 1.043 0.209 0.967 0.957 0.0392000-2008 10 1.096 1.026 0.183 1.118 1.109 0.036

This table compares betas computed using equal- and value-weighted marketreturns computed using CRSP methodology (including all securities in themarket return measure) vs. our betas computed with market returns con-structed using common stocks . Panel A presents mean CRSP betas (βall)and our proposed betas computed with common stocks (βcs) and the 95th

percentile of the difference between these measures.

46

Table 12

Returns of Large Stocks vs. Small Securities

Panel A: Average Returns of Large Stocks and Small SecuritiesEqual-Weighted Value-Weighted ETF, CEF

D10cs D1cs D1etf D10cs D1cs D1etf Frac. D1all

1995 0.343 0.408 0.024 0.374 0.368 0.231 0.0061998 0.134 0.011 0.091 0.275 −0.036 0.019 0.0272000 0.001 −0.107 0.076 −0.094 −0.124 0.089 0.0362001 −0.084 0.430 −0.065 −0.132 0.336 −0.063 0.0192002 −0.191 −0.041 −0.170 −0.217 −0.042 −0.151 0.0372003 0.375 1.705 0.362 0.299 1.631 0.378 0.0322004 0.172 0.398 0.457 0.112 0.343 0.170 0.1002005 0.119 0.058 0.088 0.070 0.060 0.086 0.1532006 0.154 0.182 0.166 0.160 0.145 0.158 0.2872007 0.098 −0.068 0.035 0.086 −0.049 0.033 0.458

Panel B: Return Correlations between Large Stocks and Small SecuritiesEqual-Weighted Returns Value-Weighted ReturnsCorrelation of D10cs and Correlation of D10cs and ETF, CEF

Year D1all D1cs D1etf D1all D1cs D1etf Frac. D1all

1995 0.340 0.347 0.146 0.316 0.325 0.164 0.0061998 0.547 0.534 0.670 0.523 0.508 0.693 0.0272000 0.518 0.516 0.245 0.468 0.463 0.257 0.0362001 0.399 0.390 0.725 0.477 0.470 0.705 0.0192002 0.479 0.464 0.745 0.470 0.452 0.736 0.0372003 0.497 0.490 0.722 0.452 0.444 0.615 0.0322004 0.581 0.564 0.211 0.572 0.542 0.719 0.1002005 0.569 0.506 0.811 0.529 0.460 0.789 0.1532006 0.588 0.493 0.884 0.565 0.464 0.884 0.2872007 0.789 0.614 0.979 0.742 0.579 0.967 0.458

This table compares (for equal- and value-weighted yearly rebalanced returns)returns of securities in the lowest and highest size deciles. The highest sizedecile includes only common stocks (D10cs). The following partitions of thelowest size decile are compared with large stocks: all securities in the low-est size decile (D1all), only common stocks in the lowest size decile (D1cs),and only exchange-traded funds and closed-end funds in the lowest size decile(D1etf). Panel A presents average yearly returns computed from yearly re-balanced portfolios of large common stocks, small common stocks and smallETFs/CEFs. Panel B presents the correlation between yearly rebalanced dailysecurity returns of large common stocks and the following securities in thesmallest size decile: all securities, only common stocks, and only ETFs/CEFs.For reference, both panels report the fraction of the lowest size decile repre-sented by ETFs and CEFs. Daily security returns including delistings are usedto compute all indices used in this table.

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Table 13

Index Composition and Dividend Yields

Panel A: Dividend Yields over TimeEqual-Weighted Value-Weighted

Period ALL STOCK Diff ALL STOCK Diffa b |b − a|/b c d |d − c|/d

1930-39 0.036 0.035 0.01 0.048 0.048 0.001940-49 0.062 0.063 0.00 0.060 0.060 0.001950-59 0.058 0.058 0.00 0.056 0.056 0.001960-69 0.030 0.031 0.01 0.033 0.033 0.001970-79 0.032 0.031 0.03 0.042 0.041 0.001980-89 0.024 0.021 0.12 0.047 0.046 0.021990-94 0.019 0.013 0.45 0.031 0.030 0.041995-99 0.017 0.010 0.66 0.022 0.021 0.07

2000 0.014 0.008 0.73 0.010 0.009 0.082001 0.018 0.010 0.78 0.012 0.011 0.102002 0.014 0.007 0.89 0.013 0.012 0.102003 0.030 0.016 0.83 0.024 0.022 0.092004 0.021 0.011 0.82 0.022 0.020 0.082005 0.018 0.009 0.93 0.019 0.018 0.102006 0.021 0.012 0.80 0.022 0.020 0.092007 0.018 0.010 0.86 0.020 0.018 0.08

This table shows average annual dividend yields by time period. Yields arecomputed using monthly rebalanced equal- and value-weighted market returns.Dividend yields are calculated as (Ix,t−1Rt − Ix,t)/Ix,t−1 where Ix,t is the ex-dividend index level and Rx,t (Rt) is the ex-dividend (cum-dividend) marketreturn. Panel A compares yields of samples of all security types (ALL) andonly common shares (STOCK).

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Table 14

Types of ETFs over Time

Panel A: Number of ETFs over Time by TypeEquity

Year Bond Commodity Global U.S. n1993 . . . 1.00 11995 . . . 1.00 21996 . . 0.89 0.11 191998 . . 0.59 0.41 291999 . . 0.55 0.45 312000 . . 0.26 0.74 922001 . 0.01 0.29 0.70 1162002 0.06 0.01 0.31 0.62 1282003 0.05 0.01 0.32 0.63 1332004 0.04 0.02 0.26 0.68 1682005 0.03 0.02 0.23 0.72 2202006 0.02 0.05 0.22 0.72 3742007 0.08 0.07 0.24 0.61 6342008 0.07 0.09 0.30 0.54 717

Panel B: ETF Investment StrategySPDR Size Leveraged

Year Index Bond Equity Industry BM Bond Commodity Equity1993 . . 1.00 . . . . .1995 . . 1.00 . 0.50 . . .1996 0.89 . 0.11 . 0.05 . . .1998 0.59 . 0.38 0.31 0.07 . . .1999 0.55 . 0.35 0.35 0.06 . . .2000 0.64 . 0.18 0.42 0.32 . . .2001 0.67 . 0.15 0.46 0.29 . . .2002 0.62 . 0.13 0.40 0.31 . . .2003 0.62 . 0.13 0.38 0.32 . . .2004 0.58 . 0.10 0.36 0.40 . . .2005 0.46 . 0.10 0.38 0.42 . . .2006 0.33 . 0.09 0.41 0.36 . 0.005 0.042007 0.26 0.02 0.07 0.35 0.32 . 0.005 0.152008 0.26 0.02 0.08 0.34 0.28 0.01 0.03 0.17

This table presents descriptive statistics of the number and type of ETFslisted in the U.S. Panel A shows the number of ETFs over time and showsthe fraction of ETFs represented by four broad categories: Bond, Commodity,Global Equity, and U.S. Equity. Panel B shows the fraction of ETFs with thefollowing strategies: Equity ETFs that include the word Index in the title;ETFs with the term SPDR in the title; ETFs based on a specific industry;ETFs that follow size, value or growth-based (BM) strategies; and ETFs thattake short or leveraged positions.

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