Biases in decomposing holding period portfolio returns
Weimin Liu∗
Norman Strong∗
University of Manchester
We thank Martin Walker and Ian Garrett and participants at the 18th Australasian Finance and Banking Conference, Sydney, Australia, December 2005 for helpful comments.
∗ Manchester
Business School, University of Manchester, Manchester, M15 6PB, UK. Tel: +44(0)161–275–
3936. Email: [email protected].
Biases in decomposing holding period portfolio returns
ABSTRACT
A growing number of studies in finance decompose multi-period buy-and-hold portfolio returns into a series of single period returns. The method used to decompose
these returns is important because researchers use them in tests of asset pricing models and market efficiency and in evaluating the returns to investment strategies such
as those based on momentum and value–growth. We provide a formal analysis of
how to calculate portfolio returns for each single period over a horizon of several
periods. Crucially, we argue and provide empirical evidence that some methods researchers use involve portfolios that nobody would seriously consider ex ante, that
transactions costs associated with such portfolios make them poor investment vehicles, and that they can lead to spurious statistical inferences.
1
Calculating portfolio returns over an investment period is a fundamental requirement in many areas
of research in finance. A basic precept of this calculation is that the returns should measure the
wealth effect to an investor who holds the portfolio over the investment period.1 Inferences drawn
from an inaccurate calculation of returns have little economic relevance. The return calculation
is also important in testing market efficiency. Tests of market efficiency based on inaccurately
calculated returns are potentially misleading.
There is little problem in calculating returns over an entire holding period. However, an increasing
number of studies in finance draw inferences based on single-period portfolio returns over a multiperiod holding interval.2,3 Crucially, some researchers simplify the calculation of single-period portfolio returns. The current paper addresses this return metric issue and shows that it has important
implications in empirical finance such as in testing the market efficiency hypothesis, assessing investment strategies, and evaluating risk-adjusted portfolio performance.
Several studies report mean portfolio returns for monthly intervals over a multi-month investment
horizon. Jegadeesh and Titman (1993) measure monthly portfolio returns over an intermediateterm investment period (3 to 12 months) to examine the momentum effect. A series of subsequent
studies of price momentum use monthly portfolio returns over an intermediate-term holding period.4 Chan, Hamao, and Lakonishok (1991) use monthly portfolio returns over a one-year horizon
to examine the relation between expected returns and earnings yield, size, book to market ratio, and
cash flow yield in Japan. Some studies report mean portfolio returns for annual intervals over a
multi-year investment horizon. For instance, Lakonishok, Shleifer, and Vishny (1994) use annual
portfolio returns over a long-term investment horizon of five years to examine contrarian investment strategies. Many studies use monthly portfolio returns over a multi-month holding period to
estimate risk-adjusted portfolio performance using an asset pricing model such as the CAPM or
the Fama–French three-factor model. For example, Fama and French (1996) test their three-factor
model using monthly returns over a one-year holding period of portfolios sorted on a number of firm
characteristics known to be associated with average asset returns. Chan, Jegadeesh, and Lakonishok
1 Earlier
papers that apply the same principle in a related context are Roll (1983) and Blume and Stambaugh (1983).
2 We explain and illustrate this further below, but a simple example is where the researcher examines return momentum
of a stock portfolio for a six-month holding period but bases statistical tests on monthly portfolio returns.
3 Roll (1983), Blume and Stambaugh (1983), and Conrad and Kaul (1993) explain the bias in mean returns from
using arithmetic or re-balanced portfolio returns rather than buy-and-hold returns.
4 See, for example, Rouwenhorst (1998) for international momentum strategies, Moskowitz and Grinblatt (1999)
for the role of industries in explaining momentum, Chordia and Shivakumar (2002) for the role of time-varying expected returns as an explanation of US momentum profits, and Griffin, Ji, and Martin (2003) for a study of whether
macroeconomic risk variables can explain momentum profits across 40 countries.
2
(1996) use monthly returns over a six-month holding period to test whether size and book-to-market
effects can explain the profitability of investment strategies based on return and earnings momentum. Carhart (1997) uses monthly returns for portfolios of mutual funds reformed annually to
examine performance relative to the CAPM and to an extended Fama–French model incorporating
an additional momentum factor.
Although the existing literature is extensive, it does not provide an explicit description or formula
for how to decompose multi-period buy-and-hold portfolio returns into single-period portfolio returns. In this paper we examine the method of calculating monthly returns from a multi-month
holding period; the procedure generalizes automatically to other intervals. Based on the rationale
that these monthly returns should be the returns earned by an investor who holds the portfolio, we
show that the monthly portfolio return in each holding-period month is a weighted average with
the weight assigned to each stock in the portfolio depending upon the stock’s performance over
previous holding-period months. In contrast to this approach, some empirical studies employ a simplified approach and calculate the monthly return of an equally-weighted portfolio in each holdingperiod month as the arithmetic average of all individual stock returns in that holding-period month.5
However, equal-weighting involves re-balancing the portfolio at the start of every month in a way
that no investor would seriously consider ex ante and, furthermore, the transactions costs typically
associated with such portfolios make them poor investment vehicles. While many investors undoubtedly do revise their investment weights over time, in practice irregular new information flows,
fund flows, or liquidity requirements are as likely to determine these revisions as regular reviews
of optimal weights. Revisions in practice are unlikely to restore initial weights. From a theoretical viewpoint, re-balancing confounds the return to an investment strategy under examination with
the effect of, and underlying reason for, portfolio revision. From an empirical viewpoint, we show
that re-balanced methods are inaccurate in reflecting investor wealth and, crucially, they can result
in spurious statistical inferences. Empirically, the re-balanced methods result in an upward bias
to finding a premium associated with size and value effects—increasing the probability of making
Type I errors—and a downward bias to finding a momentum premium—increasing the likelihood of
making Type II errors. In results over the period 1951–2003 that we report below, the difference in
mean and risk-adjusted monthly returns between the re-balanced and the decomposed buy-and-hold
5 In
the less popular case of value-weighting, the simplified approach calculates the monthly portfolio return in each
holding-period month as the weighted average of all individual stock returns in that month with each weight kept constant
at the value determined at the beginning of the holding period by each stock’s market capitalization.
3
methods is generally significant especially for low-price, small, value, and loser stocks. The bias in
mean monthly returns over a multi-month holding period can be as high as 0.694% per month (over
8% per year) and can lead to spurious statistical inferences. For example, the re-balanced method
gives a significant size effect over the sample period 1951 to 2003, while the decomposed buyand-hold method shows no size effect, and while the decomposed buy-and-hold method reveals a
significant momentum profit of 1.017% per month over the period January 1978 to December 2003,
the re-balanced method reduces this by 44% to an insignificant 0.569% per month. Consistent with
re-balancing requiring more trading, adjusting for transactions costs has the greater effect in reducing the profitability of investment strategies using the the re-balanced method. Realistic levels of
transactions costs can result in the re-balanced method underestimating the size effect and generating a significantly negative momentum effect. By formalizing the calculation of monthly portfolio
returns over a multi-month holding period, we show how to evaluate portfolio performance on a single period basis, and remind researchers of the possible biases that can result from using re-balanced
methods in research.
In addition to examining portfolio returns, we also describe how to calculate the monthly arbitrage
profits of a zero-investment portfolio over a multi-month investment horizon. This calculation differs from the commonly used monthly return difference between long and short portfolios. The
simple return differences in each holding period month tend to underestimate the arbitrage profits
that an investor can earn if the phenomenon under examination holds. The method of computing
monthly arbitrage profits also applies to the calculation of benchmark-adjusted monthly returns such
as those that Hou and Moskowitz (2003) use to examine portfolios of stocks ranked each year by
the speed with which stock price responds to information.
We begin by describing how to calculate monthly portfolio returns over a multi-month holding
period and discussing the possible biases from using re-balanced methods.
1. Decomposing multi-month holding period returns
In this section we explain how to calculate individual month portfolio returns over a multi-month
holding period. We present the corresponding re-balanced methods and discusses the biases that
arise. Finally, we describe methods of calculating arbitrage profits. Although we focus on monthly
4
portfolio returns over a multi-month holding period, the methods in this section apply to other
frequencies such as daily, weekly, and annual.
1.1. Decomposed buy-and-hold methods
An individual stock i’s return in month t is,
rit =
Pit + Dit − Pi,t−1
,
Pi,t−1
(1)
where Pit is stock i’s price per share at the end of month t, and Dit is stock i’s dividend per share
with the ex-dividend date falling in month t.6
Suppose an investor holds a standard portfolio of N stocks with weight wi invested in stock i at the
start of the holding period and ∑Ni=1 wi = 1. To begin the analysis we ask, what is the investor’s
m-month buy-and-hold portfolio return given the monthly returns for each stock in the portfolio?
The answer is,
N
rP, 1→m = ∑ wi (1 + ri1 )(1 + ri2 ) · · · (1 + rim ) − 1,
(2)
i=1
where rit (t = 1, . . . , m) is stock i’s return in month t. Equation (2) gives the portfolio return that the
investor actually obtains over the m-month holding period.7
In empirical tests, researchers often decompose the m-month buy-and-hold portfolio return into m
monthly portfolio returns. For example, studies usually estimate asset pricing models such as the
CAPM and the Fama–French three-factor model using monthly observations. Similarly, studies
of momentum strategies over intermediate holding periods typically base their statistical tests on
average monthly returns. Based on the principle that the return should equal the amount the investor
earns, the monthly portfolio return over an m-month holding period is,
τ−1
(1 + rit )
wi ∏t=1
riτ ,
τ−1
N
i=1 ∑ j=1 w j ∏t=1 (1 + r jt )
N
rPτ = ∑
τ = 2, . . . , m;
N
rP1 = ∑ wi ri1 .
i=1
m
Equation (3) follows from ∏t=1
(1 + rPt ) − 1 = ∑Ni=1 wi (1 + ri1 )(1 + ri2 ) · · · (1 + rim ) − 1.
6 As
7 We
we focus on portfolios, we restrict attention to discrete returns.
ignore transactions costs and other microstructure effects at this stage.
5
(3)
We can identify two special cases of equation (3): the monthly returns of an equally-weighted
portfolio and of a value-weighted portfolio over an m-month holding period.
(i) An equally-weighted portfolio
Equation (3) with wi = 1/N gives the return of an equally-weighted portfolio in an individual
month τ from an m-month holding period as,
τ−1
(1 + rit )
∏t=1
riτ ,
τ−1
N
i=1 ∑ j=1 ∏t=1 (1 + r jt )
N
ew
=∑
rPτ
τ = 2, . . . , m;
ew
=
rP1
1 N
∑ ri1 .
N i=1
(4)
Note that for the first month of the holding period, the equally-weighted portfolio return
equals the arithmetic average of individual stock returns, but this is not the case from month
2 onwards. Specifically, the month-τ (τ = 2, 3, . . . , m) portfolio return of an equally-weighted
m-month (m > 1) buy-and-hold investment strategy is the weighted average of the month-τ
stock returns with the weight depending upon return performance over the previous holding
period months.
(ii) A value weighted portfolio
Equation (3) with wi = MVi, 0 / ∑Nj=1 MV j, 0 , where MVi, 0 is stock i’s market value at the beginning of the holding period, gives the return of a value-weighted portfolio in an individual
month τ from an m-month holding period as,
N
MVi,τ−1
riτ ,
N
i=1 ∑ j=1 MV j,τ−1
vw
rPτ
=∑
τ = 1, 2, ..., m.
(5)
Equation (5) shows that the month-τ return of a value-weighted portfolio over an m-month
holding period is a weighted average of the month-τ returns of all stocks in the portfolio with
the weights determined by the market values at the start of the month.
1.2. Re-balanced methods
Instead of the calculations given by equations (3)–(5), some studies calculate returns for individual
months within a multi-month holding period using an investment strategy that involves monthly
re-balancing. These procedures calculate each monthly portfolio return as a weighted average of
stock returns in that month with the weight in all holding-period months held constant at the value
6
determined at the beginning of the holding period. This gives the monthly portfolio return over an
m-month holding period as,
N
r(rb)Pτ = ∑ wi riτ ,
τ = 1, 2, ..., m,
(6)
i=1
where rb denotes re-balanced.
For an equally-weighted portfolio, which is the most popular approach, the portfolio return in each
holding-period month is the arithmetic average of all stock returns in that month,
r(rb)ew
Pτ =
1 N
∑ riτ ,
N i=1
τ = 1, 2, ..., m.
(7)
For a value-weighted portfolio the portfolio return in each holding-period month is a weighted
average of all stock returns in that month with the weight in all holding-period months being a
stock’s market value relative to the total market value of all stocks at the beginning of the holding
period,
N
MVi0
riτ ,
N
i=1 ∑ j=1 MV j0
r(rb)vw
Pτ = ∑
τ = 1, 2, ..., m.
(8)
These re-balanced methods are inaccurate in reflecting investor wealth in individual months over a
multi-month holding period, unless investors re-balance their portfolios back to the initial weights
at the beginning of every month. The standard approach to measuring investment performance is
to assume a passive, buy-and-hold strategy over the investment holding period. This serves the
purpose of isolating the performance of the investment strategy itself. If in practice investors rebalance their portfolios, this is due to new information, to fund inflows or outflows, or to liquidity
requirements and occurs at irregular intervals. Assuming investors re-balance at regular discrete
intervals to maintain their initial portfolio weights fixed is unrealistic in practice and confounds the
theoretical hypothesis under examination. Crucially, re-balancing can result in different inferences
compared with the approach proposed in equation (3).
7
Taking the equally weighted approach as an example, from equations (4) and (7) the month-τ bias
in using the re-balanced method is,
N
Biasτ = ∑
i=1
(
#)
"
τ−1
(1 + r̃it )
1
∏t=1
r̃iτ
,
E [r̃iτ ] − E
τ−1
N
(1 + r̃ jt )
∑Nj=1 ∏t=1
τ = 2, . . . , m.
(9)
Clearly, there is no bias if the expected returns of individual stocks in the portfolio are all equal.
However, this is not the case in practice, and expected returns vary across individual stocks due to
their different risk exposures. In general, the bias can be either positive or negative depending on the
time-series properties of both portfolio and individual stock returns. Without loss of generality, consider the bias when τ = 2. After some manipulation and rearrangement and using the approximation
1/(1 + r̃τ−1 ) ≈ 1 − r̃τ−1 , which ignores higher order moments from the Taylor’s series expansion,
we can rewrite equation (9) as,
Bias2 ≈ E[r̃1 r̃2 ] −
1 N
∑ E[(1 − r̃1 )r̃i,1 r̃i,2 ],
N i=1
where r̃τ is the average return in month τ of stocks in the portfolio.
Assuming r̃τ−1 is uncorrelated with individual stock returns, the bias is,
Bias2 ≈ E(r̃1 )E(r̃2 ) + Cov(r̃1 , r̃2 ) −
1 N
∑ E(1 − r̃1 ) [E(r̃i,1 )E(r̃i,2 ) + Cov(r̃i,1 , r̃i,2)] .
N i=1
(10)
The two covariance terms show that the sign of the bias depends on the autocovariance in portfolio returns and in the average autocovariance in individual stock returns. Empirical studies have
found that portfolio returns are positively autocorrelated (see Lo and Mackinlay (1990)), and hence
the auto-covariance in portfolio returns could contribute to a positive bias from the use of the rebalanced method. Researchers have documented that individual stock returns are negatively autocorrelated due to non-synchronous trading (e.g., Fisher (1966)) or transaction costs/bid-ask spreads
(e.g., Roll (1984), Jegadeesh and Titman (1995)). Lo and Mackinlay (1990) report negative
average auto-covariances for individual stocks, with the effect most pronounced in small stocks.
We therefore expect to observe this negative serial correlation particularly in small and low-price
stocks. For example, Liu (2006) shows that infrequently traded stocks are highly correlated with
size and bid-ask spread. Branch and Freed (1977) and Conrad and Kaul (1993) find a negative
relation between price and bid-ask spread. Therefore, we are more likely to find a positive bias in
8
small and low-price stock portfolios. On the other hand, Kaul and Nimalendran (1990) find that
stock returns are positively autocorrelated once they extract the bid-ask effect. Therefore, we may
observe a negative bias in large and high price stock portfolios. A negative bias could also arise if
expected stock returns are constant over time but vary cross-sectionally, as higher (lower) expected
returns translate into higher (lower) expected weights in the bracketed term in equation (9) while
re-balancing to equal weights undoes this effect. Finally, equation (10) shows that even if returns
are independent, the bias is non-zero.
It is clear that how these biases affect the returns to specific investment strategies is ultimately
an empirical issue. To the extent that investment strategies examined in the research literature
typically consider a long–short strategy in portfolios polarized on some variable hypothesized to
predict return, the negative bias due to cross-sectional variation in expected stock returns is unlikely
to be a serious issue. In contrast, given the nature of the microstructure effects involved, the biases
arising from autocovariances in portfolio and individual stock returns are likely to be serious in low
price, small stock portfolios. It is a well-known fact that many return anomalies are related to small
and low-price stocks. For example, in considering a significance test of the size effect calculated as
the difference between a long position in small stocks and a short position in large stocks over an mmonth holding period, we show in the following section that the long portfolio (small stocks) largely
determines the bias. Re-balancing to maintain the portfolio weights constant at equal weights as in
equation (7) places more weight on stocks in the small stock portfolio that have higher observed
returns in the holding period and less weight on stocks that have lower observed returns in the
holding period compared with equation (4). This results in an upward bias to the returns to the long
side and consequently to the difference between the long and short sides, overestimating the true size
effect. The following section also shows the bias of re-balancing methods when testing value and
momentum effects. Sorting on book-to-market, re-balancing has little effect on the holding period
return estimates to the short-side, low book-to-market portfolio, but does overestimate the returns
to almost all other deciles. As a result, re-balancing imparts an upward bias to the estimated value
effect. Sorting on past 6-month return, the short side causes re-balancing to impart a downward
bias to the estimated momentum effect. Results using re-balancing methods therefore contaminate
the true underlying investment returns with measurement error due to market microstructure effects.
The error can in some cases be a substantial component of the estimated return. In contrast, using the
decomposed buy-and-hold approach that we present above minimizes these microstructure effects,
9
enabling the researcher to focus on the true returns to the underlying investment strategy. In effect,
the researcher can either estimate the size (book-to-market, momentum) effect or estimate the size
(book-to-market, momentum) effect contaminated by microstructure effects.
1.3. Arbitrage profits
Empirical tests also frequently examine the returns to a zero-investment (or arbitrage) portfolio. To
test the momentum effect, for example, we can form a zero-investment portfolio by buying one
dollar of past intermediate-term winning stocks and shorting one dollar of past intermediate-term
losing stocks, and hold this position for 3 to 12 months. If this strategy produces consistently positive profits, this is evidence of a momentum effect. The principle of calculating arbitrage portfolio
profits also applies, in a straightforward fashion, to calculating benchmark-adjusted portfolio returns for single periods within a multi-period test period. That is, the test portfolio corresponds to
the long side and the benchmark portfolio corresponds to the short side.
So how should we calculate the monthly profits of a zero-investment portfolio over a multi-month
holding period? Suppose that we go long $1 in portfolio or stock L for m months, and at the same
time we short $1 of portfolio or stock S for m months. The monthly profits of this zero-investment
strategy over the m-month holding period are,
τ−1
τ−1
t=1
t=1
APτ = rL, τ ∏ (1 + rL,t ) − rS, τ ∏ (1 + rS,t ),
τ = 2, . . . , m;
AP1 = rL, 1 − rS, 1 ,
(11)
where rL,t is the return of portfolio or stock L in month t and rS,t is the return of portfolio or stock
S in month t. Equation (11) follows from the expression for the m-month arbitrage profit,
m
m
m
t=1
t=1
t=1
∑ APt = ∏(1 + rL,t ) − ∏(1 + rS,t ).
Equation (11) corresponds to a natural investment strategy because the initial investment of $1 in
the long and short sides increases or decreases with changes in the market values of the long and
short portfolios.
10
However, empirical studies commonly calculate the arbitrage profit in each holding-period month
as the simple return difference between the long and short portfolios in that month,
τ = 1, 2, ..., m.
AP′τ = rL, τ − rS, τ ,
(12)
As we report in the text below, the monthly return difference between the long and short portfolios held for a multi-month period (with the portfolio monthly returns calculated using appropriate
equally- or value-weighted methods) generally underestimates the true arbitrage profits if the phenomenon under examination holds. More importantly, this monthly return difference does not correspond to a natural investment strategy over a multi-month holding period because it requires the
investor to restore investment in the long and short portfolios to an equal weighting every month.
2. Empirical evidence
This section provides empirical evidence on the possible biases caused by using re-balanced methods, by examining three well-known anomalies—size, book-to-market (B/M), and price momentum.
The data are from the CRSP/COMPUSTAT merged (CCM) database over the period 1951, when
book values are first available on the CCM database, to 2003 for all NYSE/AMEX/NASDAQ ordinary common stocks. The sample excludes negative B/M and financial stocks when examining the
book-to-market effect, but we impose no exclusion criteria when examining the size or momentum
effects. The calculation of equity book value follows Fama and French (1996), and we assume a
minimum 5-month gap between the fiscal year end and the market reporting date. The three risk
factors—market, size, and book-to-market—are from Kenneth French’s website.8
The analysis involves ranking stocks into decile portfolios based on market capitalization (MV ),
book-to-market ratio (B/M), or prior 6-month buy-and-hold return (r−6 ), using NYSE breakpoints.
We denote the extreme decile portfolios as S(hort) and L(ong), where this designation indicates the
subsequent investment strategy. S stands for the largest-MV decile, lowest-B/M decile, or lowestr−6 decile. L is the smallest-MV decile, highest-B/M decile, or highest-r−6 decile. We denote
intermediate decile portfolios as D2 to D9. In examining the size and book-to-market effects we
follow the approach of Fama and French (1993, 1996) and reform deciles annually and hold them
8 The
website is http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/.
11
for 12 months. In examining momentum we follow the standard approach in the literature and
sort eligible stocks into portfolios at the beginning of each month and hold these portfolios for
six months. Momentum portfolios therefore involve overlapping ranking and holding periods on a
monthly basis. To be eligible, stocks must have their MV (or B/M, or 6-month returns) available
immediately before portfolio formation when examining the MV (or B/M, or momentum) effect.
Also, we require one monthly return immediately after portfolio formation. These requirements
ensure that we can form stocks into portfolios and can hold eligible stocks for at least a month. For
a stock delisted in the holding period, we assume its post-delisting monthly returns are zero over
the remaining holding period months. We find similar results when substituting the risk-free rate
for the post-delisting returns.
We calculate monthly portfolio returns for each of the m months in the holding period (m = 6 for
the momentum effect, and m = 12 for MV and B/M portfolios). For the momentum effect, the nonoverlapping monthly returns over the full testing period correspond to the technique of Jegadeesh
and Titman (1993), which is equivalent to revising the weights of 1/m of the portfolio each month
and carrying over the rest from the previous month. The following three subsections examine the
basic evidence on the size, book-to-market, and momentum effects. In a final subsection we provide
a further analysis of the bias by examining the size, book-to-market, and momentum effects in pricestratified sub-samples.
2.1. The size effect
Table 1 reports the holding-period performances and characteristics of MV -classified decile portfolios. At the beginning of July of each year, we sort eligible stocks in descending order based on their
market capitalizations (MV ), we group stocks into equally-weighted decile portfolios using NYSE
breakpoints, and we hold these portfolios for 12 months. The top decile is the largest-MV portfolio
(S) and the bottom decile is the smallest-MV portfolio (L). The full sample period is July 1951 to
June 2003, while the two sub-periods are July 1951 to June 1977 and July 1977 to June 2003.
Looking at the results over the full sample period 1951 to 2003, the bias in the equally-weighted
re-balanced portfolio method is immediately apparent from Table 1. Using the decomposed buyand-hold method, Panel A shows that the estimated size effect, measured as the return difference
between portfolios L and S, is an insignificant 0.313% (t = 1.65) per month. In contrast, Panel B
12
shows that with re-balancing the estimated effect is 43% higher at 0.448% per month (t = 2.34),
suggesting a significant size effect. The Fama–French three-factor model performs well against the
decomposed buy-and-hold returns of the MV portfolios with the sole exception that the largest-MV
portfolio loses significantly after adjusting for the three-factor model. Similar to the raw return
result, there is no size effect after adjusting for the three factors. Again, the re-balanced method
leads to a different inference. The three-factor-adjusted re-balanced returns show that seven out of
ten decile portfolios lose significantly, and the three-factor model is less successful at explaining
the re-balanced return difference between the smallest-MV and largest-MV portfolios of 0.197%
(t = 1.90) per month, a figure that is 71% higher than its decomposed buy-and-hold counterpart
(0.115%, t = 1.14).
Inspecting the individual decile returns shows that both methods of calculation give similar estimates
for decile portfolios S and D2–D9, although re-balancing typically gives the lower estimated raw
and three-factor-adjusted returns. This is consistent with our previous conjecture that a negative
bias is possible within larger stock portfolios. But for the smallest stocks in the long portfolio
this difference is reversed with re-balancing overstating the buy-and-hold return and the difference
being an order of magnitude larger than for the other nine decile portfolios. It is this that drives the
difference between the two estimated size effects.9 Characteristics of the MV portfolios reported in
Panel C of Table 1 shows that price is monotonically decreasing from the highest-MV decile (S) to
the lowest-MV decile (L) with L having a substantially lower average stock price than S. This is why
we observe a large positive bias in the lowest-MV portfolio—low-price stocks are very sensitive to
microstructure effects as Ball, Kothari, and Shanken (1995) show.
The patterns in the sub-period results in Table 1 mirror the full period patterns. Consistent with
other studies, the significant size effect associated with the re-balanced method is primarily due
to the earlier period. The sub-period results also confirm that the re-balanced method overstates
the magnitude and significance of the size effect, although the bias over the recent sub-period does
not result in any difference in statistical inference, since all estimates indicate an insignificant size
effect.
Turning to the arbitrage profits of buying $1 of the smallest MV portfolio (L) and shorting $1 of
the largest MV portfolio (S), the (untabulated) mean arbitrage profits calculated according to equa9 We
examine the differences between the re-balanced and decomposed buy-and-hold returns in detail at the end of
this section.
13
tion (11) are 0.412% per month for equally-weighted portfolios over the full sample period, with
corresponding estimates of 0.549% and 0.276% over the 1951–1977 and 1977–2003 sub-periods.
These estimates are up to one-third higher than the estimates for L − S calculated using unbiased
equal-weights.
2.2. The B/M effect
Table 2 reports the holding-period performances of B/M-classified decile portfolios. At the beginning of July each year from 1951 to 2002, we sort eligible stocks in ascending order based on their
book-to-market ratios (B/M). We group stocks into equally-weighted decile portfolios using NYSE
breakpoints and hold these portfolios for 12 months. The top decile is the lowest-B/M portfolio (S)
and the bottom decile is the highest-B/M portfolio (L). The sample includes all non-negative-B/M
and non-financial NYSE/AMEX/NASDAQ common stocks over the period 1951 to 2003.
Results of the decomposed buy-and-hold method presented in Panel A of Table 2 show a clear B/Mrelated value premium over the full sample period and over both sub-periods. Raw returns for the
full sample period increase monotonically in moving from the lowest to the highest book-to-market
portfolios, and the raw return difference between L and S is 0.708% (t = 4.52) per month. The
Fama–French three-factor model fails to subsume the value premium over the full and the recent
sample periods.
In this case, using the re-balanced method does not result in mistaken inferences, but it does overstate the magnitude of the value effect by between 25.4% and 29.1% for the raw return, and by
between 31.8% and 240.0% for the three-factor-adjusted return. Across the individual decile portfolios re-balancing gives higher raw returns in 26 of the 30 cases (ten portfolios for the full sample
period and for two sub-periods). The difference between the re-balanced and the decomposed buyand-hold methods is stronger in the higher book-to-market portfolios. Panel C shows that these
portfolios tend to include lower priced, smaller stocks. Untabulated results show that arbitrage profits are higher than the L−S differences in the full sample period (0.726 v. 0.708) and the second
sub-period (1.008 v. 0.966) and marginally lower in the first sub-period (0.444 v. 0.449).
14
2.3. The momentum effect
Table 3 presents the holding-period performance of decile portfolios classified by past six-month
buy-and-hold returns (r−6 ). In this table, S denotes the past 6-month poorest price performing
decile (the subsequent short side), and L denotes the past 6-month best price performing decile (the
subsequent long side). We form decile portfolios at the beginning of each month, and hold them for
6 months.10 The full testing period is July 1951 to December 2003, while the two sub-periods are
July 1951 to December 1977 and January 1978 to December 2003.
The decomposed buy-and-hold results reported in Panel A of Table 3 show that the momentum effect
is present over the full sample period and in both sub-periods. The mean return difference between
the equally-weighted long and short portfolios is 0.875% per month (t = 4.91) over the full sample
holding period, 0.736% per month (t = 3.27) over the earlier sub-period, and 1.017% per month
(t = 3.66) over the later sub-period. The three-factor-adjusted returns on the L−S portfolio are
also significantly positive. Panel B shows re-balancing imparts a downward bias to the momentum
effect, and can turn the significant return differences between L and S insignificant. For the rebalanced equally-weighted portfolios, the mean return difference between L and S is 0.583% per
month (t = 3.01) over the full sample period, and 0.597% per month (t = 2.58) and 0.569% per
month (t = 1.82) over the two sub-periods. These figures are as much as 44.1% lower than the
unbiased estimates and in the case of the second sub-period using the re-balanced method gives a
spuriously insignificant momentum effect.
Examining the individual portfolios shows that, similar to the B/M effect, re-balancing tends to
overstate the portfolio returns. However, this effect is most marked in the return to the short side of
the momentum portfolio, which drives the above results. Panel C shows that this portfolio has lower
priced and smaller capitalization stocks. Finally, untabulated results show that arbitrage profits are
higher than the L−S differences for the full sample period and for both sub-periods.
2.4. Results in price-stratified sub-samples
The above results show that the bias of the re-balanced method tends to be negative in larger stock
portfolios, but it is positive, and much greater in magnitude in small, value, and loser portfolios.
10 Jegadeesh
and Titman (1993) show that the results for the 6 × 6 strategy (6-month ranking and holding periods) are
representative.
15
The results also indicate that small, value, and loser stocks tend to be low-priced stocks, which are
more sensitive to microstructure effects. We therefore investigate the bias further by conducting an
analysis within price-stratified sub-samples of the three anomalies with results untabulated. While
the re-balanced method generally imparts a downward bias in monthly portfolio returns within
medium- and high-price sub-samples, almost all the biases across the decile portfolios classified
by MV , B/M, or past 6-month returns are large and significantly positive. For example, the bias to
the past-6-month loser portfolio is 0.694% (t = 10.7) per month (over 8% per annum) within the
1/3 lowest-priced stocks. These results further confirm our earlier conjecture that the bias can be
positive or negative depending upon the time-series properties of security returns. The bias is particularly severe within low-priced stocks, supporting to the findings of Kaul and Nimalendran (1990)
that negative autocorrelations in security returns, which cause the positive bias in the re-balanced
method, are due to bid-ask spreads, which have a strong negative relation with price.
2.5. Transactions costs
Table 4 reports the effect of incorporating transactions costs into the analysis. For this purpose,
we need to consider the amount of trading involved in re-balancing portfolios during each holdingperiod month and in revising portfolios from one holding period to the next. An investor incurs costs
associated with re-balancing in the case of the re-balanced method but not in the case of the decomposed buy-and-hold method, while portfolio revision necessarily occurs with both methods. Panel
A reports the average traded value per month involved in longing $1 of portfolio L and shorting
$1 of portfolio S for each of the size, book-to-market, and momentum investment strategies implemented over the period 1951–2003. Appendix A explains the details behind these calculations. We
report these average traded values separately for the stock purchases and stock sales involved in
each strategy and we report results for the decomposed and re-balanced methods and for the difference between them. The re-balanced method applied to size and book-to-market strategies results
in average traded values increasing by between 59% and 140% compared with the decomposed
method. This increase is proportionately smaller in the case of momentum strategies as these involve six-month rather than twelve-month horizons and winning and losing stocks rarely retain their
status in successive holding periods. Nevertheless, the difference in average monthly traded values
associated with momentum strategies between re-balanced and decomposed methods are similar to
those for size and book-to-market strategies and all differences are significant. For the size strategy,
16
the average traded value per month for the long–short portfolio is more than 11% higher for the
re-balanced method than for the decomposed buy-and-hold method. The corresponding figure is
more than 15% for the B/M and momentum strategies.
Panel B of Table 4 reports the returns after transactions costs to portfolio L, portfolio S, and L−S
for the three investment strategies (Appendix B explains the calculation of transaction-cost-adjusted
returns). We consider four alternative estimates of transactions costs. The first uses the model of
transactions costs and empirical results of Keim and Madhavan (1997) (Appendix C provides details of the estimation). The other three alternative transactions costs we assume are 0.5%, 1.0%,
and 1.5% for both buying and selling. The results clearly show that transactions costs have a much
greater effect on the performance of the re-balanced strategy than the decomposed buy-and-hold
strategy. With the transactions costs adjustment using the results of Keim and Madhavan (1997), the
size, value, and momentum premia from the re-balanced strategy are all less than the corresponding ones from the decomposed buy-and-hold strategy. The value premium using the re-balanced
method is reduced by 46.41% (from 0.905% to 0.485%) after adjusting for transaction costs, while
the reduction is 25.85% (from 0.708% to 0.525%) using the decomposed buy-and-hold method.
The reduction of the size premium is similar to the reduction of the value premium, but the momentum premium is reduced by 147.5% with the re-balanced method and 70.4% with the decomposed
buy-and-hold method. The dramatic decrease in momentum profits from both strategies is due to
winning and losing stocks rarely retaining their positions in consecutive holding periods. Looking
at other results, with a moderate level of transaction costs of 1% for both buying and selling, we
do not observe a significant size premium even with the re-balanced method. Transactions costs
again have the greatest effect on momentum. The re-balanced method gives no significantly positive momentum profits at any transaction cost level. Even with the decomposed method, a 1% level
of transaction costs for buying and selling is high enough to eliminate the significant momentum
premium. The book-to-market effect survives the transaction-cost adjustment for the decomposed
buy-and-hold strategy and the re-balanced strategy.11 Overall, the transaction-cost-adjusted results
serve to emphasise that a re-balancing strategy is a poor, impractical investment vehicle.
11 Liu
(2006) finds that the book-to-market effect is due to liquidity risk, and his two-factor model can account for the
value premium.
17
3. Conclusion
The criterion for calculating returns should be based on an investment perspective. The return should
correspond to the amount entering an investor’s account. Although we do not know exactly how
investors manage their portfolios, the finance literature commonly assumes a multi-month holding
period but adopts a re-balancing strategy to calculate monthly portfolio returns. However, a rebalancing strategy does not correspond to the returns that investors achieve under the hypothesized
holding period investment strategy, unless they re-balance their portfolios back to the initial weights
at the beginning of each discrete interval. In practice, new information flows and new fund flows
will determine revisions of portfolio weights and these are unlikely to occur at regular intervals. In
addition, the transactions costs of regular re-balancing are prohibitive. In contrast, a buy-and-hold
investment assumption focuses on the theoretical hypothesis under examination. As an increasing
number of studies in finance base their statistical tests on individual period returns from a multiperiod portfolio return, we show in this paper how to decompose multi-period buy-and-hold returns
into consistent individual period returns. By comparing re-balanced returns with decomposed buyand-hold returns, we predict and find that re-balancing imparts an upward bias to the size or value
premium, and a downward bias to the momentum effect. Empirically, the two methods can produce
a portfolio return difference of more than 8% per year, and can lead to different statistical inferences.
The dangers of using re-balanced methods are relevant for tests of asset pricing models, of market
efficiency, and for the analysis of investment strategies, such as momentum and value–glamor, implemented in real time. As a minimum requirement, researchers should report single-period return
results based on the approach proposed in this paper in addition to results based on a re-balancing
assumption.
18
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20
Table 1
Holding-period performance of portfolios classified by MV
S
D2
D3
D4
D5
D6
D7
D8
D9
L
Panel A: Holding-period performance measured on a monthly basis using equation (4)
Testing period: July 1951 to June 2003
Raw (%) 0.922
1.045
1.067
1.095
1.067
1.148
1.138
1.168
1.164 1.235
(5.39) (5.95) (5.65) (5.72) (5.45) (5.58) (5.28) (5.30) (5.07) (5.22)
α̂3F (%) −0.070 −0.008 −0.031 −0.008 −0.076 −0.022 −0.037 −0.039 −0.049 0.044
(−2.79) (−0.20) (−0.69) (−0.18) (−1.72) (−0.54) (−0.89) (−0.99) (−0.95) (0.48)
Testing period: July 1951 to June 1977
Raw (%) 0.771
0.935
0.971
0.949
1.017
1.049
1.055
1.107
1.064 1.179
(3.44) (4.03) (3.88) (3.66) (3.85) (3.85) (3.70) (3.70) (3.47) (3.50)
α̂3F (%) −0.076 0.012 −0.010 −0.040 0.002
0.000
0.011
0.005 −0.050 0.028
(−2.49) (0.22) (−0.17) (−0.77) (0.05) (0.00) (0.24) (0.11) (−0.91) (0.30)
Testing period: July 1977 to June 2003
Raw (%) 1.073
1.154
1.163
1.240
1.117
1.247
1.221
1.228
1.265 1.292
(4.16) (4.37) (4.10) (4.41) (3.86) (4.04) (3.78) (3.79) (3.70) (3.89)
α̂3F (%) −0.033 −0.017 −0.062 0.026 −0.155 −0.069 −0.111 −0.113 −0.097 0.033
(−0.90) (−0.29) (−0.90) (0.38) (−2.19) (−1.05) (−1.63) (−1.87) (−1.17) (0.22)
L−S
0.313
(1.65)
0.115
(1.14)
0.408
(1.60)
0.104
(0.99)
0.218
(0.78)
0.066
(0.41)
Panel B: Holding-period performance measured on a monthly basis using equation (7)—re-balanced method
Testing period: July 1951 to June 2003
Raw (%) 0.920
1.037
1.055
1.103
1.064
1.133
1.105
1.134
1.120 1.368 0.448
(5.24) (5.72) (5.42) (5.60) (5.21) (5.28) (5.00) (5.01) (4.80) (5.67) (2.34)
α̂3F (%) −0.108 −0.068 −0.110 −0.065 −0.149 −0.111 −0.154 −0.157 −0.191 0.089 0.197
(−3.31) (−1.52) (−2.34) (−1.43) (−3.12) (−2.51) (−3.86) (−4.14) (−3.81) (0.93) (1.90)
Testing period: July 1951 to June 1977
Raw (%) 0.762
0.928
0.956
0.928
0.980
1.026
1.030
1.092
1.076 1.267 0.505
(3.34) (3.93) (3.75) (3.51) (3.62) (3.67) (3.53) (3.55) (3.42) (3.65) (1.92)
α̂3F (%) −0.098 −0.009 −0.041 −0.077 −0.057 −0.047 −0.043 −0.045 −0.074 0.075 0.173
(−3.08) (−0.17) (−0.73) (−1.40) (−1.10) (−0.92) (−0.96) (−0.98) (−1.30) (0.79) (1.63)
Testing period: July 1977 to June 2003
Raw (%) 1.079
1.146
1.154
1.278
1.149
1.240
1.180
1.176
1.164 1.470 0.391
(4.04) (4.16) (3.92) (4.37) (3.75) (3.81) (3.55) (3.53) (3.37) (4.38) (1.40)
α̂3F (%) −0.073 −0.095 −0.158 −0.026 −0.222 −0.184 −0.262 −0.274 −0.316 0.110 0.184
(−1.41) (−1.40) (−2.12) (−0.36) (−2.80) (−2.58) (−4.04) (−4.58) (−3.84) (0.69) (1.08)
UP
112.91
MV ($m) 9348.2
Panel C: Price and MV of portfolios over July 1951 to June 2003
46.18
38.10
34.98
31.25
27.54
24.61
20.75
16.83
1904.8 995.8
593.4
378.3
246.7
159.6
100.4
56.4
9.14
15.1
−103.78
−9333.1
At the beginning of July each year, stocks are sorted in descending order based on their MVs, and grouped
into decile portfolios using NYSE breakpoints. These portfolios are equally-weighted and held for 12 months.
S(hort) denotes the largest-MV decile, L(ong) is the smallest-MV decile, and L − S is the difference between
L and S. α̂3F is the intercept estimate of the Fama–French three-factor model. UP is unadjusted price per
share. The sample includes all NYSE/AMEX/NASDAQ ordinary common stocks over the period 1951 to
2003. Numbers in parentheses are t-statistics.
21
Table 2
Holding-period performance of portfolios classified by B/M
S
Raw (%)
α̂3F (%)
Raw (%)
α̂3F (%)
Raw (%)
α̂3F (%)
D2
D3
D4
D5
D6
D7
D8
D9
L
Panel A: Holding-period performance measured on a monthly basis using equation (4)
Testing period: July 1951 to June 2003
0.847
1.020
1.115
1.146
1.240 1.290 1.294 1.352 1.462 1.554
(3.19)
(4.32)
(4.82)
(5.38) (5.76) (6.06) (6.35) (6.77) (7.07) (6.87)
−0.225 −0.075
0.014
−0.019 0.119 0.139 0.128 0.174 0.267 0.279
(−2.87) (−1.32) (0.21) (−0.33) (1.80) (2.03) (2.15) (2.81) (3.72) (3.20)
Testing period: July 1951 to June 1977
1.012
0.994
0.984
1.076
1.085 1.222 1.204 1.341 1.464 1.461
(3.13)
(3.31)
(3.40)
(3.82) (3.90) (4.33) (4.35) (4.68) (4.85) (4.46)
0.185
0.087
−0.011
0.047
0.032 0.132 0.086 0.233 0.309 0.230
(2.13)
(1.24) (−0.16) (0.71) (0.52) (2.09) (1.39) (3.17) (4.06) (2.73)
Testing period: July 1977 to June 2003
0.681
1.046
1.245
1.216
1.396 1.358 1.383 1.363 1.460 1.647
(1.62)
(2.87)
(3.45)
(3.80) (4.25) (4.26) (4.61) (4.89) (5.15) (5.26)
−0.659 −0.262 −0.039 −0.111 0.129 0.076 0.101 0.097 0.204 0.311
(−5.45) (−3.12) (−0.38) (−1.26) (1.18) (0.67) (1.07) (1.02) (1.78) (2.11)
L−S
0.708
(4.52)
0.504
(4.92)
0.449
(2.25)
0.045
(0.37)
0.966
(4.02)
0.970
(6.45)
Panel B: Holding-period performance measured on a monthly basis using equation (7)—re-balanced method
Testing period: July 1951 to June 2003
Raw (%)
0.843
1.045
1.121
1.202
1.278 1.310 1.347 1.435 1.542 1.748
0.905
(3.11)
(4.33)
(4.91)
(5.52) (6.03) (6.21) (6.56) (7.04) (7.34) (7.42)
(5.72)
α̂3F (%)
−0.322 −0.145 −0.088 −0.037 0.067 0.068 0.102 0.192 0.277 0.394
0.716
(−3.48) (−2.18) (−1.52) (−0.65) (1.21) (1.14) (1.85) (3.12) (4.10) (4.32)
(6.65)
Testing period: July 1951 to June 1977
Raw (%)
0.960
0.996
0.962
1.095
1.100 1.211 1.238 1.372 1.510 1.524
0.563
(2.92)
(3.24)
(3.26)
(3.79) (3.89) (4.22) (4.39) (4.66) (4.90) (4.48)
(2.75)
α̂3F (%)
0.094
0.055
−0.065
0.036
0.023 0.092 0.101 0.236 0.329 0.247
0.153
(1.13)
(0.78) (−0.96) (0.53) (0.37) (1.48) (1.57) (3.11) (4.11) (2.71)
(1.24)
Testing period: July 1977 to June 2003
Raw (%)
0.725
1.094
1.281
1.308
1.457 1.410 1.456 1.497 1.574 1.972
1.247
(1.68)
(2.94)
(3.67)
(4.01) (4.61) (4.55) (4.87) (5.31) (5.50) (6.04)
(5.19)
α̂3F (%)
−0.733 −0.336 −0.132 −0.109 0.083 0.024 0.078 0.158 0.231 0.545
1.278
(−4.70) (−3.10) (−1.42) (−1.17) (0.93) (0.24) (0.87) (1.67) (2.19) (3.50)
(8.09)
UP
MV ($m)
38.25
1402.2
Panel C: Price and MV of portfolios over July 1951 to June 2003
32.15
28.82
27.14
26.67 24.65 21.94 19.76 17.04
953.4
761.8
566.6
509.2 440.9 359.2 330.1 238.2
12.93
111.6
−25.31
−1290.6
At the beginning of July each year, stocks are sorted in ascending order based on their B/M ratios, and grouped
into decile portfolios using NYSE breakpoints. These portfolios are equally-weighted and held for 12 months.
S(hort) denotes the lowest-B/M decile, L(ong) is the highest-B/M decile, and L − S is the difference between
L and S. α̂3F is the intercept estimate of the Fama–French three-factor model. UP is unadjusted price
per share. The sample includes all non-negative-B/M and non-financial NYSE/AMEX/NASDAQ ordinary
common stocks over the period 1951 to 2003. Numbers in parentheses are t-statistics.
22
Table 3
Holding-period performance of portfolios classified by past 6-month buy-and-hold returns
S
Raw (%)
α̂3F (%)
Raw (%)
α̂3F (%)
Raw (%)
α̂3F (%)
D2
D3
D4
D5
D6
D7
D8
D9
L
Panel A: Holding-period performance measured on a monthly basis using equation (4)
Testing period: July 1951 to December 2003
0.768
0.989
1.142
1.177
1.246 1.279 1.317 1.359 1.448 1.643
(2.60)
(4.39)
(5.70)
(6.28) (6.90) (7.27) (7.44) (7.47) (7.48) (6.86)
−0.760 −0.337 −0.102 −0.024 0.078 0.130 0.192 0.238 0.342 0.540
(−5.94) (−5.04) (−1.99) (−0.53) (1.83) (3.15) (4.32) (4.87) (5.81) (6.17)
Testing period: July 1951 to December 1977
0.711
0.889
0.992
0.990
1.064 1.116 1.130 1.177 1.244 1.447
(1.84)
(2.77)
(3.42)
(3.66) (4.13) (4.47) (4.57) (4.72) (4.82) (4.86)
−0.623 −0.309 −0.130 −0.090 0.014 0.085 0.116 0.170 0.250 0.457
(−5.55) (−4.00) (−2.21) (−2.00) (0.37) (2.62) (2.94) (3.44) (3.62) (4.01)
Testing period: January 1978 to December 2003
0.826
1.090
1.295
1.369
1.431 1.446 1.507 1.545 1.656 1.843
(1.84)
(3.45)
(4.68)
(5.28) (5.66) (5.83) (5.95) (5.82) (5.72) (4.89)
−0.896 −0.354 −0.055
0.061
0.153 0.185 0.267 0.293 0.405 0.551
(−3.86) (−3.32) (−0.70) (0.83) (2.09) (2.52) (3.42) (3.55) (4.37) (4.37)
L−S
0.875
(4.91)
1.300
(8.24)
0.736
(3.27)
1.080
(5.49)
1.017
(3.66)
1.447
(5.91)
Panel B: Holding-period performance measured on a monthly basis using equation (7)—re-balanced method
Testing period: July 1951 to December 2003
Raw (%)
1.063
1.073
1.187
1.208
1.263 1.291 1.328 1.363 1.453 1.646 0.583
(3.40)
(4.64)
(5.81)
(6.35) (6.92) (7.26) (7.46) (7.49) (7.55) (6.98) (3.01)
α̂3F (%)
−0.541 −0.300 −0.093 −0.022 0.067 0.118 0.175 0.215 0.317 0.510 1.050
(−3.67) (−4.11) (−1.72) (−0.47) (1.58) (2.96) (4.18) (4.76) (5.92) (6.36) (6.13)
Testing period: July 1951 to December 1977
Raw (%)
0.853
0.958
1.039
1.024
1.086 1.137 1.151 1.192 1.261 1.450 0.597
(2.14)
(2.91)
(3.50)
(3.71) (4.13) (4.48) (4.59) (4.73) (4.85) (4.86) (2.58)
α̂3F (%)
−0.513 −0.262 −0.101 −0.071 0.023 0.094 0.123 0.172 0.255 0.442 0.956
(−4.16) (−3.08) (−1.56) (−1.40) (0.55) (2.81) (3.25) (3.73) (3.96) (4.11) (4.74)
Testing period: January 1978 to December 2003
Raw (%)
1.277
1.191
1.339
1.396
1.445 1.449 1.508 1.538 1.649 1.846 0.569
(2.63)
(3.65)
(4.77)
(5.33) (5.69) (5.82) (5.97) (5.84) (5.80) (5.03) (1.82)
α̂3F (%)
−0.554 −0.315 −0.054
0.055
0.135 0.160 0.236 0.256 0.364 0.518 1.072
(−2.04) (−2.68) (−0.66) (0.76) (1.89) (2.31) (3.23) (3.35) (4.30) (4.55) (3.88)
UP
MV ($m)
12.55
163.5
Panel C: Price and MV of portfolios over July 1951 to June 2003
19.26
23.27
26.09
27.86 29.19 30.82 30.32 30.21
400.3
560.5
674.9
726.5 779.8 808.7 812.7 736.9
27.28
444.4
14.73
280.9
At the beginning of each month, stocks are sorted in ascending order based on their past 6-month buyand-hold returns, and grouped into decile portfolios using NYSE breakpoints. These portfolios are equallyweighted and held for 6 months. The non-overlapping monthly portfolio returns over the entire testing period
are worked out using the technique of Jegadeesh and Titman (1993). S(hort) is the poorest past 6-month price
performing portfolio, L(ong) is the best past 6-month price performing portfolio, and L−S is the difference
between L and S. α̂3F is the intercept estimate of the Fama–French three-factor model. UP is unadjusted
price per share. The sample includes all NYSE/AMEX/NASDAQ ordinary common stocks over the period
1951 to 2003. Numbers in parentheses are t-statistics.
23
Table 4
Traded values associated with portfolios and transaction-cost-adjusted performance
L (%)
S (%)
L (%)
S (%)
L (%)
S (%)
Panel A: Average traded value per month in longing $1 of L and shorting $1 of S
Decomposed buy-and-hold method
Re-balanced method
Re-balanced minus decomposed
Value bought
Value sold
Value bought
Value sold
Value bought
Value sold
MV -classified portfolio with 12-month holding period
2.710 (8.03)
4.098 (6.58)
6.403 (19.3)
7.771 (23.5)
3.693 (19.9)
3.673 (8.35)
2.480 (6.50)
1.504 (6.69)
4.532 (19.5)
3.611 (15.1)
2.052 (7.93)
2.108 (19.5)
B/M-classified portfolio with 12-month holding period
4.122 (8.17)
5.824 (7.05)
7.485 (14.5)
9.233 (18.1)
3.363 (22.1)
3.409 (7.95)
5.012 (6.82)
4.036 (7.88)
9.212 (18.1)
8.369 (15.4)
4.200 (10.4)
4.333 (24.6)
Past-6-month-return-classified portfolio with 6-month holding period
13.904 (11.2)
15.618 (11.0)
17.625 (14.0) 19.161 (15.9)
3.721 (27.0)
3.543 (9.47)
15.360 (10.7)
14.335 (12.4)
19.385 (17.1) 18.285 (15.8)
4.026 (8.42)
3.950 (17.9)
Panel B: Transaction-cost-adjusted return per month
Decomposed buy-and-hold method
MV portfolio
B/M portfolio
Re-balanced method
r−6 portfolio
MV portfolio
B/M portfolio
r−6 portfolio
Transaction cost is estimated based on the results of Keim and Madhavan (1997)
L (%)
1.154 (4.88)
1.442 (6.37)
1.422 (5.95)
1.143 (4.73)
1.505 (6.39)
1.336 (5.66)
S (%)
0.923 (5.40)
0.918 (3.46)
1.163 (3.91)
0.923 (5.26)
1.020 (3.76)
1.613 (5.09)
L−S (%)
0.231 (1.22)
0.525 (3.36)
0.259 (1.44)
0.219 (1.14)
0.485 (3.09)
−0.277 (−1.39)
Transaction cost level is assumed at 0.5% for buy and 0.5% for sell
L (%)
1.205 (5.10)
1.510 (6.67)
1.500 (6.28)
1.297 (5.38)
1.664 (7.07)
1.461 (6.20)
S (%)
0.940 (5.50)
0.890 (3.35)
0.914 (3.09)
0.961 (5.47)
0.931 (3.43)
1.253 (3.99)
L−S (%)
0.264 (1.39)
0.620 (3.97)
0.586 (3.28)
0.336 (1.75)
0.733 (4.65)
0.208 (1.07)
L (%)
1.174 (4.96)
1.465 (6.47)
1.358 (5.69)
1.227 (5.09)
1.581 (6.72)
1.276 (5.43)
S (%)
0.959 (5.60)
0.933 (3.51)
1.061 (3.58)
1.002 (5.70)
1.019 (3.76)
1.444 (4.59)
L−S (%)
0.215 (1.14)
0.532 (3.40)
0.298 (1.66)
0.225 (1.17)
0.562 (3.56)
−0.168 (−0.86)
Transaction cost level is assumed at 1% for buy and 1% for sell
Transaction cost level is assumed at 1.5% for buy and 1.5% for sell
L (%)
1.144 (4.83)
1.421 (6.27)
1.217 (5.10)
1.156 (4.80)
1.497 (6.37)
1.093 (4.65)
S (%)
0.978 (5.71)
0.976 (3.68)
1.208 (4.07)
1.043 (5.93)
1.107 (4.08)
1.636 (5.19)
L−S (%)
0.167 (0.88)
0.445 (2.83)
0.009 (0.05)
0.113 (0.59)
0.390 (2.47)
−0.543 (−2.76)
S(hort) denotes the largest-MV decile, the lowest-B/M decile, or the lowest-r−6 decile, where r−6 stands
for the past 6-month buy-and-hold return. L(ong) is the smallest-MV decile, the highest-B/M decile, or the
highest-r−6 decile. L−S is the difference between L and S. We form MV - and B/M-based portfolios at the
beginning of July each year and hold these portfolios for 12 months. The r−6 -based portfolios are formed each
month and held for 6 months. We classify portfolios using NYSE breakpoints, and portfolios are equallyweighted. The sample includes all NYSE/AMEX/NASDAQ ordinary common stocks over the period 1951 to
2003. For the B/M-sorted portfolios, the sample is restricted to non-financial and non-negative-B/M stocks,
but there is no such restriction imposed for MV - and r−6 -sorted portfolios. Numbers in parentheses are
t-statistics.
24
APPENDICES
A. Portfolio re-balancing
Any investment strategy potentially requires the investor to re-balance a portfolio within individual
holding periods for a fixed set of stocks and to revise the stocks in the portfolio from one holding
period to the next. In order to deal with transactions costs we need to calculate for all stocks in a
portfolio the degree of re-balancing in terms of the traded values bought and sold for each month
throughout the investment period. We explain below how we calculate the traded values bought and
sold using the re-balanced and decomposed buy-and-hold methods in reference to a long portfolio.
The same calculations apply to a short portfolio with the traded values bought (sold) in a short
portfolio corresponding to the traded values sold (bought) in a long portfolio.
(i) Traded value bought: re-balanced method, long portfolio
For each stock i in portfolio L during holding period h, we calculate the traded value bought in each
month over the m-month holding period (m > 1). For month t (t = 2, 3, . . . , m), we calculate the
value bought at the end of month t necessary to adjust to the appropriate weight for the following
month. For the first holding-period month, in order to preserve the same number of trading amounts
as months in the holding period, we sum the values bought at the beginning and end of the month.
In the first holding-period month, we have to consider whether a stock was part of the portfolio of
the previous holding period. If so, the traded value on the stock at the end of the previous holding
period includes the necessary revision to adjust to the new weight at the beginning of the first month
of the new holding period and the value bought is zero. If not, the value bought at the start of the
first month is the new portfolio weight. At the end of all holding period months there is a further
adjustment to restore original weights. We achieve this by returning the amount invested in each
stock to $wi,h at the start of every month, where wi,h is the portfolio weight of stock i in holding
period h. Therefore, with the re-balanced method, the traded value bought of stock i in portfolio L
in the first holding-period month is,

 w
/ L in h − 1,
i,h + max[−ri,1 wi,h , 0] if i ∈
BUY(L, rb)i,h,1 =
 0
+ max[−ri,1 wi,h , 0] if i ∈ L in h − 1,
where ri,1 is stock i’s return in the first holding-period month (unadjusted for transaction cost).
25
(13)
The traded value bought of stock i at the end of month t is,
BUY(L, rb)i,h,t = max[−ri,t wi,h , 0],
t = 2, . . . , m − 1,
(14)
which corresponds to the second term in equation (13).
Finally, the traded value bought of stock i at the end of the final holding-period month is,
BUY(L, rb)i,h,m = max[wi,h+1 − (1 + ri,m )wi,h , 0],
(15)
where ri,m is the month-m return of stock i (unadjusted for transaction costs) and wi,h+1 is stock i’s
weight in portfolio L in holding period h + 1, which is zero if L does not include stock i in holding
period h + 1.12,13
(ii) Traded value sold: re-balanced method, long portfolio
The traded value sold of stock i in portfolio L at the end of month t is,
SELL(L, rb)i,h,t = max[ri,t wi,h , 0],
t = 1, . . . , m − 1.
(16)
The traded value sold of stock i at the end of the final holding-period month, m, is,
SELL(L, rb)i,h,m = max[(1 + ri,m )wi,h − wi,h+1 , 0].14
(17)
(iii) Traded value bought: decomposed buy-and-hold method, long portfolio
With the decomposed buy-and-hold method, the only transactions are at the beginning and end of
the holding period. Therefore, the traded value bought of stock i in portfolio L in holding-period
month t, t = 2, 3, . . . , m − 1, is zero. The traded value bought of stock i in the first holding-period
month (at the beginning) is,
12 If

 w
/ L in h − 1,
i,h if i ∈
BUY(L, dc)i,h,1 =
 0
if i ∈ L in h − 1.
(18)
the holding period is one month (m = 1), equation (13) with max[wi,h+1 − wi,h (1 + ri,1 ), 0] replacing the original
second term gives the traded value bought of stock i for both the re-balanced and the decomposed buy-and-hold methods.
13 In this and subsequent equations we do not repeat notation defined for earlier equations.
14 If the holding period is one month (m = 1), max[w (1 + r ) − w
i,1
i,h
i,h+1 , 0] gives the traded value sold of stock i for
both the re-balanced and the decomposed buy-and-hold methods.
26
The traded value bought of stock i at the end of the final holding-period month is,
BUY(L, dc)i,h,m = max[wi,h+1 − wi,h (1 + ri,1 )(1 + ri,2 ) · · · (1 + ri,m ), 0].
(19)
(iv) Traded value sold: decomposed buy-and-hold method, long portfolio
With the decomposed buy-and-hold method, the traded value sold of stock i in portfolio L in
holding-period month t, t = 1, 2, . . . , m − 1, is zero. The traded value sold of stock i at the end
of the final holding-period month, m, is,
SELL(L, dc)i,h,m = max[wi,h (1 + ri,1 )(1 + ri,2 ) · · · (1 + ri,m ) − wi,h+1 , 0].
(20)
Based on the monthly traded values of each stock in the portfolio, the traded value (either bought or
sold) of the portfolio in each holding-period month is the sum of the traded values of the constituent
stocks in that month.
B. Transaction-cost-adjusted returns
To account for transaction costs (TC) in holding a portfolio for m months (m > 1), we calculate
portfolio returns based on transaction-cost-adjusted returns of individual stocks in the portfolio.
(i) Transaction-cost-adjusted returns: decomposed buy-and-hold method, long portfolio
With the decomposed buy-and-hold method, we apply the transaction cost adjustment to the first
and last holding-period month returns of each stock. In the first month stock i’s transaction cost
adjusted return is,
r(L, TC, dc)i,1 =
wi,h (1 + ri,1 )
− 1,
wi,h + TCi,h,0
(21)
where TCi,h,0 is stock i’s estimated transaction cost for the traded value bought of wi,h at the beginning of the holding period. TCi,h,0 equals zero if stock i is in portfolio L in holding period h − 1,
since the transaction cost adjustment then occurs at the end of holding period h − 1.
Stock i’s final month transaction-cost-adjusted return is,
r(L, TC, dc)i,m = ri,m −
TCi,h,m
,
wi,h (1 + ri,1 )(1 + ri,2 ) · · · (1 + ri,m−1 )
27
(22)
where TCi,h,m is stock i’s estimated transaction cost at the end of the holding period for the traded
value of | wi,h+1 − wi,h (1 + ri,1 )(1 + ri,2 ) · · · (1 + ri,m ) | .15 If wi,h+1 − wi,h (1 + ri,1 ) · · · (1 + ri,m ) > 0,
the investor buys stock i, and sells otherwise.
Given the transaction-cost-adjusted monthly returns for individual stocks in equations (21) and
(22), equation (3) gives the portfolio’s transaction-cost-adjusted monthly returns over the m-month
holding period.
(ii) Transaction-cost-adjusted returns: re-balanced method, long portfolio
The re-balanced method requires a transaction cost adjustment every month of the m-month holding
period (m > 1). For stock i in portfolio L, its transaction-cost-adjusted return in the first holdingperiod month is,
r(L, TC, rb)i,1 =
wi,h (1 + ri,1 ) − TCi,h,1
− 1,
wi,h + TCi,h,0
(23)
where TCi,h,1 is the transaction cost at the end of the first holding-period month for the traded value
of |−wi,h ri,1 | to re-balance the portfolio weight to wi,h . If −wi,h ri,1 < 0, the investor sells |−wi,h ri,1 |
of stock i. If −wi,h ri,1 > 0, the investor buys | −wi,h ri,1 | of stock i. TCi,h,0 is the transaction cost at
the beginning of the holding period. If stock i is in portfolio L in holding period h − 1, TCi,h,0 = 0.
Otherwise, TCi,h,0 is the transaction cost for the traded value bought of wi,h .
For month t, the transaction-cost-adjusted return is,
r(L, TC, rb)i,t =
TCi,h,t
wi,h (1 + ri,t ) − TCi,h,t
− 1 = ri,t −
,
wi,h
wi,h
t = 2, 3, ..., m − 1,
(24)
where TCi,h,t is the transaction cost for the traded value of | −wi,h ri,t |. If −wi,h ri,t < 0, the investor
sells $| −wi,h ri,t | of stock i. If −wi,h ri,t > 0, the investor buys $| −wi,h ri,t | of stock i.
For the final holding period month, stock i’s transaction-cost-adjusted return is,
r(L, TC, rb)i,m =
wi,h (1 + ri,m ) − TCi,h,m
TCi,h,m
− 1 = ri,m −
,
wi,h
wi,h
(25)
where TCi,h,m is the transaction cost at the end of holding period h for the traded value | wi,h+1 −
wi,h (1 + ri,m ) |. If wi,h+1 − wi,h (1 + ri,m ) > 0, the investor buys stock i, and sells otherwise.
15 If the holding period is one month (m = 1), the transaction-cost-adjusted return of stock i over the one-month holding
period is [wi,h (1 + ri,1 ) − TCi,h,m ]/(wi,h + TCi,h,0 ) − 1, where TCi,h,m is for the traded value of | wi,h+1 − wi,h (1 + ri,1 ) |.
28
Using equations (23)–(25), the portfolio’s transaction-cost-adjusted monthly returns over the mmonth holding period under the re-balanced method are,
N
r(TC, rb)L,τ = ∑ wi,h r(L, TC, rb)i,τ ,
τ = 1, 2, ..., m,
(26)
i=1
where N is the number of stocks in the long portfolio L in holding period h.
(iii) Transaction-cost-adjusted returns: decomposed buy-and-hold method, short portfolio
Under the decomposed buy-and-hold method, the transaction cost adjustment again applies only
to the first and the final holding-period monthly returns. For stock i in portfolio S, its first month
transaction cost adjusted (negative) return is,16
r(S, TC, dc)i,1 =
wi,h (1 + ri,1 )
− 1,
wi,h − TCi,h,0
(27)
where TCi,h,0 is stock i’s estimated transaction cost for the traded value sold of wi,h at the beginning
of the holding period. TCi,h,0 equals zero if stock i is in portfolio S in holding period h − 1, since
the transaction cost adjustment then occurs at the end of holding period h − 1.
The transaction cost adjusted (negative) return of stock i in the final holding-period month is,
r(S, TC, dc)i,m = ri,m +
TCi,h,m
,
wi,h (1 + ri,1 )(1 + ri,2 )...(1 + ri,m−1 )
(28)
where TCi,h,m is the transaction cost for the traded value of | wi,h (1 + ri,1 )(1 + ri,2 ) · · · (1 + ri,m ) −
wi,h+1 | .17 If wi,h (1 + ri,1 )(1 + ri,2 ) · · · (1 + ri,m ) − wi,h+1 < 0, the investor sells, and buys otherwise.
Given the transaction-cost-adjusted monthly returns in equations (27) and (28), equation (3) gives
the portfolio’s transaction-cost-adjusted monthly (negative) returns over the m-month holding period.
(iv) Transaction-cost-adjusted monthly returns: re-balanced method, short portfolio
16 To
be consistent with the common convention in the literature, we present the negative return from shorting a stock
or portfolio.
17 If the holding period is one month (m = 1), the transaction-cost-adjusted (negative) return from shorting stock i for
both decomposed and re-balanced methods is [wi,h (1+ri,1 )+TCi,h,m ]/(wi,h −TCi,h,0 )−1, where TCi,h,m is for the traded
value of | wi,h (1 + ri,1 ) − wi,h+1 | .
29
The re-balanced method again requires a transaction cost adjustment in every month of the m-month
(m > 1) holding period. For the first month of the holding period, stock i’s transaction-cost-adjusted
(negative) return is,
r(S, TC, rb)i,1 =
wi,h (1 + ri,1 ) + TCi,h,1
− 1,
wi,h − TCi,h,0
(29)
where TCi,h,1 is the transaction cost for a traded value of | wi,h ri,1 |. If wi,h ri,1 < 0, the investor sells,
and buys otherwise.
For holding-period month t, stock i’s transaction-cost-adjusted (negative) return is,
r(S, TC, rb)i,t = −
TCi,h,t
wi,h − [wi,h (1 + ri,t ) + TCi,h,t ]
= ri,t +
,
wi,h
wi,h
t = 2, 3, ..., m − 1,
(30)
where TCi,h,t is the transaction cost at the end of the month for a traded value of | wi,h ri,t |. If
wi,h ri,1 < 0, the investor sells, and buys otherwise.
For the final holding-period month m, stock i’s transaction-cost-adjusted negative return is,
r(S, TC, rb)i,m = −
wi,h − [wi,h (1 + ri,m ) + TCi,h,m ]
TCi,h,m
= ri,m +
,
wi,h
wi,h
(31)
where TCi,h,m is based on a traded value of | wi,h (1 + ri,m ) − wi,h+1 |. If wi,h (1 + ri,m ) − wi,h+1 < 0,
the investor sells, and buys otherwise.
Given equations (29)–(31), the short portfolio’s re-balanced transaction-cost-adjusted monthly negative returns over the m-month holding period are,
N
r(TC, rb)S,τ = ∑ wi,h r(S, TC, rb)i,τ ,
τ = 1, 2, ..., m,
(32)
i=1
where N is the number of stocks in the short portfolio S in holding period h.
C. Estimating transaction costs
Apart from assuming different transaction cost levels, we also estimate the transaction cost for an
individual stock based on the results of Keim and Madhavan (1997) as follows,
TC(buy)it = 0.767 + 0.336DNASDAQ − 0.084 ln(MV it ) + 13.807/Pit ,
(33)
TC(sell)it = 0.505 + 0.058DNASDAQ − 0.059 ln(MV it ) + 6.537/Pit ,
(34)
30
where TC(buy)it is the transaction cost (as a percentage of traded value) of buying stock i in month
t, TC(sell)it is the transaction cost of selling stock i in month t, DNASDAQ = 1 for NASDAQ stocks
and DNASDAQ = 0 for NYSE/AMEX stocks, MVit is the market capitalization of stock i at time t in
thousands of dollars, and Pit is the price of stock i at t. Wermers (2000) and Cooper, Gutierrez Jr.,
and Marcum (2005) also use the results of Keim and Madhavan (1997) with a time-adjustment
factor.
The above estimates ignore the effects of technical and indexing trades as well as trade size. In
addition, we set upper and lower limits on the estimates from equations (33) and (34) of zero and
2.5%. Barber and Odean (2000) find that the average round-trip trading cost for individual investors
is 4.03% over the period 1991 to 1996, suggesting a half-trip cost of less than 2.5%. In addition,
the results of Keim and Madhavan (1997) are for the period 1991–1993. Given the decline in
transaction costs over time, it is likely that equations (33) and (34) overestimate transaction costs
over 1994–2003, and underestimate them over 1951–1990. Taken together, our procedures are
more likely to be conservative in estimating transaction costs.
31