MACROECON & INT'L FINANCE WORKSHOP
presented by Murray Carlson
FRIDAY, April 19, 2013
3:30 pm – 5:00 pm, Room: HOH-706
Heterogeneous Information Diffusion and Horizon Effects in
Average Returns
Oliver Boguth, Murray Carlson, Adlai Fisher, and Mikhail Simutin∗
April 15, 2013
ABSTRACT
We show that when stocks react to fundamentals with heterogeneous delay, the observed
short-horizon mean return of a buy-and-hold portfolio is downward biased relative to its
average fundamental return. Our theory predicts distinct patterns in portfolio mean returns calculated at different horizons, depending on the degree of information delay for
stocks within the portfolio. Consistent with our model, average daily returns of portfolios
of small, illiquid, and volatile stocks, rescaled to longer horizons by compounding, show
downward bias on the order of 6% annually. Evidence of substantial bias remains in the
average monthly returns of some U.S. style portfolios and in the majority of international
indices. The direction and magnitude of these findings cannot be explained by standard
microstructure frictions such as bid-ask bounce, iid measurement error, or asynchronous
trade. Our results contribute to growing evidence of delayed price adjustment as an important friction with broad impacts. The theory and findings also have practical implications
for benchmarking and performance evaluation.
∗
We thank Hank Bessembinder, Carole Comerton-Forde, Cam Harvey, Andrew Karolyi, and seminar participants at the University of New South Wales, University of North Carolina at Chapel Hill, the University
of Toronto, SUNY-Buffalo, Wilfred Laurier University, the Finance Down Under Conference, the Northern
Finance Association Meetings, and the Pacific Northwest Finance Conference for helpful comments. This
paper previously circulated under the title “On Horizon Effects and Microstructure Bias in Average Returns and Alphas.”
Boguth: oliver.boguth@asu.edu; W. P. Carey School of Business, Arizona State
University, PO Box 873906, Tempe, AZ 85287-3906. Carlson and Fisher: murray.carlson@sauder.ubc.ca; adlai.fisher@sauder.ubc.ca; Sauder School of Business, University of British Columbia, 2053 Main Mall, Vancouver, BC, V6T 1Z2. Simutin: mikhail.simutin@rotman.utoronto.ca; Rotman School of Management, University
of Toronto, 105 St. George Street, Toronto ON, Canada, M5S 3E6. Support for this project from the Social
Sciences and Humanities Research Council of Canada and the UBC Bureau of Asset Management is gratefully
acknowledged.
Heterogeneous Information Diffusion and Horizon Effects in
Average Returns
Abstract
We show that when stocks react to fundamentals with heterogeneous delay, the observed short-horizon mean return of a buy-and-hold portfolio is downward biased
relative to its average fundamental return. Our theory predicts distinct patterns
in portfolio mean returns calculated at different horizons, depending on the degree
of information delay for stocks within the portfolio. Consistent with our model,
average daily returns of portfolios of small, illiquid, and volatile stocks, rescaled to
longer horizons by compounding, show downward bias on the order of 6% annually. Evidence of substantial bias remains in the average monthly returns of some
U.S. style portfolios and in the majority of international indices. The direction
and magnitude of these findings cannot be explained by standard microstructure
frictions such as bid-ask bounce, iid measurement error, or asynchronous trade.
Our results contribute to growing evidence of delayed price adjustment as an important friction with broad impacts. The theory and findings also have practical
implications for benchmarking and performance evaluation.
1. Introduction
Over the last two decades, considerable evidence has accumulated that stock prices
react to new information with differential delays. In influential early work, Lo and
MacKinlay (1990) present evidence of predictability from large stocks to small stocks.
An impressive subsequent literature shows that analyst coverage, degree of firm complexity, investor attention depending on day of the week, and other factors can influence the speed with which stocks react to news.1 When stock prices relate to news at
different rates, well-known consequences include strong positive autocorrelations and
cross-autocorrelations in short-horizon portfolio returns.
We demonstrate new implications of heterogeneity in information diffusion for average portfolio returns. We first develop a theoretical environment in which stocks react
to fundamentals with different delays. We show that the observed short-horizon mean
return of a buy-and-hold portfolio is downward biased relative to its average fundamental return. This result may seem surprising. Prior literature has focused on earning
positive profits by active trading strategies that take advantage of slow information
diffusion. Our result applies to the average short-horizon returns of a buy-and-hold
strategy, and can be understood by considering the portfolio weights implied by a buyand-hold strategy.
Consider measuring portfolio returns following a positive shock to fundamentals.
Stocks that react slowly to the positive information will be underweighted at the beginning of the measurement interval relative to their fundamental values. At the same
time, their observed short-horizon returns are expected to be high. Similarly, follow1
See, for example, Brennan, Jegadeesh, and Swaminathan (1993), Badrinath, Kale, and Noe (1995),
Klibanoff, Lamont, and Wizman (1998), Chordia and Swaminathan (2000), Hong, Lim, and Stein (2000),
Huberman and Regev (2001), Hirshleifer and Teoh (2003), Hou and Moskowitz (2005), Cohen and Frazzini
(2008), Dellavigna and Pollet (2009), Hirshleifer, Lim, and Teoh (2009), Menzly and Ozbas (2010), Chordia,
Sarkar, and Subrahmanyam (2011), Hirshleifer, Lim, and Teoh (2011), Tetlock (2011), and Cohen and Lou
(2012).
1
ing a negative shock to fundamentals, slow-adjusting stocks are overweighted in the
portfolio, while their future returns will tend to be low. The negative cross-sectional
correlation between portfolio weights and future observed returns has implications for
portfolio returns. In particular, observed average portfolio returns become downward
biased relative to their fundamental return at short horizons. These effects are less
important when returns are measured over longer intervals, since the effects of slow information diffusion become smaller. Thus, theory predicts specific patterns in average
returns measured over different horizons.
The setting for our analysis differs fundamentally from prior literature, which has
considered bias in average returns arising from iid measurement error in prices. Blume
and Stambaugh (1983) and Roll (1983) explain how iid measurement errors in prices
cause upward bias in the mean returns of individual stocks and equally-weighted,
periodically-rebalanced portfolios, due to Jensen’s inequality. Conrad and Kaul (1993),
Canina, Michaely, Thaler, and Womack (1998), and Liu and Strong (2008) provide additional empirical analysis and recommendations to empirical researchers in the presence
of iid measurement error. Most recently, Asparouhova, Bessembinder, and Kalcheva
(“ABK”, 2010, 2012) show how biases arise in Fama-MacBeth regressions and average
returns of equal-weighted style portfolios under iid measurement error, and propose
empirical corrections. In contrast to this literature, we focus on delayed price adjustment as a pricing friction. Unlike the standard assumption of iid measurement error,
our setting implies price deviations that are correlated with fundamentals. ABK (2012,
p. 46) anticipate that the case we investigate, where measurement errors are correlated
with fundamentals, should be a profitable direction for research.
Our implications for biases in measured mean returns also differ fundamentally from
prior literature. Under the iid measurement errors studied previously, equal-weighted
portfolio returns are upward biased, and the mean returns of well-diversified value-
2
weighted portfolios remain unbiased. Indeed, the weighting schemes proposed by ABK
capture the beneficial aspects of allowing portfolio weights to vary with past returns.
In contrast, under the case of heterogeneous information diffusion that we focus on,
portfolio returns are downward biased rather than upward biased, and value-weighted
portfolios are more severely impacted than equal-weighted or otherwise rebalanced portfolios. We can thus sharply distinguish the empirical predictions for mean returns of
iid measurement error versus heterogeneity in information diffusion.
An existing literature analyzes differential price adjustment, but focuses on impacts to higher moments of returns, such as individual stock and portfolio variances,
autocorrelations, and cross-autocorrelations. Most notably, delayed reaction of some
securities to fundamentals causes portfolio returns to be positively autocorrelated and
observed portfolio variances to be downward biased (Scholes and Williams, 1977; Lo
and MacKinlay, 1990). These same studies conclude, in contrast to our findings, that
the implications for observed mean returns are innocuous.2 However, these studies
base their conclusions on the properties of logarithmic returns. Empirical researchers
are more often concerned with simple returns, as for example when calculating an alpha
from a time-series regression or carrying out a standard cross-sectional asset pricing test.
Unlike logarithmic returns, which are additive, simple returns involve compounding. In
the presence of asynchronous price adjustment, positive autocorrelations, generated by
heterogeneity in information diffusion rather than fundamentals, inflates the importance of compounding. The average single-period return must be downward biased to
compensate.
To provide a more complete framework for empirical analysis, we develop an approximation for the difference between average returns calculated over arbitrary different
2
Scholes and Williams write, “. . .expectations of measured returns [. . .] always equal true mean returns” (p.
113). Lo and MacKinlay state, “. . . nontrading does not affect the mean of observed returns” (p. 187).
3
time scales.3 Our formula shows that when the variance ratio of long-horizon to shorthorizon returns exceeds one, indicating persistence, the average subperiod return scaled
by multiplying or compounding is downward biased relative to the buy-and-hold return.
We verify that this approximation is empirically accurate for a variety of standard style
portfolios. Building on this result, we show that alpha differences across different horizons can be largely explained by applying our formula. Thus, alphas across horizons are
not arbitrarily different, but to a close approximation are explained by a few specific
moments of returns, most notably variance ratios across horizons.
Consistent with our theoretical predictions, we show empirically that short-horizon
portfolio return averages, alphas, and Fama-MacBeth coefficients are often not reliable,
even in the case of value-weighting or other non-rebalanced strategies. For all portfolios we consider, performance metrics based on daily return measures significantly
understate longer-horizon buy-and-hold returns. As theory predicts the biases are particularly strong for illiquid and volatile portfolios such as those containing small stocks
and momentum losers. The differences in return measures reach up to 6% annually for
U.S. style portfolios. Even monthly returns of U.S. style portfolios show meaningful
biases, and only at a quarterly horizon do average returns reliably show small horizon
effects.
The evidence in international indices is even stronger. Average monthly returns,
rescaled by compounding to an annual horizon, consistently understate longer-horizon
returns. The magnitude of the effect in emerging and frontier markets, where heterogeneity in information arrivals is most severe, exceeds 10% annually in some cases.
Exchange Traded Funds (ETFs) which track a subset of the international indices are
available, for which daily closing prices are arguably less succeptible to the microstruc3
Horizon effects in returns have been a fundamental topic in finance since Blume (1974). More recent
developments focus on the subtle effects of estimation error (Jacquier, Kane, and Marcus, 2003, 2005) and
parameter uncertainty (Pastor and Veronesi, 2003; Pastor and Stambaugh, 2012).
4
ture biases that affect the individual stocks prices used to compute index returns. Although horizon effects are somewhat smaller for ETFs than for the associated indices,
rescaled average monthly returns of ETFs remain consistently below the annual average
returns. The differences between short and long-horizon mean returns in international
markets thus remain economically significant even in data that is likely to reflect the
price impact of trading activity at the individual stock level.
Section 2 provides theoretical analysis of the link between heterogeneity in information diffusion and average portfolio returns. Section 3 demonstrates implications for
horizon effects in standard empirical methods. Section 4 provide empirical evidence
in U.S. style portfolios. Section 5 gives international evidence. All proofs are in the
Appendix.
2. A Model of Heterogeneity in Information Diffusion
In this section we develop a model in which stocks react to fundamental information with
heterogeneous delays. We provide considerable generality in the model. The delay type
of an individual firm follows a stationary Markov-switching process, permitting that the
speed at which an individual firm adjusts to systematic information may change over
time. The model also allows heterogeneous adjustment to multiple sources of systematic
news.
We first develop an analytical result showing that in the presence of heterogeneity
in information diffusion, individual stock and portfolio mean returns are downward biased relative to the mean returns of fundamentals. We then provide a parsimonious
calibration showing parameters that permit the model to quantitatively match in a
portfolio of small stocks the following empirically observed moments: 1) a slowly decaying (hyperbolic) pattern of loadings on current and lagged daily market returns,
2) a slowly decaying pattern of return autocorrelations, and 3) observed mean returns
5
at different horizons, including apparent downward bias at short-horizons. We also
show the parameters under which the model matches empirically observed moments of
a value-weighted large-stock portfolio. These parameters imply negligible amounts of
slow price adjustment.
The setting we analyze complements a set of models developed in earlier literature. First, iid measurement error (e.g., Blume and Stambaugh (1983)) produces effects in value-weighted portfolio returns that differ qualitatively from what we observe
in the data. Second, models based on asynchronous trade (e.g., Lo and MacKinlay
(1990)) produce qualitatively similar patterns in portfolio autocorrelations to the data.
However, prior literature has already shown that even aggressive calibrations of asynchronous trade models cannot match the level and persistence of observed portfolio
autocorrelations (Boudoukh, Richardson, and Whitelaw (1994)). We add to prior literature by showing theoretically the implications of asynchronous trade for downward
bias and horizon effects in average returns. We also show, complementing Boudoukh,
Richardson, and Whitelaw (1994), that even aggressive calibrations of asynchronous
trade models cannot capture quantitatively observed horizon effects in mean returns.
For the convenience of the reader, the iid measurement error and asynchronous trade
models, as well as our new theoretical results for the asynchronous trade model and
calibrations, are summarized in the Internet Appendix.
We now focus attention on our model of heterogeneity in information diffusion. This
is the only model among the broad set we have considered that is able to jointly match
empirical moments of value-weighted portfolio betas and lagged betas, autocorrelations,
and horizon effects in mean returns.
6
2.1. Model Setup
We assume fundamental asset returns over one period have a K-dimensional factor
structure, and denote the vector of factor realizations at time t by ft = (f1,t ...fK,t )0 , for
t ∈ N. For simplicity, we assume the factors are independent and identically normally
distributed with mean µf and diagonal covariance matrix Σf .4 The fundamental value
of an individual stock has instantaneous logarithmic returns given by
rt∗ = rf t + β 0 ft + εt ,
(1)
where we use an asterisk to denote the fundamental return, rf t is the riskless rate, β
is a K × 1 vector of constant factor exposures, and εt is an iid normal variable with
zero mean and standard deviation σ. To simplify notation, we suppress subscripts to
distinguish individual assets throughout our exposition.
To capture the idea of slow information diffusion, we permit that observed stock
returns may reflect innovations in lagged as well as contemporaneous fundamentals.
We stochastically assign stocks into one of Θ groups. Each group θ ∈ 1, ..., Θ differs by
the speed δθ at which fundamental information incorporates into observed asset prices.
Observed logarithmic stock returns follow
" Θ
#
X
rt = rf t + β
δθ (Dθt−1 + ρθt ft ) + εt ,
(2)
θ=1
where, for each θ, δθ is a scalar, Dθt−1 is a K × 1 vector, and ρθt is a diagonal K × K
matrix. The state variables Dθt track “information deficits” with respect to all factors
in each decay group θ, and the elements of ρθt randomly match information arriving
from the K factors to a group θ.
The random variables ρθt are required to be stationary, non-negative, and sum to
P
one: Θ
θ=1 ρθt = 1. These restrictions ensure that new information arrival cannot make
4
The analysis can easily be modified to accommodate features such as non-normal returns or stochastic
volatility.
7
prices “catch up” on old information faster, and that all information is eventually
incorporated into prices. In the implementation, we will further assume that combinations of period−factor information are be assigned to only one decay group, i.e., that
ρθt ∈ {0, 1} for all θ, t.
Information deficits Dθt accumulate underreaction to past and current factor realizations according to the state equations
D1t = (1 − δ1 )(D1t−1 + ρ1t ft )
D2t = (1 − δ2 )(D2t−1 + ρ2t ft )
..
.
DΘt = (1 − δΘ )(DΘt−1 + ρΘt ft ),
(3)
where 1 = δ1 > δ2 > ... > δΘ > 0 are non-stochastic, stock specific information delays.
Both the states Dθt and the random matrices ρθt are assumed to be unobservable.
The impact of a current factor realization on stock returns is determined by the terms
PΘ
θ=1 δθ ρθt ft in equation (2). The date−t information delay is therefore determined by
the evolution of the family of random matrices ρθt . To model these processes, we
define the associated Markov chains skt ∈ {1, 2, ..., Θ} on the “delay groups” skt . The
delay group skt = 1 is associated with the parameter δ1 = 1 and therefore produces a
price process that reacts instantaneously to factor k information, while the delay state
skt = Θ, with δΘ < δθ ∀θ, corresponds to the slowest possible reaction stock prices
to current factor−k realizations. The transition matrices for the delay states for each
factor k are assumed to be independent and are given by

p
p
· · · pkΘ
 k1 k2

 pk1 pk2 · · · pkΘ

Pk = e−λ∆ IΘ + (1 − e−λ∆ ) 
..

.


pk1 pk2 · · · pkΘ
8









(4)
where λ is the arrival intensity of a change in state and IΘ is the identity matrix
of dimension Θ. This form of the transition matrix for skt conveniently distinguishes
between changes in the state variable, which occur according to the intensity parameter
λ, and the steady-state distribution of skt as defined by the parameters pkθ . By further
defining the diagonal elements of the matrices ρθt (k, k) = 1{skt =θ} , we assume that in
delay state skt = θ a constant fraction δθ of the current factor realization immediately
impacts on returns, and that the remaining information is released into the market at
a geometrically declining rate δθ (1 − δθ )s−t .
2.2. The Impact of Slow Information Diffusion on Return Measurement
We now consider a special case of our model of slow information diffusion where returns
are driven by a single factor. We consider two classes of assets, one which we refer to as
the leader assets for which factor realizations are immediately incorporated into prices
and one which we refer to as the lagger assets for which there is a single delay state.
Logarithmic returns of the leader assets are assumed to be generated by equation (1)
with rf t = 0 and β = 1,
rt∗ = p∗t − p∗t−1 = ft ,
(5)
where p∗t is the log price of the fundemental asset. The factor realizations ft are independently normally distributed with mean µf and standard deviation σf . Lagger
logarithmic returns rt = pt − pt−1 given by the system of equations
rt = δ(Dt−1 + ft )
Dt = (1 − δ)(Dt−1 + ft ).
(6)
(7)
Straightforward manipulation of the dynamic equation for the information deficit Dt
shows that it is a stationary random variable.
9
Lemma 1 The information deficit follows an AR(1) process
Dt − Dt−1
1−δ
= −δ Dt −
µf + (1 − δ)σf ξt+1 ,
δ
(8)
where ξt are independent standard normal random variables. The unconditional distribution for Dt is normal with E(Dt ) =
1−δ
µf
δ
and Var(Dt ) =
(1−δ)2
σ2 .
1−(1−δ)2 f
If we further assume that there is a known date t = 0 when p∗0 = p0 = D0 = 0 we
can conveniently relate the leader prices to the lagger prices.
Lemma 2 Leader and lagger logarithmic prices are cointegrated:
p∗t = pt + Dt .
(9)
The fact that leader and lagger prices are cointegrated leads to the intuitive conclusion
that short-run return distributions of the two series will be different but that their
long-run return distributions will converge.
It is possible to obtain closed-form expressions for characteristics of lagger returns
within this simplified setting. The following proposition characterizes the unconditional
distribution of short-run returns.
Proposition 1 Lagger short-run log returns rt are unconditionally normal with E(rt ) =
µf and Var(rt ) =
1
δ
δ
σ2
2−δ f
1
eµf + 2 [ 2−δ ]σf < eµf + 2 σf
2
2
< σ 2 . Mean short-run simple returns are given by E (ert ) =
∗
= E ert .
This proposition makes several notable points that are relevant to applied empirical research. First, the proposition shows that the unconditional mean of logarithmic returns
of leader and lagger stocks are equal. Second, although slow diffusion of information
has no implications for mean returns, the short-run lagger log returns have lower unconditional variance than the leader log returns. This is a direct reflection of the fact
that short run lagger returns are exposed to only a fraction 0 < δ < 1 of the current
10
factor realization. Third, a consequence of these two facts is that mean simple returns
of lagger stocks are a downward biased estimate of the fundamental mean simple return
that can be measured from leader stocks. Although the first two points have been made
in prior work (e.g., Lo and MacKinlay (1990)), the downward bias in expected simple
returns has not been emphasized.
Conventional wisdom is that value-weighted portfolios produce returns that are less
susceptible to bias than reweighted portfolios such as equal-weighted portfolios. We
now show that when some stocks respond slowly to factor information, value-weighted
portfolios produce returns with an additional source of downward bias for the fundamental mean return. Assume that a fraction π of stocks are leaders and that the
remaining fraction 1 − π are laggers. The following proposition characterizes the simple
returns for a value weighted portfolio in this setting.
Proposition 2 The simple return on a value-weighted portfolio of leader and lagger
stocks is given by
Rt = wt−1 eft + (1 − wt−1 )eδ(Dt−1 +ft ) ,
(10)
where
wt−1 =
πeDt−1
.
πeDt−1 + 1 − π
(11)
The mean portfolio return is given by
E(Rt ) = E(wt−1 )e
where Cov
µf + 12 σf2
δ
µf + 21 [ 2−δ
σf ]
+(1−E(wt−1 ))e
1−π
, eδ(Dt−1 +ft )
πeDt−1 +1−π
2
+Cov
1−π
δ(Dt−1 +ft )
,e
,
πeDt −1 + 1 − π
(12)
< 0.
This proposition makes several important points. Equation (11) shows that the value
weights are a function of the lagged information deficit Dt−1 and, therefore, stationary.
This result follows from the fact that leader and lagger stock prices are cointegrated.
Equation (12) shows that there are three potential sources of bias when attempting to
11
measure fundamental mean returns using a value weighted portfolio. First, the mean
weight does not equal the proportion of leaders E(wt−1 ) 6= π. Although there is no
closed-form expression for this bias, it can be shown to be small. Second, the lagger
mean stock return is biased downwards relative to the fundamental mean return, as
was demonstrated in Proposition 1. The third source of bias arises because of the
negative correlation between lagger value weights and lagger stock returns as given
by Cov(1 − wt−1 , eδ(Dt−1 +ft ) ). This additional downward bias in the portfolio simple
return, which is caused by downweighting past high factor returns in the lagger stocks,
can be highly significant. Interestingly, the use of exogenous portfolio weights, such as
equal weights, will eliminate this source of bias and will produce higher mean portfolio
returns than does value weighting when slow information diffusion governs a subset of
the returns. The next subsection quantifies the magnitude of these biases within the
context of a calibrated version of the full model.
2.3. Parameterization and Calibration
We provide a calibrated version of our model to quantify the magnitude of heterogenous
information diffusion in small stocks. Heterogeneity across stocks can arise from differences in decay rates δθ as well as from the probabilities that any particular information
is assigned to a given decay rate. In order to parameterize our model in a parsimonious
way, we focus on only two factors, a market and non-market factor, and restrict the
set of allowable delay states and transition probabilities. Small stock returns display
long memory with significant autocorrelations that persist beyond one trading month
(21 days). To capture the slowly-decaying autocorrelations that we find in the data,
we choose the information delay parameters δθ , θ ∈ {1, 2, ...7}, to produce a geometric
progression {0, 20 , 21 , ..., 25 } of shock half lives. To limit parameters in the transition
matrix for delay states, shocks to factor k ∈ {1, 2} are assumed to be entirely incorpo-
12
rated in contemporaneous returns with an unconditional probability pk1 = ak and to
affect returns with a lag chosen from the six possible values δθ , θ ∈ {2, 3, ..., 7}, with
equal unconditional probabilities (1 − ak )/6. These restrictions on the parameter space
allow us to characterize slow reaction to market and non-market factors using only two
parameters: a1 is low when returns react slowly to market return innovations, and a2
is low when returns react slowly to non-market news.
We obtain daily returns produced by the smallest decile of US stocks during the
period 1926-2012 from the data library of Ken French. Two sets of moments are used
to calibrate the model: 1) Betas from univariate regressions of daily excess returns on
the excess market return with lags from 0 to 42 days, and 2) Autocorrelations with lags
from one to 42 days. These moments allows us to determine the slow adjustment of
small stock prices to both market news and undiversifiable non-market news.
Panel A in Figure 1 shows that positive exposure to lagged market shocks persists
for up to two trading months (dashed line). Our model can produce a similar pattern
in lagged market betas (solid line) when a1 = 0.35. Thus, our calibration implies that
market information is instantaneously incorporated into small stock returns only 35%
of the time and that 65% of the time the shock is delayed, potentially so severely that
the half-life of the news is 32 days.
Panel B in Figure 1 shows that positive autocorrelations in small stock returns
persist for up to two trading months (dashed line). To calibrate our model to this
pattern, a second, undiversifiable, non-market factor with the same variance as the
market factor is required. The small value of the parameter a2 = 0.1 indicates that,
unconditionally, small stocks respond slowly to non-market news: Small stocks react
instantaneously to non-market news only 5% of the time. In the vast majority of cases
(95% of the time) non-market factor information diffuses only very slowly into small
stock prices.
13
For comparison, Panels C and D of Figure 1 show lagged betas and autocorrelations
for the largest decile of CRSP stocks. Neither the betas nor the autocorrelations display
evidence of slow information diffusion, and a calibration of the model with a1 = a2 = 1
provides the best fit to this data.
The magnitude of information delay required to explain lagged beta and autocorrelation for small stocks is much higher than can be produced by models of microstructure
frictions. Boudoukh, Richardson, and Whitelaw (1994) consider the impact of nonsynchronous trading and show that such models produce geometrically declining autocorrelations, and that even extreme heterogeneity in daily non-trading probabilities ranging
from 0 − 85% produces autocorrelations of only 18%. Our model, calibrated to small
stock returns, shows that heterogeneity in information diffusion is necessary to explain
both the relatively low short-horizon dependencies in returns and the high long-horizon
dependencies. This pattern in returns is most plausibly generated by slow information
diffusion into observed small-stock prices.
3. Horizon Effects in Empirical Applications
We provide an analytical formula that shows how average returns scale across horizons.
We then extend our analysis to the scaling of abnormal performance as measured by
Jensen’s (1968) alpha and Fama and MacBeth (1973) coefficients, and develop a simple
diagnostic tool that shows the impact of microstructure effects at different horizons.
3.1. Horizon Effects in Average Returns
Consider the problem of an empiricist evaluating the performance of an investment
strategy. A standard return decomposition identifies abnormal return as measured by
alpha, the systematic market component, and a residual:
R̄in − R̄f n = αin + βin R̄M n − R̄f n + εin ,
14
(13)
where R̄jn , j ∈ {i, M, f } , denotes the unconditional average n-period gross return
R̄jn = E(Rj,t+1 . . . Rj,t+n )
for all t,
(14)
the index i denotes an arbitrary portfolio, M the market, f the risk-free rate, βin is the
market exposure calculated from n-period returns, and εin is the component of returns
that is uncorrelated with the market. Setting the base period to one day, R̄i1 gives the
average return using daily returns, while R̄i,21 would reflect an average return calculated
from n = 21 day periods, or approximately monthly returns.
A standard convention for reporting mean returns is to rescale linearly to a different
horizon, typically either a month or a year. We correspondingly define the rescaled
mean returns
RS
R̄in
≡ 1 + n R̄i1 − 1 ,
(15)
and the associated ratio of the linearly rescaled to buy-and-hold mean return
RS
νjn ≡ R̄jn
/R̄jn .
Rescaling a short-horizon mean to approximate a longer-horizon mean introduces a bias
whenever the average net returns of the portfolio do not rescale one-for-one with time.
For iid returns, the ratio νjn will be less than one due to compounding, but we
generally expect this effect to be small. To see this, define the short-horizon return
rescaled by compounding as
n
RSg
R̄in
≡ R̄i1
(16)
and the related ratio
g
RSg
νjn
≡ R̄jn
/R̄jn .
g
Under independence of returns, νjn
= 1. Consider the rescaling of daily alphas to
monthly, quarterly, or annual frequencies, or the rescaling of monthly alphas to quarterly
or annual frequencies. In these cases, the effects of compounding alone are small, and
15
RSg
g
RS
R̄in
≈ R̄in
or equivalently νjn
≈ νjn .5 Hence, when the returns of the portfolio i are
iid, linearly rescaling a daily to a monthly or annual alpha or a monthly to an annual
alpha is innocuous.
To identify and quantify the primary source of horizon effects in mean returns, we
consider normally distributed, single period logarithmic returns: ri1,t = ln (Ri1,t ) ∼
N (µi , σi2 ). Assume that log returns aggregated over n periods have a normal distribu2
tion with variance σin
, where n is the relevant horizon. These assumptions are exactly
satisfied if ri1,t is a stationary ARMA(p, q) process with Gaussian innovations, and
approximately hold in more general cases. We show:
Proposition 3 The ratio of rescaled to buy-and-hold mean returns satisfies
g
νin ≈ νin
≡
RSg
R̄in
2
= enσi (1−V Rin )/2 ,
R̄in
2
/ (nσi2 ) is the variance ratio. The net return ratios are
where V Rin ≡ σin
net
νin
g,net
≈ νin
RS
R̄in
−1
νin − 1
≡
= νin +
R̄in − 1
R̄in − 1
g
RSg
−1
νin
R̄in − 1
g
= νin +
.
≡
R̄in − 1
R̄in − 1
The bias in rescaled returns relative to the buy-and-hold return is thus determined by
the short-horizon variance σi2 and the variance ratio V Rin . When the variance ratio is
net
one, for example if returns are iid, then νin = νin
= 1 and rescaled short-horizon return
means approximate well longer-horizon buy-and-hold averages. Empirically, individual
asset returns are typically negatively autocorrelated, implying V Rin < 1, consistent
with the upward bias in Blume and Stambaugh (1983). For portfolios, the tendency for
net
positive spurious autocorrelations suggests νin
< νin < 1, implying a downward bias
in short-horizon average returns.
RSg
RS
For example, assuming a one percent average monthly return, R̄i12
= 1.1268 and R̄i12
= 1.12. The
magnitudes of the biases in (18) will then be tiny. Since νin = 1.12/1.1268 = 0.9939, the bias in alpha due to
the first term of (18) will be (νin − 1)R̄in = −0.0061 ∗ .01, less than a basis point per month in absolute terms.
5
16
3.2. Horizon Effects in Alphas
Horizon effects in mean returns give rise to horizon effects in performance measures.
Consider the problem of an empiricist evaluating the performance of an investment
strategy using Jensen’s (1968) alpha αin . A standard convention for reporting alphas
is to rescale linearly to a different horizon, typically either a month or a year. For
example, Ang and Kristensen (2011), Barber (2007), Lewellen and Nagel (2006), and
Li and Yang (2011), compute alphas from daily returns, and rescale linearly to longer
horizons. We correspondingly define the rescaled alpha
RS
αin
≡ nαi1 ,
(17)
and show
Proposition 4 The difference between the linearly rescaled and buy-and-hold alphas is
RS
αin
− αin ≈ (νin − 1)R̄in − β1 (νM n − 1)R̄M n − (β1 − βn ) R̄M n − R̄f n .
(18)
Rescaling a short-horizon alpha to approximate a longer-horizon alpha introduces a bias
whenever the average net returns of the portfolio or the market index do not rescale
one-for-one with time, or the betas calculated using different return frequencies are
not identical. A substantial literature investigates the measurement of betas across
different horizons (e.g., Dimson, 1979; Scholes and Williams, 1977), and our paper does
not address this issue. Rather, we focus on the scaling of mean returns across horizons.
In practice, however, horizon effects in alphas can be substantial. To preview our
empirical results, for the portfolio of small stocks, the daily alpha rescaled to a monthly
RS
frequency is αsmall,21
= 43 basis points, while the alpha obtained from using buy-
and-hold returns of 21-day “months” is almost 50% larger, αsmall,21 = 61 basis points.
Rescaling the alpha from a monthly to a quarterly frequency also substantially increases
the alpha.
17
A common view in the empirical finance literature is that alphas calculated at a daily
frequency represent the profits available to an investor with a daily investment horizon,
that monthly alphas represent performance from the perspective of an investor with a
monthly horizon, and so forth. This view implicitly assumes that investors can transact
at observed daily prices. Following arguments in ABK, even if a subset of investors is
able to implement such daily rebalancing, gains and losses from the trades sum to zero
across all investors. Consequently, rescaled alphas can provide an unbiased measure of
profitability for at best only a subset of short-horizon investors who successfully transact
at observed prices. Importantly, such performance measures will not properly capture
profitability of investors in aggregate. As is clear from (13), any bias in measured
average returns of the portfolio or the index due to microstructure effects will bias the
calculation of alpha.
3.3. Horizon Effects in Fama-MacBeth Regressions
The choice of return horizon also impacts Fama and MacBeth (1973, FM) coefficients
and their interpretation. Consider cross-sectional regressions:
Rin,t − 1 = ant + bnt Xit + int
(19)
where for simplicity Xit is a univariate characteristic for stock i at time t, Rin,t is an nperiod gross return on stock i starting at time t, and int is uncorrelated and mean zero.
Fama (1976, p. 328) shows that the time-series of FM coefficients bnt can be interpreted
as payoffs on a zero-cost investment strategy with portfolio weights proportional to the
characteristic.6
To see how the estimated FM coefficients change with time scale, consider estimating
(19) using n = 1 and n = 2 period returns. The conditional expectation of a 1-period
6
For example, suppose Xit is a small stock indicator. Then, ant measures the realized return of a large
stock portfolio, and bnt represents the realized size premium.
18
return is
E (Rin,t | Xit ) = 1 + a1 + b1 Xit .
(20)
The conditional expectation of the two-period return is
E (Rin,t Rin,t+1 | Xit ) = E (1 + a1t + a1t+1 + b1t Xit + b1t+1 Xit+1 | Xit )
+E (a1t a1t+1 | Xit )
+E (a1t b1t+1 Xit+1 + a1t+1 b1t Xit | Xit )
+E (b1t b1t+1 Xit Xit+1 ) | Xit ) .
(21)
If the time-series of FM coefficients are serially independent and uncorrelated with the
characteristic Xit , only the first term of Equation (21) is substantially different from
zero. In this case, provided the characteristic Xit is approximately constant across periods, the coefficient means scale approximately linearly, i.e., a2 ≈ 2a1 and b2 ≈ 2b1 ,
where an and bn respectively denote the time-series averages of the regression coefficients ant and bnt . Serial correlations in the time-series of FM coefficients can be driven
by microstructure noise. For example, in the case where positive autocorrelations and
cross-serial correlations are driven by asynchronous price adjustment, we expect the
short-horizon coefficients to be downward biased, 2a1 < a2 , 2b1 < b2 .7 Generally, different types of microstructure noise could lead to different patterns in the autocorrelations
of the FM coefficients.
In recent work, ABK focus on the specific case of noise in prices that is uncorrelated
with fundamentals. They show that weighting observations i by their lagged returns
can correct the bias caused by iid noise in prices. The ABK weighting scheme would
not work, however, to correct bias in FM coefficients caused by asynchronous price
adjustment, as such a weighting scheme does not address the autocorrelation terms in
(21). To account for both iid measurement error, as in ABK, and asynchronous price
7
For example, Xit inversely related to size ensures cross-serial correlations will be positive since large stocks
typically lead small stocks.
19
adjustment, one would need to use the ABK weights as well as choose a return horizon
long enough to minimize the importance of autocorrelations and cross-autocorrelations
in the FM coefficients.
3.4. Diagnosing Microstructure Frictions and Choosing the Return Horizon
We propose a simple procedure to diagnose biases in average returns and choose a horizon for empirical analysis. Modifying and extending the notation developed in Section
3.1 to accommodate greater empirical flexibility, consider geometrically rescaling an
n-period return to a horizon of m periods:
m/n
R̄inm ≡ [E(Ri,t+1 . . . Ri,t+n )]m/n ≡ R̄in .
(22)
If returns are iid,
R̄inm = R̄imm ≡ R̄im
for all n, m > 0.
(23)
This suggests using a plot of R̄inm versus the return-period length n as a diagnostic.
When this plot is approximately flat, then rescaling effects have a small impact.
Figure 2 demonstrates this diagnostic technique using data simulated from our Section 2 model using both the small and large stock calibrations. Panel A clearly shows
that for small stocks, where evidence of slow information diffusion is strong, value
weighted portfolio return means are downward biased at short horizons. This bias is
highly economically significant, with rescaled daily mean returns a full 40 basis points
per month lower than the rescaled annual mean returns. On the other hand, Panel B
shows that for the large stocks, where lagged betas and return autocorrelations provided no evidence of slow information diffusion, no horizon effects are present in average
holding period returns.
20
4. Empirical Evidence in U.S. Indices and Style Portfolios
We show the magnitudes of the rescaling biases in attribute-sorted portfolios and market
indices, and implement empirical methods to avoid these biases.
4.1. Data
Using data from CRSP and Compustat, we form portfolio returns on the basis of the
following characteristics:
Market Equity: The market value of the equity of the firm at the end of calendar
year τ − 1 is used to form portfolios beginning in July of year τ .
Book-to-Market: The ratio of the book value of equity to the market value of
equity, using the end of the previous calendar year book and market equity values.8
Similar to Fama and French (1993), the book-to-market ratio at the end of calendar
year τ − 1 is used to form portfolios starting in July of year τ .
Momentum: The cumulative return of individual stocks in the calendar months
t − 12 to t − 2 is used to form portfolios starting in month t.
Price: The price per share at the end of month t − 2 is used to construct portfolios
beginning in month t.
Short-term reversal: The return in month t − 2 is used to form portfolios starting
in month t.
Volatility: The standard deviation of daily stock returns estimated from months
t − 13 to t − 2 is used to form portfolios starting in month t.
8
Book equity used to calculate the book-to-market ratio is defined as stockholders’ book equity plus balance
sheet deferred taxes plus investment tax credit less the redemption value of preferred stock. If the redemption
value of preferred stock is not available, we use its liquidation value. If the stockholders’ equity value is not on
Compustat, we compute it as the sum of the book value of common equity and the value of preferred stock.
Finally, if these items are not available, stockholders’ equity is measured as the difference between total assets
and total liabilities.
21
Illiquidity: The Amihud (2002) price impact measure defined as average ratio of
absolute daily returns to daily dollar volume, estimated in year τ − 1 is used to
construct portfolios beginning in year τ .
Z-Score: The bankruptcy predictor introduced in Altman (1968) and measured
in fiscal year ending in calendar year τ − 1 is used to form portfolios beginning in
July of year τ .
For each attribute, we construct decile portfolios and study the returns of the top
and bottom groups. We calculate daily time series of each decile under three weighting
schemes.9 First, we compute initially equal-weighted (IEW) returns by investing $1 in
each stock at the beginning of each rebalancing period and holding the resulting portfolio until the next rebalancing period.10 Second, we calculate initially value-weighted
(IVW) returns, where, at the beginning of each rebalancing period, we invest in each
stock an amount proportional to its most recent market capitalization.11 Under the first
two weighting schemes, market equity, book-to-market, illiquidity, and Z-score portfolios are rebalanced annually, and the other portfolios are rebalanced monthly. Finally,
we compute equal-weighted (EW) returns by rebalancing stocks in a portfolio to equal
weights daily.
4.2. Horizon Effects in Average Returns and Alphas
Table 1 compares average daily, monthly (21 day), quarterly (63 day), semi-annual (126
day), and annual (252 day) returns, rescaled geometrically to different horizons. We
9
The sample periods are 1964-2009 for the book-to-market and Z-score portfolios, 1927-2009 for the momentum portfolio, and 1926-2009 for the remaining portfolios.
10
See Asparouhova, Bessembinder, and Kalcheva (2010) for discussion of initially equal-weighted portfolios.
Many empirical studies construct IEW portfolios by allocating stocks to portfolios according to some stock
characteristic, with equal portfolio weights at the initial formation (e.g., Greenwood (2005); Brennan and
Wang (2010); Sadka (2010); Kaniel, Ozoguz, and Starks (2012)). Event studies evaluating BHARs also use
this weighing scheme (e.g., Loughran and Ritter (1995)).
11
Note that IVW differs from conventional value-weighting if the market capitalization changes for reason
unrelated to stock performance, such as share issuance and repurchases.
22
consider the CRSP market index, and decile 1 and 10 portfolios of stocks sorted by
market equity, book-to-market ratio, and momentum.
Following our theoretical analysis, we expect rescaled daily returns to be smaller
than buy-and-hold returns of longer horizons, with the downward bias being largest for
portfolios of illiquid and volatile stocks. The results are consistent with this prediction.
For the initially equal-weighted small stock portfolio, the geometrically rescaled daily
means are substantially below the buy-and-hold averages for monthly returns (1.41 percent versus 1.63), quarterly returns (4.30 versus 5.73), and annual returns (18.34 versus
25.87). The biases are even larger for initially equal-weighted momentum losers, with
geometric rescaled daily versus buy-and-hold returns of 0.67 versus 1.00 at a monthly
horizon, and 8.34 versus 14.97 at an annual horizon. The differences between rescaled
and buy-and-hold returns are smaller, although still meaningful, for portfolios of large
stocks and initially value-weighted portfolios. These results confirm our prediction of
systematic biases in mean returns calculated from daily returns.
Importantly, these biases do not completely disappear at a monthly return horizon.
For example, the annualized monthly return of the initially equal-weighted momentum
loser portfolio is 12.72, while the corresponding measures of quarterly, semi-annual,
and annual returns are all relatively similar between 14.64 and 14.97. The effects of
rescaling are thus strong in daily returns and remain relevant in monthly returns.
In the Appendix we show robustness of the results of Table 1 to linear instead of
geometric rescaling. As anticipated in Section 3, the results are very similar to those
with geometric rescaling. Since linear rescaling easily accommodates statistical tests
based on means, is more common empirically, and as shown in Section 3 is implicit in
performance regressions that calculate alphas at different horizons, in the remainder of
tables we use linear rescaling.
In Table 2 we test the significance of differences in linearly rescaled (RS) versus
23
buy-and-hold (BH) returns, and demonstrate the accuracy of our analytical approxnet
net
is close to the
developed in Section 3. The statistic νin
imation of the ratio νin
estimated RS/BH ratio for all portfolios, validating the accuracy of the approximation
given in Proposition 3. The results also show that the rescaled returns are below the
buy-and-hold returns for all portfolios and horizons considered. Even apparently small
biases, for example for the large stock portfolio (−0.02 monthly, −0.15 quarterly) are
statistically significant, indicating that the return difference is stable over time. Many
of the portfolios show much larger differences between rescaled short-horizon returns
and longer-horizon returns. For example, in Panel B the rescaled monthly return average consistently understates the quarterly averages, with the largest differences for
portfolios of low-priced stocks (1.21 quarterly) and small stocks (0.82), and sizeable
differences for other portfolios (e.g., 0.24 for value stocks).
Interestingly, portfolios with the highest autocorrelation do not necessarily have the
largest differences between rescaled and buy-and-hold returns. For example, the low
liquidity portfolio has high autocorrelations (0.20 in daily returns and 0.17 in monthly
returns), but the low volatility of this portfolio keeps horizon effects contained to about
0.40 quarterly. The portfolio of liquid stocks exhibits similar horizon effects, achieved
through a higher return volatility paired with lower autocorrelation, so that the return spread (high illiquidity minus low illiquidity) is nearly unaffected by the choice of
horizon. In contrast, the high- and low-volatility portfolios have comparable daily autocorrelations (0.20 vs 0.17), but the standard deviation of the high-volatility portfolio
is about three times larger than the low-volatility portfolio. As a result, an apparently
small difference in means based on daily returns (3.03 − 2.62 = 0.42 when rescaled
to a quarterly frequency) translates to a large difference in actual quarterly returns of
4.52 − 2.79 = 1.73.
Table 3 clarifies how the rescaling bias we focus on in this paper differs from the ef-
24
fects of iid measurement error. Focusing on the small stock portfolio, the average daily
return rebalanced to equal weights (RB = 0.24) substantially exceeds the average daily
return of the buy-and-hold portfolio (BH = 0.07). This is the measurement error bias of
Blume and Stambaugh (1983), and is highly significant. At monthly and annual horizons, the difference between the returns of rebalanced portfolios and the buy-and-hold
portfolio continues to grow. For example, the annual return of the rebalanced portfolio
is 138.8, while the buy-and-hold return is 25.87. The biases caused in multi-period
returns by frequent portfolio rebalancing are discussed by Roll (1983) and subsequent
authors. The rescaling bias that we focus on can be seen in the difference between
the rescaled (RS) and buy-and-hold (BH) returns at monthly and annual horizons. In
contrast to the upward biases caused by iid measurement error and portfolio rebalancing, the rescaled returns are downward biased, as indicated by the negative signs
and statistically significant t-statistics in all entries of the row RS-BH for long-only
portfolios.
We show how horizon effects in mean returns translate into horizon effects in alphas.
Table 4 presents linearly rescaled daily alphas and monthly alphas for the attributesorted portfolios. We decompose the difference in the two alphas into return bias, factor
bias, and beta bias, using Equation (18). The largest estimates for factor and beta bias
are 0.05 and 0.06 respectively. The average return biases are larger, in the range of
0.10 to 0.40 monthly. Table 5 similarly decomposes the difference in alphas calculated
from monthly and quarterly data. The raw return biases are again the most substantial
contributor to the alpha bias. Thus, horizon effects in portfolio mean returns, which
are the focus of our paper, are the dominant component driving alpha differences across
horizons.
25
4.3. Choosing a Horizon
In Figure 3, we follow the method proposed in Section 3 of plotting the rescaled returns
R̄inm versus the return-period length n = 1, ..., 252 for fixed m = 21. All of the plots
show the characteristic shape associated with asynchronous trade and partial price
adjustment demonstrated in Section 3. For low n, the plots slope upward, and for larger
n the plots flatten out and stabilize. In a number of portfolios (small stocks, momentum
losers, high volatility, high inverse price, low reversal), the difference between R̄inm for
n = 1 (daily) and the flat part of the graph is 40-50 basis points monthly, or more. Most
of the portfolios show a horizon effect of at least 10 basis points monthly for n = 1. In
many of the plots a strong upward slope is still apparent for monthly returns (n = 21).
Only at a quarterly horizon do the plots reliably flatten out for all portfolios.
Plotting the return-measurement interval versus rescaled returns, as shown in Figure
3, provides a useful diagnostic tool for empiricists. Choosing a measurement interval
sufficiently long that it is on the flat portion of the rescaled return plot is a simple way
to alleviate concerns about the impact of microstructure frictions on average returns.
Conversely, using a return-measurement interval on the sloping portion of the rescaled
return graph should suggest consideration of the effects of measurement errors. In future
work we anticipate that the effects of microstructure frictions demonstrated here could
be directly incorporated into empirical moment conditions when using higher-frequency
data.
5. International Evidence
International portfolios exhibit considerable heterogeneity in liquidity.12 In this section,
we study horizon effects among region and country portfolios, as well as among style
12
Summarizing a large empirical literature, Bekaert, Harvey, and Lundblad (2007) note in their abstract:
“Given the cross-sectional and temporal variation in their liquidity, emerging equity markets provide an ideal
setting to examine the impact of liquidity on expected returns.”
26
portfolios constructed from international equities.
From Datastream, we identify all available country and regional US-dollar-denominated
MSCI indices. To be included in our sample, we require indices to have at least 10 years
of valid data and eliminate overly redundant regions13 to produce our international return sample of 56 country and 49 regional indices. We supplement this international
data with developed market style portfolios from Ken French’s website. We download 6 portfolios formed on size and book-to-market ratio and 6 portfolios formed on
size and momentum for five markets: Asia Pacific excluding Japan, European, Global,
Japanese, and North American. Following the methodology outlined on the website,
we compute returns on small, big, high book-to-market, low book-to-market, winner,
and loser portfolios for each market.
Tables 6, 7, and 8 compare average monthly returns rescaled to annual frequency
(RS) with average buy-and-hold annual returns (BH).14 Several observations from the
Tables are particularly noteworthy. First, rescaling biases are significant. For example,
the average difference between BH and RS returns is 4.1% per year in country portfolios
and 2.4% in regional index portfolios. These magnitudes are economically significant
and clearly call for caution in interpreting statistics calculated using average monthly
returns.
Second, rescaling biases in the international portfolios are systematic. Rescaled
returns are lower than buy-and-hold returns for every country, every region, and every
style portfolio. Figure 1 plots the histogram of the ratios of RS and BH returns and
confirms this observation graphically: The ratio is always below 1 and for several indices
approaches 0.5.
Third, consistent with our theoretical predictions, horizon effects are stronger in
13
MSCI maintains a number of regional indices (e.g., Europe Excluding Ireland) that closely overlap with
other broader indices that we study (e.g., Europe). For brevity, we exclude such regional indices.
14
Our focus on monthly returns is due to their wide use in the international finance.
27
smaller, less liquid markets where information diffusion is likely to be slow. For example,
Table 6 shows that the average difference between BH and RS returns in countries
classified as Developed by MSCI is 2% annually, whereas for the Emerging and Frontier
countries the difference is 5% on average and exceeds 10% for several countries. Regional
indices in Table 7 and international style portfolios in Table 8 exhibit similar patterns,
with stronger biases in regions with emerging markets. For example, the bias in the
small stock portfolio in Asia Pacific (ex. Japan), at 4.84% per year, is nearly six times
larger than it is in North America.
Finally, Table 8 also shows that the biases in the long and the short sides of the
factor portfolios partially offset, and the bias in the net long-short factor portfolios is
smaller. This suggests that biases in the average returns of country and regional indices will closely translate into biases in alphas. An empiricist measuring performance
of international portfolios, particularly those in smaller, less liquid markets can inadvertently introduce substantial biases in average returns and alphas, even when monthly
returns are used.
To determine whether the differences between rescaled and buy-and-hold returns
reflect an econometric bias or tradable profit opportunities, we compare horizon effects in non-investable market indices and investable exchange-traded funds. Table
9 summarizes the results for MSCI country indices from Datastream and the corresponding iShares country ETFs from CRSP. The average (median) difference between
rescaled and buy-and-hold returns amounts to −1.87% (−1.33%) per year for indices.
The magnitude is lower by more than a third for ETFs: −1.17% (−0.81%). Smaller
horizon effects in ETFs suggest that their prices reflect information more fully than do
index prices. The results presented in the table indicate that a considerable portion
of the difference between rescaled and buy-and-hold returns of the indices is due to
a non-tradable econometric bias. Importantly, the horizon effects are sizeable even in
28
ETFs, suggesting some room for profitable investment strategies as a consequence of
slow information diffusion into stock prices.
6. Conclusion
We establish theoretically, using standard models of asynchronous trading and partial price adjustment, that short-horizon average returns of value-weighted and other
non-rebalanced portfolios are biased. In contrast to the existing literature on iid measurement error, the bias under these sources of microstructure friction is downward
rather than upward, and impacts value-weighted portfolios more than equal-weighted
portfolios.
To explain these biases, we develop an analytical approximation linking the average
returns of a portfolio over arbitrary horizons. Average returns over a long horizon
are closely approximated using only a few simple moments of returns, namely, the
short-horizon average return and variance, and the variance ratio of long- to shorthorizon returns. The formula explains why bid-ask bounce, which generates a variance
ratio of long-horizon to short-horizon returns less then one, produces an upward bias
in short-run returns. On the other hand, asynchronous price adjustment produces a
variance ratio greater than one, and a downward bias in short-horizon portfolio returns.
We show that, consistent with theory, buy-and-hold portfolios of small, illiquid, and
volatile stocks have average returns that are substantially downward biased at a daily
horizon. These biases remain significant in many cases even for monthly returns.
We propose a simple solution to account for these biases. Plotting rescaled average
returns versus the return horizon permits easy diagnosis of market microstructure frictions by a sharp slope at short-horizons. Choosing a sufficiently long return horizon on
the flat portion of this graph ensures that pricing frictions should not cause a problem
in measured return means, alphas, or Fama-MacBeth coefficients. Given the increas-
29
ing importance of short-horizon returns in empirical work, awareness of these biases is
important, and we anticipate continued advances in empirical methods that explicitly
account for microstructure frictions.
30
7. Appendix: Proofs
Proof of Lemma 1
Let ft = µf + σf ξt where the random variables ξt are independent standard normals.
Substituting this expression for ft in equation (7) produces equation (8). Since 0 <
δ < 1, the process for Dt is AR(1) and the formulas for the unconditional mean and
variance can be obtained by applying the expectation and variance operators to both
sides of equation (8).
Proof of Lemma 2
By assumption, p∗0 = p0 + D0 . To apply the inductive proof of the result, assume
that p∗t−1 = pt−1 + Dt−1 . Equations (5), (6), and (7) yield
pt + Dt = pt−1 + δ(Dt−1 + ft ) + (1 − δ)(Dt−1 + ft )
= pt−1 + Dt−1 + ft
= p∗t−1 + ft
= p∗t .
Proof of Proposition 1
Formulas for the unconditional mean and variance of lagger returns follow by applying the expectation and variance operator to both sides of equation (6) and then
substituting the expressions for E(Dt ) and Var(Dt ) from Lemma 1. The upper bound
on lagger variance follows from the fact that 0 < δ < 1 < 2 − δ implies that
Proof of Proposition 2
31
δ
2−δ
< 1.
Formula (11) is derived as follows:
∗
wt−1
πept−1
=
∗
πept−1 + (1 − π)ept−1
πept−1 +Dt−1
=
πept−1 +Dt−1 + (1 − π)ept−1
πeDt−1
.
=
πeDt−1 + (1 − π)
(24)
(25)
(26)
Expression (12) follows from applying the expectation operator to equation (10) and
applying the definition of covariance.
Proof of Proposition 3
RSg
We first observe that R̄in
= en(µi +σi /2) . The RS measure depends only on the first
2
and second moments of daily returns. By contrast, buy-and-hold returns over a monthly
horizon depend on how daily returns aggregate. In particular, by assumption of joint
normality of daily log returns, the monthly log returns are also normally distributed, i.e.,
2
2
BH
ri,1 + · · · + ri,n ∼ N (nµi , σin
). As a consequence, the BH statistic is R̄in
= enµi +σin /2 .
The ratio of the two statistics is
νin ≡
RSg
R̄in
2
2
2
= e(nσi −σin )/2 = enσi (1−V Rin )/2 .
BH
R̄in
(27)
The ratio in the net returns is
net
νin
≡
RSg
R̄in
−1
νin − 1
= νin + BH
.
BH
R̄in − 1
R̄in − 1
(28)
Proof of Proposition 4
Scaling the short-horizon Jensen’s alpha, given by equation (13) in the case where
n = 1, to its n-period long-horizon value yields the expression
RS
nαi1 ≡ αin
= nR̄i1 − nR̄f 1 + β1 nR̄M 1 − nR̄f 1
= (nR̄i1 − 1) − (nR̄f 1 − 1) + β1 (nR̄M 1 − 1) − (nR̄f 1 − 1)
= (1 + (nR̄i1 − 1)) − (1 + (nR̄f 1 − 1)) + β1 (1 + (nR̄M 1 − 1)) − (1 + (nR̄f 1 − 1))
= νin R̄in − νf n R̄f n + β1 νM n R̄M n − νf n R̄f n ,
32
RS
where νjn ≡ (1 + n(R̄j1 − 1))/R̄jn = R̄jn
/R̄jn . Subtracting the long-horizon buy-and-
hold alpha αin produces
RS
− αin =
αin
νin R̄in − νf n R̄f n + β1 νM n R̄M n − νf n R̄f n
− R̄in − R̄f n + βn R̄M n − R̄f n
= (νin − 1)R̄in − (νf n − 1)R̄f n − β1 νM n R̄M n − νf n R̄f n + βn R̄M n − R̄f n .
This expression can be equivalently written
RS
αin
− αin = (νin − 1)R̄in − β1 (νM n − 1)R̄M n + (βn − β1 ) R̄M n − R̄f n
−(1 − β1 )(νf n − 1)R̄nf .
(29)
The approximation (18) follows if rescaling has an insignificant impact on the average
return of the risk-free asset, νf n ≈ 1.
33
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36
Table 1. Horizon Effects in Style Portfolios
Performance
Metric
Daily
Monthly
Quarterly
Semi-Annual
Annual
Daily
Monthly
Quarterly
Semi-Annual
Annual
Daily
Monthly
Quarterly
Semi-Annual
Annual
Daily
Monthly
Quarterly
Semi-Annual
Annual
Daily
Monthly
Quarterly
Semi-Annual
Annual
Daily
Monthly
Quarterly
Semi-Annual
Annual
Daily
Monthly
Quarterly
Semi-Annual
Annual
1 day
0.04
0.04
0.04
0.05
0.06
0.07
0.08
Holding Horizon
1 mo
3 mo
6 mo
1 year
1 day
A. CRSP Value-Weighted Index
0.84
2.55
5.17
10.61
0.86
2.62
5.30
10.89
2.62
5.31
10.90
5.33
10.94
10.97
Holding Horizon
1 mo
3 mo
6 mo
B. Market Capitalization, Initially Value-Weighted
Big
0.80
2.43
4.92
10.09
0.06
1.26
0.81
2.45
4.95
10.14
1.46
2.49
5.04
10.32
5.06
10.37
10.49
C. Market Capitalization, Initially Equal-Weighted
Big
0.81
2.46
4.97
10.20
0.07
1.41
0.83
2.52
5.10
10.46
1.63
2.58
5.22
10.71
5.19
10.64
10.69
D. Book-to-Market,
Value
1.16
3.52
7.17
1.20
3.63
7.39
3.64
7.42
7.31
Initially Value-Weighted
E. Book-to-Market,
Value
1.36
4.13
8.42
1.49
4.52
9.25
4.70
9.62
9.65
Initially Equal-Weighted
1.51
1.51
1.71
1.80
14.85
15.33
15.38
15.16
15.37
17.55
19.36
20.16
20.24
19.91
0.04
0.02
0.77
0.78
0.48
0.59
F. Momentum, Initially Value-Weighted
Winners
4.60
9.41
19.71
-0.01
-0.15
4.61
9.44
19.77
0.03
4.72
9.67
20.27
9.76
20.47
20.26
G. Momentum, Initially Equal-Weighted
Winners
5.23
10.74
22.62
0.03
0.67
5.51
11.32
23.93
1.00
5.74
11.82
25.04
12.05
25.56
24.69
Small
3.83
4.44
5.12
Small
4.30
4.98
5.73
Growth
2.33
2.37
2.42
Growth
1.44
1.79
1.97
Losers
-0.46
0.09
0.37
Losers
2.02
3.04
3.47
1 year
7.80
9.07
10.49
10.43
16.22
18.97
22.09
21.95
22.65
8.78
10.21
11.78
11.78
18.34
21.46
24.94
24.94
25.87
4.72
4.79
4.90
4.96
9.65
9.80
10.04
10.16
10.52
2.90
3.62
3.98
3.82
5.89
7.37
8.11
7.79
7.82
-0.91
0.18
0.75
0.73
-1.81
0.37
1.51
1.47
1.82
4.09
6.17
7.07
7.21
8.34
12.72
14.64
14.93
14.97
Notes: This table reports average buy-and-hold returns (BH, on the diagonal) as well as geometrically rescaled short
horizon returns (RSg, off-diagonal). Returns are calculated using periods of n = 1, 21, 63, 126, and 252 days corresponding to Daily, Monthly, Quarterly, Semi-Annual, and Annual frequencies, and are scaled to the corresponding
Holding Horizon. For all calculations, Panel A uses the value-weighted CRSP index, and the remaining use either initially equally-weighed (IEW) or initially value-weighted (IVW) daily returns for size, value, and momentum
portfolios. A complete description of the portfolios is provided in Section 4 of the text.
Table 2. Decomposing Horizon Effects
A. Daily Performance Metric, Monthly Holding Horizon
RS
BH
RS-BH
σRS
ρRS
VR
RS/BH
net
νin
RS
BH
RS-BH
σRS
ρRS
VR
RS/BH
net
νin
Market Capitalization
H
L
HL
0.81
1.40 -0.59
0.83
1.63 -0.80
-0.02
-0.23
0.21
[-2.02] [-4.53] [4.47]
1.09
1.45
0.08
0.13
1.17
1.91
0.97
0.86
0.97
0.88
Book-to-Market
H
L
HL
1.35
0.48
0.87
1.49
0.59
0.89
-0.14
-0.12
-0.02
[-9.24] [-6.62] [-1.25]
0.88
1.22
0.25
0.19
2.59
1.77
0.91
0.80
0.91
0.80
Short-Term Reversal
H
L
HL
1.01
1.01
0.00
1.16
1.25 -0.09
-0.15
-0.25
0.10
[-6.34] [-7.12] [4.42]
1.30
1.41
0.18
0.21
1.83
2.12
0.87
0.80
0.87
0.81
H
1.01
1.35
-0.34
[-7.75]
1.73
0.20
2.04
0.75
0.76
Volatility
L
HL
0.87
0.14
0.91
0.44
-0.04
-0.30
[-6.03] [-7.52]
0.66
0.17
1.75
0.96
0.96
Momentum
H
L
1.70
0.67
1.80
1.00
-0.10
-0.34
[-5.83] [-7.25]
1.35
1.47
0.15
0.25
1.49
2.40
0.94
0.67
0.95
0.68
H
1.04
1.12
-0.08
[-6.44]
0.92
0.20
1.83
0.93
0.93
HL
1.03
0.80
0.23
[5.82]
Illiquidity
L
HL
0.97
0.08
1.04
0.09
-0.07
-0.01
[-3.68] [-0.48]
1.48
0.08
1.30
0.93
0.93
Inverse Price
H
L
HL
1.80
0.96
0.84
2.21
1.00
1.21
-0.41
-0.04
-0.37
[-7.14] [-5.06] [-6.71]
1.47
0.96
0.23
0.13
2.58
1.37
0.81
0.96
0.83
0.96
H
1.16
1.23
-0.07
[-6.54]
0.87
0.17
1.83
0.94
0.95
Z-Score
L
0.43
0.66
-0.23
[-7.65]
1.16
0.27
2.64
0.65
0.65
HL
0.73
0.57
0.16
[6.05]
B. Monthly Performance Metric, Quarterly Holding Horizon
RS
BH
RS-BH
σRS
ρRS
VR
RS/BH
net
νin
RS
BH
RS-BH
σRS
ρRS
VR
RS/BH
net
νin
Market Capitalization
H
L
HL
2.50
4.90 -2.40
2.58
5.73 -3.15
-0.08
-0.82
0.75
[-1.80] [-2.33] [2.31]
5.39
9.18
0.08
0.22
1.14
1.44
0.97
0.86
0.98
0.90
Book-to-Market
H
L
HL
4.46
1.78
2.67
4.70
1.97
2.73
-0.24
-0.19
-0.06
[-3.10] [-1.89] [-0.66]
6.49
7.42
0.29
0.14
1.27
1.21
0.95
0.91
0.96
0.91
Short-Term Reversal
H
L
HL
3.49
3.76 -0.28
3.80
4.31 -0.50
-0.32
-0.54
0.22
[-2.97] [-2.42] [1.56]
8.03
9.41
0.23
0.19
1.26
1.30
0.92
0.87
0.93
0.90
H
4.06
4.52
-0.46
[-1.69]
11.31
0.19
1.17
0.90
0.92
Notes: Table continues on the next page.
Volatility
L
HL
2.73
1.33
2.79
1.73
-0.06
-0.40
[-2.34] [-1.56]
4.01
0.08
1.18
0.98
0.98
Momentum
H
L
5.41
3.01
5.74
3.47
-0.33
-0.47
[-3.37] [-1.92]
7.53
10.41
0.15
0.20
1.26
1.21
0.94
0.87
0.96
0.90
H
3.37
3.54
-0.17
[-3.14]
5.72
0.17
1.25
0.95
0.96
HL
2.40
2.27
0.13
[0.62]
Illiquidity
L
HL
3.11
0.26
3.31
0.23
-0.19
0.02
[-2.49] [0.48]
7.73
0.12
1.17
0.94
0.95
Inverse Price
H
L
HL
6.62
2.99
3.63
7.83
3.07
4.76
-1.21
-0.08
-1.13
[-2.68] [-2.70] [-2.59]
10.84
5.12
0.23
0.10
1.41
1.15
0.85
0.97
0.90
0.98
H
3.70
3.82
-0.13
[-2.82]
5.37
0.13
1.19
0.97
0.98
Z-Score
L
1.98
2.25
-0.27
[-1.52]
8.64
0.23
1.21
0.88
0.89
HL
1.72
1.57
0.15
[0.93]
Table 2. Decomposing Horizon Effects, Continued
C. Daily Performance Metric, Quarterly Holding Horizon
RS
BH
RS-BH
σRS
ρRS
VR
RS/BH
net
νin
RS
BH
RS-BH
σRS
ρRS
VR
RS/BH
net
νin
Market Capitalization
H
L
HL
2.43
4.21 -1.78
2.58
5.73 -3.15
-0.15
-1.51
1.37
[-2.67] [-3.16] [3.12]
1.09
1.45
0.08
0.13
1.33
2.76
0.94
0.74
0.95
0.79
Book-to-Market
H
L
HL
4.04
1.43
2.61
4.70
1.97
2.73
-0.65
-0.54
-0.12
[-6.87] [-4.67] [-1.17]
0.88
1.22
0.25
0.19
3.30
2.14
0.86
0.73
0.88
0.73
Short-Term Reversal
H
L
HL
3.03
3.02
0.01
3.80
4.31 -0.50
-0.78
-1.29
0.51
[-5.14] [-4.09] [2.66]
1.30
1.41
0.18
0.21
2.30
2.77
0.80
0.70
0.81
0.73
H
3.03
4.52
-1.49
[-4.61]
1.73
0.20
2.40
0.67
0.70
Volatility
L
HL
2.62
0.42
2.79
1.73
-0.18
-1.31
[-5.43] [-4.36]
0.66
0.17
2.06
0.94
0.95
Momentum
H
L
HL
5.10
2.00
3.10
5.74
3.47
2.27
-0.64
-1.47
0.83
[-4.89] [-4.09] [2.62]
1.35
1.47
0.15
0.25
1.88
2.90
0.89
0.58
0.91
0.62
H
3.13
3.54
-0.40
[-5.00]
0.92
0.20
2.29
0.89
0.90
Illiquidity
L
HL
2.90
0.24
3.31
0.23
-0.41
0.01
[-3.46] [0.09]
1.48
0.08
1.52
0.88
0.89
Inverse Price
H
L
HL
5.39
2.87
2.52
7.83
3.07
4.76
-2.44
-0.20
-2.24
[-3.91] [-5.26] [-3.73]
1.47
0.96
0.23
0.13
3.63
1.57
0.69
0.94
0.75
0.94
H
3.48
3.82
-0.34
[-5.50]
0.87
0.17
2.17
0.91
0.92
Z-Score
L
1.29
2.25
-0.96
[-5.02]
1.16
0.27
3.20
0.57
0.58
HL
2.20
1.57
0.63
[3.84]
Notes: This table reports and decomposes the horizon effect bias in the initially equally weighted buy-and-hold
style portfolios. The rescaled (RS) performance measures are obtained from daily (in Panels A and C) or monthly
(in Panel B) returns, and are rescaled and compared to monthly
(in Panel
A) or quarterly (in Panels B and C)
1 PNn
Rnt − 1, where Rnt is the portfolio gross
buy-and-hold (BH) returns. BH returns are computed as
Nn t=1
n-day return, t indices n-day periods, Nn is the number of n-day intervals in the sample, and n = 1, 21, 63 correspond
to daily, monthly, and quarterly periods, respectively. RS returns are computed by rescaling the daily (n1 = 1 in
Panels A and C) or monthly (n1 = 21 in Panel B) portfolio returns to a corresponding monthly (n2 = 21 in Panel
i
1 PNn2 hPτ n2 /n1
A) or quarterly (n2 = 63 in Panels B and C) frequency:
τ =1
t=(τ −1)n2 /n1 +1 (Rn1 t − 1) . t-statistics for
N n2
the difference between RS and BH returns are in square brackets. Also reported are the standard deviation σRS
and autocorrelation ρRS of the short horizon (daily in Panels A and C, monthly in Panel B) returns, the ratio
V R of variance of performance-measurement-horizon returns variance to variance of holding-horizon returns, as well
net
as the empirically measured bias (RS/BH) and its analytical approximation νin
from Proposition 3. A complete
description of the portfolios is provided in Section 4 of the text.
Table 3. Blume and Stambaugh (1983), Roll (1983), and Horizon Effects
Portfolio
Rule
D
High
M
A
BH
0.04
[5.25]
0.83
[5.00]
10.69
[4.87]
RB
0.04
[5.69]
0.91
[5.27]
11.66
[5.11]
0.24
[25.7]
5.53
[13.3]
RS
0.04
[5.25]
0.81
[4.90]
9.71
[4.54]
0.07
[6.80]
RB-BH
0.00
[3.51]
0.08
[3.23]
0.97
[2.81]
0.17
[30.7]
-0.02
[-2.02]
-0.98
[-2.68]
RS-BH
Low
D
M
A
A. Market Capitalization
0.07
1.63
25.87
[6.80]
[5.22]
[4.00]
D
High – Low
M
A
-0.03
[-3.22]
-0.80
[-3.41]
-15.18
[-2.85]
138.80
[3.70]
-0.19
[-23.0]
-4.62
[-13.1]
-127.14
[-3.48]
1.40
[4.90]
16.84
[3.78]
-0.03
[-3.22]
-0.59
[-2.84]
-7.13
[-2.27]
3.90
[17.6]
112.93
[3.21]
-0.17
[-30.3]
-3.82
[-17.6]
-111.96
[-3.21]
-0.23
[-4.53]
-9.03
[-3.37]
0.21
[4.47]
8.05
[3.14]
0.04
[5.53]
0.89
[4.62]
12.08
[4.04]
B. Book-to-Market
0.02
0.59
7.82
[2.00]
[1.89]
[1.81]
BH
0.06
[7.79]
1.49
[5.30]
19.91
[4.55]
RB
0.15
[18.0]
3.37
[10.8]
51.74
[7.20]
0.05
[4.58]
1.22
[3.84]
16.08
[3.56]
0.10
[12.9]
2.15
[10.5]
35.66
[6.56]
RS
0.06
[7.79]
1.35
[4.86]
16.18
[4.47]
0.02
[2.00]
0.48
[1.52]
5.72
[1.40]
0.04
[5.53]
0.87
[4.54]
10.46
[3.60]
RB-BH
0.09
[34.2]
1.88
[16.8]
31.83
[5.88]
0.03
[14.6]
0.63
[9.68]
8.26
[5.23]
0.06
[21.3]
1.26
[14.8]
23.58
[5.68]
-0.14
[-9.24]
-3.73
[-3.54]
-0.12
[-6.62]
-2.10
[-2.62]
-0.02
[-1.25]
-1.62
[-2.21]
RS-BH
BH
0.08
[8.93]
1.80
[7.78]
24.69
[6.81]
C. Momentum
0.03
1.00
[3.19]
[2.92]
14.97
[2.60]
0.05
[6.45]
0.80
[3.40]
9.72
[2.35]
RB
0.10
[11.4]
2.30
[9.74]
32.79
[7.29]
0.15
[14.9]
3.59
[9.39]
65.37
[5.33]
-0.05
[-5.94]
-1.29
[-4.73]
-32.58
[-3.14]
RS
0.08
[8.93]
1.70
[7.42]
20.40
[6.60]
0.03
[3.19]
0.67
[2.06]
8.01
[1.83]
0.05
[6.45]
1.03
[4.77]
12.39
[4.73]
RB-BH
0.02
[22.4]
0.49
[13.6]
8.10
[4.15]
0.12
[58.9]
2.59
[23.1]
50.39
[5.82]
-0.09
[-45.6]
-2.10
[-21.1]
-42.29
[-5.46]
-0.10
[-5.83]
-4.29
[-5.26]
-0.34
[-7.25]
-6.96
[-3.23]
0.23
[5.82]
2.68
[1.39]
RS-BH
Notes: This table reports average returns (in percent) computed over daily (D), monthly (M), and annual (A) holding
horizons for portfolios formed on the basis of market capitalization, book-to-market ratio, and momentum. BH
are
1 PNn
average returns of the annually-rebalanced initially equal-weighted buy-and-hold portfolio,
Rnt − 1,
Nn t=1
where Rnt is the portfolio gross n-day return, t indices n-day periods, Nn is the number of n-day intervals in the
sample, and n = 1, 21, 252 correspond to daily, monthly, and annual periods, respectively. RB returns are computed
similarly for portfolios whose components are rebalanced daily to equal weights. RS are average daily returns of the
annually-rebalanced initially equal-weighed portfolio rescaled to monthly (n = 21) or annual (n = 252) horizons,
i
1 PNn hPτ n
τ =1
t=(τ −1)n+1 (R1t − 1) . Also shown are the average differences between RB and BH, and between RS
Nn
and BH performance measures. Corresponding t-statistics are in square brackets. Results are reported for the
portfolios of high and low deciles as well as for their difference. A complete description of the portfolios is provided
in Section 4 of the text.
Table 4. Horizon Effects in Alphas: Daily vs. Monthly Horizon
Alpha
Daily
t(Alpha)
Beta
Small
Big
S-B
0.43
-0.01
0.44
[2.54]
[-0.38]
[2.53]
1.31
1.01
0.30
Value
Growth
V-G
0.45
-0.49
0.94
[3.96]
[-4.27]
[7.11]
1.18
1.36
-0.18
0.64
-0.38
1.02
Winners
Losers
W-L
0.74
-0.52
1.25
[7.23]
[-3.67]
[8.00]
1.21
1.61
-0.40
0.69
0.15
0.54
[4.14]
[3.37]
[3.12]
1.48
0.95
0.54
Winners
Losers
W-L
-0.01
-0.13
0.12
[-0.12]
[-1.02]
[0.87]
1.31
1.52
-0.21
High
Low
H-L
-0.23
0.20
-0.43
[-1.36]
[5.34]
[-2.54]
1.70
0.69
1.01
0.07
0.23
-0.16
F. Volatility
[0.29] 1.70
[4.57] 0.67
[-0.66] 1.03
High
Low
H-L
0.19
-0.06
0.25
[2.77]
[-0.67]
[2.59]
1.02
1.32
-0.30
0.26
-0.02
0.28
High
Low
H-L
0.30
-0.55
0.85
[4.18]
[-3.58]
[6.23]
1.08
1.41
-0.33
0.38
-0.30
0.68
High
Low
H-L
Monthly
t(Alpha) Beta
A. Size
0.61
[2.69] 1.36
-0.01
[-0.39] 1.02
0.62
[2.69] 0.34
Alpha
Comp. 1
Decomposition
Comp. 2 Comp. 3
Alpha Bias
-0.23
-0.02
-0.21
0.03
0.02
0.01
0.02
0.00
0.02
-0.18
0.00
-0.18
B. Book-to-Market
[3.29] 1.02
[-2.34] 1.35
[5.59] -0.33
-0.14
-0.12
-0.02
0.01
0.01
0.00
-0.06
0.00
-0.06
-0.19
-0.11
-0.08
0.83
-0.22
1.05
C. Momentum
[6.47] 1.19
[-0.97] 1.61
[4.61] -0.42
-0.10
-0.34
0.23
0.03
0.04
-0.01
-0.01
0.00
-0.01
-0.09
-0.30
0.21
1.04
0.18
0.86
D. Inverse Price
[3.74] 1.54
[3.84] 0.92
[3.00] 0.62
-0.41
-0.04
-0.37
0.03
0.02
0.01
0.03
-0.02
0.05
-0.35
-0.03
-0.31
E. Short-Term Reversal
0.11
[0.80] 1.31
-0.15
0.09
[0.50] 1.50
-0.25
0.02
[0.12] -0.19
0.10
0.03
0.03
0.00
0.00
-0.01
0.01
-0.12
-0.22
0.10
-0.34
-0.04
-0.30
0.05
0.02
0.03
0.00
-0.01
0.01
-0.30
-0.03
-0.27
G. Illiquidity
[3.55] 0.99
[-0.15] 1.32
[2.45] -0.33
-0.08
-0.07
-0.01
0.02
0.03
-0.01
-0.02
0.00
-0.02
-0.07
-0.04
-0.03
H. Z-score
[3.74] 1.04
[-1.16] 1.33
[3.12] -0.29
-0.07
-0.23
0.16
0.01
0.01
0.00
-0.01
-0.03
0.02
-0.08
-0.25
0.17
Notes: This table reports the rescaled daily alpha, the one-month buy-and-hold alpha, as well as the decomposition of
net
BH
the difference following Equation (18). Component 1 is the bias in portfolio returns, Comp. 1 ≡ (νin
− 1)(R̄in
− 1),
net
BH
component 2 is the bias in factor returns, Comp. 2 ≡ −βi (νM n − 1)(R̄M n − 1), and component 3 is the beta bias,
BH
Comp. 3 ≡ −(βi − βin )(R̄M
n − 1). Daily betas are estimated with 10 Dimson (1979) lags. A complete description
of the test portfolios is provided in Section 4 of the text.
Table 5. Horizon Effects in Alphas: Monthly vs. Quarterly Horizon
Alpha
Monthly
t(Alpha)
Beta
Small
Big
S-B
0.93
0.02
-0.91
[1.42]
[0.25]
[-1.36]
1.90
0.99
-0.91
Quarterly
t(Alpha) Beta
A. Size
1.58
[1.72] 1.91
-0.02
[-0.28] 1.01
-1.61
[-1.71] -0.90
Value
Growth
V-G
1.42
-1.40
2.82
[2.59]
[-2.90]
[5.17]
1.47
1.59
-0.12
1.87
-1.17
3.04
Winners
Losers
W-L
2.18
-1.02
3.20
[5.65]
[-1.53]
[4.67]
1.36
1.82
-0.46
2.25
0.49
1.76
[2.74]
[3.43]
[2.07]
2.03
0.95
1.08
Winners
Losers
W-L
-0.14
-0.08
-0.06
[-0.34]
[-0.14]
[-0.14]
1.58
1.72
-0.13
High
Low
H-L
-0.34
0.69
-1.03
[-0.49]
[4.55]
[-1.43]
2.01
0.67
1.34
0.00
0.68
-0.69
F. Volatility
[-0.01] 2.04
[3.76] 0.69
[-0.80] 1.35
High
Low
H-L
0.66
-0.09
0.75
[2.98]
[-0.27]
[2.16]
1.06
1.35
-0.29
0.73
-0.07
0.80
High
Low
H-L
1.03
-1.37
2.39
[3.46]
[-1.76]
[3.66]
1.12
1.73
-0.61
1.12
-1.00
2.12
High
Low
H-L
Alpha
Comp. 1
Decomposition
Comp. 2 Comp. 3
Alpha Bias
-0.82
-0.08
0.75
0.15
0.08
-0.07
0.02
0.04
0.02
-0.66
0.04
0.70
B. Book-to-Market
[3.06] 1.22
[-2.31] 1.49
[4.87] -0.26
-0.24
-0.19
-0.06
0.05
0.06
0.00
-0.29
-0.12
-0.17
-0.47
-0.25
-0.23
2.46
-0.96
3.41
C. Momentum
[5.69] 1.33
[-1.29] 1.96
[4.40] -0.63
-0.33
-0.47
0.13
0.10
0.14
-0.03
-0.06
0.25
-0.31
-0.29
-0.08
-0.21
2.91
0.58
2.33
D. Inverse Price
[2.44] 2.25
[3.52] 0.90
[1.90] 1.35
-1.21
-0.08
-1.13
0.15
0.07
0.08
0.39
-0.09
0.48
-0.67
-0.11
-0.57
E. Short-Term Reversal
0.18
[0.43] 1.51
-0.32
0.05
[0.08] 1.86
-0.54
0.13
[0.27] -0.35
0.22
0.12
0.13
-0.01
-0.13
0.26
-0.39
-0.33
-0.16
-0.18
-0.46
-0.06
-0.40
0.06
0.02
0.04
0.05
0.04
0.00
-0.35
0.00
-0.35
G. Illiquidity
[2.98] 1.07
[-0.20] 1.38
[2.04] -0.31
-0.17
-0.19
0.02
0.08
0.10
-0.02
0.02
0.05
-0.04
-0.07
-0.04
-0.03
H. Z-score
[3.60] 1.12
[-1.21] 1.58
[2.89] -0.46
-0.13
-0.27
0.15
0.04
0.06
-0.02
-0.01
-0.18
0.17
-0.09
-0.38
0.30
Notes: This table reports the rescaled monthly alpha, the quarterly buy-and-hold alpha, as well as the decomposition
net
BH
of the difference following Equation (18). Component 1 is the bias in portfolio returns, Comp. 1 ≡ (νin
−1)(R̄in
−1),
net
BH
component 2 is the bias in factor returns, Comp. 2 ≡ −βi (νM n − 1)(R̄M n − 1), and component 3 is the beta bias,
BH
Comp. 3 ≡ −(βi − βin )(R̄M
n − 1). Monthly betas are estimated with 3 Dimson (1979) lags. A complete description
of the test portfolios is provided in Section 4 of the text.
Table 6. Horizon Effects in Country Index Portfolios
Country
Argentina
Australia
Austria
Belgium
Brazil
Canada
Chile
China
Colombia
Croatia
Czech Republic
Denmark
Egypt
Estonia
Finland
France
Germany
Greece
Hong Kong
Hungary
India
Indonesia
Ireland
Israel
Italy
Japan
Jordan
Kenya
Malaysia
Mauritius
Mexico
Lebanon
Morocco
Netherlands
New Zealand
Nigeria
Norway
Pakistan
Peru
Philippines
Poland
Portugal
Russia
Singapore
Slovenia
South Africa
South Korea
Spain
Sri Lanka
Sweden
Switzerland
Taiwan
Thailand
Turkey
United Kingdom
United States
MSCI Category
Frontier
Developed
Developed
Developed
Emerging
Developed
Emerging
Emerging
Emerging
Frontier
Emerging
Developed
Emerging
Frontier
Developed
Developed
Developed
Developed
Developed
Emerging
Emerging
Emerging
Developed
Developed
Developed
Developed
Frontier
Frontier
Emerging
Frontier
Emerging
Frontier
Emerging
Developed
Developed
Frontier
Developed
Frontier
Emerging
Emerging
Emerging
Developed
Emerging
Developed
Frontier
Emerging
Emerging
Developed
Frontier
Developed
Developed
Emerging
Emerging
Emerging
Developed
Developed
RS
25.50
12.35
11.07
13.12
30.50
11.66
19.83
6.35
21.77
11.92
16.18
14.37
20.17
18.03
12.99
12.27
12.08
9.41
20.33
18.68
13.71
22.71
5.88
8.14
8.54
11.07
4.70
25.42
12.80
24.64
24.13
16.95
11.56
13.43
8.84
20.80
14.94
13.51
22.34
13.21
24.02
5.62
28.75
15.29
8.45
15.75
14.25
11.47
12.72
16.02
12.75
12.48
16.06
26.66
12.26
10.45
BH
32.71
12.99
13.88
15.28
32.58
12.29
23.57
7.99
26.30
16.94
17.21
16.65
29.95
22.35
16.52
13.43
13.53
12.74
23.66
20.72
18.15
30.89
8.22
9.46
10.19
13.41
6.44
37.39
16.57
30.24
28.37
18.79
13.07
14.30
10.39
25.09
18.96
18.92
24.72
18.49
46.62
6.91
40.29
19.16
15.64
16.90
18.47
13.05
17.19
17.85
13.87
14.64
21.50
48.30
13.85
11.19
RS/BH
0.78
0.95
0.80
0.86
0.94
0.95
0.84
0.79
0.83
0.70
0.94
0.86
0.67
0.81
0.79
0.91
0.89
0.74
0.86
0.90
0.76
0.74
0.72
0.86
0.84
0.83
0.73
0.68
0.77
0.81
0.85
0.90
0.88
0.94
0.85
0.83
0.79
0.71
0.90
0.71
0.52
0.81
0.71
0.80
0.54
0.93
0.77
0.88
0.74
0.90
0.92
0.85
0.75
0.55
0.89
0.93
RS-BH
-7.21 [-0.87]
-0.64 [-0.86]
-2.81 [-1.53]
-2.16 [-2.38]
-2.08 [-0.55]
-0.63 [-1.26]
-3.74 [-2.16]
-1.64 [-0.82]
-4.53 [-1.71]
-5.03 [-1.24]
-1.02 [-0.56]
-2.28 [-2.49]
-9.78 [-2.69]
-4.32 [-1.71]
-3.54 [-1.31]
-1.16 [-1.64]
-1.46 [-1.29]
-3.32 [-1.89]
-3.33 [-1.59]
-2.04 [-0.78]
-4.44 [-2.11]
-8.18 [-1.63]
-2.33 [-1.34]
-1.32 [-1.49]
-1.65 [-1.24]
-2.34 [-2.00]
-1.75 [-2.16]
-11.97 [-1.15]
-3.78 [-1.65]
-5.61 [-1.63]
-4.24 [-2.28]
-1.85 [-0.50]
-1.51 [-1.69]
-0.87 [-1.86]
-1.55 [-1.46]
-4.29 [-0.98]
-4.02 [-1.96]
-5.41 [-1.72]
-2.39 [-1.29]
-5.28 [-2.52]
-22.59 [-0.94]
-1.29 [-1.58]
-11.54 [-1.99]
-3.87 [-1.71]
-7.19 [-1.96]
-1.15 [-1.02]
-4.21 [-2.46]
-1.58 [-1.66]
-4.47 [-1.24]
-1.83 [-2.41]
-1.12 [-1.43]
-2.16 [-1.47]
-5.43 [-1.61]
-21.64 [-1.60]
-1.59 [-2.14]
-0.74 [-2.57]
Years
25
43
43
43
25
43
25
20
20
10
18
43
18
10
31
43
43
25
43
18
20
25
25
20
43
43
25
10
25
10
25
10
18
43
31
10
43
20
20
25
20
25
18
43
10
20
25
43
20
43
43
25
25
25
43
43
Notes: This table reports the horizon effect bias in the MSCI country index portfolios. Rescaled (RS) returns are
average monthly returns mutiplied by 12. Buy-and-hold (BH) returns are average annual returns. t-statistic for
the difference between RS and BH returns is shown in square brackets. Sample period ends in 2012 and spans the
number of years shown in the last column. MSCI category is as of January 2013.
Table 7. Horizon Effects in Regional Index Portfolios
Regional Index
All Country Americas
All Country Asia
All Country Asia Excluding Japan
All Country Asia Pacific
All Country Asia Pacific Excluding Japan
All Country EAFE + Emerging Markets
All Country Europe
All Country Europe + Middle East
All Country Far East
All Country Far East Excluding Japan
All Country Far East Excluding Japan and Hong Kong
All Country Pacific
All Country Pacific Excluding Japan
All Country Pacific Excluding Japan and Hong Kong
All Country World
All Country World Excluding United States
Brazil, Russia, India and China (BRIC)
EAFE
EAFE + Canada
EAFE Excluding Japan
EAFE Excluding United Kingdom
Emerging Markets
Emerging Markets Asia
Emerging Markets Eastern Europe
Emerging Markets Europe
Emerging Markets Europe + Middle East
Emerging Markets Europe, Middle East and Africa
Emerging Markets Excluding Asia
Emerging Markets Far East
Emerging Markets Latin America
EMU
Europe
European Union
Far East
Frontier Markets
Frontier Markets Africa
Frontier Markets Central and Eastern Europe + CIS
Frontier Markets Europe, Middle East and Africa
Frontier Markets Excluding Gulf Cooperation Council
G7
Golden Dragon
Nordic
North America
Pacific
Pacific Excluding Japan
South East Asia
World
World Excluding United States
Zhong Hua
RS
10.65
3.10
12.31
4.80
11.61
7.48
10.14
5.94
3.98
12.33
11.16
4.73
11.65
10.58
8.39
7.61
13.42
10.83
10.78
11.27
10.87
15.00
11.33
13.22
14.33
12.71
13.10
16.41
11.26
22.87
10.16
11.46
9.37
11.36
10.33
19.41
10.27
10.75
11.97
10.63
7.83
14.67
10.66
11.13
12.71
5.81
10.16
10.91
13.19
BH
11.66
4.61
15.44
5.95
14.29
8.37
11.08
7.42
4.98
15.17
14.32
5.80
14.15
12.98
9.22
8.53
18.58
12.10
11.98
12.43
12.22
18.38
15.11
16.56
18.42
16.64
16.05
20.36
14.76
27.46
11.00
12.58
10.38
13.90
14.49
24.91
16.26
14.82
16.51
11.44
9.61
16.54
11.40
13.30
14.62
9.02
11.03
12.06
16.60
RS/BH
0.91
0.67
0.80
0.81
0.81
0.89
0.91
0.80
0.80
0.81
0.78
0.82
0.82
0.82
0.91
0.89
0.72
0.90
0.90
0.91
0.89
0.82
0.75
0.80
0.78
0.76
0.82
0.81
0.76
0.83
0.92
0.91
0.90
0.82
0.71
0.78
0.63
0.73
0.73
0.93
0.82
0.89
0.94
0.84
0.87
0.64
0.92
0.90
0.79
RS-BH
-1.02 [-2.68]
-1.51 [-1.95]
-3.13 [-2.21]
-1.15 [-2.00]
-2.68 [-2.36]
-0.89 [-1.81]
-0.94 [-1.73]
-1.48 [-1.40]
-1.00 [-1.71]
-2.85 [-2.00]
-3.16 [-2.09]
-1.07 [-1.93]
-2.50 [-2.25]
-2.40 [-2.34]
-0.83 [-2.20]
-0.92 [-1.87]
-5.17 [-2.48]
-1.26 [-2.60]
-1.20 [-2.65]
-1.15 [-2.31]
-1.36 [-2.36]
-3.38 [-2.61]
-3.79 [-2.43]
-3.33 [-1.61]
-4.09 [-2.05]
-3.92 [-2.18]
-2.95 [-1.88]
-3.95 [-1.95]
-3.50 [-2.10]
-4.58 [-1.79]
-0.84 [-1.49]
-1.12 [-2.05]
-1.01 [-1.67]
-2.54 [-2.09]
-4.16 [-1.96]
-5.50 [-1.68]
-5.99 [-1.81]
-4.07 [-1.89]
-4.53 [-2.10]
-0.81 [-2.54]
-1.78 [-1.33]
-1.86 [-2.71]
-0.74 [-2.64]
-2.17 [-2.29]
-1.90 [-2.13]
-3.20 [-1.54]
-0.87 [-2.99]
-1.15 [-2.54]
-3.41 [-1.78]
Years
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
18
43
43
43
43
25
25
18
25
25
16
25
25
25
25
43
25
43
10
10
10
10
10
36
16
43
43
43
43
18
43
43
20
Notes: This table reports the horizon effect bias in the MSCI regional index portfolios. Rescaled (RS) returns are
average monthly returns mutiplied by 12. Buy-and-hold (BH) returns are average annual returns. t-statistic for
the difference between RS and BH returns is shown in square brackets. Sample period ends in 2012 and spans the
number of years shown in the last column.
Table 8. Horizon Effects in Developed Market Factor Portfolios
RS
BH
RS-BH
RS/BH
RS
BH
RS-BH
RS/BH
RS
BH
RS-BH
RS/BH
RS
BH
RS-BH
RS/BH
RS
BH
RS-BH
RS/BH
Small
Big
SMB
High
Low
HML
A. Asia Pacific Excluding Japan
-2.79
16.35
8.95
7.40
-0.68
20.46
12.82
7.64
-2.11
-4.12
-3.87
-0.24
[-2.42]
[-2.40]
[-2.58]
[-0.32]
0.80
0.70
Winners
Losers
WML
11.21
16.04
-4.84
[-2.58]
0.70
14.00
16.72
-2.72
[-2.19]
0.84
8.92
12.66
-3.74
[-1.54]
0.70
17.63
22.34
-4.71
[-3.09]
0.79
-8.71
-9.68
0.97
[0.51]
9.04
10.92
-1.89
[-2.29]
0.83
9.86
10.86
-0.99
[-1.75]
0.91
-0.83
0.07
-0.89
[-2.32]
B. Europe
11.80
6.82
13.48
8.50
-1.68
-1.69
[-1.94]
[-2.74]
0.88
0.80
4.98
4.97
0.01
[0.01]
4.39
6.18
-1.80
[-1.71]
0.71
15.35
17.62
-2.26
[-3.29]
0.87
-10.96
-11.43
0.47
[0.53]
9.42
10.64
-1.22
[-1.88]
0.89
8.47
9.21
-0.74
[-1.81]
0.92
0.95
1.43
-0.48
[-1.30]
C. Global
11.18
6.39
12.20
7.73
-1.01
-1.34
[-1.76]
[-2.42]
0.92
0.83
4.80
4.47
0.33
[0.70]
5.81
7.00
-1.19
[-1.35]
0.83
13.26
14.85
-1.59
[-2.17]
0.89
-7.45
-7.85
0.40
[0.45]
2.90
4.21
-1.31
[-1.24]
0.69
3.10
3.83
-0.72
[-0.87]
0.81
-0.20
0.39
-0.59
[-0.73]
D. Japan
5.96
0.29
6.56
2.51
-0.59
-2.21
[-0.71]
[-1.39]
0.91
0.12
5.67
4.05
1.62
[1.12]
2.82
3.44
-0.62
[-0.56]
0.82
4.07
6.20
-2.13
[-1.24]
0.66
-1.25
-2.76
1.51
[0.80]
12.68
13.52
-0.84
[-1.21]
0.94
10.29
11.25
-0.96
[-2.18]
0.91
2.38
2.27
0.11
[0.25]
8.98
10.07
-1.09
[-1.15]
0.89
16.47
18.04
-1.57
[-1.76]
0.91
-7.49
-7.98
0.48
[0.46]
E. North America
13.22
9.51
3.71
14.41
10.58
3.83
-1.19
-1.07
-0.13
[-2.20]
[-1.61]
[-0.25]
0.92
0.90
Notes: This table reports the horizon effect bias in the developed market factor portfolios. Rescaled (RS) returns
are average monthly returns mutiplied by 12. Buy-and-hold (BH) returns are average annual returns. t-statistic for
the difference between RS and BH returns is shown in square brackets. Sample period is 1991-2012.
Table 9. Horizon Effects in Non-Investable and Investable Country Portfolios
Country
Australia
Brazil
Canada
France
Germany
Hong Kong
Italy
Japan
Malaysia
Singapore
South Africa
South Korea
Spain
Sweden
Switzerland
Taiwan
United Kingdom
RS
12.34
21.71
12.16
8.66
9.79
9.09
7.31
1.62
8.19
9.96
18.07
20.92
10.92
13.01
9.63
9.95
6.59
MSCI Index
BH RS/BH
14.11
28.73
13.85
9.48
10.94
10.60
8.11
2.69
12.06
12.85
18.97
22.81
11.66
15.71
10.29
10.86
7.92
0.87
0.76
0.88
0.91
0.90
0.86
0.90
0.60
0.68
0.78
0.95
0.92
0.94
0.83
0.94
0.92
0.83
RS-BH
-1.76
-7.03
-1.68
-0.82
-1.15
-1.52
-0.80
-1.06
-3.87
-2.89
-0.89
-1.89
-0.74
-2.70
-0.66
-0.91
-1.33
[-1.31]
[-1.54]
[-1.58]
[-1.13]
[-1.20]
[-1.06]
[-0.75]
[-1.30]
[-1.27]
[-1.44]
[-0.84]
[-1.23]
[-0.66]
[-1.68]
[-1.23]
[-0.52]
[-1.48]
RS
11.72
19.98
11.48
8.04
9.47
8.17
6.68
1.15
10.10
8.17
17.48
19.87
10.33
11.47
8.64
8.04
6.01
iShares ETF
BH RS/BH
12.79
26.39
12.66
8.51
10.27
9.23
7.15
1.96
10.85
9.74
17.77
21.23
10.61
13.30
8.86
8.40
6.96
0.92
0.76
0.91
0.94
0.92
0.89
0.94
0.59
0.93
0.84
0.98
0.94
0.97
0.86
0.98
0.96
0.86
RS-BH
-1.06
-6.41
-1.18
-0.47
-0.80
-1.06
-0.46
-0.81
-0.75
-1.56
-0.28
-1.35
-0.28
-1.83
-0.22
-0.36
-0.95
[-0.92]
[-1.51]
[-1.27]
[-0.67]
[-0.80]
[-0.74]
[-0.46]
[-0.92]
[-0.26]
[-0.99]
[-0.34]
[-0.83]
[-0.26]
[-1.43]
[-0.44]
[-0.24]
[-1.18]
Index(RS-BH)
vs ETF(RS-BH)
-0.70
-0.62
-0.51
-0.35
-0.35
-0.46
-0.34
-0.25
-3.12
-1.32
-0.61
-0.54
-0.46
-0.87
-0.44
-0.55
-0.38
[-2.14]
[-1.09]
[-2.66]
[-2.28]
[-1.91]
[-1.88]
[-1.80]
[-1.39]
[-1.10]
[-0.95]
[-1.53]
[-2.26]
[-1.76]
[-1.75]
[-1.89]
[-1.35]
[-1.92]
Years
16
12
16
16
16
16
16
16
16
16
9
12
16
16
16
12
16
Notes: This table reports the horizon effect bias in the non-investable MSCI country indices and the corresponding exchange-traded
funds. It also shows the difference in biases in the indices and ETFs. Rescaled (RS) returns are average monthly returns mutiplied
by 12. Buy-and-hold (BH) returns are average annual returns. t-statistics are shown in square brackets. The sample covers the
number of years shown in the last column, ending in 2012.
A. Beta vs Lag
0.5
B. Autocorrelation vs Lag
0.3
Simulated
Small Stock
0.4
Simulated
AR(1) matching first order AR
Hyperbolic
Small Stock
0.25
0.2
0.3
0.15
0.2
0.1
0.05
0.1
0
0
−0.1
−0.05
0
5
10
15
20
25
30
35
40
C. Beta vs Lag
−0.1
0
5
10
Simulated
Large Stock
0.15
0.2
25
30
35
40
Simulated
AR(1) matching first order AR
Hyperbolic
Large Stock
0.25
0.2
0.3
20
D. Autocorrelation vs Lag
0.3
0.4
15
0.1
0.05
0.1
0
0
−0.1
−0.05
0
5
10
15
20
Lag
25
30
35
40
−0.1
0
5
10
15
20
Lag
25
30
35
40
Figure 1. Panels A and C plot slope coefficients from univariate regressions of daily excess returns on
lagged excess market returns versus lag in days. Panels B and D plot autocorrelation coefficients versus lag
in days. Red dashed lines are generated by returns from the smallest and largest deciles of US stocks during
the period 1926-2012. Solid blue lines are generated by returns simulated from equation (2) with parameters
chosen to match the empirical moments: σ1 = σ2 = 0.2, λ = 12, δ = (1, 0.5, 0.29, 0.156, 0.08, 0.04, 0.02),
(pik1 , pik2 , ..., pik7 ) = (ak , (1 − ak )/6, (1 − ak )/6, ...(1 − ak )/6). Small Stocks: µ1 = µ2 = 0.06, β = [1, 1.5],
a1 = 0.35, a2 = 0.1. Large Stocks: µ1 = µ2 = 0.05, β = [1, 1], a1 = 1, a2 = 1.
A. Small Stocks
2
1.9
Mean Return
1.8
1.7
1.6
1.5
1.4
1.3
1.2
0
50
100
150
200
250
100
150
Holding Period (days)
200
250
B. Large Stocks
1.6
1.5
Mean Return
1.4
1.3
1.2
1.1
1
0.9
0.8
0
50
Figure 2. Panels A and B plot average simple returns versus holding period in days. Returns are simulated
from equation (2) with parameters chosen to match the empirical moments: σ1 = σ2 = 0.2, λ = 12, δ =
(1, 0.5, 0.29, 0.156, 0.08, 0.04, 0.02), (pik1 , pik2 , ..., pik7 ) = (ak , (1 − ak )/6, (1 − ak )/6, ...(1 − ak )/6). Small Stocks:
µ1 = µ2 = 0.06, β = [1, 1.5], a1 = 0.35, a2 = 0.1. Large Stocks: µ1 = µ2 = 0.05, β = [1, 1], a1 = 1, a2 = 1.
Monthly Return, Percent
Big
Small
1.20
2.00
1.10
1.90
1.00
1.80
0.90
1.70
0.80
1.60
0.70
1.50
0.60
1.40
0.50
1.30
0.40
0
21
42
63
84 105 126 147 168 189 210 231 252
1.20
0
21
42
63
84
Monthly Return, Percent
Value
Growth
1.90
1.00
1.80
0.90
1.70
0.80
1.60
0.70
1.50
0.60
1.40
0.50
1.30
0.40
1.20
0.30
1.10
0
21
42
63
84 105 126 147 168 189 210 231 252
0.20
0
21
42
63
84
Monthly Return, Percent
Winner
1.40
2.10
1.30
2.00
1.20
1.90
1.10
1.80
1.00
1.70
0.90
1.60
0.80
1.50
0.70
21
42
63
84 105 126 147 168 189 210 231 252
0.60
0
21
42
Monthly Return, Percent
High Inverse Price
1.40
2.50
1.30
2.40
1.20
2.30
1.10
2.20
1.00
2.10
0.90
2.00
0.80
1.90
0.70
21
42
63
84 105 126 147 168 189 210 231 252
Days in Compounding Period
Figure 3. This figure continues on the following page.
63
84
105 126 147 168 189 210 231 252
Low Inverse Price
2.60
1.80
0
105 126 147 168 189 210 231 252
Loser
2.20
1.40
0
105 126 147 168 189 210 231 252
0.60
0
21
42
63
84
105 126 147 168 189 210 231 252
Days in Compounding Period
Monthly Return, Percent
High Short-Term Reversal
Low Short-Term Reversal
1.60
1.60
1.50
1.50
1.40
1.40
1.30
1.30
1.20
1.20
1.10
1.10
1.00
1.00
0.90
0.90
0.80
0
21
42
63
84 105 126 147 168 189 210 231 252
0.80
0
21
42
63
Monthly Return, Percent
High Volatility
1.30
1.60
1.20
1.50
1.10
1.40
1.00
1.30
0.90
1.20
0.80
1.10
0.70
1.00
0.60
21
42
63
84 105 126 147 168 189 210 231 252
0.50
0
21
42
63
84
Monthly Return, Percent
Illiquid
1.40
1.40
1.30
1.30
1.20
1.20
1.10
1.10
1.00
1.00
0.90
0.90
0.80
0.80
0.70
21
42
63
84 105 126 147 168 189 210 231 252
0.60
0
21
42
63
Monthly Return, Percent
High Z-Score
1.00
1.50
0.90
1.40
0.80
1.30
0.70
1.20
0.60
1.10
0.50
1.00
0.40
0.90
0.30
21
42
63
84 105 126 147 168 189 210 231 252
Days in Compounding Period
84
105 126 147 168 189 210 231 252
Low Z-Score
1.60
0.80
0
105 126 147 168 189 210 231 252
Liquid
1.50
0.70
0
105 126 147 168 189 210 231 252
Low Volatility
1.70
0.90
0
84
0.20
0
21
42
63
84
105 126 147 168 189 210 231 252
Days in Compounding Period
Figure 3. This figure plots for different style portfolios average rolling n-day buy-and-hold returns scaled to a
monthly equivalent, [E (Rt−n+1 · · · Rt )]21/n − 1. The number of days n in a compounding horizon is shown on
the x-axis. The y-axis in each panel is scaled to accommodate a bias of up to 0.80%. A complete description
of the portfolios is provided in Section 4 of the text. Computation details are in the Appendix.
A. MSCI Country Indices
Percent of Observations
20
15
10
5
0
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
0.85
0.90
0.95
1.00
0.90
0.95
1.00
B. MSCI Regional Indices
Percent of Observations
40
30
20
10
0
0.50
0.55
0.60
0.70
0.75
0.80
C. Developed Markets Style Portfolios
30
Percent of Observations
0.65
20
10
0
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
Ratio of Average Rescaled Monthly Returns to Average Buy-and-Hold Annual Returns
Figure 4. This figure plots histograms of the ratios of average monthly returns mutiplied by 12 (RS) to
average buy-and-hold (BH) annual returns for three sets of portfolios: MSCI country index portfolios in Panel
A, MSCI regional index portfolios in Panel B, and developed market style portfolios in Panel C.