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The term structure of CAPM alphas and betas

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The term structure of CAPM alphas and betas

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Content The Term Structure of CAPM Alphas and Betas
Wayne Chang
∗
Dissertation for Doctor of Philosophy in Business Administration
Finance and Business Economics, Marshall Business School
University of Southern California
August 8, 2017
Abstract
Using monthly returns to estimate portfolio alphas and betas is inappropriate for long-horizon
investors or for discounting long-term cashflows. Alphas and betas do not necessarily have flat
term structures. This paper develops a novel beta estimation method that is non-parametric,
allows for time-variation, and better captures long-horizon estimates compared to the realized
volatility or rolling regression approaches. Long-short portfolios sorted on size, value, and
momentum have CAPM betas that can reverse sign with longer horizons. At multi-year horizons,
the average alpha associated with size increases while momentum’s decreases. These differences
are economically significant for long-term investors.
Keywords: Risk-return, Investment horizon, CAPM, Conditional alpha, Time-varying beta, Size
effect, Value premium, Momentum effect
JEL Classification: G11, G12, G17
∗
I would especially like to thank my dissertation chair, Professor Wayne Ferson, for his many hours of patient
guidance and advice. For providing valuable input, I’m also grateful to Christopher Jones, Kenneth Ahern, K.R.
Subramanyam, David Solomon, Mark Soliman, Oguzhan Ozbas, Larry Harris, David Harris, Louis Piccotti and to
participants in seminars at the 2016 SWFA Finance Conference, 2016 FMA Annual Conference, USC, Nanyang
Technological University, National Taiwan University, National Central University, Hong Kong City University, and
Research Affiliates. All errors are my own. The USC Provost’s Ph.D. Fellowship and the USC Graduate School Final
Year Fellowship has provided much appreciated financial support. Comments welcome. For contact information,
please visit my website at sites.google.com/site/waynelinchang/home.
1
Contents
1 Introduction 3
2 Related Literature and Paper Contribution 7
3 Theoretical Sources of Term Structure Effects 11
4 Data and Methods Used for Estimating Term Structures 16
5 The Presence of Auto and Lead-Lag Correlations 26
6 The Shape of Alpha and Beta Average Term Structures 33
7 Stationarity of Conditional Alphas and Betas 42
8 Conclusion 44
Appendices
A The Term Structures Shape of Expected Returns and Risk Premiums 46
B Summary Descriptions 49
C Comparing Alphas and Betas across Two Horizons 56
D Robustness of Term Structure Results 58
E Robustness of Unconditional Term Structure Results to Different Reformation
and Rebalancing Periods 66
F Sensitivity of Optimal Portfolios and DCF Valuations to Horizon Mismatch 71
References 73
2
1 Introduction
The risk and return of owning equities depend on the investment horizon. However, the
standard approach uses monthly returns with the implicit assumption that monthly alphas and
betas are appropriate. This paper explores investment horizons spanning 1-month to 10-years, a
large range motivated by evidence on actual investor holding periods. Cella et al. (2013) find for
institutional investors a 5%-to-95% range of average stock holding periods that covers 4 months to
12 years. Retail investors saving for retirement tend to have little account activity, implying very
long holding periods. Ameriks and Zeldes (2004) find that 70% of defined contribution retirement
accounts don’t make any asset allocation changes in the course of 10 years. This paper shows that
alpha and beta term structures are not flat. Buy-and-hold investors with multi-year horizons may
face different risk-return characteristics than those implied by monthly returns. In some cases, an
investment’s beta can reverse sign, indicating a hedge instead of adding risk. I find that momentum
investing should be more attractive to shorter-horizon investors, while small-cap stocks appear more
attractive to those with longer horizons.
The paper studies size, value, and momentum long-short portfolios. These portfolios are
periodically rebalanced according to the standard Fama and French (1993) approach. Rebalancing
ensures that the strategies remain exposed to the desired characteristics, even over many years.
The paper focuses on CAPM alphas and betas for these characteristics because of their central
importance in the asset pricing literature and to simplify the analysis of the drivers of sloped alpha
and beta term structures. However, the paper’s message and methods are applicable more generally.
The paper focuses on the empirical documentation of term structure shapes while offering some
analysis of the economic forces driving these patterns.
Sloped alpha and beta term structures occur because the market and portfolios exhibit lead-lag
correlations with each other. Long-horizon betas aggregate short-horizon contemporaneous betas
plus short-horizon lead-lag correlations. Negative correlations produce long-horizon betas smaller
than short-horizon ones, resulting in a downward-sloping term structure. I observe this for the size
effect when horizons exceed one year. On the other hand, positive lead-lag correlations result in
an upward-sloping term structure. I observe this for momentum. In addition, alphas and betas
3
are time-varying and mean-reverting. Even with zero lead-lag correlations, an abnormally high
short-horizon beta will likely be followed by less extreme betas, resulting in a lower long-horizon
estimate. The term structures will tend to be downward-sloping given abnormally high short-horizon
estimates and upward-sloping given abnormally low ones. Alphas and betas should thus properly
reflect the investment strategy’s horizon just like future cashflows should use discount rates with
the appropriate maturity. Indeed, betas are inputs into computing discount rates so the wrong beta
can lead to misvaluation.
The paper offers two novel contributions. First, it develops a new conditional covariance
estimation method that uses returns observed at a higher frequency than the investment horizon of
interest. For example, to estimate a one-year beta, the method can use daily or monthly returns.
This approach is non-parametric, avoiding the use of instruments to capture conditionality (e.g.
Ferson and Schadt (1996), Ferson and Harvey (1999)) and of models to describe return behavior.
Using higher-frequency data is standard in the realized volatility literature (e.g. Andersen and
Bollerslev (1998), Fleming et al. (2003), Barndorff-Nielsen and Shephard (2004), Andersen et al.
(2006)). However, the realized approach ignores the conditional mean effect on the conditional
covariance needed for betas. It ignores lead-lag correlations among higher-frequency returns, and
cannot capture sloped average term structures. Compared to the classical rolling regression approach,
the new conditional method I propose is more sensitive to period-to-period variation and better
estimates long-horizon conditional moments.
The paper’s second contribution is the documentation of several novel empirical findings
regarding long-short portfolios sorted on size (SMB), value (HML), and momentum (UMD). First, it
finds significant market-portfolio lead-lag correlations at monthly and annual horizons. They show in-
sample and in some cases, out-of-sample predictability. Second, the three portfolios exhibit different
average term structure shapes. Size’s beta term structure mostly slopes downward, momentum’s
slopes upward, and value’s stays mostly flat. Beta term structure effects directly affect alpha term
structures since conditional mean return estimates remain similar across horizons. Short-horizon
conditional alphas and betas are stationary and mean-revert. Consequently, the term structure
at a particular point in time can still be sloped even without lead-lag correlations. Overall, these
empirical findings have important implications for the estimation and use of alphas and betas,
4
pointing to a more nuanced approach that accounts for the investment horizon. This impacts
investor’s optimal portfolio holdings. I find that maximum Sharpe-ratio or minimum variance
portfolios estimated using the wrong horizon returns can cost investors 2-6% annually. Corporate
investment and financing decisions that use beta-dependent discount rates can also be affected.
Long-horizon cashflows that load heavily on size, value, or momentum factors can suffer misvaluation
by as much as -20% to +120%.
Figure 1 previews the paper’s main results and shows the average conditional term structure
estimated using the new higher-frequency approach. Size’s beta peaks at the 1-year horizon,
occurring with positive monthly lead-lag correlations with the market. Given a further lengthening
of the horizon, beta’s slope reverses and turns downward as negative correlations dominate at annual
horizons. When size’s beta falls, market comovement explains less of its excess returns. Average
alpha thus increases from an insignificant 0.5% at the annual horizon to a significant 3.2% per year
at the 10-year horizon. Value and momentum do not share these term structure shapes, as they
exhibit different lead-lag correlation patterns. Value tends to have insignificant correlations and
thus flat alpha and beta term structures. On the other hand, momentum has significantly positive
correlations at annual horizons. This produces an upward-sloping beta term structure and alphas
that fall from 6.8% at the annual horizon to 5.3% per year at the 10-year horizon. These results
demonstrate that alpha and beta term structures are not always flat.
The rest of the paper is organized as follows. Section 1 reviews the related literature and
shows how this paper contributes. Section 2 describes the theoretical sources of term structure
effects. Section 3 covers data and methods used for estimating alphas and betas, including the
new higher-frequency approach. Section 4 offers evidence of significant lead-lag correlations in the
portfolios studied. Section 5 contains the main results on term structure shapes and highlights
investor implications. Section 6 finds that alphas and betas tend to be stationary and mean-reverting.
Section 7 concludes.
5
Figure 1: Average Alpha and Beta Conditional Term Structures (New Higher-
Frequency Method). Data from 1926-2015 used to form non-overlapping monthly, quarterly, annual
returns and annual-overlapping 3-year and 10-year returns. Conditional alpha and beta estimated using the
new higher-frequency approach described in Section 4. Dotted lines contain the 95% confidence interval
calculated using Newey-West ’94 standard errors.
6
2 Related Literature and Paper Contribution
2.1 Contribution to Equity Term Structure Literature
“Term structure” as applied to equities has taken two different meanings in the literature. The
first usage refers to changing risk or return characteristics due to different holding periods, reflecting
a “holding” term structure of the same stock or portfolio. The second usage decomposes segments
of equities that differ by their cashflow duration, reflecting a “maturity” term structure. These two
perspectives offer distinct dimensions for analyzing financial assets. The holding term structure
requires only a single series compounded over different horizons. In contrast, the maturity term
structure requires a maturity decomposition with multiple return series that determine different
points on the term structure curve.
This paper’s analysis of how alphas and betas change as a function of a portfolio’s investment
horizon thus applies the holding term structure perspective. It builds on a large literature. Samuelson
(1969) and Merton (1969) are early papers that study how the market portfolio’s risk-return dynamics
depend on the investment horizon. More recent work includes Campbell and Viceira (2002, 2005),
Bandi and Perron (2008), Colacito and Engle (2010), Rua and Nunes (2012), Diris et al. (2014),
Chaudhuri and Lo (2015). These papers consider investing in the equity market as whole, relative
to cash or bonds. I study alphas and betas, thus assuming the investor already holds the market
but is considering additional investments in equity sub-portfolios. My results show that optimal
allocations to size, value, and momentum depend on the investor’s investment horizon.
The tendency for equity alphas and betas to change with the return horizon is known as the
intervalling effect (e.g. Levhari and Levy (1977), Hawawini (1983), Cohen et al. (1983), Handa et al.
(1989, 1993), Longstaff (1989), Gencay et al. (2005), Gilbert et al. (2014)). Past papers usually
focus on unconditional alphas and betas and not the time-varying ones I study. I also analyze
a much broader range of horizons, from 1-month to 10-years. For example, Gilbert et al. (2014)
find differences between daily and quarterly betas that are mostly induced by positive lead-lag
correlations. In contrast, I find the most dramatic effects at multi-year horizons, with negative
correlations playing a key role. Betas also play a key role in discount rates, critical for evaluating
7
corporate financing or investing projects. Brennan (1997) and Ang and Liu (2004) investigate the
term structure of discount rates but they focus on the challenges associated with time-varying
risk-free rates and market risk premiums. I study beta’s horizon-dependence and document large
misvaluations given poor horizon-matching between the cashflow and the discount rate beta.
Recentworkstudiesbeta-relatedhorizonissuesforthesize, value, andmomentumcharacteristics
I examine (e.g. Cohen et al. (2009), Bandi et al. (2010), In et al. (2010), In et al. (2011), Jurek
and Viceira (2011), Ang and Kristensen (2012), Gonzalez et al. (2012), Brennan and Zhang (2013),
Cenesizoglu and Reeves (2015), Kamara et al. (2016)). I differ by investigating the underlying lead-
lag correlations that drive alpha and beta horizon effects and by also developing a new conditional
estimation approach. For example, Kamara et al. (2016) find term structure effects for different
factors’ cross-sectional prices of risk. They do not offer an explanation for their empirical findings,
but my paper provides one in terms of lead-lag correlation patterns. Taking the market price
of risk, they find that it peaks at the semi-annual to annual horizons. This can be driven by
high-returning stocks having betas that peak at that horizon, implying that these stocks have
positive lead-lag correlations with the market at shorter horizons and negative ones at longer
horizons. Indeed, I find this pattern for small caps. They have betas that peak at the semi-annual to
annual horizons, consistent with Kamara et al. (2016)’s findings given a cross section of size-sorted
portfolios. Appendix A details the intuition connecting the term structure of betas with the term
structure of the price of risk.
Recent work on holding term structures extend to other equity concepts, including average
returns (Boguth et al. (2016)), market-implied expected returns (van Binsbergen et al. (2012), Ang
and Ulrich (2012)), cost-of-capital estimates (Levi and Welch (2016)), investor risk-aversion (Andries
et al. (2015a)), dividend volatility (Belo et al. (2015)), tail risks (Guidolin and Timmermann (2006)),
macroeconomic risks (Boons and Tamoni (2015)), inflation risk prices (Ang et al. (2008)), variance
risk prices (Andries et al. (2015b)), and consumption risk prices (Croce et al. (2014), Bryzgalova
and Julliard (2015), Dew-Becker and Giglio (2016)). I add to this literature by dissecting alphas
and betas.
As opposed to the holding term structure, there’s also a literature on the maturity term
structure of the equity market as whole. For example, Dechow et al. (2004) and Lettau and Wachter
8
(2007) note that value stocks, as opposed to growth stocks, tend to have fewer growth opportunities
and thus have shorter duration. Given value’s higher returns, this implies a downward-sloping
term structure of equity returns. A series of papers (van Binsbergen et al. (2012), van Binsbergen
et al. (2013), van Binsbergen and Koijen (2017)) uses traded derivatives to decompose the market’s
expected returns across different maturities and also finds a downward sloping term structure. My
paper’s documentation of positive HML alphas are consistent with this.
Papers usually focus on either the holding or maturity term structure dimension but these two
perspectives should be related theoretically. For bonds, the Expectations Hypothesis postulates
that investors are indifferent to holding bonds of different maturities. For example, under the
Return to Maturity version of the Expectation Hypothesis, the expected return from compounding
short-term Treasury Bills for 10-years should equal the yield-to-maturity of 10-year Treasury Bonds.
For equities, the analogue predicts HML expected returns that converge to zero as the horizon
lengthens. This does not occur, and my findings show that adjusting for market risk does not
help. HML alphas remain significantly positive at the longest 10-year horizons. For bonds, much of
the literature grew around studying Expectations Hypothesis deviations. Perhaps the equity term
structure literature progresses similarly, and this paper is one step in that direction.
2.2 Contribution to Cross-Sectional Anomalies Literature
The literature’s explanations for cross-sectional anomalies like size, value, and momentum
is large but can be broadly classified into four groups: data mining, empirical misspecification,
missing risks, or mispricing. First, data-mining argues that the anomalies are merely chance patterns
dredged out of the historical record through repeated testing. This argument is most persuasive for
size since value and momentum have received more out-of-sample support in more recent data and
in other asset classes and geographies (e.g. Asness et al. (2013)). My paper contributes by finding
that at 10-year horizons, size’s alphas are actually much larger and more significant. Long-horizon
returns merely compound short-horizons ones so this is not an independent out-of-sample test.
Nevertheless, larger alphas and t-statistics lowers data snooping concerns while introducing the
possibility that size is truly anomalous at long-horizons even if it is not at short ones. The reverse
applies to momentum. Although alphas decline and in some cases become insignificant at 10-year
9
horizons, this need not imply a data-mined short-horizon momentum. Heterogeneous investors
operate at different time scales so looking at the entire term structure opens up an additional
holding-period dimension for anomalies to occur or to be mined.
Second, empirical tests of the asset pricing model may be misspecified for various reasons.
Microstructure noise and other market imperfections generate prices that are measured with error.
This paper contributes by seriously investigating long-horizon returns, where true signal likely
constitutes a larger share relative to noise. The Roll (1977) Critique argues against using only a
US stock index to proxy for what should be a more comprehensive market portfolio that includes
human capital and other assets. Since the validity of the critique depends on the extent to which
the proxy and the true market portfolio are correlated (Stambaugh (1983), Kandel and Stambaugh
(1987), Shanken (1987)), this paper contributes to the extent stocks and other asset classes covary
more strongly at longer horizons that are less subject to noise and other frictions. The stock
market then becomes a more appropriate proxy for the true market portfolio as the investment
horizon lengthens. The Hansen and Richard (1987) Critique argues against testing a conditional
model using unconditional moments. Lewellen and Nagel (2006) and Ang and Kristensen (2012)
explore approaches to test the conditional CAPM but they focus on only short monthly horizons. I
contribute by developing a new conditional estimation method that applies even to multiyear returns,
thus extending their results to much longer horizons. Overall, findings of large alphas for size,
value, and momentum that persist across horizons challenges this group of model misspecification
explanations.
Third, anomalies may proxy for missing risk factors that investors care about (e.g. APT) or
for state variables that forecast future investment opportunities (e.g. ICAPM). Yet, these rational
theories provide little guidance on the precise time horizon for empirically testing the data. Focusing
exclusively on monthly returns is arbitrary, so this paper contributes by assessing a range of
alternative return horizons. It finds, for example, that the term structure of average alphas need
not be flat or monotonic. Size alphas are insignificant at annual horizons but significant at both
monthly and multi-year horizons. These results expand the range of empirical patterns that any
theoretical model must fit. In some cases, risk-based models have predictions about lead-lag return
patterns that this paper’s analyses directly evaluate. For example, the ICAPM introduces priced
10
state variables that should forecast future market performance. This paper finds, however, that the
strongest correlations are for market returns predicting portfolio returns, not the other way around.
In other cases, risk-based models are better evaluated using long-horizon returns. For example,
duration-based theories of the value effect (e.g. Brennan and Xia (2006), Lettau and Wachter (2007),
Campbell et al. (2010)) argue that short-horizon beta poorly captures priced risk since it contains
components that investors don’t care about. In particular, long term investors are likely indifferent
to discount rate fluctuations that cancel over time. If so, evaluating beta using long horizon returns
should better capture what investors price. Campbell and Vuolteenaho (2004) explicitly suggests
testing their model by “estimating beta over long horizons.” I find, however, that HML betas don’t
change with horizon and alphas remain robust. Overall, my results cast doubt on at least some
risk-based theories. By engaging in tests of different horizon returns, it’s also an example of how
theories with alternative horizon implications can be evaluated.
Finally, the fourth group of anomaly explanations rely on some form of mispricing. Investors
may suffer from behavioral biases or have information processing limits, and this can occur in
the presence of institutional constraints that prevent effective arbitrage. The finding in Section
6.2 that greater initial illiquidity is associated with larger long-horizon betas and thus with lower
long-horizon alphas could be consistent with this explanation. If illiquidity is risky, it’s cost should
be most acute in the short-run. But times of greater illiquidity correspond to relatively higher, not
lower, short-term alphas. Illiquidity may inhibit arbitrage in the same way short-selling costs do (e.g.
Miller (1977), Stambaugh et al. (2012)) and thus contribute to mispricing. Furthermore, my finding
of alphas even at multi-year horizons is also consistent with mispricing. Long horizon inefficiencies
may be especially difficult to exploit and eliminate. Arbitraging over longer time periods involves
larger holding costs while idiosyncratic and career risks also take longer to resolve (e.g. Shleifer and
Vishny (1990), Shleifer and Vishny (1997), Pontiff (2006)).
3 Theoretical Sources of Term Structure Effects
This section investigates the drivers of CAPM alpha and beta term structure dynamics. Alpha
estimates follow from betas, so the section begins with an investigation of the determinants of beta
11
term structures. A decomposition of long-horizon betas into short-horizon moments makes clear
that beta term structures are only flat under special conditions. The dual presence of lead-lag
correlations among short-horizon returns and of mean-reversion in beta estimates induces sloped
term structures. Given horizon-dependent betas, alphas are usually horizon-dependent too.
For clarification, “horizon,” as used in this paper, refers to an investment strategy’s anticipated
period of implementation. The strategy need not be passive but can involve active rebalancing and
other preset rules, such as signal-dependent trades, stop-loss rules, or even volatility management.
The key requirement is that alpha and beta are estimated using the strategy’s past returns and the
same strategy remains in place over the anticipated investment period. The horizon is assumed
exogenous, perhaps due to investor preferences or costs. Money may also be invested to fund a
future expenditure with a known timeline. These assumptions are consistent with the empirical
literature documenting highly persistent stock holding periods (e.g. Cremers and Pareek (2015)).
3.1 Decomposing Long-Horizon Beta Into Short-Horizon Betas
A long-horizon beta estimate can be decomposed into short-horizon market autocorrelation and
market-portfolio lead-lag terms. Specifically, the conditional market beta estimate given information
at time t and log returns of portfolio i spanning a period of horizon h can be expressed as follows.
ˆ
β
t→t+h
≡
ˆ
Cov
t
(r
i
t→t+h
,r
m
t→t+h
)
ˆ
Var
t
(r
m
t→t+h
)
=
ˆ
Cov
t
(
P
h
τ=1
r
i
t+τ−1→t+τ
,
P
h
υ=1
r
m
t+υ−1→t+υ
)
ˆ
Cov
t
(
P
h
τ=1
r
m
t+τ−1→t+τ
,
P
h
υ=1
r
m
t+υ−1→t+υ
)
=
(ˆ ρ
im
t,h,lag=−h+1
+... + ˆ ρ
im
t,h,lag=0
+... + ˆ ρ
im
t,h,lag=h−1
)(ˆ σ
i
t,h,lag=0
ˆ σ
m
t,h,lag=0
)
(ˆ ρ
m
t,h,lag=−h+1
+... + ˆ ρ
m
t,h,lag=0
+... + ˆ ρ
m
t,h,lag=h−1
)(ˆ σ
m2
t,h,lag=0
)
=
(ˆ ρ
im
t,h,lag=0
+
P
h−1,l6=0
l=−h+1
ˆ ρ
im
t,h,lag=l
)(
ˆ σ
i
t,h,lag=0
ˆ σ
m
t,h,lag=0
)
1 + 2
P
h−1
l=1
ˆ ρ
m
t,h,lag=l
(1)
=
ˆ
β
t,h,l=0
+
P
h−1,l6=0
l=−h+1
ˆ
β
t,h,lag=l
1 + 2
P
h−1
l=1
ˆ ρ
m
t,h,lag=l
(2)
The hat “ ˆ” symbol denotes sample estimates. Without loss of generality, I assume the short-
horizon return spans a period of 1 unit while the long-horizon one spans h units, where h is an
integer larger than 1. In line 1, I can thus disaggregate period h returns into the higher-frequency
12
returns indexed by τ and υ. I then assume for line 2 that returns are conditionally covariance
stationary, permitting me to consistently estimate the h-horizon covariance terms as a series of
higher-frequency correlations.
1
The numerator consists of lead-lag cross-correlations (ˆ ρ
im
t,h,lag=l
)
while the denominator consists of market autocorrelations (ˆ ρ
m
t,h,lag=0
). The σ scaling terms are
the higher-frequency portfolio and market standard deviations. Subscripts t and h mean these
terms are estimated using information conditional at t for the period spanning horizon h while
the numeric index l denotes the number of higher-frequency leads or lags. Equation (1) simply
groups together all non-contemporaneous lead-lag correlations. A long-horizon beta equals the
properly-scaled sum of higher-frequency market-portfolio cross-correlations divided by the sum of
higher-frequency market autocorrelations. Equation (2) converts correlations into betas.
The result is a more general case of the classic Scholes and Williams (1977) non-synchronous
beta, which adjusts betas for lead-lag returns induced by microstructure noise. As this derivation
shows, Scholes-Williamsbetascanbeinterpretedasalonger-horizonbetabyaddingparticularchoices
of lagged betas and market autocorrelations. In the presence of short-term lead-lag correlations that
are spurious and not tradeable, a longer-horizon beta better captures the true relationship between
the market and the portfolio. Dimson (1979) betas are also used to correct for microstructure noise
but its definition differs from Scholes-Williams and equation (2) since Dimson betas suffer from
inconsistency (Fowler and Rorke (1983)).
Microstructure noise is well-known for inducing lead-lag return correlations but it is not the only
source. As Section 5 shows, significant lead-lags correlations occur at monthly and annual horizons,
with magnitudes often larger than those found at daily intervals. These results may be surprising
since substantive lead-lag correlations contain predictive and potentially profitable information that
invite arbitrage. However, predictability need not imply market inefficiency (Fama and French
(1989)), and the literature has consistently documented both market return autocorrelation (e.g.
Fama and French (1988), Poterba and Summers (1988), Conrad and Kaul (1989)) and portfolio
cross-correlations (e.g. Lo and MacKinlay (1990), Chordia and Swaminathan (2000), Hou (2007),
1
Despite having more covariance terms, closer lead-lag correlations are not weighted more heavily in equations (1)
or (2). The definition of lead-lag correlation estimates already accounts for the different number of product terms,
ˆ ρ
im
t,h,lag=l
≡
P
h−l
τ=1
(r
i
τ+l
r
m
τ
)
q
P
h
τ=1
(r
i2
τ
)
q
P
h
τ=1
(r
m2
τ
)
.
13
Hong et al. (2007), Cohen and Frazzini (2008), Menzly and Ozbas (2010), Chordia et al. (2011),
Cohen and Lou (2012)). Different explanations have been proposed, including time-varying risk
aversion, liquidity shocks, differences in the speed of information diffusion, and supply-chain links.
Section 6 explores some of these as possible drivers of size and momentum term structure effects.
Returning to equation (2), the long-horizon beta equals the short-horizon zero-lag beta (i.e.
ˆ
β
t→t+h
=
ˆ
β
t,h,lag=0
) only if all numerator and all denominator lead-lag correlations cancel. This
occurs, for example, if all lead-lag correlations equal zero. Furthermore, only if this special case holds
across all possible horizons is the average beta term structure flat. I emphasize average because
ˆ
β
t,h,lag=0
can be viewed as the average short-horizon beta estimated for the period spanning t to
t +h. Therefore, the first driver of sloped beta term structures is the significant presence of market
autocorrelations or market-portfolio lead-lag correlations. This idea is old and dates back to at least
Levhari and Levy (1977). Lead-lag cross-correlations in the numerator shift the long-horizon beta
away from the short-horizon one to the point where the sign can even reverse, as is the case for size
and momentum. On the other hand, market autocorrelations in the denominator preserve the sign
but modulate the magnitude of the long-horizon beta. Empirically, this denominator effect exhibits
more limited impact.
In addition to the presence of lead-lag correlations, mean-reverting short-horizon betas are the
second driver of sloped beta term structures. Notice that the short-horizon lag-0 beta discussed
above,
ˆ
β
t,h,lag=0
, may not equal the short-horizon conditional beta estimated at timet,
ˆ
β
t→t+1
. Both
are of the same frequency with a short-horizon of 1 unit, and both are conditional on information at
time t. But the former is an unconditional estimate of all zero-lag short-horizon betas from time
t to t +h while the latter concerns only the first period following time t. The two estimates are
similar only if short-horizon betas remain constant over period h or if betas have unit roots. Unit
root betas mean further away beta estimates are martingales and thus have an expectation equal to
the same horizon beta next period. Therefore, even without auto or lead-lag cross-correlations (such
that
ˆ
β
t→t+h
=
ˆ
β
t,h,lag=0
), the time t conditional long-horizon beta (
ˆ
β
t→t+h
) differs from the time
t conditional short-horizon beta (
ˆ
β
t→t+1
) if short-horizon betas are time-varying and stationary.
Indeed, the literature does view betas as stochastic and mean-reverting processes (e.g. Andersen
et al. (2006)). Abnormally high betas tend to be followed by lower ones, such that a high
ˆ
β
t→t+1
will
14
tend to be followed by a lower
ˆ
β
t+1→t+2
. The beta term structure will thus be downward-sloping
since
ˆ
β
t→t+h
is then likely lower than
ˆ
β
t→t+1
. The opposite occurs given abnormally low betas,
which tend to be followed by higher betas and a upward-sloping term structure.
Understanding the difference between the term structure at a specific point in time and the
average term structure over all time helps differentiate between the two term structure drivers.
The first driver, auto and cross-correlations, can produce both date-specific and average sloped
term structures. A non-zero ˆ ρ
im
t,h,lag=1
affects the time t term structure while a non-zero
ˆ
E[ρ
im
t,h,lag=1
]
affects the average term structure. The latter is a much stronger condition, requiring multiple time
t ˆ ρ
im
t,h,lag=1
in the historical record to push consistently in the same direction. On the other hand,
the second driver, beta mean-reversion, can produce date-specific sloped term structures but not
average ones. Without lead-lag correlations, mean-reversion inducing term structure effects will be
averaged out across time because
ˆ
E[β
t→t+1
]≈
ˆ
E[β
t,h,lag=0
]. The two unconditional average betas
are similar because both are the same horizon, estimate contemporaneous comovement, and use the
same historical data. Section 6 analyzes the historical average term structure where only historical
average auto and lead-lag correlations can explain their slopes. Section 7 assumes away all lead-lag
correlations and analyzes how beta mean-reversion induces date-specific sloped term structures.
3.2 How Beta Determines Alpha
Given conditional beta and mean return estimates, the conditional alpha estimate follows.
ˆ α
t→t+h
=
ˆ
E
t
[r
i
t→t+h
]−
ˆ
β
t→t+h
ˆ
E
t
[r
m
t→t+h
] (3)
ˆ
E[α
t→t+h
] =
ˆ
E[r
i
t→t+h
]−
ˆ
E[β
t→t+h
]
ˆ
E[r
m
t→t+h
]−
ˆ
Cov[β
t→h
,
ˆ
E
t
[r
m
t→t+h
]] (4)
Equation (3) expresses the date-specific conditional alpha estimate while equation (4) expresses
the historical average estimate. The literature has shown how the covariance term could play a
critical role (Grant (1977), Jagannathan and Wang (1996)). Avoiding the overconditioning bias of
Boguth et al. (2011) requires either the conditional beta or conditional market mean estimate in
the covariance term to use only ex-ante information available to investors. The average conditional
15
alpha term structure is thus a result of the term structures of its four components. The paper’s
use of log returns produces flat expected mean return term structures.
2
Section 6 estimates a flat
term structure for the covariance between betas and market premiums. This leaves the average
beta term structure driving the average alpha term structure.
4 Data and Methods Used for Estimating Term Structures
This section covers the data and the empirical methods for estimating CAPM alphas and
betas. For each portfolio, three different term structures are estimated: an unconditional, a
rolling regression, and a new higher-frequency approach. For each portfolio and each moment, the
unconditional approach estimates a single constant across all time while the latter two approaches
allow for time-varying moments.
4.1 Data Source
The paper uses monthly data from Ken French’s website and collects market, size (SMB), value
(HML), and momentum (UMD) excess-return portfolios for the period from 1927 through 2015.
It begins with monthly returns since higher-frequency returns are more subject to microstructure
noise, which when corrected for, effectively results in a longer horizon estimate. Simple returns are
converted to log returns and compounded by adding them.
3
To get a sense of the term structure, I
study monthly, quarterly, annual, 3-year, and 10-year horizons. Multi-year results use overlapping
annual returns to increase power. For the shortest monthly horizon, the new higher-frequency beta
estimation approach uses weekly returns (while accounting for possible microstructure noise through
incorporation of lead-lag correlations). Longer horizons use monthly returns as the higher-frequency
return.
4
2
See Appendix A for a discussion of the mechanical drivers behind the expected return term structure under a
variety of specifications.
3
Directly adding excess log returns is equivalent to directly compounding excess simple returns and implies monthly
rebalancing between the portfolio’s long and short legs. I do not compound long and short legs separately because I
analyze size, value, and momentum as dynamic strategies and not as fixed assets. See Appendix A for a detailed
discussion.
4
All empirical analyses is done with R (RStudio and Microsoft R Open), and I grateful acknowledge the authors of
the packages used: data.table (Dowle et al. (2015)), dplyr (Wickham and Francois (2016)), fPortfolio (Diethelm Wuertz
(2014)), gmm (Chausse (2010)), magrittr (Bache and Wickham (2014)), rmarkdown (Allaire et al. (2016)), sandwich
16
4.2 Unconditional Alpha and Beta
Unconditional alphas and betas simply use OLS regression of portfolio excess returns on market
excess returns. Each horizon h requires a separate regression.
r
i
t→t+h
= ˆ α
U
h
+
ˆ
β
U
h
r
m
t→t+h
∀t = 1,h + 1, 2h + 1,...,T−h (5)
T denotes the total number of monthly observations and equals 1,068 in this paper (1927 through
2015). The five horizons I study translate to h values of 1, 3, 12, 36, and 120 months.
4.3 Rolling Regression Alpha and Beta
The rolling regression approach, is old and often used to estimate time-varying CAPM alphas
and betas (Fama and MacBeth (1973)). Alpha and beta estimates use an OLS regression with
the previous J observations of h-horizon returns. Each horizon h and time t requires a separate
regression.
r
i
t−τ→t+h−τ
= ˆ α
Roll
t→t+h
+
ˆ
β
Roll
t→t+h
r
m
t−τ→t+h−τ
∀τ =h, 2h,...,Jh (6)
The method assumes a stable alpha and beta during the window period Jh. Backward looking
windows avoid look-ahead bias, and the base set of results uses window lengths of 5, 10, 20, 30, and
40 years for monthly, quarterly, annual, 3-year, and 10-year horizons, respectively. Five years for
monthly returns is standard in the literature, and window lengths rise for longer horizon returns
under the constraint that only 89 years of data exists. This constraint means longer horizon returns
have windows with fewer observations (i.e. J = 60, 40, 20, 10, and 4, respectively) even as they are
longer in duration. Appendix D explores alternative specifications for robustness, and average term
structure results remain similar. Different window lengths affect the persistence and volatility of
moment estimates but the average level remains similar because an unconditional mean is taken
(Zeileis (2004)), tidyr (Wickham (2016)), urca (Pfaff (2008)), xtable (Dahl (2016)), zoo (Zeileis and Grothendieck
(2005)).
17
over all the data.
4.4 Relation Between Realized Volatility and New Higher-Frequency Estima-
tion Method
The paper develops a novel way to estimate time-varying covariances by using returns of
a higher frequency than the horizon of interest. The method is based on the realized volatility
approach but makes changes to incorporate the contribution of the expected return process. To
begin, I highlight the difference between conditional variance and quadratic variation, and it is the
latter that realized volatility estimates. This can be seen using a simple decomposition of a return
under a continuous-time model without jumps.
5
r
t→t+h
=
Z
t+h
t
μ
s
ds +
Z
t+h
t
σ
s
dW
s
≡ μ
t→t+h
+M
t→t+h
μ
s
is the continuous-time expected return process, σ
s
is the instantaneous conditional volatility, and
W
s
is standard Brownian motion. The sum of all σ
s
dW
s
innovations over horizon h are aggregated
into the martingale M
t→t+h
.
Quadratic variation is the realized square of the martingale innovation, M
2
t→t+h
. In the absence
of jumps, it equals the integrated volatility
R
t+h
t
σ
2
s
ds because Brownian innovations are uncorrelated,
Cov[dW
s
dW
v
] = 0 ∀s6=v. The expected return process is usually considered to have bounded
variation so adds no quadratic variation. In contrast, the discrete horizon conditional variance is
defined as Var
t
[r
t→t+h
]≡ E
t
[(r
t→t+h
−E
t
[r
t→t+h
])
2
], with E
t
[r
t→t+h
] = E
t
[μ
t→t+h
] as the conditional
mean over horizon h. Expected quadratic variation, E
t
[M
2
t→t+h
], is a component of conditional
variance as the following decomposition shows.
5
This setup follows Andersen et al. (2010). They use “expected volatility” to refer to conditional variance and
“notional volatility” to refer to quadratic variation.
18
Var
t
[r
t→t+h
] ≡ E
t
[(r
t→t+h
− E
t
[r
t→t+h
])
2
]
= E
t
[((μ
t→t+h
− E
t
[r
t→t+h
]) +M
t→t+h
)
2
]
= Var
t
[μ
t→t+h
] + 2Cov
t
[μ
t→t+h
,M
t→t+h
] + E
t
[M
2
t→t+h
] (7)
E
t
[·] denotes the conditional expectation using information available at time t. Standard realized
measures that estimate quadratic variation deliver the third term of equation (7). Thus, they
are adequate approximations of conditional variance if the first two terms of (7) are negligible.
This approximation works given a sufficiently short horizon when the magnitude of unexpected
innovations dominates that of expected returns. In the limit as the horizon shrinks to zero,
conditional variance approaches expected quadratic variation, with both equal to the instantaneous
variance σ
2
s
. The mean return process contributes no variance or covariance, and the Var
t
[μ
t→t+h
]
and Cov
t
[μ
t→t+h
,M
t→t+h
] terms can be ignored.
With a lengthening of the horizon, the difference between quadratic variation and conditional
variance increases as the relative contribution of the expected return component grows. To get a
sense of the magnitude, I follow Andersen et al. (2010) and assume a detrended Ornstein-Uhlenbeck
process r
s
=−φp
s
+σdW
s
, where p
s
is the log price and φ is the mean reversion parameter. I
calibrate φ to annual market data and arrive at an estimate of 0.11. For each unit of quadratic
variation, this implies a conditional variance smaller by 10% at annual horizons and by 26% at
3-year horizons (e.g. if quadratic variation were 10%, conditional variance would be 9% at annual
horizons). These discrepancies affect the variance term in the beta denominator but what about
the covariance numerator term? Applying the same model to two returns, the market’s and the
portfolio’s, yields the following.
Cov
t
[r
i
t→t+h
,r
m
t→t+h
]≡ Cov
t
[μ
i
t→t+h
,μ
m
t→t+h
] + Cov
t
[μ
i
t→t+h
,M
m
t→t+h
]
+ Cov
t
[μ
m
t→t+h
,M
i
t→t+h
] + Cov
t
[M
i
t→t+h
,M
m
t→t+h
] (8)
19
The realized approach applied to covariances continues to ignore the conditional mean and sums
the product of higher-frequency returns. It estimates covariation by capturing only the last term
of equation (8) while ignoring the first three terms. Analogous to quadratic variation, covariation
sufficiently approximates conditional covariance when the covariance from the innovations, dW
i
s
and dW
m
s
, dominates. But unlike variances, this approximation may not be sufficiently accurate at
any horizon. To see this, picture a portfolio whose innovations are completely uncorrelated with
the market’s, such that the last term of equation (8) equals 0. Any covariance must therefore
involve the expected return process, regardless of the horizon. Indeed asset pricing models like the
CAPM are models about the conditional mean, which in continuous time predicts μ
i
=β
i
μ
m
. Using
high-frequency covariation estimates that capture only comovement in the innovation term may
therefore not help in tests of asset pricing models.
4.5 New Higher-Frequency Estimation Method
This motivates the paper’s development of a new covariance estimation method. Like realized
covariance, it uses higher-frequency returns, but it makes changes to enable the estimation of
conditional covariance rather than the innovations’ covariation. These changes are necessary for
capturing the contribution of the mean process. The method is non-parametric and thus requires no
model of returns, of return moments, or use of instruments. Specifically, the horizon h conditional
covariance formed using information up to time t for the period from t to t +h is estimated using
higher-frequency realized returns from t to t +h. Use of realized returns, however, means estimates
are not available ex-ante for forecasting purposes.
6
The new method includes higher-frequency contemporaneous covariances while allowing for the
inclusion of lead-lag covariances. This flexibility allows full capture of the conditional mean process,
which could respond with delays (the first three terms in equation (8) include the possibility of
lead-lags within t and t +h). Lead-lag covariances are scaled through a user-specified weighting
kernel. The kernel can be determined based on the user’s priors about the relative importance
6
Using realized beta estimates may incur the over-conditioning alpha bias of Boguth et al. (2011). I also estimate
unconditional and ex-ante rolling regressions and arrive at qualitatively similar term structure results. For the
high-frequency approach, the over-conditioning bias can be avoided if the conditional mean estimate uses only ex-ante
information. In the Appendix, I use a backward-looking conditional mean process and also arrive at qualitatively
similar results.
20
of certain lead-lags.
7
Existing methods, like the covariance terms in realized betas and Scholes-
Williams betas, are special cases in which the weighting kernel unit weights some covariances while
assigning zero weight to other ones. Taking the product of long-horizon returns without working with
higher-frequency ones is also a special case in which all lead-lags are weighted equally. Essentially,
higher-frequency returns provides users the option to down-weight certain regions of the return
product. This may be desirable and lead to more efficient estimates if these regions are more likely
to have no truly cross-correlated terms.
Figure 2 provides a visual comparison of the realized covariance and the new covariance
estimation methods. Looking at the realized approach in panels (a) and (b) reveals that the average
estimated covariance term structure will always be flat by construction since no lead-lag terms are
included. In contrast, panels (c) and (d) show my base case Gaussian weighting kernel where more
distant lead-lags are down-weighted. Based on the magnitude of lead-lag terms and how they’re
weighted, the covariance term structure can now be sloped to reflect the data.
The new higher-frequency estimation method makes the following two changes to realized
covariance.
1. Assume a conditional expected value for the horizon return, ¯ μ
t→t+h
, that proxies for
E
t
[r
t→t+h
] = E
t
[μ
t→t+h
]. The conditional mean process μ
s
can be stochastic and be left
unspecified but its expected value over horizon h must be proxied for. In my base case, I use
a rolling-average of returns centered around the period of interest.
8
2. Provide a weighting kernel for the higher-frequency lead-lag covariances. My base case
Gaussian kernel specifies weights for lead-lag l and horizon h of w(l,h) = e
−
(πl/h)
2
2
. The
concurrent term receives a weight of 1 while more distant terms receive declining weights. This
reflects a prior where return shocks gradually dissipate over time such that true correlations
are stronger at closer lead-lags. Standard return models, like an AR(1), have this property.
I implement no priors regarding the relative importance of leads versus lags so treat them
7
This new method differs from realized kernel methods (e.g. Barndorff-Nielsen et al. (2008, 2009, 2011)). Like
other realized methods, realized kernels also estimates quadratic variation or covariation despite accounting for some
lead-lags. As the number of higher-frequency returns goes to infinity, realized kernels’ optimal lead-lag bandwidth as
a proportion of all lead-lags shrinks toward zero. In contrast, the new method proposed here estimates variance or
covariance, weighting all lead-lags regardless of the number of higher-frequency observations.
8
I use the same window lengths as the rolling regression approach: 5, 10, 20, 30, and 40 years for monthly, quarterly,
annual, 3-year, and 10-year horizons respectively.
21
Figure 2: Realized Covariance vs. New Higher-Frequency Approach. Figure illus-
trates differences between the standard realized approach shown in panels (a)-(b) and the paper’s new
higher-frequency approach shown in panels (c)-(d). The following estimates conditional covariance:
ˆ
Cov
t
[r
i
t→t+h
,r
m
t→t+h
] = ˜ r
i
t→t+h
˜ r
m
t→t+h
, where ˜ r is the realized return with the conditional mean removed.
These estimates are shown as large boxes with numeric side labels counting the periods constituting the
h-period horizon. Each interval on an axis depicts one period, with market returns on the x-axis and
portfolio returns on the y-axis. Panels (a) and (c) depict a sequence of five 5-period horizons while panels (b)
and (d) depict a single 25-period horizon. The standard realized approach captures only higher-frequency
contemporaneous cross-products, represented by the sequence of diagonal black boxes. In contrast, the new
approach weights all lead-lags based on a specified weighting kernel, with darker shades representing larger
weights. I use a Gaussian kernel. The realized approach’s average covariance term structure (comparing
5-period versus 25-period horizons, with proper scaling) is flat by construction since the two horizons capture
the same black squares. This is not true for the new approach. More lead-lags exist for the 25-period horizon,
and the squares are also weighted differently. The average term structure will thus be upward-sloping if the
sign of the sum of these different weighted terms is positive.
(a) Realized Approach (Horizon = 5)
(b) Realized Approach (Horizon = 25)
22
(c) New Higher-Frequency Approach
(Horizon = 5)
(d) New Higher-Frequency Approach
(Horizon = 25)
23
symmetrically under the base case.
The following example illustrates the new method’s estimation of the time t conditional co-
variance of a 5-period horizon return,
ˆ
Cov
t
[r
i
t→t+5
,r
m
t→t+5
]. It may be helpful to think of a single
period as one month. Realized covariance estimates covariation as
ˆ
M
i
t→t+5
ˆ
M
m
t→t+5
=
P
5
τ=1
r
i
τ
r
m
τ
but as equation (8) shows, realized covariation does not equal conditional covariance. An unbi-
ased estimate of conditional covariance takes the product of the conditionally demeaned returns:
ˆ
Cov
t
[r
i
t→t+5
,r
m
t→t+5
] = ˜ r
i
t→t+5
˜ r
m
t→t+5
, where ˜ r
t→t+5
≡r
t→t+5
− ¯ μ
t→t+5
. But higher-frequency returns
can be used by noting that ˜ r
i
t→t+5
˜ r
m
t→t+5
= (
P
5
τ=1
˜ r
i
τ
)(
P
5
υ=1
˜ r
m
υ
), where ˜ r
i
τ
and ˜ r
m
υ
are demeaned
higher-frequency returns. I demean through an even split of ¯ μ
t→t+5
to get ˜ r
τ
=r
τ
−
1
5
¯ μ
τ
.
9
The
new estimation method simply weights these higher-frequency lead-lag terms with a specified kernel.
The result equals
ˆ
Cov
t
[r
i
t→t+5
,r
m
t→t+5
] =
P
5
τ=1
w(0, 5)˜ r
i
τ
˜ r
m
τ
+
P
4
τ=1
w(1, 5)(˜ r
i
τ+1
˜ r
m
τ
+ ˜ r
i
τ−1
˜ r
m
τ
) +
... +w(4, 5)(˜ r
i
τ+4
˜ r
m
τ
+ ˜ r
i
τ−4
˜ r
m
τ
), where under the Gaussian kernel w(0, 5) = 1, w(1, 5)≈ 0.82, ...
w(4, 5)≈ 0.04.
The standard realized method is a special case of the new higher-frequency method, with a
conditional mean assumption of ¯ μ
t→t+h
= 0 and only the lag-0 term receiving unit weights while
others are ignored. If these assumptions are appropriate, the realized approach successfully estimates
conditional covariance. For example, demeaning may be trivial when horizon h is sufficiently short.
However, including lead-lags can be important. The next section shows that lead-lag correlations
are often significant and can even dominate contemporaneous correlation. Assigning zero weights to
these terms may thus inaccurately capture conditional covariance. The Scholes-Williams beta is a
special case where only preselected lags are uniformly unit-weighted. This is traditionally used to
correct for microstructure noise, so included lags are ones where the user believes these effects to be
important. Other priors can be also be enforced through the weighting kernel. For example, if the
user does not believe the portfolio forecasts the market, all lower triangular lead-lags can have zero
weights.
To estimate betas, the new higher-frequency method can be used to estimate both the
9
The decomposition holds regardless of how ¯ μ
t→t+h
is allocated among the higher-frequency returns. Different
allocations can affect individual higher-frequency cross-terms though. Since the new method potentially weights these
cross-terms differently, it’s possible that the resulting estimate may be sensitive to how ¯ μ
t→t+h
is allocated. I choose
an even allocation because it is simplest.
24
covariance term in the numerator and the variance term in the denominator. Alphas directly follow
using equation (3). I add an useful option to place a lower limit, Min.Den, on the denominator
term. Relative over-weighting of negative autocorrelations may substantially shrink and even
turn the denominator negative. But this variance term should clearly be positive so having a
positive Min.Den enforces this. Larger limits prevent the denominator from being too small and
inadvertently blowing up the beta estimate. Under the base case, Min.Den = 0.3, which constrains
the long-horizon market variance to be at least 30% of the scaled short-horizon variance.
10
The
following summarizes the higher-frequency approach for estimating alphas and betas of horizon h
using information conditional at time t.
ˆ
β
HF
t→t+h
≡

P
h−1
l=−h+1
w(l,h)ˆ ρ
im
t,h,lag=l

ˆ σ
i
t,h,lag=0
ˆ σ
m
t,h,lag=0

Max

P
h−1
l=−h+1
w(l,h)ˆ ρ
m
t,lag=l
,Min.Den
(9)
ˆ α
HF
t→t+h
≡ ¯ μ
i
t→t+h
−
ˆ
β
HF
t→t+h
¯ μ
m
t→t+h
where
˜
r
i
τ
≡ r
i
τ
−
1
h
¯ μ
i
t→t+h
ˆ σ
i
t,h,lag=0
≡
v
u
u
t
h
X
τ=1
(˜ r
i2
τ
)
ˆ ρ
im
t,h,lag=l
≡
P
h−l
τ=1
(˜ r
i
τ+l
˜ r
m
τ
)
ˆ σ
i
t,h,lag=0
ˆ σ
m
t,h,lag=0
w(l,h) ≡ weighting kernel for lead-lagl
Each horizon h and time t requires a separate estimate. I use weekly returns as the higher-
frequency return for the monthly horizon and monthly returns for longer ones. As long as the
higher-frequency horizon isn’t too coarse, the specific choice has little impact. To see this, look at
Figure 2 panel (d) and note that given a specific kernel, the weight at specific locations of the
grid is largely unaffected by the grid’s granularity. For robustness, Appendix D tests a variety of
alternative specifications and finds qualitatively similar results.
10
Equivalently, this constrains to -0.7 the 1-lag autocorrelation coefficient of returns with horizon one-half that of
the long-horizon period (Variance-Ratio(h) = 1−ρ(
h
2
)). This limit does not usually bind. For comparison, Fama and
French (1988) find that 5-year returns have an unadjusted 1-lag autocorrelation parameter of -0.37 which translates to
a 10-year variance ratio of 0.63. This is more than twice my base limit.
25
5 The Presence of Auto and Lead-Lag Correlations
Persistent market-portfolio cross-correlations or market autocorrelations are the only way to
produce a sloped average beta term structure. This section empirically evaluates their presence.
I ignore daily returns because microstructure noise leading to daily or weekly lead-lags is already
well-known and does not play a significant role in the longer-horizon results I examine.
5.1 Significance of Individual Auto and Lead-Lag Correlations
Figure3showsstandardautoandcross-correlogramsusingallhistoricaldata, withexceedances
beyond the dotted lines indicating a 95% significance level. Significance is based on a single test, but
the figure shows results for multiple lead-lags. For every 20 lead-lags shown, therefore, we should
expect only one exceedance occurring by chance given independent tests. Left and right panes show
monthly and annual correlations, respectively. Panel (a) depicts excess-market autocorrelations that
are adjusted for finite-sample bias (Kendall (1954)). Although there is some evidence of significant
monthly correlations, no exceedances occur at the annual horizon. What can cause sloped beta term
structure are sequential correlations of nearby lags that push in the same direction. Otherwise, if
some correlations are negative while others are positive, they cancel and leave no net impact. This
directional consistency, however, is not apparent in market returns, suggesting a limited contribution
from the market denominator component of term structure effects (see equation (1)).
Panels (b)-(d) show market-portfolio correlations for size, value, and momentum, with positive
lags indicating market returns that occur before and thus anticipate portfolio ones. The finite-sample
Kendall bias becomes the Stambaugh (1999) bias when cross-correlations are estimated. I do not
adjust for this since the bias has a negligible impact due to the small regression coefficient of the
lead-lag residuals on the auto-regressive residuals (Stambaugh’s φ). Results here are interesting and
foreshadow beta term structure patterns in the next section. Monthly lag-0 correlations can be easily
converted into monthly betas (simply scale by
σ
i
month
σ
m
month
) while annual lag-0 correlations similarly reflect
annual betas. Ignoring market denominator effects, equation (1) shows that long-horizon betas
are simply the sum of higher-frequency lead-lag correlations. Visually summing these correlations
tells us the direction in which average longer-horizon betas slope. The same scale for monthly and
26
Figure 3: Auto and Lead-Lag Correlations at Individual Lead-Lags. Data from 1926-2015
used to form non-overlapping monthly (left panes) and annual returns (right panes). Panel (a) shows market
excess return autocorrelations at individual lead-lags, with adjustments for finite-sample bias using Kendall
’54. Panels (b)-(d) show market-portfolio lead-lag correlations. Dotted lines contain the 95% confidence
interval under the null of iid returns leading to zero correlation at all lags. This means ˆ ρ
l
is asymptotically
normal with mean zero and variance
1
T
.
(a) Market Autocorrelations
(b) Market-Size Cross-Correlations
27
(c) Market-Value Cross-Correlations
(d) Market-Momentum Cross-Correlations
28
annual results is maintained to facilitate this exercise.
Let’s begin with size in panel (b). Monthly exceedances abound, with the 1-month lagged
correlation being the most striking. Gilbert et al. (2014) argue that small stock prices incorporate
information more slowly because they are likely less-understood and more opaque. It thus takes time
for market changes to catch up to small stocks, resulting in rising betas as the horizon lengthens
and fundamental co-movement is eventually reflected more fully. Interestingly, this pattern reverses
itself as the number of lags increase, with negative correlations becoming more prevalent and having
greater significance. Annual results make this pattern apparent and show a sequence of 5 negative
correlated lags. The second is especially significant and large enough to counteract the positive
lag-0 correlation. This anticipates the next section when we see SMB’s historical-average beta term
structure peak at the annual horizon and then decline to become eventually negative at multi-year
horizons. Gradual information diffusion cannot explain this finding since negative correlations reflect
reversal and not return continuation. Instead, over-reaction to new information is more consistent.
The results for value in panel (c) are interesting in a different way. The lag-0 monthly correlation
is significantly positive, but the magnitude shrinks by half and becomes insignificant for the lag-0
annual correlation. Indeed, at the annual horizon, non-contemporaneous lead-lag correlations have
much larger magnitudes. But these correlations have little significance and mostly cancel. This
suggests that estimation noise will dominate at the annual horizon, making it difficult to reject the
flat beta term structure hypothesis. Momentum’s results in panel (d) are like size’s but in reverse.
At the monthly horizon, the lag-0 correlation is very negative, but significant exceedances at further
lead-lags go the other way and tend to be positive. By the annual horizon, the lag-0 correlation
loses its significance and is in fact, dwarfed by highly positive lag-1 and lag-2 values. This points to
a term structure pattern that’s opposite to size, where monthly betas tend to be negative but then
slope upward to become positive at longer horizons.
These correlations, though substantial, are not necessarily tradeable in a meaningful way.
This suggests that they may not be easily arbitraged and could therefore persist in the future.
Transaction costs can be overwhelming, especially when trading small or less liquid stocks that are
in the size (SMB) and momentum (UMD) long-short portfolios. Frazzini et al. (2015) estimate for
these portfolios one-way one-leg costs of 23 bps and 21 bps, respectively. But their estimates are
29
substantially lower than other studies, possibly because they use more recent data. With an older
sample, Hasbrouck (2009) estimates 60 bps median one-way costs and 110 bps average costs. How
do these costs compare to simple strategies that try to take advantage of the significant lead-lag
correlation in Figure 3? Utilizing the strong monthly correlation between SMB and lagged MktRF,
a strategy that longs SMB if lagged MktRF returns are positive and shorts it otherwise yields 45
bps/month more than just being long SMB all the time. However, this strategy requires reversing
SMB in 45% of the months, leading to transaction costs of at least 41 bps/month (= 23 bps/trade
x 4 trades/reversal x 45% reversal/month) under Frazzini et al. (2015) and several times that
under Hasbrouck (2009). The analogous UMD strategy taking advantage of its strong 2-month
lagged correlation actually generates no benefit, even before costs. For strategies that use annual
correlations, SMB’s delivers an extra 7 bps/year while UMD’s generates 137 bps/year. The latter
strategy could cover Frazzini et al. (2015) costs of 37 bps/year (=21 bps/trade x 4 trades/reversal
x 44% reversal/year) but not the higher Hasbrouck (2009) costs. An arbitrageur could of course
implement cost mitigation controls and other more sophisticated strategies but overcoming these
sizable trading costs likely remains challenging.
5.2 Significance of Cumulative Auto and Lead-Lag Correlations
Beta term structure effects require adjacent lead-lag correlations to push in the same direction
consistently. While Figure 3 looks at individual lags, Table 1 sums them cumulatively to capture
better when correlations of neighboring lead-lags agree and are more likely to have an impact. Leads
and lags of the same distance are summed since their impact on the beta term structure occur
together. Panel (a) shows in-sample results that use all historical data, like the Figure 3 results
we already looked at. Market autocorrelations are positive at monthly horizons, in some cases
significantly so, but they become negative by Year-2. Cumulative autocorrelations peak at -25% by
Year-5, reflecting the mean-reversion tendencies of 3 to 5-year returns found in Fama and French
(1988). The market denominator impact is thus to pull betas toward zero at monthly horizons (when
the denominator becomes larger) but magnifies them at multi-year horizons (when the denominator
shrinks). These results, however, have little to no statistical significance.
For size, we see cumulative positive correlations that are significant up to a lag of 3-months, but
30
Table 1: Auto and Lead-Lag Correlations at Cumulative Lead-Lags. Data from 1926-2015 used to form monthly and annual
non-overlapping returns. Panel (a) assesses the extent of in-sample predictability. Using all historical data, excess-market autocorrelations and
market-portfolio lead-lag correlations are calculated. Correlations of the same distance are combined and closer correlations are summed together
P
l=lag
l=−lag,l6=0
ρ
l
. This indicates the total directional impact on longer horizon returns of all correlations up to the indicated lag. A significant difference
from zero at a two-sided 95% level is boldfaced and assessed using tstat
lag
=
P
l=lag
l=−lag,l6=0
ρ
l
√
2lag/T
. Standard errors are calculated under the null of iid returns
leading to zero correlation at all lags. Panel (b) assesses the extent of out-of-sample predictability by showing the OOS R
2
lag
= 1−
MSE
lag,alternative
MSE
lag,null
,
with MSE
lag,forecast
=
P
t=T
t=11
(
P
l=lag
l=−lag,l6=0
ρ
t,l,forecast
−
P
l=lag
l=−lag,l6=0
ρ
t,l,actual
)
2
. The first 30 periods form the training period. For monthly and
annual correlations, ρ
t,l,actual
is calculated using the next 12-month and 10-year period, respectively. The null predicts zero correlation at all lags while
the alternative predicts they equal their historical value. Predictability is assessed on a one-step-ahead rolling basis with continuously updated forecasts.
Boldfaced R
2
values are positive and indicate historical out-performance by the alternative over the null. Market autocorrelations are adjusted for
finite-sample bias using Kendall ’54.
(a) In-Sample Cumulative Lead-Lag Correlations (%)
Months Years
(Cum Lags and Leads) (Cum Lags and Leads)
1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9
Market Auto 11 10 2 4 12 10 12 16 21 22 22 8 -6 -9 -16 -25 -20 -11 -4 1
Market-Size Cross 25 26 23 7 -5 -10 -12 -16 -4 5 6 -7 -30 -49 -57 -67 -80 -88 -84 -95
Market-Value Cross 12 10 -3 -8 -5 -16 -15 -15 -0 1 -5 -0 -14 2 -21 -34 -38 -21 -28 11
Market-Momentum Cross -17 -4 17 23 25 28 35 41 28 29 33 17 49 35 15 27 18 30 41 51
(b) Out-Of-Sample R
2
of Cumulative Lead-Lag Correlations (%)
Months Years
(Cum Lags and Leads) (Cum Lags and Leads)
1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9
Market Auto -11 -9 -1 -0 1 5 7 10 10 17 15 -46 -34 -28 -16 -20 -23 -20 -18 -14
Market-Size Cross -13 1 2 -1 -11 -0 -5 -19 -85 -85 -72 -18 9 -11 -80 -617 -214 -512 -313 -117
Market-Value Cross -25 6 8 -3 -7 0 -3 -12 -46 -59 -78 -13 -17 -48 -32 -557 -274 -326 -320 -106
Market-Momentum Cross 3 -4 -20 -8 -8 -1 -12 -20 -51 -97 -101 -30 -78 -23 -276 -367 -442 -466 -224 -81
31
the sign then reverses and becomes significantly negative by Year-5. Value shows little significance
and only small magnitudes, implying a mostly flat term structure. Momentum sees mostly positive
cumulative lead-lag correlations with significance beginning at Month-3 and also showing up in
Year-2. Like size, momentum has large annual lagged correlations but where it differs, as Figure 3
shows, is that its leads offset much of these effects. Size’s leads are relatively smaller in magnitude,
and thus the lags dominate. This means that multi-year term structure effects will be more muted
for momentum, and we see this in lower cumulative annual correlations that are less significant.
Unlike in-sample panel (a) results, Table 1 panel (b) assesses how investors could have
anticipated the direction of future cumulative correlations in real time. Investors have access only to
backward-looking information, so the results avoid look-ahead bias by using a pseudo out-of-sample
approach. In each period, I form two forecasts, with the null assuming zero cumulative correlations
at all lags and the alternative assuming correlations that reflect the history up to that point. The
null hypothesizes iid returns while the alternative assumes past lead-lag patterns are persistent
and repeat themselves. I compare these two forecasts to actual realized future led-lags using next
12-month returns for monthly results and next 10-year returns for annual results. Squared forecast
errors are then calculated, and the exercise is repeated on a one-step-ahead basis for the next
month or year. Correlation forecasts are updated recursively at every step using the latest available
information given an expanding historical window. When squared forecast errors are calculated
for all periods, they are averaged to form the mean-squared-error (MSE) for each forecast. Panel
(b) shows the OOS R
2
, where positive numbers mean the alternative hypothesis has outperformed
while negative ones reflect null out-performance. Out-of-sample predictability is rare compared to
in-sample predictability, and Goyal and Welch (2008) show that nearly all market mean-return
predictors lack out-of-sample performance. Nevertheless, panel (b) shows that positive OOS R
2
occur at the monthly horizon for market autocorrelations and at the annual horizon for market-SMB
correlations at two years. This indicates strong persistence in some lead-lag patterns such that
they are possibly actionable in real-time. For SMB especially, investors could have anticipated long-
horizon betas that deviate from short-horizon ones and formed adjusted conditional expectations
accordingly.
32
6 The Shape of Alpha and Beta Average Term Structures
This section presents the paper’s main empirical findings and shows the average term structure
shape for size, value, and momentum. For each portfolio and for alpha and beta separately, I
compare the three methods described in the previous section: unconditional, rolling regression, and
the new higher-frequency approaches. Table 2 displays coefficient estimates and corresponding
Newey-West ’94 standard errors.
6.1 Discussion of Size, Value, and Momentum Results
Size’s betas increase and peak at the quarterly or annual horizon, and then decline until they’re
negative by the 10-year horizon. Beta term structures impact alpha term structures directly. They
are positive at the monthly horizon and significantly so under both conditional approaches. As the
horizon lengthens, alpha changes follow beta changes but with the opposite sign, falling as betas rise
and then rising as betas fall. Conditional alphas shrink and lose their significance at the quarterly
and annual horizons. Handa et al. (1989) and Handa et al. (1993) find that the size anomaly
disappears under annual instead of monthly returns. They examine only unconditional results over
a more limited time period, however, and I provide updated, confirming evidence for conditional
alphas as well. In addition and more strikingly, the pattern reverses at multi-year horizons, with
alphas becoming highly significant by 10-years with magnitudes roughly triple that of monthly ones.
To my knowledge, this is the first time this pattern has been documented. Recent work (e.g. Hou
and Van Dijk (2014), Asness et al. (2015)) has shown ways to resurrect the size effect given its poor
performance in the past three decades. It seems that long-horizon returns are another place where
the size effect is alive and well.
For each estimation approach and for alphas and betas separately, Table 2 also shows the
statistical deviation from a flat term structure. The last column’s number in parenthesis is the
p-value under the null hypothesis that the coefficients are equal across all horizons. I use a Wald
test that accounts for all horizon coefficients’ estimates, standard errors, and correlations with
each other. All estimates use the same underlying historical data so incorporating correlations is
crucial. Each pairwise correlation is obtained from an exactly-identified GMM with a Newey-West
33
Table 2: Alpha and Beta Term Structures. Data from 1926 - 2015 used to form non-overlapping
monthly, quarterly, annual returns and annual-overlapping 3-year and 10-year returns. Tables show alpha and
beta term structures using average higher-frequency, average rolling regression, and unconditional approaches.
Alphas are annualized percentages. These methods are described in Section 4. T-statistics shown in brackets
use Newey-West ’94 standard errors. Boldfaced coefficients denote significant differences from zero at the
two-sided 95% level. Pval shows Wald Test results under the joint hypothesis of equal coefficients under all 5
horizons (a small value supports sloped term structures).
(a) Size (SMB)
Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Higher 1.9 1.3 0.5 1.2 3.2 0.09 0.19 0.31 0.19 -0.12
Frequency [6.4] [2.3] [0.8] [2.3] [4.3] (0.03) [4.1] [4.8] [5.1] [2.5] [-0.9] (0.00)
Rolling 1.3 1.3 0.0 1.3 4.7 0.18 0.22 0.26 0.20 -0.26
Regression [0.6] [1.9] [0.0] [2.3] [1.8] (0.33) [2.8] [8.6] [5.6] [0.5] [-1.5] (0.00)
Unconditional 0.9 0.4 0.7 1.2 3.9 0.19 0.27 0.23 0.16 -0.24
[0.9] [0.4] [0.6] [0.5] [3.1] (0.34) [6.0] [4.6] [3.6] [1.3] [-1.7] (0.00)
(b) Value (HML)
Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Higher 3.7 4.0 3.5 4.0 4.5 -0.01 0.02 0.02 -0.01 -0.03
Frequency [12.0] [7.8] [6.1] [7.7] [5.8] (0.59) [-0.3] [0.2] [0.2] [-0.1] [-0.3] (0.97)
Rolling 3.8 4.2 3.8 4.4 5.1 -0.00 -0.02 0.02 -0.01 -0.06
Regression [4.5] [8.4] [5.3] [6.3] [8.1] (0.03) [-0.0] [-0.3] [0.1] [-0.0] [-0.2] (0.61)
Unconditional 3.1 3.1 3.5 3.8 4.9 0.12 0.13 0.07 0.04 -0.07
[2.2] [2.2] [2.8] [2.5] [7.9] (0.86) [1.8] [1.1] [0.8] [0.4] [-0.5] (0.54)
(c) Momentum (UMD)
Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Higher 6.2 6.4 6.8 6.7 5.3 -0.06 -0.05 -0.11 -0.02 0.24
Frequency [12.7] [7.9] [7.9] [7.4] [3.5] (0.64) [-2.7] [-0.8] [-1.2] [-0.3] [2.7] (0.00)
Rolling 6.2 7.4 7.8 6.0 4.1 -0.12 -0.21 -0.09 0.06 0.34
Regression [2.8] [8.4] [5.1] [3.9] [0.5] (1.00) [-0.8] [-3.8] [-1.9] [1.1] [0.5] (0.00)
Unconditional 8.6 9.1 7.4 6.1 5.1 -0.32 -0.41 -0.13 0.07 0.26
[5.4] [5.6] [3.7] [2.8] [1.8] (0.96) [-2.4] [-2.6] [-1.5] [1.0] [1.5] (0.05)
‘94 kernel robust to heteroskedasticity and autocorrelation. I find significant rejection of a flat
beta term structure under all three approaches and a significant rejection of a flat alpha term
34
structure under the higher-frequency approach. Alpha significance is more difficult because standard
errors include variation from both betas and expected returns. In Appendix C, I make pairwise
comparisons and find larger t-statistics for alpha increases at multi-year horizons compared to alpha
decreases at multi-month horizons. Prior works’ finding that the size effect becomes less anomalous
at multi-month horizons is thus completely reversed by the opposite effect at multi-year horizons.
Size becomes more anomalous at these longer horizons, and these differences are significant.
From equation (4), the mean conditional alpha term structure is determined not just by beta
effects but also by the covariance between conditional betas and the market risk-premium. Figure
4 plots the covariance’s term structure on the same scale as Figure 1. Under both conditional
approaches, the level is near zero and the term structure is essentially flat. This term thus plays
no major role in either the level or shape of the alpha term structure, and it must be beta effects
that drive alpha results. The result is consistent with Lewellen and Nagel (2006)‘s and Ang and
Kristensen (2012)’s findings that this covariance term is simply too small to explain these anomalies’
alpha. They focus on only monthly horizons so the results here extent their findings to much longer
horizons.
Turning to value, Table 2 shows consistent results among all three estimation approaches.
We see insignificant betas and highly significant alphas across all horizons examined. Value’s beta
is indistinguishable from zero regardless, translating to significant alphas at all horizons. Alpha’s
flat term structure reflects beta’s flat term structure since the covariance term between beta and
the market-premium plays a negligible role (Figure 4). The null hypothesis that the value term
structures are flat cannot be rejected under any estimation approach. These patterns reflect value’s
lack of significant lead-lag correlations that consistently push in the same direction (Table 1), which
is necessary for sloped average term structures.
These results contrast with Cohen et al. (2009) who find that value stocks have rising betas
and declining alphas over multi-year horizons. Our approaches to value investing, however, are
different. Cohen et al. (2009) hold a fixed basket of stocks that start out as value but may drift
over time and not be considered value after a few years. In contrast, I hold a value strategy that
automatically rebalances toward new value stocks each year and thus avoids style drift. Cohen et al.
(2009) calculate cashflow betas whereas my betas reflect period-end total returns from all sources,
35
Figure 4: Term Structure of Covariance Term between Time-Varying Betas and the
Market Risk Premium. Data from 1926 - 2015 used to form non-overlapping monthly, quarterly,
annual returns and annual-overlapping 3-year and 10-year returns. Time-varying conditional mean and beta
estimation methods described in Section 4. Figures show the unconditional covariance between time-varying
betas and the market risk premium (i.e. the market conditional mean return, which I estimate with a
rolling-window average).
36
including dividends assumed to be reinvested along the way. Their results are thus more relevant for
holding an initially-cheap asset for a long time whereas mine are more applicable for implementing
a consistent value stock investing strategy. Since the portfolios differ, the results may differ too.
Nevertheless, one central point of agreement is that alpha and beta estimates can vary depending
on the investment horizon.
Momentum results tend to be the reverse of size’s. Monthly betas are significantly negative
but increase and turn positive at multi-year horizons. Alphas are strongly significant at short and
medium term horizons but significance weaken by 10-years. Increasing betas drive down alphas but
widening standard errors also contribute. Under all three estimation methods, we can reject the
hypothesis of a flat beta term structure but not a flat alpha one. Like size and value, Figure 4
also shows that the term structure of the covariance term between momentum beta and the market
risk-premium is mostly flat. There is a slight ~1% elevation at shorter horizons, suggesting some
correlation without which short-term alphas would be even higher. Positive correlations increase
the market’s ability to explain momentum’s excess returns and thus lower the anomalous alpha.
Since the correlation diminishes at longer horizons, this also dampens the degree to which the alpha
term structure slopes downward.
The paper’s momentum results are consistent with Daniel and Moskowitz (2015). They find
that momentum crashes tend to follow large market declines and coincide with subsequent market
rebounds. This pattern is partly forecastable, and proper risk management can lead to much better
momentum performance. Negative momentum returns in a rising market environment contribute
to negative contemporaneous short-term betas. When this follows market declines, they add to
positive lagged correlations. These patterns result in an upward-sloping momentum beta term
structure, which is exactly what I find. Longer horizons pick up lead-lag patterns short-horizons
avoid, including the tendency for momentum to crash after a market crash. I confirm this tendency
and find that the bottom quartile of market annual returns, when loses exceed 10%, drives entirely
the strong 1-year and 2-year lagged correlations between the market and momentum. Lagged
positive correlations thus raise beta and lower alpha. The momentum strategy I study assumes
automatic rebalancing toward momentum regardless of market conditions. Daniel and Moskowitz
(2015) as well as Barroso and Santa-Clara (2015) argue that this isn’t optimal when crashes may be
37
anticipated. But this paper’s purpose is not to develop optimal trading strategies but rather to
document the potential for risk-return expectations to vary according to investment horizon. The
forecastability of momentum crashes thus implies that investors can form conditional expectations
about these lead-lag correlations and do so on an ex-ante basis.
Overall, all three estimation approaches lead to similar conclusions regarding average term
structure shapes. The two time-varying approaches have similar average alpha and beta coefficients,
but the Newey-West standard errors can differ. Table 2 show that the new higher-frequency
approach has much smaller standard errors at multi-year horizons. By construction, the rolling
regression approach has more persistent estimates over time (see Appendix B for summary statistics),
and this high autocorrelation makes precise estimation of its mean difficult. In contrast, the higher-
frequency approach uses only data within each period, producing independent observations whose
mean can be more confidently estimated. Take the longest 10-year horizon with the given 89-year
dataset. The rolling regression approach has effectively 2 independent 40-year observations (base
case window for 10-year horizons) while the higher-frequency approach has nearly 9. Long horizons
thus limit the rolling regression approach’s usefulness since the approach requires sampling across
adjacent periods. Thankfully, the new higher-frequency approach samples within periods instead,
making it an especially useful method for long-horizon alpha and beta estimation.
Finally, Appendix D explores a variety of alternative specifications for the unconditional, rolling
regression, and higher-frequency estimation approaches. This includes using simple instead of log
returns, using annual instead of monthly returns, using only non-overlapping data, varying the
conditional mean estimation method, varying the window size under the rolling regression approach,
varying the kernel for the higher-frequency approach, and many other scenarios. The main results
are robust to these alternatives, and the basic shape of the alpha and beta term structures do not
change. In addition, Appendix E analyzes alternative ways of forming SMB, HML, and UMD that
depart from the standard Fama-French approach. Specifically, I explore varying the reformation
frequency (the time it takes for stocks in a particular portfolio to be refreshed) and the rebalancing
frequency (the time it takes for different portfolios that constitute the long-short factors to be
rebalanced to equal weights). Lengthening the reformation period lowers turnover but permits more
style drift. Alpha levels consequently decline but alpha and beta term structure shapes remain
38
qualitatively similar. Changing the rebalancing frequency from monthly to annually results in only
minor differences. Lastly, both Appendix D and E also analyze the portfolios’ long and short legs
separately. I find that small caps are the primary drivers of SMB term structure dynamics while
winners and losers both contribute to UMD’s dynamics although loser effects are generally stronger.
6.2 Economic Drivers and Implications for Investors
Section 3 outlines the mechanical drivers of beta term structure shapes, but what are the
economic sources behind the especially large size and momentum beta changes that occur over
multi-year horizons? I hypothesize that slow information diffusion causes a delayed portfolio price
response to the broader market, thus inducing positive cross-correlations and increasing betas with
increasing horizons. This effect would need years to resolve and should be stronger in times of
greater initial market illiquidity when frictions inhibit contemporaneous price movement. Gilbert
et al. (2014) hypothesize a similar mechanism for changes in monthly betas while Chordia et al.
(2011) do indeed link illiquidity with portfolio lead-lag effects.
Table 3 addresses this hypothesis. It shows OLS coefficients and Newey-West ’94 standard
errors of annual rolling time-series regressions. The dependent variable is the future 10-year beta
minus the 1-year beta of the same starting year, with betas estimated using the new higher-frequency
method.
ˆ
β
HF
t→t+10yr
−
ˆ
β
HF
t→t+1yr
=γ
0
+γ
1
X
t→t+1yr
+
t→t+1yr
(10)
Models 1 and 2 use as regressors the annual versions of market illiquidity proxies: the Amihud
illiquidity index and the return of the short-term reversal factor (suggested by Nagel (2012)). The
two have a 51% correlation. Table 3 shows they are statistically significant and have comparable
magnitudes that positively contribute to both size and momentum beta increases as the horizon
lengthens. The regressors are standardized to have mean zero and standard deviation one, so a one
standard deviation increase in market illiquidity during the year of the annual beta is associated
with a subsequent 0.1-0.2 increase for size and a 0.2 increase for momentum in the 10-year beta fully
39
realized nine years later. These changes are large in comparison to average changes over the full
sample. SMB betas decline on average by around 0.4 while UMD betas increase by around 0.3.
11
Daniel and Moskowitz (2015) document momentum crashes that tend to occur when the
market and especially past losers rebound sharply following a prolonged market decline. Models 3
and 4 connect beta changes with market return levels. Model 3 contains a single dummy variable
indicating when the market return falls in the bottom quartile of all historical market returns (below
11
In results not shown, I find that small caps and winners in the long legs contribute most to the regression results.
Small caps are much more illiquid than large caps while winners and losers have comparable liquidity levels.
Table 3: Drivers of Multi-year Horizon Beta Changes Data from 1927-2015. Conditional
betas estimates use the new higher-frequency method. Each model is a variant of the following regression:
ˆ
β
HF
t→t+10yr
−
ˆ
β
HF
t→t+1yr
=γ
0
+γ
1
X
t→t+1yr
+
t→t+1yr
, where X are the regressors. Standardized regressors
(with mean 0 and standard deviation 1) include the average across all months of the average across all stocks
of the square root of the monthly version of the Amihud illiquidity index (
q
|return|
price×volume
), and the sum of
monthly short-term reversal returns as calculated by Ken French. Non-standardized regressors are the market
excess return (in %), the market excess return interacted with a dummy indicating a negative excess market
return over the past two years, the above plus an additional interaction with a dummy indicating a positive
current market excess return, and a dummy indicating a current market excess return in the bottom quintile
of annual returns (less than -9%). Panel (a) applies to size (SMB) beta differences while panel (b) applies to
momentum (UMD) differences. Newey-West ’94 standard errors are in parentheses. Boldfaced coefficients
denote significant differences from zero at the two-sided 95% level.
(a) Size (SMB) 10yr-1yr Beta Difference
Model 1 Model 2 Model 3 Model 4 Model 5 Model 6
Intercept −0.44 −0.44 −0.45 −0.52 −0.45 −0.49
(0.12) (0.12) (0.15) (0.12) (0.14) (0.12)
Amihud.Illiq 0.19 0.18 0.16
(0.05) (0.07) (0.08)
ShrtTrm.Reversal 0.11 0.03 0.02
(0.05) (0.07) (0.11)
Mkt.WorstQuint 0.04 0.04
(0.16) (0.16)
Mkt −0.00 −0.00
(0.00) (0.00)
Mkt*PstDwn −0.00 −0.00
(0.00) (0.00)
Mkt*PstDwn*CrntUp 0.02 0.01
(0.01) (0.01)
Adj. R
2
0.07 0.01 -0.01 0.01 0.04 0.04
Num. obs. 80 80 80 78 80 78
40
(b) Momentum (UMD) 10yr-1yr Beta Difference
Model 1 Model 2 Model 3 Model 4 Model 5 Model 6
Intercept 0.28 0.28 0.16 0.25 0.14 0.32
(0.10) (0.11) (0.13) (0.12) (0.09) (0.10)
Amihud.Illiq 0.23 0.09 0.09
(0.08) (0.10) (0.12)
ShrtTrm.Reversal 0.24 0.25 0.19
(0.05) (0.08) (0.06)
Mkt.WorstQuint 0.58 0.69
(0.20) (0.21)
Mkt −0.01 −0.01
(0.00) (0.00)
Mkt*PstDwn −0.02 −0.01
(0.01) (0.01)
Mkt*PstDwn*CrntUp 0.04 0.01
(0.01) (0.02)
Adj. R
2
0.08 0.09 0.08 0.09 0.22 0.14
Num. obs. 80 80 80 78 80 78
-9% annually). This variable is insignificant for size but highly significant for momentum. Indeed,
adding it shaves off enough of the intercept that momentum beta changes are no longer significant.
Model 4 borrows three market-related regressors from Daniel and Moskowitz (2015). In Table 3, I
find that both SMB and UMD long-horizon betas do tend to subsequently increase when markets
rebound after a two-year decline. A positive 10% market return during these times is associated
with a subsequent increase in 10-year horizon betas of 0.2 for SMB and 0.1 for UMD, an effect
which sums across all three terms.
Models 5 and 6 combine previous regressors to identify whether some are subsumed by others.
For size, Model 1 containing only the Amihud illiquidity index has the highest adjusted R
2
of 7%.
For momentum, Model 5 containing the illiquidity proxies and a market bottom quintile dummy
is the best with an adjusted R
2
of 22%. The R
2
’s are notable given regressors that use only data
from one year to forecast beta differences that span ten years. Overall, periods of illiquidity add to
both size and momentum beta increases, consistent with the the hypothesis that illiquidity slows
information diffusion and induces positive cross-correlations between the market and the portfolios.
41
Controlling for illiquidity, momentum’s upward slope is also related to poor market performance
while size’s is not. These findings explain instances of upward sloping beta term structures but,
however, cannot explain why size’s beta term structure slopes downward on average.
Overall, how should the findings in this paper inform investors seeking to invest in size, value,
or momentum or exposed to SMB, HML, or UMD factors? CAPM alpha and beta term structures
are not necessarily flat so optimal holdings should be horizon dependent. Size is risky at short-term
horizons but a market hedge at multi-year horizons, suggesting it would be more valuable to long-
term investors. Momentum has the opposite term structure shape, implying greater attractiveness
to short-term investors. Using a standard mean-variance setup to determine optimal portfolios for
investors of different horizons, Appendix F indeed finds large portfolio differences that are consistent
with these intuitions. This predicted behavior, however, finds only mixed support in the literature
on actual institutional holdings. Long-term investors do tend to avoid momentum, but they also
avoid small stocks (e.g. Lewellen (2011), Cella et al. (2013), Cremers and Pareek (2015)). The
literature has yet to establish whether institutional investors take advantage of (e.g. Yan and Zhang
(2009)), are indifferent to (e.g. Lewellen (2011)), or contribute to (Edelen et al. (2016)) existing
anomalies. The extent to which these behaviors vary across institutional investment horizons is a
topic for future research. Appendix F also illustrates the misvaluation of long-horizon cashflows
if short-horizon betas are improperly applied. Corporate investing and financing decisions can be
heavily impacted, with valuation deviations that range from -20% to +120% given sufficiently long
horizons and heavy loadings on size or momentum.
7 Stationarity of Conditional Alphas and Betas
The paper has so far focused exclusively on the average term structure across all time periods
andfoundsometobesignificantlysloped. Thiscanonlybeinducedbypersistentlead-lagcorrelations,
and I’ve only investigated this first driver of term structure effects so far. In this section, I cover the
second driver, namely the mean-reverting nature of conditional alphas and betas. Stationarity implies
mean-reversion after stochastic shocks, such that high alphas or betas tend to be followed by lower
ones. In the absence of lead-lag correlations, the long-horizon conditional beta approximates the
42
average conditional short-horizon beta over the horizon (see equation (2)). This means abnormally
high short-horizon betas at a particular time tend to occur along with downward-sloping conditional
term structures. Given the difficulty in estimating conditional moments precisely, however, it’s hard
to demonstrate that the term structure at a particular time is sloped in a statistically significant
way. As seen in the previous section, long-horizon standard errors are already large for average
conditional alphas and betas. Obtaining significant differences across horizons for a term structure
at a particular point in time is even more difficult. This is because we at least have multiple
observations to estimate the average but essentially only a single observation to estimate each
instance. The resulting standard errors are simply too large for meaningful inference.
Iovercomethisdifficultybyfocusingonthedemonstrationofalphaandbetastationarity. Sloped
conditional term structures are then a direct implication, thus avoiding the need to demonstrate
this explicitly for individual cases at different time periods. I conduct Augmented Dickey-Fuller
tests to assess the null that alphas or betas have unit-roots such that rejection of the null implies
stationarity. Table 4 shows the resulting test statistics under both conditional estimation methods
and under alphas and betas estimated using monthly and quarterly returns. The rolling regression
approach’s base case window length has 60 and 40 observations, respectively. These highly persistent
estimates have near unit roots by construction, making rejection of the unit root null difficult even
if the true underlying process is stationary. Therefore, I explore an alternative specification where
the window length is half that of the base case. This reduces the mechanical persistence of the
estimates and thus increases the power of the test.
Under the new higher-frequency method, significant rejection of unit-roots occurs in all cases.
Under the rolling regression approach, they occur in most cases, with HML betas being the main
exception. As expected, halving the window length leads to stronger results, but the findings are
broadly consistent with base case windows. Consequently, stationary alphas and betas imply sloped
conditional term structures when these moments receive an especially large shock or when they are
estimated with especially large errors. Even in the absence of lead-lag correlations, the conventional
practice of using conditional monthly alphas and betas for long-term investors is thus incorrect.
In the absence of these correlations, long-horizon conditional moments are better approximated
by unconditional rather than conditional short-horizon moments. The long-term investor is better
43
Table 4: Augmented Dickey-Fuller Statistics to Test for Unit Roots In Time-Varying
Alphas and Betas. Data from 1926 - 2015 used to form non-overlapping monthly and quarterly returns
used to estimate time-varying alphas and betas (methods described in Section 4). Alphas are annualized
percentages. Rolling (Half-Window) uses 2.5 and 5 year windows to estimate monthly and quarterly moments,
which are half that of the base case Rolling (Full-Window) specification. The table shows Augmented
Dickey-Fuller test statistics using intercept but no trend and with BIC-specified lags. Boldfaced values denote
a significant rejection of the unit root null at the 95% level.
Alpha Beta
Mn Qt Mn Qt
Size Higher Frequency -6.0 -9.3 -11.6 -12.1
(SMB) Rolling (Half-Window) -3.6 -4.9 -3.6 -4.2
Rolling (Full-Window) -2.5 -4.0 -2.7 -4.5
Value Higher Frequency -5.2 -8.7 -6.5 -9.6
(HML) Rolling (Half-Window) -4.9 -3.7 -2.3 -1.9
Rolling (Full-Window) -4.2 -3.2 -1.6 -1.6
Momentum Higher Frequency -6.7 -6.7 -12.1 -9.5
(UMD) Rolling (Half-Window) -4.7 -4.3 -3.5 -3.5
Rolling (Full-Window) -3.6 -3.4 -2.6 -3.0
served with regression estimates that use all historic data rather than the most recent rolling
window. Using recent data estimates conditional results that lack persistence while using all data
better captures the long-horizon after conditional information gradually dissipates. Therefore, even
if the evidence presented in the previous section for lead-lag correlations and for sloped average
term structures is unpersuasive, sloped conditional term structures still occur because of stationary
conditional alphas and betas.
8 Conclusion
This paper examines the unconditional and conditional term structures of CAPM alphas and
betas for long-short portfolios sorted on size, value, and momentum. The literature traditionally uses
monthly returns to estimate alphas and betas even though they are inappropriate for long-horizon
investors or for discounting long-term cashflows. Many investors have horizons that span years and
are more interested in long-horizon moments. In addition, firms’ investing or financing decisions
44
should use discount rates matched to cashflows and thus require betas of the right horizon. When
appropriate, alternative return periods should be considered, and monthly returns should not be
relied on exclusively.
Long-horizon betas can deviate from short-horizon ones for two reasons. First, portfolios
exhibit lead-lag correlations that are significant and persistent across time. Long-horizon betas
can be dominated by these correlations rather than simply reflect short-term contemporaneous
comovement. I find this for size, which exhibits strongly negative annual lead-lag correlations with
the market. This leads to size’s beta sign reversal such that size turns from a risk into a hedge
at multi-year horizons. Value has lead-lag effects that are weak and largely cancel, resulting in a
mostly flat beta term structure. Momentum is the opposite of size. The second source of sloped
beta term structures is stationary and mean-reverting short-horizon conditional betas. Abnormally
large short-horizon betas tend to be followed by smaller ones such that a period-specific conditional
term structure can be sloped even in the absence of lead-lag correlations.
These beta term structure effects directly impact alphas. A higher beta that explains more
of the portfolio’s excess return will thus diminish unexplained alpha. The other possible alpha
driver, the covariance term between conditional beta and the market risk-premium, shows little
term structure dynamics. Therefore, size’s downward-sloping beta term structure translates into
an upward-sloping alpha term structure. Momentum’s is the opposite while value’s is flat. By
the 10-year horizon, all three portfolios have comparable alphas. These results have important
implications for investor portfolio formation and for valuation of corporate financing or investing
decisions. The horizon matters.
45
A TheTermStructuresShapeofExpectedReturnsandRiskPre-
miums
Since the paper investigates the term structure of alphas and betas, this appendix discusses the
related term structures of expected returns and risk premiums. It focuses on the mechanical drivers
behind how they slope. Expected returns are important because they link betas to alphas through
equation (4). The price of risk is also connected since betas play a key role in their estimation.
Throughout, I discuss the “holding” term structure (and not the “maturity” term structure), where
a portfolio is compounded over various investment horizons.
Term Structure Shape of Expected Returns
Conditional on a point in time, the holding term structure of expected returns can be upward or
downward sloping depending on what the information set forecasts. However, the term structure of
unconditional log returns must be flat by construction, regardless of the portfolio. Long horizon
returns compound the same short horizon returns, and since compounding log returns involves
simple addition, it must lead to the same result once returns are averaged and standardized.
ˆ
E
t
[r
t→t+h
]
h
=
ˆ
E
t
[r
t→t+1
+r
t+1→t+2
+... +r
t+h−1→t+h
]
h
=
ˆ
E
t
[r
t→t+1
] (11)
A flat holding term structure of log returns provides a useful starting point to investigate how
alternative specifications may lead to a sloped holding term structure instead. First, the hat
“ ˆ” symbol in the above equation denotes realized averages that estimate the true unconditional
expectation. These estimates are unbiased only if rational expectations hold. But if realized returns
differ systematically from investor’s expected returns, then a flat realized term structure may be a
biased estimate of an expected one that is actually sloped.
Second, as an empirical methodology choice, use of overlapping returns lead to long-horizon returns
that over-sample the middle portion of the historical data. If this middle portion happens to have
higher realized returns, average long-horizon log returns will exceed average short-horizon ones,
resulting in an upward-sloping holding term structure. In this paper, I use overlapping annual
returns for 3-year and 10-year horizons. Appendix B’s summary statistics show that my term
structure of log returns is consequently nearly but not exactly flat.
Third, use of simple rather than log returns must result in a downward-sloping term structure.
From Jensen’s inequality and given non-constant returns, the arithmetic mean (i.e. average of
short-horizon returns) is larger than the geometric mean (i.e. standardized long-horizon returns).
The greater the portfolio’s volatility, the greater the Jensen term (≈
1
2
Var[r]), and the more the term
structure of returns slopes downward. Some papers argue that only a single true expected return
exists and that these mechanics therefore bias returns. Asparouhova et al. (2013) worry that noise
adds volatility and generates an upward bias while Boguth et al. (2016) worry that slow information
diffusion dampens volatility and generates a downward bias. However, an alternative perspective
recognizes the entire term structure of expected returns. Section 5 shows that lead-lag patterns
exist at both monthly and annual horizons and vary according to different stock characteristics.
Pinpointing a horizon when returns are “unbiased” may thus be impractical since prices may never
46
be free of frictions and noise. It is also undesirable when investors do actually trade over a range of
horizons.
Term Structure Shape of Excess Returns
The term structure of excess returns depends on portfolio compounding rules governing how excess
returns cumulate. In this paper, I directly compound short-horizon excess returns to obtain long
horizons ones. This approach implicitly assumes consistent rebalancing to equal weights of the long
and short legs of the portfolio. Although rebalancing incurs transaction costs, it prevents style
drift and ensures term structure dynamics don’t simply reflect the increasing dominance of one
leg over another. The alternative approach avoids rebalancing and compounds the long and short
legs separately and only differences at the end of the horizon. Ken French uses this approach for
the different horizon factor returns he posts on his website. So does Kamara et al. (2016). The
following equations compare two period simple excess returns compounded under each method. R
denotes simple gross returns.
r
sim,ex,reb
0→2
≡ (1 +r
sim,ex
0→1
)(1 +r
sim,ex
1→2
)− 1 (12)
r
sim,ex,sep
0→2
≡ (R
long
0→1
R
long
1→2
)− (R
short
0→1
R
short
1→2
) (13)
Compounding excess returns under the rebalancing approach is much easier since it parallels
compounding gross returns. Only a single sequence of returns is used for any horizon. In contrast,
separate compounding of the long and short legs requires tracking two return series. For a given
period, the excess return is not unique but depends on when compounding began. It is a function
of each legs’ performance history from the start of compounding up to that period.
Under the rebalancing approach, the term structure of log excess returns is flat in the same way it is
for log gross returns in equation (11). Since Jensen’s equality applies too, we can also conclude that
the term structure of simple excess returns must be downward sloping. Keeping the assumption of a
two-period return, equations (12) and (13) can be differenced and rearranged to yield the following.
r
sim,ex,reb
0→2
−r
sim,ex.sep
0→2
= (1−R
short
0→1
)(R
long
1→2
−R
short
1→2
) + (1−R
short
1→2
)(R
long
0→1
−R
short
0→1
) (14)
The value of equation (14) allows the inference of the excess return term structure shape under
separate compounding. A positive value means rebalancing outperforms separate compounding.
Since log excess returns under rebalancing is flat, log excess returns under separate compounding
must be downward-sloping. Similarly, a negative value implies an upward slope. What about simple
instead of log excess returns? It will be downward sloping if log returns are already so since Jensen’s
inequality can only diminish long-horizon returns. However, if equation (14) is sufficiently negative
such that log excess returns is upward sloping, it may be sufficient to overcome the Jensen term
(≈
1
2
Var[r
sim,ex
]) and produce an upward slope. This is possible for separate compounded but not
rebalanced excess returns.
The two compounding methods discussed apply to long-short portfolios but also to any gross return
portfolios with multiple assets. Equal weighting, for example, can occur with rebalancing after each
period (which is the standard approach) or it can be applied initially, with each asset separately
compounded thereafter. The former requires working with only a single return sequence while the
47
latter produces for any given period, different returns that depends on when compounding began.
Value weighting is a distinct methodology that allows dynamic instead of static weights. Like the
rebalanced approach, it also requires only a single return series for compounding to any horizon.
Term Structure Shape of Factor Risk Premiums
The term structure of risk premiums addresses whether factors are priced differently depending on
the investment horizon. Given a single market factor, the cross-sectional risk premium for a specific
time and horizon, λ
t→t+h
, is determined by the estimated expected returns and CAPM betas of the
N test assets (indexed by i).
ˆ
E
t
[r
i
t→t+h
] =λ
t→t+h
ˆ
β
i
t→t+h
+α
i
t→t+h
λ
t→t+h
can be estimated using a cross-sectional GLS or OLS regression, without an intercept under
the no alpha restriction. If the factor is traded, as it is for the market, then the model further
requires
ˆ
λ
t→t+h
=
ˆ
E[r
m
t→t+h
]. The risk premium for horizon h must equal the expected market
return for the corresponding horizon. This restriction means the term structure of the price of risk
equals the term structure of the factor’s expected return. With log returns, it must be flat and have
the same price across all horizons. The cross-sectional regression is unnecessary. But if the factor
return restriction is relaxed, the price of risk can be estimated using OLS and equals the following.
ˆ
λ
t→t+h
=
Cov(
ˆ
E
t
[r
i
t→t+h
],
ˆ
β
i
t→t+h
)
Var(
ˆ
β
i
t→t+h
)
(15)
Under the no alpha restriction, the covariance and variance terms should avoid demeaning. Varying
h in equation (15) allows a mapping of each point on the term structure. Using log returns that are
horizon invariant, changes in the price of risk will be driven by changing betas. If the variance in
the denominator stays the same, then the price of risk will increase if higher returning assets have
higher betas. As discussed in Section 3, higher betas from a lengthening of the horizon occurs when
stocks have positive lead-lag correlations with the market.
For example, what happens when the market price of risk is estimated with a cross-section of
size-sorted portfolios? Small-cap stocks have higher expected returns. If a lengthening of the horizon
increases small-cap betas, the price of risk also increases (assuming the denominator term doesn’t
offset the numerator effect). The term structure of the price of risk will then reflect SMB’s beta term
structure: first rising, peaking at around the annual horizon, and then decline thereafter. When
SMB’s beta becomes negative at multi-year horizons, the sign of the price of risk reverses too. In
results not shown, I find broadly consistent term structure patterns among size-sorted portfolios.
As discussed in Section 2, this intuition can also be applied to Kamara et al. (2016) results.
48
B Summary Descriptions
In this appendix, I provide basic summary statistics with a brief discussion. Table B1 panel
(a)’s left-hand side contains summary statistics for each portfolio’s excess return. This is done at
horizons ranging from 1-month to 10-years. All returns are log returns, so long-horizon returns
simply sum short-horizon ones. Since different horizons use the same historical data, returns of
different horizons should all have the same mean statistic when annualized. However, the table
shows slight deviations at multi-year horizons because these horizons use overlapping annual data.
This effectively underweights the beginning and end portion of the dataset, deviating from the
uniform weighting of non-overlapping returns at short-horizons. Table B1 panel (a)’s right-hand
side contains summary statistics for each portfolio’s conditional mean return, estimated using rolling
windows.
The mean return term structure for log returns should be flat, but the same is not necessarily
true for simple returns. Long-horizon simple returns are not the sum of short-horizon returns but
instead, require geometric compounding. Long-horizon mean returns will thus be smaller than
short-horizon ones given volatility and Jensen’s inequality. However, I find a nearly flat mean return
term structure for simple returns too (results not shown). The Jensen inequality term turns out not
to be important because I study only excess-return portfolios. These portfolios have Jensen terms
that are driven by long-short leg volatility differences, and these differences tend to be small.
Table B1 panel (b) contains summary statistics of the conditional alpha and beta estimates under
both the rolling regression and the higher-frequency approaches. The rolling regression approach
smooths across periods whereas the higher-frequency approach does not. This means the new higher-
frequency approach has lower beta autocorrelations and higher standard deviations, leading to
sizable time-series fluctuations roughly comparable to those estimated by Lewellen and Nagel (2006).
Beta variation reflects not only changing market sensitivities but also changing equity compositions
since these excess return portfolios are reconstituted annually (or monthly in momentum’s case).
As for alphas, the higher-frequency approach also has larger standard deviations than the rolling
regression approach.
Figure B1 plots alpha and beta time-series under all three estimation approaches. Unconditional
values are constant and thus represented by a straight line. Rolling regressions are backward looking
so time t results reflect past information. In contrast, time t conditional expectations for the new
higher-frequency approach uses future realized returns and is thus forward looking. These dating
conventions mean the two approaches use very different data for estimates at any particular point
in time. This explains why correlation between the two approaches’ estimates are so low.
49
Table B1: Summary Statistics. Monthly portfolio excess returns from Ken French’s website for the
period 1927-2015. I convert to log returns (%/year) so long-horizon returns are the simple sum of monthly
returns. Panel (a) shows summary statistics for portfolio returns and for rolling-window averages using 5,
10, 20, 30, and 40 years for monthly, quarterly, annual, 3-year, and 10-year returns, respectively. I center
the window on the date and permit partial windows during the beginning and end of the dataset. Monthly,
quarterly, and annual returns are non-overlapping while 3-year and 10-year ones are annual-overlapping
(autocorrelation statistics use non-overlapping returns in all cases though). Mean statistics are in percent and
annualized (e.g. x12 for monthly). Standard deviations are also in percent and annualized but done differently
for returns (scale by square root, e.g. x
√
12) versus for rolling averages (scale directly, e.g. x12). This
accounts for the different time-series properties of returns (little autocorrelation) versus of rolling averages
that proxy for the expected mean (highly persistent). Panel (b) shows summary statistics for time-varying
alphas (%/year) and betas estimated using two different methods. The rolling regression approach is the
OLS regression alpha and beta using a backward-looking window of returns of the same horizon (with same
window lengths as conditional mean estimates above). The new higher-frequency approach uses data within
each horizon and weights lead-lag covariance terms according to its lead-lag. See Section (4) for details of the
estimation approaches. The Rolling-HF Corr row shows the estimate’s correlation across the two conditional
approaches. All single-lag autocorrelations (Auto1) are adjusted for finite sample bias where ρ is back-solved
given ˆ ρ and the bias term, E[ˆ ρ−ρ] =−
1+4ρ
T
(Kendall (1954)).
(a) Portfolio Returns and Rolling Averages
Average Return Average Conditional Mean
Mn Qt Yr 3Y 10Y Mn Qt Yr 3Y 10Y
Obs Yr Overlap 1068 356 89 87 80 1068 356 89 87 80
No Overlap 30 9 30 9
Market Mean 6.0 6.0 6.0 5.7 6.2 5.9 5.8 5.9 5.9 6.4
(ExMkt) SD 18.7 21.2 20.0 19.8 16.0 8.2 5.7 3.5 2.4 1.9
Auto1 0.1 -0.0 0.1 -0.3 0.1 1.0 1.0 1.0 0.9 1.0
Size Mean 2.0 2.0 2.0 2.1 2.4 2.0 2.2 2.2 2.3 2.3
(SMB) SD 10.9 11.8 12.3 14.7 11.2 6.2 3.4 1.9 1.3 1.1
Auto1 0.1 -0.0 0.3 0.0 -0.5 1.0 1.0 1.0 0.7 0.5
Value Mean 3.9 3.9 3.9 4.1 4.5 3.9 3.9 4.0 4.3 4.6
(HML) SD 11.9 13.8 12.3 11.2 7.6 4.7 2.9 1.8 1.1 0.9
Auto1 0.2 -0.0 -0.0 -0.1 -0.2 1.0 0.9 0.9 1.0 1.0
Momentum Mean 6.7 6.7 6.7 6.5 6.7 6.6 6.4 6.4 6.5 7.1
(UMD) SD 18.2 20.6 18.5 17.6 17.0 7.2 5.5 3.9 3.3 2.7
Auto1 0.1 -0.1 -0.0 -0.2 0.3 1.0 0.9 1.0 1.0 1.0
50
(b) Time-Varying Alphas and Betas
Average Alpha Average Beta
Mn Qt Yr 3Y 10Y Mn Qt Yr 3Y 10Y
Obs No Overlap 1067 355 88 29 8 1067 355 88 29 8
Yr Overlap 84 70 84 70
Size Higher Mean 1.9 1.3 0.5 1.2 3.2 0.1 0.2 0.3 0.2 -0.1
(SMB) Frequency SD 2.3 3.3 4.1 5.2 10.1 0.6 0.7 0.6 0.5 0.6
Auto1 0.5 0.3 0.4 0.3 -0.4 0.1 0.1 -0.0 0.5 -0.0
Rolling Mean 1.3 1.3 0.0 1.3 4.7 0.2 0.2 0.3 0.2 -0.3
Regression SD 1.7 3.3 5.1 4.3 5.8 0.2 0.4 0.2 0.3 0.2
Auto1 1.0 0.6 0.7 0.7 1.0 1.0 0.4 0.8 1.0 0.2
Higher Mean 3.7 4.0 3.5 4.0 4.5 -0.0 0.0 0.0 -0.0 -0.0
Value Frequency SD 2.2 3.1 4.3 5.2 8.8 0.6 0.8 0.7 0.5 0.4
(HML) Auto1 0.4 0.3 0.2 0.2 -0.3 0.3 0.2 0.3 0.2 0.1
Rolling Mean 3.8 4.2 3.8 4.4 5.1 -0.0 -0.0 0.0 -0.0 -0.1
Regression SD 1.5 1.8 2.5 3.0 3.6 0.3 0.3 0.3 0.3 0.3
Auto1 0.9 0.7 0.8 1.0 1.0 1.0 0.9 1.0 1.0 1.0
Higher Mean 6.2 6.4 6.8 6.7 5.3 -0.1 -0.0 -0.1 -0.0 0.2
Frequency SD 3.1 3.9 4.7 7.5 14.7 0.8 0.8 0.7 0.5 0.4
Momentum Auto1 0.7 0.5 0.5 0.5 1.0 0.4 0.3 0.1 0.2 -0.2
(UMD) Rolling Mean 6.2 7.4 7.8 6.0 4.1 -0.1 -0.2 -0.1 0.1 0.3
Regression SD 2.2 4.3 5.0 7.7 17.9 0.3 0.5 0.2 0.2 0.3
Auto1 1.0 0.4 0.8 1.0 1.0 1.0 0.4 0.7 0.9 0.5
51
Figure B1: Times Series of Market Volatility and of Portfolio Alphas and Betas. Data
from 1927-2015 used to form non-overlapping monthly, annual returns and annual-overlapping 10-year returns.
Panel (a) shows the time-series of market excess-returns (%/year) and of its standard deviation (%/year).
Monthly, annual, and 10-year rolling means average the previous 5, 20, and 40-year windows, respectively.
Panels (b)-(d) show the time-series plots of size, value, and momentum alphas (%/year) and betas, estimated
under the rolling regression and the higher-frequency approaches. These methods are described in Section 4.
(a) Market (ExMkt)
52
(b) Size (SMB)
53
(c) Value (HML)
54
(d) Momentum (UMD)
55
C Comparing Alphas and Betas across Two Horizons
This appendix supplements Table 2. Instead of a joint test of equal coefficients across all horizons,
it tests pairwise equality between just two horizons. These results show which horizons are most
likely different from each other.
Table C1: T-Statistic for Alpha and Beta Term Structure Equality. Data and method as in
Table 2. Table shows the t-statistic of equal alpha or beta coefficients across two different horizons (denoted
in the rows versus columns). For example, Tstat(β
Qt
−β
Mn
) =
β
Qt
−β
Mn
√
se(β
Qt
)
2
+se(β
Mn
)
2
−2ρ
Qt,Mn
se(β
Qt
)se(β
Mn
)
where ρ
Qt,Mn
denotes the correlation between the coefficient estimates calculated using pairwise GMM.
Positive numbers denote an increase in alpha or beta with a lengthening of the horizon. Boldfaced t-stats
denote a significant difference at the two-sided 95% level.
(a) Size (SMB)
Alpha Beta
Qt Yr 3Y 10Y Qt Yr 3Y 10Y
Higher Mn -1.4 -2.1 -1.3 1.5 3.2 3.9 1.5 -1.6
Frequency Qt -1.2 -0.2 2.1 2.1 -0.0 -2.4
Yr 1.3 2.9 -1.9 -3.3
3Y 2.6 -2.6
Rolling Mn -0.0 -0.6 0.0 1.0 0.8 0.9 0.1 -2.2
Regression Qt -1.6 0.1 1.3 0.6 -0.1 -2.9
Yr 2.0 1.5 -0.2 -3.2
3Y 1.2 -1.4
Unconditional Mn -1.3 -0.6 0.1 1.7 2.2 0.8 -0.2 -3.0
Qt 0.6 0.4 1.9 -0.8 -0.9 -3.6
Yr 0.3 1.7 -0.7 -3.3
3Y 1.1 -3.4
56
(b) Value (HML)
Alpha Beta
Qt Yr 3Y 10Y Qt Yr 3Y 10Y
Higher Mn 0.6 -0.4 0.6 1.0 0.6 0.4 0.0 -0.2
Frequency Qt -1.0 0.1 0.6 0.1 -0.3 -0.4
Yr 1.3 1.1 -0.5 -0.4
3Y 0.6 -0.2
Rolling Mn 0.6 0.0 0.7 1.5 -0.2 0.2 -0.0 -0.3
Regression Qt -0.7 0.4 1.7 0.3 0.0 -0.2
Yr 2.9 2.6 -0.0 -0.7
3Y 1.8 -0.1
Unconditional Mn -0.2 0.4 0.5 1.3 0.2 -0.8 -0.8 -1.7
Qt 0.5 0.6 1.4 -0.8 -0.9 -1.6
Yr 0.4 1.3 -0.3 -1.2
3Y 0.8 -0.9
(c) Momentum (UMD)
Alpha Beta
Qt Yr 3Y 10Y Qt Yr 3Y 10Y
Higher Mn 0.3 0.8 0.6 -0.7 0.2 -0.6 0.6 3.4
Frequency Qt 0.6 0.3 -0.8 -1.0 0.4 2.6
Yr -0.3 -1.3 1.3 2.8
3Y -1.3 2.3
Rolling Mn 0.8 0.9 -0.1 -0.2 -0.8 0.2 1.0 0.7
Regression Qt 0.4 -1.1 -0.4 2.1 3.4 0.8
Yr -1.4 -0.4 1.7 0.6
3Y -0.3 0.5
Unconditional Mn 0.9 -0.8 -1.2 -1.3 -1.8 1.1 2.4 2.3
Qt -1.1 -1.3 -1.6 1.4 2.5 2.6
Yr -0.7 -1.0 2.2 2.0
3Y -0.7 1.5
57
D Robustness of Term Structure Results
This appendix evaluates the robustness of the main empirical results on the term structure of alphas
and betas (Table 2). Each scenario involves a single change while preserving all other specifications
under the base case. Overall, alphas’ and betas’ sign and degree of significance remain qualitatively
similar under nearly all scenarios. The paper’s main findings are robust.
Beginning with the unconditional approach, Table D1 presents the following alternative specifica-
tions. I forgo using overlapping annual returns and use only the 30 3-year and 9 10-year observations
that are available. To obtain longer-horizon returns, I use annual rather than monthly returns (also
obtained from Ken French’s website), thus reducing the implicit rebalancing that occurs between
the long and short legs. I use simple instead of log returns. Using simple returns, I also examine
the long and short legs separately. Alphas and betas are linearly additive when expressed as simple
(but not log) returns. Therefore, coefficients for a particular specification equals the difference
between the corresponding coefficients of the long and short legs if the examined horizon is that of
the base case monthly rebalancing frequency (and before annualizing which may distort this exact
relationship). At longer horizons this linear relationship does not necessarily hold because monthly
rebalancing occurs between the long and short legs (which implicitly happens when compounding
monthly excess returns). See Appendix E for a more detailed exploration of the difference between
portfolio rebalancing, reformation, and investment periods.
Turning to the robustness of the rolling regression approach, Table D2 shows results under
alternative specifications. In addition to the alternatives explored for unconditional results, different
ways of forming rolling-windows are also assessed. First, instead of backward-looking windows, I
consider centered or forward-looking ones. Although not implementable in real-time, they may
better proxy for investors’ conditional expectations under rational expectations. I also consider
window sizes that are half the base case’s. In all results, I allow partial windows as long as they
have at least 2 non-overlapping observations. Overall results are qualitatively similar except for
momentum’s non-overlapping 10-year beta, which is actually negative when all other specifications
are positive. The beta estimate has as its first observation a partial window that contains only two
periods. This leads to an extreme result that swamps others when averaged together. Requiring a
minimum of three instead of just two periods restores the base case pattern. Other specifications
follow the base case’s use of overlapping annual data for 10-year horizons, preventing a single
extreme outlier from dominating.
Table D3 completes the robustness section and contains results for the new higher-frequency
approach. Alternative specifications change the weighting kernel for lead-lag higher-frequency
returns, using triangular (Bartlett) instead of Gaussian weights or using uniform weighting that
includes only the closest half of all lead-lag terms. Other specifications vary the conditional mean
assumption, by using forward-looking or backward-looking windows (instead of centered ones) or
by using an expanding historical window. I also alter the choice of higher-frequency returns, using
daily instead of weekly for the monthly horizon, weekly instead of monthly for the quarterly horizon,
quarterly instead of monthly returns for the 3-year horizon, and annual instead of monthly for the
10-year horizon. Finally, I alter the limit in which market autocorrelations can impact the beta
denominator, using a denominator minimum of 0.5 or 0.1 versus the base case’s 0.3. The main term
structure patterns remain robust.
58
Table D1: Robustness of Unconditional Approach. Data from 1927-2015 used to form non-
overlapping monthly, quarterly, annual returns and annual-overlapping 3-year and 10-year returns. Alphas
are annualized percentages. The table shows different specifications for estimating unconditional alphas and
betas, where estimates are from regressing portfolio excess-returns on the market excess-return using all
historical data. No Overlap uses non-overlapping returns (affects only 3-year and 10-year horizons which use
annual overlapping horizons under the base case). Annual Returns compound annual rather than monthly
returns. Simple Returns uses simple rather than log returns. Long-Leg and Short-Leg looks at the two legs
of the excess return portfolios. I use simple returns since alphas and betas are then linearly additive if the
horizon equals the monthly rebalancing frequency (as is the case for all 3 portfolios) and before annualizing
(see Appendix E for details). T-statistics are in brackets using Newey-West ’94 standard errors. Boldfaced
coefficients denote significant differences from zero at the two-sided 95% level. P-value reflects Wald Test of
null hypothesis that all 5 horizon coefficients are equal (if specification includes blank horizons, assume those
equal to base case).
(a) Size (SMB)
Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Base Case 0.9 0.4 0.7 1.2 3.9 0.19 0.27 0.23 0.16 -0.24
[0.9] [0.4] [0.6] [0.5] [3.1] (0.34) [6.0] [4.6] [3.6] [1.3] [-1.7] (0.00)
No Overlap 1.2 5.5 0.14 -0.58
[0.8] [4.6] (0.03) [1.0] [-3.6] (0.00)
Annual 0.9 1.6 4.5 0.24 0.16 -0.27
Returns [0.7] [0.6] [3.5] (0.04) [3.9] [1.1] [-1.8] (0.00)
Simple 1.1 0.3 0.9 2.0 4.4 0.19 0.29 0.24 0.17 -0.17
Returns [1.1] [0.2] [0.6] [0.8] [2.8] (0.39) [6.0] [6.7] [3.7] [1.0] [-2.3] (0.00)
Long-Leg 1.8 0.8 1.7 2.2 5.4 1.25 1.49 1.32 1.37 0.96
(Simple Ret) [1.5] [0.5] [1.0] [0.8] [2.5] (0.29) [25.0] [9.2] [13.7] [4.0] [3.5] (0.17)
Short-Leg 0.6 0.3 0.7 0.9 0.8 1.06 1.12 1.04 1.04 1.12
(Simple Ret) [1.3] [0.5] [1.6] [1.8] [1.9] (0.62) [31.7] [14.6] [21.8] [32.0] [26.4] (0.08)
(b) Value (HML)
Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Base Case 3.1 3.1 3.5 3.8 4.9 0.12 0.13 0.07 0.04 -0.07
[2.2] [2.2] [2.8] [2.5] [7.9] (0.86) [1.8] [1.1] [0.8] [0.4] [-0.5] (0.54)
No Overlap 3.3 5.4 0.08 -0.24
[3.6] [5.6] (0.09) [0.9] [-1.3] (0.42)
Annual 3.4 3.6 4.3 0.07 0.08 0.04
Returns [2.4] [2.1] [4.7] (0.93) [0.8] [0.7] [0.3] (0.93)
Simple 3.5 3.3 4.2 4.6 5.1 0.15 0.20 0.07 0.03 -0.02
Returns [2.4] [2.1] [3.1] [2.6] [6.3] (0.78) [1.7] [1.4] [0.8] [0.3] [-0.2] (0.61)
Long-Leg 2.8 1.9 3.0 3.5 5.3 1.26 1.47 1.27 1.29 1.35
(Simple Ret) [2.1] [1.1] [1.9] [1.9] [3.0] (0.48) [15.0] [7.8] [13.4] [4.9] [3.5] (0.55)
Short-Leg -0.7 -1.1 -0.9 -0.8 0.3 1.11 1.20 1.16 1.12 0.86
(Simple Ret) [-1.0] [-1.3] [-1.2] [-0.9] [0.3] (0.81) [46.6] [38.3] [41.9] [10.0] [16.6] (0.00)
59
(c) Momentum (UMD)
Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Base Case 8.6 9.1 7.4 6.1 5.1 -0.32 -0.41 -0.13 0.07 0.26
[5.4] [5.6] [3.7] [2.8] [1.8] (0.96) [-2.4] [-2.6] [-1.5] [1.0] [1.5] (0.05)
No Overlap 4.8 4.2 0.30 0.39
[2.1] [1.2] (0.27) [2.2] [1.5] (0.01)
Annual 8.1 6.7 5.8 -0.12 0.07 0.27
Returns [4.4] [2.0] [0.7] (0.86) [-1.1] [0.5] [1.0] (0.03)
Simple 11.0 11.8 9.6 8.0 6.8 -0.30 -0.34 -0.15 0.02 0.23
Returns [7.3] [8.0] [5.2] [3.4] [2.2] (0.09) [-2.7] [-3.6] [-1.8] [0.2] [1.5] (0.00)
Long-Leg 5.4 5.2 5.0 4.6 6.5 1.08 1.12 1.21 1.42 1.60
(Simple Ret) [6.2] [6.4] [4.7] [1.8] [3.2] (0.90) [27.1] [32.5] [16.4] [7.9] [8.2] (0.01)
Short-Leg -5.1 -5.8 -4.8 -4.1 -2.0 1.37 1.56 1.28 1.10 0.63
(Simple Ret) [-4.9] [-4.1] [-3.7] [-3.0] [-1.2] (0.24) [18.2] [11.2] [20.8] [7.0] [4.1] (0.00)
60
Table D2: Robustness of Rolling Regression Approach. Data from 1927-2015 used to form
non-overlapping monthly, quarterly, annual returns and annual-overlapping 3-year and 10-year returns. Alphas
are annualized percentages. The table shows different specifications for estimating the rolling regression
approach, where estimates are from regressing portfolio excess-returns on the market excess-return using
a backward-looking window, with no look-ahead bias. Base Case uses windows of 5 years for the monthly
horizon, 10 years for the quarterly horizon, 20 years for the annual horizon, 30 years for the 3-year horizon,
and 40 years for the 10-year horizon. Partial windows of at least 2 independent observations are allowed. No
Overlap uses non-overlapping returns (affects only 3-year and 10-year horizons which use annual overlapping
horizons under the base case). Annual Returns compounds annual rather than monthly returns. Look Ahead
uses a forward-looking window instead of the base case’s backward looking window. Half Window uses half
the window length compared to the base case (i.e. 2.5, 5, 10, 15, 20 years, respectively). Simple Returns uses
simple rather than log returns. Long-Leg and Short-Leg looks at the two legs of the excess return portfolios.
I use simple returns since alphas and betas are then linearly additive if the horizon equals the monthly
rebalancing frequency (as is the case for all 3 portfolios) and before annualizing (see Appendix E for details).
T-statistics shown in brackets use Newey-West ’94 standard errors. Boldfaced coefficients denote significant
differences from zero at the two-sided 95% level. P-value reflects Wald Test of null hypothesis that all 5
horizon coefficients are equal (if specification includes blank horizons, assume those equal to base case).
(a) Size (SMB)
Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Base Case 1.3 1.3 0.0 1.3 4.7 0.18 0.22 0.26 0.20 -0.26
[0.6] [1.9] [0.0] [2.3] [1.8] (0.33) [2.8] [8.6] [5.6] [0.5] [-1.5] (0.00)
No Overlap 1.4 5.5 0.12 -0.44
[1.7] [7.0] (0.00) [0.2] [-3.1] (0.00)
Annual 0.3 1.9 5.4 0.27 0.20 -0.30
Returns [0.4] [2.7] [1.8] (0.12) [6.5] [0.4] [-1.8] (0.00)
Look Ahead 1.4 0.4 0.3 1.8 4.1 0.19 0.28 0.21 0.06 -0.35
Window [0.6] [0.0] [0.2] [1.3] [2.8] (0.57) [2.4] [4.2] [2.9] [0.1] [-1.4] (0.00)
Center 1.2 0.8 1.0 1.7 4.5 0.19 0.26 0.22 0.13 -0.34
Window [0.5] [0.4] [0.7] [1.0] [2.1] (0.84) [2.4] [4.6] [1.9] [0.3] [-2.4] (0.00)
Half 1.1 1.4 -0.1 1.9 4.2 0.19 0.23 0.28 0.15 -0.17
Window [0.6] [1.4] [-0.0] [1.3] [2.3] (0.46) [3.2] [7.4] [3.4] [0.4] [-0.5] (0.66)
Simple 1.6 1.3 0.4 2.2 5.1 0.18 0.23 0.26 0.20 -0.19
Returns [0.7] [2.0] [0.5] [3.3] [1.5] (0.06) [2.8] [8.9] [5.1] [0.7] [-2.2] (0.00)
Long-Leg 2.5 3.0 1.7 3.0 6.3 1.18 1.27 1.30 1.36 0.95
(Simple Ret) [1.2] [3.8] [2.0] [2.0] [3.9] (0.17) [17.1] [34.4] [8.3] [2.1] [3.3] (0.67)
Short-Leg 1.0 1.1 1.1 1.1 1.0 1.00 1.01 1.02 1.02 1.09
(Simple Ret) [4.0] [7.3] [6.4] [2.4] [1.9] (0.99) [20.4] [43.6] [21.5] [6.5] [5.5] (0.99)
61
(b) Value (HML)
Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Base Case 3.8 4.2 3.8 4.4 5.1 -0.00 -0.02 0.02 -0.01 -0.06
[4.5] [8.4] [5.3] [6.3] [8.1] (0.03) [-0.0] [-0.3] [0.1] [-0.0] [-0.2] (0.61)
No Overlap 3.7 5.4 0.03 -0.11
[3.9] [5.5] (0.38) [0.2] [-0.5] (0.94)
Annual 3.5 3.7 4.2 0.03 0.04 0.08
Returns [3.3] [2.8] [2.5] (0.46) [0.1] [0.0] [0.1] (0.72)
Look Ahead 3.5 3.2 2.9 3.9 5.4 -0.01 -0.02 0.00 -0.12 -0.23
Window [1.4] [0.6] [1.1] [2.1] [11.6] (0.41) [-0.0] [-0.1] [0.0] [-0.4] [-2.7] (0.57)
Center 3.7 3.7 3.7 4.5 5.2 -0.01 0.00 -0.00 -0.08 -0.14
Window [2.1] [3.2] [2.1] [2.9] [9.7] (0.64) [-0.0] [0.0] [-0.0] [-0.2] [-2.0] (0.30)
Half 3.3 3.9 3.7 4.4 5.7 -0.00 -0.02 0.00 -0.06 -0.17
Window [2.1] [4.3] [3.4] [4.5] [5.8] (0.60) [-0.0] [-0.3] [0.0] [-0.4] [-1.0] (0.84)
Simple 4.4 4.9 4.5 4.9 5.5 -0.00 -0.02 0.01 -0.02 -0.06
Returns [4.8] [8.3] [4.3] [5.4] [9.2] (0.51) [-0.0] [-0.2] [0.0] [-0.0] [-0.4] (0.81)
Long-Leg 3.8 4.5 3.4 3.9 6.0 1.13 1.18 1.21 1.27 1.30
(Simple Ret) [4.5] [3.7] [4.0] [2.5] [2.7] (0.83) [12.2] [8.1] [8.7] [1.0] [1.5] (0.99)
Short-Leg -0.6 -0.5 -0.7 -0.2 1.3 1.13 1.17 1.17 1.11 0.82
(Simple Ret) [-0.4] [-1.4] [-1.4] [-0.4] [0.2] (0.98) [18.6] [70.2] [75.7] [10.1] [18.9] (0.00)
(c) Momentum (UMD)
Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Base Case 6.2 7.4 7.8 6.0 4.1 -0.12 -0.21 -0.09 0.06 0.34
[2.8] [8.4] [5.1] [3.9] [0.5] (1.00) [-0.8] [-3.8] [-1.9] [1.1] [0.5] (0.00)
No Overlap 5.8 7.9 0.22 -0.40
[2.1] [4.8] (0.75) [1.2] [-1.1] (0.00)
Annual 9.5 8.7 7.8 -0.03 0.09 0.15
Returns [8.8] [5.2] [8.1] (0.45) [-0.4] [1.3] [0.4] (0.00)
Look Ahead 6.0 7.2 8.0 5.5 5.7 -0.13 -0.21 -0.21 0.09 0.31
Window [2.6] [0.9] [4.6] [1.9] [1.7] (0.56) [-0.7] [-1.3] [-1.7] [1.4] [0.7] (0.19)
Center 6.0 6.6 7.1 5.9 4.2 -0.12 -0.21 -0.14 0.07 0.41
Window [2.2] [4.8] [2.9] [1.5] [0.6] (0.97) [-0.6] [-1.5] [-0.6] [0.8] [0.7] (0.21)
Half 5.0 6.4 7.4 6.2 6.0 -0.07 -0.17 -0.10 0.06 0.24
Window [1.3] [5.2] [3.5] [2.6] [1.1] (1.00) [-0.5] [-2.7] [-1.0] [0.9] [0.8] (0.06)
Simple 8.5 10.5 10.1 8.3 6.2 -0.11 -0.18 -0.10 0.04 0.28
Returns [5.8] [13.8] [6.8] [8.3] [0.6] (0.00) [-0.7] [-3.8] [-2.1] [0.9] [0.0] (0.00)
Long-Leg 4.8 5.2 5.1 5.5 7.2 1.12 1.13 1.22 1.41 1.56
(Simple Ret) [3.5] [11.7] [12.3] [3.2] [4.6] (0.77) [14.2] [66.1] [29.5] [3.8] [7.3] (0.06)
Short-Leg -3.4 -3.8 -5.6 -4.0 -1.0 1.23 1.32 1.23 1.06 0.62
(Simple Ret) [-2.0] [-5.0] [-6.1] [-11.6] [-0.3] (0.44) [14.1] [24.7] [15.6] [7.2] [5.8] (0.00)
62
Table D3: Robustness of Higher-Frequency Approach. Data from 1927-2015 used to form
non-overlapping monthly, quarterly, annual returns and annual-overlapping 3-year and 10-year returns. Alphas
are annualized percentages. The table shows different specifications for calculating the higher-frequency
approach (see Section 4 for details). Base Case uses monthly higher-frequency returns (except the monthly
horizon uses weekly returns) with a Gaussian kernel and expected means calculated using centered rolling
averages with windows of 5, 10, 20, 30, 40 years for monthly, quarterly, annual, 3-year, and 10-year horizons,
respectively. No Overlap uses non-overlapping returns (affects only 3-year and 10-year horizons which use
annual overlapping horizons under the base case). Krnl: Bartlett uses as the weighting kernel for lead-lag
terms a triangular (Bartlett) kernel. Krnl: Unit Closest Half uses uniform weighting that includes only
the closest half of all lead-lag terms while ignoring all others. Cond Mean: Look Ahead and Cond Mean:
Backward uses forward and backward looking windows for the conditional mean (instead of the base case’s
centered window). Cond Mean: Hist All uses an expanding historical window with no look-ahead and a
minimum 20-year window. Different HF Frequency uses different higher frequency returns than the base case
(higher for monthly and quarterly horizons and lower for 3-year and 10-year horizons). Den Limit uses a
different limit on the extent to which market autocorrelations can affect the beta denominator (0.5 and 0.1
limits compared to the base case of 0.3). T-statistics shown in brackets use Newey-West ’94 standard errors.
Boldfaced coefficients denote significant differences from zero at the two-sided 95% level. P-value reflects
Wald Test of null hypothesis that all 5 horizon coefficients are equal (if specification includes blank horizons,
assume those equal to base case).
(a) Size (SMB)
Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Base Case 1.9 1.3 0.5 1.2 3.2 0.09 0.19 0.31 0.19 -0.12
[6.4] [2.3] [0.8] [2.3] [4.3] (0.03) [4.1] [4.8] [5.1] [2.5] [-0.9] (0.00)
No Overlap 0.8 3.0 0.23 -0.22
[2.1] [4.1] (0.01) [2.4] [-1.7] (0.00)
Krnl: 1.9 1.3 0.7 1.2 3.2 0.09 0.18 0.28 0.20 -0.12
Bartlett [6.6] [2.3] [1.2] [2.3] [4.9] (0.02) [3.9] [4.5] [4.6] [2.8] [-1.1] (0.00)
Krnl: Unit 1.6 -0.9 0.3 1.4 3.6 0.12 0.63 0.34 0.13 -0.11
Closest Half [4.2] [-0.8] [0.5] [2.5] [1.3] (0.02) [5.2] [4.4] [4.9] [1.7] [-0.3] (0.00)
Cond Mean: 2.0 0.3 -0.3 0.6 2.6 0.10 0.22 0.28 0.15 -0.15
Look Ahead [5.7] [0.4] [-0.3] [0.7] [4.2] (0.00) [5.2] [5.5] [4.2] [1.4] [-1.7] (0.00)
Cond Mean: 2.0 1.1 0.5 2.3 5.5 0.08 0.20 0.25 0.00 -0.38
Backward [7.3] [1.6] [0.6] [3.1] [4.1] (0.02) [3.9] [5.8] [4.1] [0.0] [-2.7] (0.00)
Cond Mean: 2.0 1.4 1.0 2.5 5.9 0.07 0.17 0.24 -0.02 -0.40
Hist All [11.4] [4.7] [1.7] [4.4] [5.6] (0.00) [2.9] [4.3] [3.1] [-0.2] [-3.4] (0.00)
Diff HF 1.8 1.4 1.2 3.3 0.10 0.18 0.18 -0.13
Frequency [6.1] [2.9] [2.3] [3.7] (0.12) [5.4] [5.6] [2.5] [-0.8] (0.00)
Den Limit: 1.9 1.4 0.7 1.3 3.1 0.08 0.18 0.27 0.17 -0.10
0.5 [6.9] [2.4] [1.1] [2.5] [4.0] (0.08) [4.2] [4.7] [4.8] [2.4] [-0.7] (0.00)
Den Limit: 1.9 1.3 0.4 1.2 3.4 0.09 0.19 0.33 0.19 -0.17
0.1 [5.9] [2.3] [0.5] [2.4] [4.9] (0.03) [4.2] [4.8] [5.1] [2.5] [-1.4] (0.00)
63
(b) Value (HML)
Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Base Case 3.7 4.0 3.5 4.0 4.5 -0.01 0.02 0.02 -0.01 -0.03
[12.0] [7.8] [6.1] [7.7] [5.8] (0.59) [-0.3] [0.2] [0.2] [-0.1] [-0.3] (0.97)
No Overlap 3.0 3.9 0.10 -0.00
[3.4] [5.2] (0.41) [0.7] [-0.0] (0.90)
Krnl: 3.7 4.0 3.6 4.0 4.8 -0.01 0.02 0.01 -0.01 -0.08
Bartlett [11.9] [7.7] [6.2] [8.6] [7.6] (0.39) [-0.4] [0.2] [0.1] [-0.1] [-0.8] (0.86)
Krnl: Unit 4.0 4.3 3.2 4.0 5.6 -0.02 0.05 0.05 -0.01 -0.15
Closest Half [12.4] [3.0] [4.6] [6.7] [1.3] (0.57) [-0.8] [0.2] [0.4] [-0.1] [-0.4] (0.94)
Cond Mean: 3.0 2.6 2.7 2.8 3.6 -0.01 0.03 0.02 -0.00 -0.02
Look Ahead [7.6] [3.9] [3.2] [2.0] [3.9] (0.72) [-0.2] [0.6] [0.3] [-0.0] [-0.1] (0.74)
Cond Mean: 4.6 4.4 3.6 4.7 5.6 -0.01 0.02 0.06 -0.03 -0.08
Backward [16.8] [9.2] [4.4] [8.9] [12.2] (0.28) [-0.2] [0.3] [0.7] [-0.4] [-1.0] (0.45)
Cond Mean: 5.2 4.8 4.6 5.3 6.0 -0.11 -0.06 -0.03 -0.11 -0.14
Hist All [30.8] [12.4] [6.0] [9.1] [10.2] (0.37) [-4.4] [-1.1] [-0.3] [-1.2] [-1.8] (0.71)
Diff HF 3.8 3.8 4.0 4.6 -0.01 0.01 -0.01 -0.05
Frequency [12.6] [7.9] [7.3] [5.0] (0.67) [-0.4] [0.3] [-0.1] [-0.4] (0.79)
Den Limit: 3.7 4.0 3.6 4.0 4.7 -0.01 0.02 0.01 0.00 -0.06
0.5 [12.2] [8.0] [6.4] [8.1] [6.2] (0.55) [-0.4] [0.2] [0.1] [0.0] [-0.5] (0.92)
Den Limit: 3.7 4.0 3.6 3.9 4.4 -0.01 0.01 0.02 0.00 -0.01
0.1 [12.0] [7.8] [6.1] [6.7] [6.3] (0.77) [-0.4] [0.2] [0.3] [0.1] [-0.1] (0.97)
64
(c) Momentum (UMD)
Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Base Case 6.2 6.4 6.8 6.7 5.3 -0.06 -0.05 -0.11 -0.02 0.24
[12.7] [7.9] [7.9] [7.4] [3.5] (0.64) [-2.7] [-0.8] [-1.2] [-0.3] [2.7] (0.00)
No Overlap 7.0 5.3 -0.01 0.21
[7.1] [3.9] (0.50) [-0.1] [2.6] (0.01)
Krnl: 6.2 6.2 6.8 6.7 5.5 -0.05 -0.03 -0.11 -0.02 0.21
Bartlett [12.7] [7.7] [8.5] [7.8] [4.1] (0.70) [-2.5] [-0.5] [-1.2] [-0.3] [2.9] (0.00)
Krnl: Unit 6.4 6.1 6.7 6.7 4.0 -0.09 -0.06 -0.12 -0.02 0.44
Closest Half [12.6] [3.5] [7.5] [7.5] [2.1] (0.44) [-3.3] [-0.3] [-1.5] [-0.2] [2.7] (0.00)
Cond Mean: 7.9 7.4 7.7 6.5 5.2 -0.05 -0.05 -0.08 -0.00 0.20
Look Ahead [20.2] [9.7] [9.0] [8.1] [2.9] (0.14) [-2.2] [-0.8] [-0.8] [-0.0] [1.8] (0.06)
Cond Mean: 6.1 6.5 7.6 7.4 5.2 -0.05 -0.02 -0.08 -0.04 0.21
Backward [12.0] [6.2] [6.5] [6.9] [2.0] (0.39) [-2.4] [-0.3] [-0.8] [-0.5] [0.9] (0.58)
Cond Mean: 6.4 6.1 6.5 6.3 4.5 -0.03 0.01 -0.04 -0.03 0.14
Hist All [26.8] [11.2] [8.0] [7.9] [2.2] (0.79) [-1.2] [0.1] [-0.4] [-0.3] [0.5] (0.81)
Diff HF 6.1 6.6 6.8 5.2 -0.05 -0.07 -0.05 0.24
Frequency [13.1] [8.1] [8.1] [3.5] (0.56) [-2.6] [-1.2] [-0.6] [2.8] (0.00)
Den Limit: 6.2 6.4 6.6 6.6 5.5 -0.06 -0.04 -0.07 -0.02 0.21
0.5 [13.5] [8.0] [8.2] [7.5] [3.3] (0.80) [-2.7] [-0.7] [-1.0] [-0.3] [2.8] (0.00)
Den Limit: 6.2 6.4 6.8 6.8 5.2 -0.07 -0.05 -0.11 -0.05 0.23
0.1 [12.5] [7.9] [8.1] [8.0] [3.4] (0.71) [-3.0] [-0.8] [-1.0] [-0.5] [2.3] (0.00)
65
E Robustness of Unconditional Term Structure Results to Differ-
ent Reformation and Rebalancing Periods
This appendix evaluates the robustness of the unconditional alpha and beta term structure to
changes in portfolio reformation and rebalancing periods. I focus on unconditional results since
they are least dependent on methodology assumptions. I also use simple returns and examine the
long and short legs separately. Alphas and betas are linearly additive when expressed as simple
(but not log) returns. This sheds light on which of the legs are driving the term structure shapes
of SMB, HML, and UMD. Overall, the paper’s main empirical results are robust to alternative
specifications of the reformation and rebalancing period. The appendix also shows that SMB term
structure patterns are almost entirely driven by small caps while UMD patterns are mostly driven
by losers although winners have interesting term structure dynamics too.
In the construction and investment of SMB, HML, and UMD, four distinct frequency concepts exist.
Briefly, the first deals with which characteristics are sorted in a particular portfolio (e.g. winners
based on prior year of returns), the second with how long stocks of that characteristic are held before
resorting occurs (e.g. past winners held for one month at a time), the third with how often different
portfolios that constitute a single long-short strategy are rebalanced (e.g monthly rebalancing
between winner and loser portfolios), and the fourth with how long that strategy is held by the
investor (e.g. 10-year investment in a long-short momentum strategy).
To begin, the sorting criteria can be frequency-related. In all cases, I follow the methodology listed
on Ken French’s website. For sorting on momentum, I use prior 2-12 month returns. For sorting on
market value, I use the latest available month-end data. For sorting on book-to-market, I use fiscal
year-end data from December of the prior year and assume it’s available by the end of June of the
current year. These methodology choices remain constant in all robustness scenarios.
Second, the reformation or holding period can change within the 4 to 6 characteristic-sorted
portfolios that constitute SMB, HML, and UMD. Half are long, and half are short. For example,
HML’s long-leg averages the small-value and large-value portfolios while it’s short leg averages the
small-growth and large-growth portfolios. These individual portfolios are reformed on a monthly
basis for UMD using updated prior 2-12 month returns and on an annual basis (at the end of June)
for SMB and HML using updated market-value and book-to-market data. In this appendix, besides
the monthly reformation base case for UMD, I also explore longer quarterly and annual periods.
This reduces turnover costs but results in a momentum strategy that’s less frequently refreshed
and is slower to capture recent winners and losers. Besides the annual reformation base case for
SMB and HML, I also explore a shorter quarterly period and a longer 3-year period. For quarterly
reformation, only market value data is updated since I use fiscal year-end book-value that’s only
available annually.
Third, periodic rebalancing occurs across the 4 to 6 characteristic-sorted portfolios that constitute
SMB, HML, and UMD. Reformation trades stocks within a portfolio while rebalancing trades
entire portfolios. Under the base case when using Ken French’s monthly returns for SMB, HML,
and UMD, monthly compounding implicitly results in monthly rebalancing. For example, 2-
month horizon returns from compounding two 1-month simple excess returns is equivalent to
implementing a long-short strategy for the first month, rebalancing to equal weights between the
long and shorts legs, and then repeating the strategy for the second month. Using simple returns:
r
ex,rebal
t→t+2
= [1 +r
ex
t→t+1
]× [1 +r
ex
t+1→t+2
]− 1 = [1 +r
long
t→t+1
−r
short
t→t+1
]× [1 +r
long
t+1→t+2
−r
short
t+1→t+2
]− 1 =
[1+(1+r
long
t→t+1
)−(1+r
short
t→t+1
)]×[1+(1+r
long
t+1→t+2
)−(1+r
short
t+1→t+2
)]−1. Beside monthly rebalancing
66
under the base case, I also consider annual rebalancing.
Finally, the investment horizon that is this paper’s focus can vary. This is the time period for which
a specific investment strategy is implemented. The strategy need not be in a fixed asset but can
be dynamic and have turnover from preset rules for periodic reformation and rebalancing. The
entire strategy must be liquidated for the investment horizon to end. As always, I explore monthly,
quarterly, annual, 3-year, and 10-year horizons for all robustness scenarios.
All results in this appendix are obtained from analyzing merged CRSP and Compustat data
(supplemented with Davis et al. (2000) book-value data) with the goal of replicating as close as
possible the SMB, HML, and UMD methodology described in Ken French’s website. Because this
replication isn’t perfect, the results under the base case using simple returns can deviate from those
shown in Appendix D Table D1 for unconditional results estimated using Ken French’s returns
directly. For SMB and HML, the base case is annual reformation with monthly rebalancing, and
the discrepancy here is most notable for SMB alphas that are smaller by 0.4~0.7% annually. This is
driven by my inability to replicate average returns of small caps. Betas are broadly consistent, and
overall term structure patterns are also very similar. For UMD, the base case monthly reformation
with monthly rebalancing alphas deviates by at most 0.2% compared to results in Appendix D
Table D1. Except for smaller small cap alphas, the long and short legs of the portfolios under the
base case reformation and rebalancing periods are also largely consistent with Appendix D Table
D1.
This appendix also shows sensitivity results for both the long and short legs of each strategy. Note
that each leg are averages of two portfolios for HML and UMD and averages of three portfolios
for SMB. I first subtract the risk-free rate before estimating alphas and betas to ensure they are
all excess returns. When using simple (and not log) returns, linearity for alphas and betas hold if
the examined investment horizon matches the rebalancing frequency. In other words, SMB results
should equal small cap minus large cap results. In the tables, this relationship may not hold exactly
due to rounding and return annualization. For longer horizons, this relationship no longer holds
theoretically because rebalancing occurs between the long and short legs in the interim. For example,
10-year SMB results may not equal 10-year small cap minus 10-year large cap results, even without
rounding or annualizing.
TableE1: RobustnesstoChangesinPortfolioReformationandRebalancingFrequency
Data from 1927-2015 used to form non-overlapping monthly, quarterly, annual returns and annual-overlapping
3-year and 10-year returns. Simple (not log) returns are used. Unconditional alphas (annualized percentages)
and betas are estimated. The table varies the reformation and rebalancing periods for constructing SMB,
HML, and UMD. Reformation occurs when trading occurs *within* the 4-6 portfolios that constitute SMB,
HML, and UMD. Rebalancing occurs when trading occurs *across* the portfolios that constitute each factor.
Under the base case (following Ken French), SMB and HML is reformed annually and rebalanced monthly
while UMD is reformed monthly and rebalanced monthly. T-statistics shown in brackets use Newey-West ’94
standard errors. Boldfaced coefficients denote significant differences from zero at the two-sided 95% level.
P-value reflects Wald Test of null hypothesis that all 5 horizon coefficients are equal (if specification includes
blank horizons, assume those equal to base case).
67
(a) Size (SMB)
Reform Rebal Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Qt Mn 0.6 -0.1 0.3 1.4 4.1 0.21 0.28 0.26 0.18 -0.15
[0.5] [-0.0] [0.2] [0.5] [2.1] (0.40) [7.5] [8.1] [4.4] [1.1] [-1.8] (0.00)
Yr Mn 0.7 -0.1 0.3 1.3 3.9 0.20 0.30 0.26 0.20 -0.14
[0.7] [-0.1] [0.2] [0.5] [2.3] (0.41) [6.4] [7.4] [4.0] [1.1] [-1.9] (0.00)
3Y Mn 1.5 0.7 1.2 2.3 4.6 0.19 0.28 0.22 0.15 -0.16
[1.2] [0.5] [1.0] [1.0] [1.9] (0.46) [5.2] [5.0] [4.7] [0.9] [-1.7] (0.00)
Qt Yr 0.6 -0.1 0.5 1.8 4.6 0.21 0.28 0.28 0.20 -0.17
[0.5] [-0.0] [0.2] [0.6] [2.2] (0.18) [7.5] [8.1] [4.8] [1.0] [-1.9] (0.00)
Yr Yr 0.7 -0.1 0.5 1.8 4.6 0.20 0.30 0.29 0.21 -0.17
[0.7] [-0.1] [0.3] [0.6] [2.5] (0.15) [6.4] [7.4] [4.1] [1.0] [-2.0] (0.00)
3Y Yr 1.5 0.7 1.3 2.6 5.2 0.19 0.28 0.26 0.18 -0.18
[1.2] [0.5] [0.9] [1.0] [1.9] (0.47) [5.2] [5.0] [4.3] [0.9] [-1.6] (0.00)
(b) Small Caps (SMB Long-Leg)
Reform Rebal Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Qt Mn 1.2 0.5 1.2 1.7 5.2 1.27 1.45 1.33 1.37 0.94
[1.1] [0.4] [0.6] [0.6] [2.2] (0.30) [36.3] [14.2] [16.9] [4.2] [3.2] (0.08)
Yr Mn 1.3 0.3 1.2 1.6 4.8 1.26 1.49 1.33 1.39 0.98
[1.1] [0.3] [0.6] [0.6] [2.2] (0.30) [27.2] [9.4] [14.3] [4.0] [3.5] (0.20)
3Y Mn 1.7 0.7 1.6 1.9 5.2 1.25 1.48 1.32 1.37 0.98
[1.6] [0.6] [1.0] [0.8] [2.0] (0.43) [26.3] [9.1] [15.2] [4.0] [3.8] (0.12)
Qt Yr 1.2 0.5 1.3 1.9 5.5 1.27 1.45 1.32 1.35 0.92
[1.1] [0.4] [0.6] [0.7] [2.1] (0.29) [36.3] [14.2] [17.3] [4.2] [3.2] (0.07)
Yr Yr 1.3 0.3 1.4 1.9 5.2 1.26 1.49 1.32 1.37 0.96
[1.1] [0.3] [0.7] [0.7] [2.2] (0.27) [27.2] [9.4] [14.2] [4.0] [3.5] (0.17)
3Y Yr 1.7 0.7 1.7 2.2 5.5 1.25 1.48 1.31 1.35 0.97
[1.6] [0.6] [1.0] [0.9] [2.1] (0.36) [26.3] [9.1] [15.5] [4.0] [4.0] (0.13)
(c) Large Caps (SMB Short-Leg)
Reform Rebal Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Qt Mn 0.6 0.3 0.8 0.9 0.8 1.06 1.12 1.04 1.03 1.12
[1.3] [0.5] [1.6] [1.7] [1.5] (0.62) [31.9] [14.7] [22.4] [31.7] [26.2] (0.03)
Yr Mn 0.6 0.3 0.7 0.9 0.8 1.06 1.12 1.04 1.04 1.12
[1.3] [0.5] [1.6] [1.7] [1.6] (0.62) [32.0] [14.9] [22.9] [32.0] [26.0] (0.05)
3Y Mn 0.3 -0.1 0.3 0.4 0.4 1.06 1.12 1.05 1.04 1.07
[0.6] [-0.1] [0.7] [0.9] [1.1] (0.63) [31.1] [14.5] [22.2] [39.8] [30.5] (0.04)
Qt Yr 0.6 0.3 0.9 1.0 0.9 1.06 1.12 1.03 1.03 1.13
[1.3] [0.5] [1.8] [1.8] [1.5] (0.45) [31.9] [14.7] [22.9] [30.0] [26.2] (0.02)
Yr Yr 0.6 0.3 0.8 1.0 0.9 1.06 1.12 1.04 1.03 1.12
[1.3] [0.5] [1.7] [1.7] [1.6] (0.46) [32.0] [14.9] [23.3] [30.4] [26.1] (0.03)
3Y Yr 0.3 -0.1 0.4 0.5 0.5 1.06 1.12 1.05 1.03 1.08
[0.6] [-0.1] [0.8] [1.0] [1.1] (0.45) [31.1] [14.5] [21.8] [39.0] [31.3] (0.05)
68
(d) Value (HML)
Reform Rebal Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Qt Mn 4.0 3.4 4.4 5.0 5.5 0.15 0.25 0.11 0.04 -0.03
[2.7] [2.0] [3.1] [3.0] [7.2] (0.78) [1.7] [1.4] [1.0] [0.4] [-0.3] (0.65)
Yr Mn 3.5 3.3 4.3 4.7 4.9 0.13 0.17 0.04 0.00 -0.02
[2.4] [2.1] [3.1] [2.6] [5.5] (0.82) [1.6] [1.3] [0.5] [0.0] [-0.2] (0.62)
3Y Mn 2.0 1.7 2.3 2.6 3.2 0.15 0.20 0.10 0.07 0.02
[1.5] [1.1] [1.7] [1.5] [3.9] (0.82) [1.8] [1.5] [1.3] [0.8] [0.2] (0.76)
Qt Yr 4.0 3.4 4.4 4.9 5.1 0.15 0.25 0.14 0.08 0.04
[2.7] [2.0] [3.0] [2.8] [5.6] (0.81) [1.7] [1.4] [1.2] [0.7] [0.3] (0.83)
Yr Yr 3.5 3.3 4.2 4.5 4.5 0.13 0.17 0.06 0.05 0.06
[2.4] [2.1] [2.8] [2.4] [3.8] (0.83) [1.6] [1.3] [0.6] [0.3] [0.5] (0.84)
3Y Yr 2.0 1.7 2.5 2.7 3.0 0.15 0.20 0.09 0.09 0.07
[1.5] [1.1] [1.9] [1.5] [2.9] (0.80) [1.8] [1.5] [1.1] [0.9] [0.5] (0.81)
(e) Value Stocks (HML Long-Leg)
Reform Rebal Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Qt Mn 2.7 1.7 2.8 3.4 5.2 1.27 1.49 1.29 1.31 1.36
[2.1] [1.0] [1.8] [1.8] [2.5] (0.53) [16.0] [7.8] [12.5] [5.0] [3.4] (0.53)
Yr Mn 2.5 1.6 2.7 3.3 4.8 1.26 1.47 1.26 1.28 1.37
[2.0] [1.0] [1.7] [1.7] [2.3] (0.57) [15.3] [7.8] [12.7] [5.2] [3.6] (0.54)
3Y Mn 1.7 0.8 1.8 2.2 3.9 1.26 1.47 1.28 1.28 1.27
[1.6] [0.5] [1.4] [1.3] [2.1] (0.57) [15.7] [7.7] [12.8] [6.7] [3.9] (0.49)
Qt Yr 2.7 1.7 3.1 3.7 5.6 1.27 1.49 1.27 1.29 1.33
[2.1] [1.0] [1.9] [1.8] [2.6] (0.41) [16.0] [7.8] [12.5] [4.9] [3.3] (0.50)
Yr Yr 2.5 1.6 3.0 3.6 5.2 1.26 1.47 1.25 1.27 1.35
[2.0] [1.0] [1.8] [1.8] [2.5] (0.46) [15.3] [7.8] [12.6] [5.3] [3.5] (0.51)
3Y Yr 1.7 0.8 2.0 2.5 4.3 1.26 1.47 1.26 1.26 1.25
[1.6] [0.5] [1.5] [1.4] [2.2] (0.43) [15.7] [7.7] [13.0] [6.6] [3.9] (0.46)
(f) Growth Stocks (HML Short-Leg)
Reform Rebal Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Qt Mn -1.3 -1.2 -1.3 -1.2 -0.1 1.12 1.13 1.13 1.08 0.84
[-1.9] [-1.5] [-1.7] [-1.1] [-0.1] (0.85) [41.5] [25.7] [32.2] [12.5] [17.3] (0.00)
Yr Mn -1.0 -1.3 -1.2 -1.2 0.1 1.13 1.22 1.19 1.16 0.87
[-1.5] [-1.6] [-1.6] [-1.3] [0.1] (0.85) [55.8] [35.5] [33.6] [7.3] [16.3] (0.00)
3Y Mn -0.2 -0.6 -0.5 -0.4 1.0 1.12 1.20 1.17 1.13 0.89
[-0.4] [-0.8] [-0.6] [-0.4] [0.7] (0.79) [60.3] [29.7] [30.1] [8.6] [11.9] (0.00)
Qt Yr -1.3 -1.2 -1.3 -1.2 -0.2 1.12 1.13 1.13 1.08 0.84
[-1.9] [-1.5] [-1.6] [-1.0] [-0.1] (0.85) [41.5] [25.7] [31.5] [12.6] [16.8] (0.00)
Yr Yr -1.0 -1.3 -1.2 -1.2 0.1 1.13 1.22 1.19 1.16 0.87
[-1.5] [-1.6] [-1.5] [-1.2] [0.1] (0.85) [55.8] [35.5] [31.6] [7.1] [15.2] (0.00)
3Y Yr -0.2 -0.6 -0.5 -0.4 1.0 1.12 1.20 1.17 1.13 0.89
[-0.4] [-0.8] [-0.6] [-0.3] [0.7] (0.80) [60.3] [29.7] [29.7] [8.6] [12.5] (0.00)
69
(g) Momentum (UMD)
Reform Rebal Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Mn Mn 11.0 11.9 9.6 8.1 6.6 -0.30 -0.34 -0.16 -0.00 0.25
[7.3] [7.8] [5.1] [3.4] [2.2] (0.04) [-2.6] [-3.5] [-1.9] [-0.0] [1.6] (0.00)
Qt Mn 9.3 10.4 8.3 6.8 5.0 -0.28 -0.35 -0.16 0.00 0.27
[6.0] [6.4] [4.2] [3.0] [1.7] (0.00) [-2.4] [-3.2] [-1.9] [0.0] [2.1] (0.00)
Yr Mn 3.9 4.4 3.0 1.6 2.1 -0.19 -0.22 -0.08 0.06 0.01
[2.8] [2.7] [2.4] [1.1] [1.0] (0.31) [-2.5] [-2.8] [-1.2] [0.6] [0.1] (0.03)
Mn Yr 11.0 11.9 10.1 8.7 8.0 -0.30 -0.34 -0.08 0.07 0.22
[7.3] [7.8] [7.7] [2.7] [2.8] (0.33) [-2.6] [-3.5] [-0.9] [0.5] [1.4] (0.00)
Qt Yr 9.3 10.4 8.5 6.9 5.6 -0.28 -0.35 -0.10 0.06 0.31
[6.0] [6.4] [6.4] [2.1] [1.8] (0.18) [-2.4] [-3.2] [-1.2] [0.4] [2.4] (0.00)
Yr Yr 3.9 4.4 3.6 2.2 2.6 -0.19 -0.22 -0.08 0.05 0.00
[2.8] [2.7] [3.1] [1.1] [1.1] (0.40) [-2.5] [-2.8] [-0.9] [0.4] [0.0] (0.02)
(h) Winners (UMD Long-Leg)
Reform Rebal Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Mn Mn 5.3 5.2 4.9 4.6 6.3 1.08 1.12 1.21 1.41 1.61
[6.2] [6.2] [4.6] [1.9] [3.3] (0.93) [27.9] [31.7] [17.6] [7.6] [7.7] (0.01)
Qt Mn 4.3 4.3 4.0 3.8 5.2 1.09 1.10 1.19 1.32 1.52
[5.1] [4.5] [4.2] [1.8] [2.5] (0.95) [28.5] [23.7] [23.2] [9.1] [6.7] (0.08)
Yr Mn 2.5 2.3 2.3 2.0 4.1 1.11 1.16 1.19 1.29 1.18
[2.9] [2.4] [2.4] [1.1] [2.4] (0.88) [33.0] [33.2] [20.4] [6.3] [4.1] (0.18)
Mn Yr 5.3 5.2 5.2 4.9 6.7 1.08 1.12 1.20 1.39 1.59
[6.2] [6.2] [4.5] [1.9] [3.3] (0.95) [27.9] [31.7] [18.5] [7.7] [7.8] (0.01)
Qt Yr 4.3 4.3 4.2 4.1 5.5 1.09 1.10 1.19 1.31 1.51
[5.1] [4.5] [4.2] [1.9] [2.5] (0.98) [28.5] [23.7] [23.7] [9.1] [6.6] (0.07)
Yr Yr 2.5 2.3 2.4 2.2 4.4 1.11 1.16 1.19 1.28 1.17
[2.9] [2.4] [2.4] [1.2] [2.5] (0.84) [33.0] [33.2] [20.9] [6.5] [4.1] (0.19)
(i) Losers (UMD Short-Leg)
Reform Rebal Alpha Beta
Mn Qt Yr 3Y 10Y Pval Mn Qt Yr 3Y 10Y Pval
Mn Mn -5.2 -5.9 -4.8 -4.1 -1.9 1.38 1.57 1.29 1.11 0.63
[-5.0] [-4.1] [-3.7] [-3.0] [-1.1] (0.25) [17.0] [10.5] [21.0] [6.6] [4.1] (0.00)
Qt Mn -4.6 -5.4 -4.3 -3.5 -0.9 1.36 1.60 1.30 1.12 0.61
[-4.2] [-3.5] [-3.0] [-2.6] [-0.5] (0.19) [17.5] [9.4] [17.6] [6.4] [4.2] (0.00)
Yr Mn -1.4 -2.0 -1.3 -0.9 0.1 1.30 1.45 1.28 1.20 0.99
[-1.4] [-1.5] [-1.0] [-0.6] [0.1] (0.73) [22.9] [11.8] [17.4] [6.2] [8.3] (0.05)
Mn Yr -5.2 -5.9 -4.9 -4.2 -2.0 1.38 1.57 1.28 1.10 0.62
[-5.0] [-4.1] [-3.6] [-3.0] [-1.1] (0.35) [17.0] [10.5] [21.0] [6.5] [3.9] (0.00)
Qt Yr -4.6 -5.4 -4.3 -3.5 -1.0 1.36 1.60 1.29 1.11 0.61
[-4.2] [-3.5] [-3.0] [-2.5] [-0.5] (0.30) [17.5] [9.4] [17.6] [6.3] [4.0] (0.00)
Yr Yr -1.4 -2.0 -1.2 -0.8 0.2 1.30 1.45 1.27 1.18 0.98
[-1.4] [-1.5] [-0.9] [-0.5] [0.1] (0.70) [22.9] [11.8] [17.3] [6.2] [7.9] (0.05)
70
F Sensitivity of Optimal Portfolios and DCF Valuations to Hori-
zon Mismatch
This appendix, assesses the impact of using horizon-matched versus non-matched returns on the
construction of optimal portfolios (Table F1) and on DCF valuations (Table F2). I use simple
rather than log returns to ensure that the investment impact is not distorted by taking logs. Overall,
the Appendix shows that optimal portfolio formation and DCF present value analysis can be highly
distorted if the standard monthly beta, rather than the correct horizon-matched beta, is used.
Table F1: Impact on Optimal Portfolios of Horizon-Matched vs Monthly Returns Data
from 1927-2015 using unconditional estimates. Simple (not log) returns are used, and the risk-free rate
is added to all four excess return portfolios: ExMkt, SMB, HML, and UMD. The unit is percentages.
Implications for two potential optimal portfolios are explored: the Global Mean Variance portfolio and the
Maximum Sharpe Ratio portfolio. For each portfolio, I consider a true investment horizon of either 1-year or
10-years and estimate the optimal portfolio using horizon-matched returns. This is done for every year and
the resulting performance at the end of the investment horizon is recorded (leading to annual overlapping
results for the 10-year horizon). The table shows annualized summary statistics across all years. Utility
denotes the certainty-equivalent utility calculated according to Campbell, Thompson ’08 and assuming a
risk-aversion parameter of 5. Alternatively, I estimate optimal portfolios using monthly data, which are not
matched to the true investment horizon. These results are shown in a separate column, while a third column
shows the difference between the correctly-matched and monthly results. Panel (a) shows the standard
in-sample approach that uses all historical data to estimate portfolio means and covariances. Panel (b) shows
an out-of-sample approach using only ex-ante data that avoids look-ahead basis. Beginning with a 30-year
training period, at each subsequent year, the optimal portfolio is re-estimated using an expanding window
that takes advantage of each new year’s data. The table also shows portfolio weights. Panel (a)’s are constant
since they are estimated only once using all historical data, whereas Panel (b)’s are the weights averaged
across all years.
(a) Horizon-Matched Estimates vs. Wrong Monthly Ones (In-Sample)
True Horizon: Yr True Horizon: 3Y True Horizon: 10Y
Yr Mn Diff 3Y Mn Diff 10Y Mn Diff
Global Mean 9.1 9.3 -0.2 8.5 8.6 -0.0 8.6 10.3 -1.7
Min SD 7.8 7.8 -0.0 10.6 11.0 -0.4 17.9 25.0 -7.1
Variance Sharpe 1.2 1.2 -0.0 0.8 0.8 0.0 0.5 0.4 0.1
Utility 7.6 7.8 -0.2 5.7 5.5 0.2 0.6 -5.4 5.9
Wgt.Mkt 8.8 9.2 -0.5 19.7 9.2 10.4 41.1 9.2 31.9
Wgt.SMB 31.7 26.6 5.1 21.1 26.6 -5.5 27.3 26.6 0.7
Wgt.HML 31.7 34.4 -2.7 40.2 34.4 5.8 55.4 34.4 21.0
Wgt.UMD 27.8 29.8 -1.9 19.0 29.8 -10.8 -23.9 29.8 -53.6
Max Mean 9.9 10.0 -0.1 9.1 9.1 -0.0 9.8 10.8 -1.0
Sharpe SD 8.1 8.2 -0.1 10.9 11.3 -0.4 19.4 25.9 -6.4
Ratio Sharpe 1.2 1.2 0.0 0.8 0.8 0.0 0.5 0.4 0.1
Utility 8.3 8.3 -0.0 6.1 5.9 0.2 0.4 -5.9 6.3
Wgt.Mkt 17.1 16.0 1.1 26.4 16.0 10.4 44.5 16.0 28.6
Wgt.SMB 17.8 14.8 3.0 10.6 14.8 -4.2 5.6 14.8 -9.2
Wgt.HML 30.6 33.9 -3.3 37.9 33.9 4.0 57.8 33.9 23.9
Wgt.UMD 34.5 35.4 -0.9 25.1 35.4 -10.3 -7.9 35.4 -43.3
71
(b) Horizon-Matched Estimates vs. Wrong Monthly Ones (Out-of-Sample)
True Horizon: Yr True Horizon: 3Y True Horizon: 10Y
Yr Mn Diff 3Y Mn Diff 10Y Mn Diff
Global Mean 10.4 10.7 -0.3 9.7 9.8 -0.1 9.2 11.6 -2.4
Min SD 8.0 7.4 0.6 11.0 11.1 -0.0 20.9 25.8 -5.0
Variance Sharpe 1.3 1.5 -0.2 0.9 0.9 -0.0 0.4 0.4 -0.0
Utility 8.8 9.3 -0.5 6.6 6.7 -0.1 -1.7 -5.1 3.4
Wgt.Mkt -0.1 7.3 -7.5 7.8 7.3 0.5 26.6 7.3 19.3
Wgt.SMB 33.0 26.4 6.6 23.0 26.4 -3.4 37.2 26.4 10.8
Wgt.HML 37.2 32.7 4.5 44.9 32.7 12.3 41.3 32.7 8.6
Wgt.UMD 30.0 33.6 -3.6 24.2 33.6 -9.4 -5.1 33.6 -38.7
Max Mean 11.0 11.2 -0.2 10.1 10.3 -0.3 10.1 12.3 -2.2
Sharpe SD 7.6 7.4 0.2 10.7 10.9 -0.2 22.4 25.6 -3.2
Ratio Sharpe 1.4 1.5 -0.1 0.9 1.0 -0.0 0.5 0.5 -0.0
Utility 9.5 9.8 -0.3 7.2 7.4 -0.2 -2.4 -4.0 1.6
Wgt.Mkt 9.5 14.4 -4.9 16.2 14.4 1.8 30.4 14.4 16.0
Wgt.SMB 19.0 13.4 5.6 12.7 13.4 -0.6 24.2 13.4 10.8
Wgt.HML 34.9 33.0 1.9 42.5 33.0 9.5 41.8 33.0 8.8
Wgt.UMD 36.6 39.2 -2.5 28.6 39.2 -10.6 3.6 39.2 -35.6
Table F2: Impact on DCF of Horizon-Matched vs Monthly Betas Data from 1927-2015
using unconditional estimates. Simple (not log) returns are used. I value an investment with a future bullet
payout that has a 100 present value if it were discounted at the risk-free rate. I consider true investment
horizons of 1, 3, or 10 years, and investments with three different risk exposures. These risk exposures are
ones that mirror SMB’s, HML’s, or UMD’s CAPM betas. The discount rate, R = 1 +RF +β× (Mkt−RF ),
yields the DCF value of
100(1+RF)
R
. The table shows results using the correct horizon-matched beta and the
incorrectly-matched monthly beta. For all results, I use the historical average risk-free rate and market return
of 3.5% and 11.8%, respectively.
True Horizon: Yr True Horizon: 3Y True Horizon: 10Y
Yr Mn Diff 3Y Mn Diff 10Y Mn Diff
SMB 98.2 98.5 -0.3 96.2 95.5 0.6 106.5 84.0 22.5
HML 99.6 98.8 0.7 99.3 96.6 2.8 95.0 87.3 7.7
UMD 101.0 102.4 -1.5 96.8 107.7 -10.9 62.1 141.4 -79.3
72
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Abstract (if available)

Abstract Using monthly returns to estimate portfolio alphas and betas is inappropriate for long-horizon investors or for discounting long-term cashflows. Alphas and betas do not necessarily have flat term structures. This paper develops a novel beta estimation method that is non-parametric, allows for time-variation, and better captures long-horizon estimates compared to the realized volatility or rolling regression approaches. Long-short portfolios sorted on size, value, and momentum have CAPM betas that can reverse sign with longer horizons. At multi-year horizons, the average alpha associated with size increases while momentum's decreases. These differences are economically significant for long-term investors.

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Asset Metadata

Creator Chang, Wayne L. (author)

Core Title The term structure of CAPM alphas and betas

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 06/16/2017

Defense Date 05/22/2017

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag CAPM,conditional alpha,investment horizon,momentum effect,OAI-PMH Harvest,risk-return,size effect,time-varying beta,value premium

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ferson, Wayne E. (committee chair), Ahern, Kenneth R. (committee member), Jones, Christopher S. (committee member), Subramanyam, K. R. (committee member)

Creator Email cyurmt@gmail.com,cyurmt@hotmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-383588

Unique identifier UC11258303

Identifier etd-ChangWayne-5412.pdf (filename),usctheses-c40-383588 (legacy record id)

Legacy Identifier etd-ChangWayne-5412.pdf

Dmrecord 383588

Document Type Dissertation

Rights Chang, Wayne L.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA