Smart Beta Is Not Monkey Business

Transcription

Smart Beta Is Not
Monkey Business
The Journal of Index Investing 2016.6.4:12-29. Downloaded from www.iijournals.com by Taras Zlupko on 03/09/16.
It is illegal to make unauthorized copies of this article, forward to an unauthorized user or to post electronically without Publisher permission.
Noël Amenc, Felix Goltz, and Ashish Lodh
Noël A menc
is a professor of finance at
EDHEC Risk Institute and
the CEO of ERI Scientific
Beta in Singapore.
noel.amenc@scientificbeta.com
Felix Goltz
is the head of applied
research at EDHEC Risk
Institute and the research
director at ERI Scientific
Beta in Nice, France.
felix.goltz@edhec.edu
Ashish L odh
is the deputy research
director at ERI Scientific
Beta in Nice, France.
ashish.lodh@scientificbeta.com
I
n the marketing of smart beta strategies,
index providers focus primarily on the
ability of these strategies to deliver outperformance over the cap-weighted
(CW) benchmark. The issue of the risk
exposure of these indexes and performance
attribution to well-defined risk factors is
rarely addressed by index providers. The
existence of so many smart beta strategies
coupled with so little information on their
sources of performance poses a risk of confusion and overgeneralizations.
Arnott et al. [2013, p. 91] claimed that
smart beta “necessarily results in value and
size tilts, regardless of the weighting method
chosen” and concluded that “the investment beliefs upon which many investment
strategies are ostensibly based play little or
no role in their outperformance.” Likewise,
Hsu, Kalesnik, and Li [2012, p. 12] wrote
that “outwardly different smart betas produce
nearly similar premiums for similar reasons.”
The argument that all smart beta strategies lead to all but identical performance and
risk factor exposures is further supported by
two claims put forward by Brightman [2013,
p. 5]. First, Brightman argued that “strategies premised on seemingly sensible investment beliefs […] add the same or more value
when inverted.” Second, the author argued
that smart beta strategies “add value […]
like Malkiel’s monkey” because their performance is similar to randomly generated
12 Smart Beta Is Not Monkey Business
portfolios, also termed monkey portfolios. Since
the idea that smart beta strategies are as good
as random deviations from cap-weighting is
at the heart of all the preceding claims, we
collectively refer to these as the monkey portfolio argument.
In this article, we report the results of a
series of straightforward tests of these claims.
To test the various claims, we distinguish the
three main claims made by monkey portfolio
proponents:
1. All smart beta strategies lead to similar
performance.
2.All smart beta strategies have unavoidable value and small cap tilts resulting
in performance that is similar across
strategies.
3.Smart beta strategies are as good as
inverse or upside-down strategies.
Our results are not supportive of the
monkey portfolio argument. We find that
various smart beta strategies display pronounced differences in performance characteristics and factor exposures. We also obtain
a reassuring finding that inverting a portfolio
strategy does not, in general, lead to the same
performance as the original.
Our f indings imply that analyzing
smart beta performance and risks is not
monkey business. For a better understanding
of smart beta strategies, it is crucial to analyze
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their construction principles, performance characteristics, and risk factor exposures—including not only value
and small-cap factors but also a variety of other welldocumented risk factors, such as momentum, profitability, investment, low risk, and possibly others.
OUR SET OF SMART BETA STRATEGIES
Obviously, whether or not the previously mentioned arguments hold may depend heavily on which
class of strategies one includes. Although their empirical
tests are limited to selected strategies, the authors putting forward the monkey portfolio arguments claim that
these results apply to smart beta strategies in general,
meaning that an analysis of any choice of smart beta
strategies should fulfill their claims.
Our selection of strategies focuses mainly on
explicit factor-tilted smart beta strategies, which correspond to indexes that providers have launched relatively
recently. In fact, the first generation of smart beta indexes
usually changed the weighting scheme from market cap
weighting, while paying no attention to explicitly controlling the exposures to systematic risk factors. Such
strategies provided implicit tilts to systematic factors.
More recently, many providers have launched factortilted indexes to extract factor premiums (explicit factor
tilts). For a discussion of this development, we refer
readers to Amenc and Goltz [2013] and Amenc, Goltz,
and Lodh [2012].
The increasing interest in factor-tilted smart beta
indexes is also due to the success of factor investing,
especially since the Norwegian Oil Fund report by Ang,
Goetzmann, and Schaefer [2009], which showed that
the returns relative to a cap-weighted benchmark of
the fund’s actively managed portfolio can be explained
by exposure to a set of well-documented alternative
risk factors.
Among possible strategies, we included a broad set
of smart beta strategies in our tests. First, we included the
popular fundamental-weighted portfolio strategy and
the equal-weighted strategy based on a broad universe.
Given that many monkey portfolio proponents (e.g.,
Hsu, Kalesnik, and Li [2012]; Arnott et al. [2013]) are
also promoters of fundamental-weighted indexes, it is
interesting to first check whether their general claims
apply to the type of smart beta strategy they promote.
Second, we included a variety of strategies that seek
explicit exposure to a given risk factor by selecting stocks
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with desired factor exposures. Such smart beta strategies
are being offered as so-called factor indexes by most major
index providers. We incorporated two approaches in
our tests: First, we included indexes that select stocks by
factor score and apply a diversification-based weighting
scheme (diversified factor indexes); and second, we included
indexes that, in addition to selecting stocks with the
highest factor exposure, also use a weighting that favors
stocks with desired characteristics (concentrated factor
indexes).
It should be noted that our set of strategies deliberately differs from the strategies tested by Arnott et al.
[2013], who did not include explicit factor-tilted strategies in their set of test portfolios. Our research can thus
be understood as a test of whether the claims of monkey
portfolio proponents apply only to the specifications of
smart beta the authors have selected for their empirical
tests or whether they apply more generally, to a broader
set of commonly used smart beta strategies. In particular,
given the recent interest in explicit factor-tilted strategies, it appears to be relevant to test the degree to which
the strong claims of monkey portfolio proponents carry
over to such strategies.
In fact, our article differs from Arnott et al. [2013]
in the three following ways. First, it considers additional
factor exposures of smart beta strategies (like low beta,
profitability, and investment). Second, it employs different portfolio construction techniques. Third, it provides a more detailed performance analysis including
factor risk contributions and dependencies of performance on market regimes. If such changes in tests lead
to differences in results with respect to those obtained
by Arnott et al. [2013], we would conclude that their
findings do not hold in general, but are specific to the
particular test setup and strategies that they tested.
Exhibit 1 provides an overview of the smart beta
strategies we have constructed for our tests. In addition to the broad fundamental-weighted strategy1 and
simple quarterly rebalanced equal-weighted strategy,
we constructed several factor-tilted smart beta strategies that select half the stocks in the universe to obtain
the desired factor tilt. For example, to obtain a value
tilt, our test portfolios select 50% of the stocks with the
highest book-to-market ratio.
In the same way, we constructed portfolios that
select stocks so as to tilt to momentum, size, low volatility,
profitability and investment. We applied two different
types of weighting schemes to these stock selections.
The Journal of Index Investing 13
Exhibit 1
Distinctions among Various Smart Beta Strategies Categorized by the Kind of Factor Exposure
(implicit vs. explicit) and Weighting Scheme
One method of designing factor-tilted portfolios is to
use a weighting scheme that aims at improved diversification. Many such weighting schemes exist, including
simple equal weighting, equal risk weighting, and optimization-based weighting schemes such as minimum
volatility. We followed an approach introduced by
Amenc et al. [2014] that applies five commonly used
alternative weighting schemes to portfolios that select
50% of stocks in the investment universe by desired
factor exposure, and then combine these different
weighting schemes with equal weights. We used this
weighting scheme, which is known as diversified multistrategy, for our first set of factor-tilted test portfolios.2
Our second set of factor-tilted smart beta strategies is
based on the same stock selections by factor exposure,
but uses the factor scores to determine constituent
weights3 among selected stocks. For example, in the case
of the value score–weighted portfolio, we selected the
top 50% value stocks from the stock universe and then
weighted them by adjusting their market cap weight by
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their value scores. Such score-weighting is employed by
factor indexes from a variety of major providers.4
The factor tilts we consider here relate to the most
widely used and documented factor tilts, namely low
volatility, momentum, size, value, high profitability,
and low investment.5 Additionally, for both diversified
multi-strategy and score-weighting, we constructed
multi-factor portfolios by combining the six factor-tilted
portfolios in equal proportion on a quarterly basis.
All factor-tilted portfolios are rebalanced quarterly. All strategies are applied to the U.S. large-cap
stock universe (500 stocks) over a period of 40 years
(December 31, 1973, to December 31, 2013). To avoid
any hindsight bias, all parameters used in selecting and
weighting the stocks are based on data observed prior
to each respective rebalancing date. All stock price data
were obtained from the Center for Research in Security Prices (CRSP) database, and fundamental data were
obtained from WorldScope.
HOW SIMILAR IS SMART BETA
PERFORMANCE ACROSS STRATEGIES?
A key argument of monkey portfolio proponents
is that all smart beta strategies are similar. They have
argued that “outwardly different smart betas produce
nearly similar premiums for similar reasons” (Hsu,
Kalesnik, and Li [2012, p. 12]) and “any one of these
strategies can be mimicked by combinations of the
others” (Chow et al. [2011, p. 37]).
A simple way of assessing such claims is to compare
returns across the different strategies. However, it is necessary to look beyond a simple comparison of long-term
average returns because concluding that all smart beta
strategies are similar to each other simply because the
long-term excess returns look similar is not justified.
In this section, we provide an overview of descriptive
performance statistics including not only unconditional
performance, but also performance in different market
conditions. We also look directly at the similarity of
strategies by assessing the correlation of relative returns
across different strategies.
Exhibit 2 shows basic descriptive statistics for the
different test portfolios. Concerning their long-term
average returns, the strategies are similar in the sense that
they all outperform cap-weighted indexes, which may
lead at first glance to the monkey portfolio conclusion
that all strategies are similar. However, it also appears
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that some performance differences exist over the long
term, especially among smart beta strategies that tilt
toward different risk factors. For example, the size-tilted
(i.e., mid cap) portfolios have much higher returns and
Sharpe ratios compared to their low volatility counterparts. In the diversified multi-strategy category, the
outperformance ranges from an annualized average of
4.72% for mid cap tilt to 2.95% for the low volatility
tilt. Similarly, in the score weighting category, the mid
cap tilt results in 4.45% outperformance compared to a
mere 0.54% for the low volatility tilt. The fundamentalweighted strategy, during the same period, returns 1.56%
of excess return over the broad cap-weighted index, and
the equal-weighted index returns 2.87% over the capweighted index.
More striking than differences across long-term
returns and Sharpe ratios are differences in conditional
returns. In particular, the results in Exhibit 2 reveal
that the performance differs across various smart beta
strategies in bull and bear market conditions depending
on the risk exposure of the strategy. From a relative
returns perspective, the fundamental-weighted strategy
is more favored in bear markets, whereas the equalweighted strategy performs better in bull markets.
Differences are even more pronounced for strategies
that provide explicit factor tilts. For example, the low
volatility–tilted score-weighted strategy generates
outperformance mainly in bear markets, whereas the
corresponding size-tilted strategy generates outperformance mainly in bull markets. To be more specific, the
low volatility score–weighted strategy returns almost
9% relative return during bear markets but underperforms by more than 5% during bull markets. In contrast,
the corresponding size tilt (mid cap score–weighted)
displays outperformance of almost 7% in bull markets
but only adds about 1% of returns during bear markets. This observation is quite intuitive because risk factors carry time varying risk premiums (Asness [1992];
Cohen, Polk, and Vuolteenaho [2003]), and therefore
all risk factors must not be expected to outperform in
the same periods or by the same magnitude. Overall,
although long-term performance may be similar across
many strategies, conditional performance is quite different across different factor-tilted smart beta indexes.
A simpler and more intuitive way to understand
the difference across the set of smart beta strategies is
to examine the correlation matrix of excess returns
over the cap-weighted benchmark. Exhibit 3 shows the
Exhibit 2
Absolute, Relative, and Conditional Performance
Notes: All statistics are annualized, and daily total returns from December 31, 1973 to December 31, 2013 are used for the analysis. The CRSP S&P 500
index is used as the cap-weighted benchmark. Yield on Secondary U.S. Treasury bills (3M) is used as a proxy for the risk-free rate. Calendar quarters with
positive cap-weighted index returns comprise bull markets, and the rest constitute bear markets. Effective number of stocks, a measure of portfolio concentration,
is the inverse of Herfindahl Index, which in turn is defined as the sum of square of portfolio weights. We show the average effective number over all quarterly
rebalancing dates.
Pearson correlation coefficients of the relative returns of
our test strategies over the cap-weighted reference index.
The results in Exhibit 3 suggest that the correlations across different smart beta strategies are far less
than one (i.e., the strategies are far from identical in
terms of their relative returns over time). Low correlations of relative returns are even observed across
strategies that do not provide explicit factor tilts and
thus correspond to strategies that were included in the
monkey portfolio proponents’ original tests. In particular, the fundamental-weighted strategy’s and the
equal-weighted portfolio’s correlation of excess returns
is a mere 0.51 (i.e., the two strategies are far from similar). Likewise, when it comes to strategies with explicit
factor tilts, the levels of correlation are far below one.
Some pairwise correlations of relative returns across
factor-tilted strategies are even negative. These findings
show that although all of these strategies outperform
cap-weighted indexes, they do so with very different
behavior over time.
Overall, although monkey portfolio proponents
claim that “the investment beliefs upon which many
investment strategies are ostensibly based play little
or no role in their outperformance” (Arnott et al.
[2013, p. 91]), it is evident from simple descriptive
statistics that different strategies based on different
investment beliefs do lead to different returns. This is
true in particular for different factor tilts, which translate
different investment beliefs and lead to indexes that have
different conditional performance and far from perfect
correlation of relative returns.
DO ALL SMART BETA STRATEGIES
OUTPERFORM SOLELY DUE TO SIZE
AND VALUE LOADINGS?
Monkey portfolio proponents have argued that,
once we deviate from selecting and weighting stocks
by their market cap, as is done in cap-weighted market
indexes, we necessarily introduce a positive value and
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Exhibit 3
Correlation of Excess Returns of Smart Betas
Notes: All statistics are annualized, and daily total returns from December 31, 1973 to December 31, 2013 are used for the analysis. The CRSP S&P 500
index is used as the cap-weighted benchmark.
a positive size factor exposure (Chow et al. [2011]).
These proponents have also argued that the same
effect would occur with randomly generated portfolios, also referred to as monkey portfolios—see also
Clare, Motson, and Thomas [2013]. Similarly, Arnott
et al. [2013, p. 91] stated that any smart beta strategy
“necessarily results in value and size tilts, regardless of
the weighting method chosen,” and Chow et al. [2011,
p. 41] claimed that smart beta strategies “outperform
because of the positive value and size loadings” given
that “none of these strategies are different from naive
equal weighting.”
Such claims seem at least surprising, given the
well-documented findings on relevant risk factors in
portfolio performance. For example, it has been shown
that risk-based strategies (e.g., minimum variance and
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equal risk contribution) take on exposures to low beta
and low idiosyncratic risk factors (Leote de Carvalho,
Lu, and Moulin [2012]; Clarke, de Silva, and Thorley
[2013]). Moreover, there is ample evidence that stock
portfolios seeking exposures to momentum, quality,
or low beta factors generate returns that cannot be
explained by the value and small cap factors (Asness,
Frazzini, and Pedersen [2013]; Asness, Moskowitz,
and Pedersen [2013]; Frazzini and Pedersen [2014]).
To assess the role of such additional factors in commonly
used smart beta strategies relative to the value and size
factor, we performed regressions using a multi-factor
model including a wide range of consensual equity factors. In particular, we used a seven-factor model that uses
the betting-against-beta (BAB) factor from Frazzini and
Pedersen [2014] 6 and the high profitability factor and
low investment factor from Fama and French [2015] in
addition to the four factors of Carhart [1997], namely
the market, value, size, and momentum factors.7 The
resulting regression coefficients are reported in Exhibit 4.
If the monkey portfolio claims hold for our test
portfolios, regression coefficients for size and value factors should be significantly positive for all strategies.
Moreover, regression coefficients for all factors but size
and value should be insignificant. Finally, in comparison
with equal-weighting, none of the strategies should provide a different direction of factor tilt. However, inspection of the results in Exhibit 4 shows that none of these
conditions is satisfied by the results.
Exhibit 4 shows that, even though most strategies
have positive exposure to size and value, some strategies
do have negative exposure, sometimes to both of these
factors. This is notably the case for the low volatility
and high profitability score–weighted strategies. Despite
deviating from the cap-weighted index, these strategies
do not lead to a value and small cap tilt, but rather to
a growth and large cap tilt. Therefore, it is not valid to
claim that all smart beta strategies have positive value
and size loadings, even though many strategies do.
Moreover, given the negative exposure of some strategies to value and small cap, it would not make sense to
claim that the outperformance of these portfolios is fully
explained by their size and value exposure.
The results also show that most strategies have significant exposures to factors other than value and small
cap, in particular to the BAB factor, the momentum
factor, the profitability factor, and the investment factor.
This is not surprising given the ample evidence on the
importance of these factors and the fact that some indexes
in our set of strategies explicitly seek exposure to such
factors. For example, the high profitability strategies
(diversified multi-strategy and score weighted) lead to
profitability factor loadings of 0.08 and 0.19, respectively;
the low-investment strategies (diversified multi-strategy
Exhibit 4
Seven-Factor Regression
Notes: The market factor is the excess returns of the CRSP S&P 500 index over the risk-free rate. The yield on secondary U.S. Treasury bills (3M) is
used as a proxy for the risk-free rate. The size, value, momentum, high profitability, and low investment factors were obtained from the Kenneth French
data library. The BAB factor was obtained from the Andrea Frazzini data library. The Newey–West (Newey and West [1987]) estimator is used to
correct for autocorrelation. Daily total returns from December 31, 1973 to December 31, 2013 are used for the analysis. Regression coefficients that have
P-values less than 5% are highlighted in bold. Alphas are annualized.
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and score weighted) lead to investment factor loadings of
0.32 and 0.52; and the low volatility strategies (diversified
multi-strategy and score weighted) lead to BAB factor
loadings of 0.10 and 0.09, respectively. The momentum
diversified multi-strategy portfolio has a momentum beta
of 0.17, whereas the momentum score–weighted strategy
has an even higher momentum beta of 0.40. Even when
these strategies also tilt to value and size, the other factor
exposures of these strategies are important in magnitude
relative to the size and value exposures. However unsurprising they may be, these results invalidate the monkey
portfolio proponents’ claim that smart beta does nothing
but tilt to small cap and value.
It is also interesting to assess whether all strategies really
behave like equal-weighting in terms of factor exposure.
In fact, the results in Exhibit 4 confirm the monkey portfolio proponents’ findings that equal-weighting leads to
positive size and value exposure. Unfortunately, though,
monkey portfolio proponents have not provided any discussion of other factor tilts implicit in equal-weighted
indexes. Exhibit 4 shows that the broad equal-weighted
index also has negative momentum exposure. It is interesting to compare this to the momentum exposures
of the other strategies. Although the fundamentalweighted portfolio also has negative momentum
exposure, seven of the twelve single factor–tilted strategies have significantly positive momentum exposure.
Our results thus provide evidence that most of these strategies are opposed to equal-weighting in terms of momentum
exposure. Moreover, the equal-weighted strategy does not
have significant exposures to the BAB factor and the profitability factor. On the contrary, many of the factor-tilted
strategies have significant exposure the BAB and high
profitability factor. In this sense, the factor-tilted indexes
offer investment opportunities that cannot be captured by
an equal-weighted index. Such factor-tilted indexes allow
investors to express investment beliefs (concerning factors) that cannot be captured through the equal-weighted
index. The factor exposures in Exhibit 4 directly invalidate
monkey portfolio proponents’ claim that no smart beta
strategy is any different from equal weighting.
To provide more intuition on the role of different
factors in the performance and risk of different smart
beta strategies, we complement our analysis of regression coefficients with attributions of the relative return
and tracking error of these strategies to their factor
exposures. The reason for this additional analysis is that
factor coefficients may not be easy to interpret because
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the contributions that a factor makes to the risk and
return of a strategy of course depend not only on the
factor loading of the strategy, but also on the risk and
return properties of the factor.
Exhibit 5 shows the breakdown of the excess
returns and tracking error of these smart beta strategies into the components derived from factors and the
unexplained part. The performance of a portfolio over
risk-free rate can be attributed to the given factors using
the following Equations:
7
R P,t − R rfr,t = α + ∑ βFi R Fi ,t + ε
i =1
(1)
7
R P − R rfr = Unexplained (alpha) + ∑ βFi R Fi
i =1
(2)
Another set of regressions described in Equation (3)
was run, and then Equation (4) was used to break down
the square of tracking error (TE) into three parts: the
unexplained (or idiosyncratic) part, the TE attributable
to each factor, and the TE attributable to the interaction of factors:
7
R + + ∑ β
R P,t − RCWT ,t = α
ε
Fi
Fi ,t
i =1
(3)
7
2 Var( R )
TE 2 = ∑ β
Fi
F,t
i =1
 7 2 2

2
(4)
+ 2 ∑ β
Fi β F j Cov(R Fi ,t , R F j ,t )  + σ ε
 i , j =1 ( i ≠ j )

R P,t represents portfolio return series, RCWT,t represents cap-weighted benchmark return series, R rfr,t
represents risk-free rate time series, and R Fi,t represents
the time series of returns of the seven factors listed in
Exhibit 4. R P , R rfr , and R Fi represent annualized average
returns. The second term in Equation (4), involving
the covariance across the factors, is labeled as an “Interaction” component; and the last term in Equation (4),
representing the unexplained TE, is labeled as an “Idiosyncratic” component of TE in Exhibit 5. Since Equation
(4) is quadratic, both sides of the equation are divided by
TE to remove the square term. This is an approximation that allows us to represent the division of TE across
factors in a simple way without affecting their relative
contribution to TE.
Exhibit 5
Seven-Factor Performance and Tracking Error Attribution
Notes: The market factor is the excess returns of the CRSP S&P 500 index over risk-free rate. The yield on secondary U.S. Treasury bills (3M) is used
as a proxy for the risk-free rate. The size, value, momentum, high profitability, and low investment factors were obtained from the Kenneth French data
library. The BAB factor was obtained from the Andrea Frazzini data library. The Newey–West (Newey and West [1987]) estimator is used to correct
for autocorrelation. Daily total returns from December 31, 1973 to December 31, 2013 are used for the analysis.
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As seen in Exhibit 5, factors other than value
and size play a major role in the returns and the risk
of many strategies. In particular, the low volatility–
and momentum-tilted portfolios, irrespective of the
weighting scheme, derive a large portion of their performance from their exposure to BAB and momentum
factors, respectively. The contributions of factors other
than value and size to portfolio risk and return again
invalidate the claim that these strategies are not different
from equal-weighting.
Although many of the strategies tested go beyond
value and small cap exposure and offer pronounced exposure to additional factors, the presence of a considerable
portion of unexplained performance suggests that the
portfolio construction of these indexes captures effects
that cannot be explained fully by the relevant factors.
One possible explanation for this unexplained part of
performance is that the improved diversification scheme
allows value to be added beyond the explicit factor tilts.
Another explanation is that other additional factors,
which are omitted from the factor model, are at work.
Our results unmistakably show that smart beta strategies can have exposure to factors other than small size
and value. As mentioned previously, this finding may not
be surprising and is fully consistent with the academic
literature, which has documented the importance of
various equity risk factors beyond value and small cap
(Leote De Carvalho, Lu, and Moulin [2012]; Asness,
Frazzini, and Pedersen [2013]; Asness, Moskowitz, and
Pedersen [2013]; Clarke, de Silva, and Thorley [2013]).
However, although our findings are in line with this
literature, they are in stark contradiction to the claims
of monkey portfolio proponents who have argued that
there is nothing beyond value and small-cap exposure
in smart beta strategies. Instead, our results suggest that
different smart beta strategies derive performance from
different exposures to several factors that may go beyond
size and value. Importantly, our results also show that
differences in product design of course lead to differences in factor exposures, and the investment philosophy
on which smart beta strategies are based is therefore
nowhere near irrelevant for performance.
Our differences in results with those documented
by monkey portfolio proponents suggest that monkey
portfolio arguments do not apply generally. Our findings
directly invalidate the monkey portfolio proponents’
claims that smart beta is simply about small cap and
value exposure. Although we do not claim to provide an
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exhaustive set of smart beta strategies used in practice,
we include some of the commonly used strategies and
come to opposite conclusions from the monkey portfolio argument. This suggests that the monkey portfolio
claims may apply only to a specific selection chosen by
the proponents, without taking into account the full
range of smart beta strategies used in practice. One
should thus be careful to refrain from overgeneralizing
these results.
DO SMART BETA STRATEGIES ADD THE
SAME OR MORE VALUE WHEN INVERTED?
Another key claim of monkey portfolio proponents
is that smart beta strategies “add the same or more value
when inverted,” as suggested by Brightman [2013, p. 5].
An even stronger version of this claim was put forward
by Arnott et al. [2013, p. 98], who stated that “popular
strategy indexes, when inverted, produce even better
outperformance.” Clearly, if one were to confirm such
results for inverted strategies, this would be strong evidence suggesting that “the investment beliefs upon which
many investment strategies are ostensibly based play
little or no role in their outperformance” (Arnott et al.
[2013, p. 91]). In this subsection, we empirically assess
the validity of such claims with respect to our set of test
portfolios.
In the preceding section, we already visited the
problem of selecting particular strategies to test claims
relating to their performance and then overgeneralizing the findings to all of smart beta. This problem, of
course, also applies to selecting strategies for the creation
of inverse portfolios. Whether a strategy behaves differently from its inverse will depend heavily on the type
of strategy that one tests. Let us take the example of an
equal-weighted (also known as 1/N) strategy, which is
the simplest form of smart beta strategies. If one inverts
its weights using the methodology of Arnott et al. [2013],
one would arrive at the original portfolio (i.e., the inverse
of the 1/N portfolio is the portfolio itself ). Therefore,
this smart beta strategy and its inverse are identical, and
both outperform the cap-weighted benchmark identically. Similarly, by using portfolios that are constrained
to correspond to some optimization objective while
keeping a close distance to equal-weighted portfolios,
one would bias the results in favor of the claim that
inverse strategies have as much merit as the original strategies. This problem may be exacerbated by the fact that
monkey portfolio arguments have been derived using
simulated strategies that do not correspond to the actual
index rules used in practice, as mentioned in the disclaimers in Arnott et al. [2013] and Chow et al. [2011].
We used the fundamental-weighted strategy and
single factor-tilted portfolios to assess the claim that the
performance of smart beta strategies remains the same
or increases if their weights are inverted. The equalweighted strategy is excluded because, as argued earlier,
the equal-weighted portfolio is its own inverse portfolio. To analyze this effect, we constructed the inverse
or upside-down portfolios for each smart beta strategy
in a manner similar to that of Arnott et al. [2013].8
In particular, we applied two methods of turning
strategy weights upside-down, which we refer to as
type-1 and type-2 upside-down strategies. It should,
however, be noted that our test portfolios include a stock
selection step for the explicit factor-tilted portfolios.
Therefore, in addition to inverting the portfolio weights,
we also invert the direction of the stock selection.
In the upside-down strategy for value, for example, we
selected the stocks with the lowest value score (i.e., the
highest inverse value score) and then tilted the weights
by the inverse of their value score. In fact, a key contribution of our article relative to Arnott et al. [2013]
is that we include a more general class of smart beta
strategies that do not only use alternative weighting
schemes to reweight a given stock universe, but also
select stocks based on factor characteristics. Contrary
to our approach, Arnott et al. [2013] only tested broad
smart beta strategies that reweight stocks from a broad
universe and often resemble equal-weighted strategies
by construction. Of course, one would expect that such
strategies, when inverted, remain quite similar to the
original strategies.
When inverting strategies that include a stock
selection, it is clear that by selecting stocks with the
opposite characteristics, the upside-down version of such
strategies is expected to display more pronounced differences. It is important to note that by omitting strategies
with a stock selection step, Arnott et al. [2013] have
omitted a practically important class of smart beta strategies. Our article fills this gap by considering additional
types of smart beta strategies, which are widely used in
practice. To illustrate how the creation of upside-down
strategies works, we provide a simple example.
Exhibit 6 shows the weights of the risk-weighting
scheme and its upside-down type-2 version, with and
without low volatility stock selection, applied to the
top 10 stocks by total market cap as of December 2013.
This is an illustrative example, which aims to show
the differences that arise when one does or does not
Exhibit 6
Illustrative Examples of Portfolio Weights of Risk-Weighted, Equal-Weighted, and Their Upside-Down
Type-2 Versions
Notes: The portfolios are constructed using a sample universe consisting of the 10 largest U.S. stocks by total market capitalization as of December 20,
2013. Risk weighting strategy weights the stocks in the proportion of the reciprocal of their past two-year volatilities. The low volatility selection picks five
stocks with the lowest past two-year volatilities.
Spring 2016
use stock selection in designing a smart beta strategy
and its inverse. We can see that inverting not only the
weights but also the stock selection naturally leads to
more pronounced differences in weight between the
strategy and its inverse. The illustration also shows that
even when using a weighting scheme that targets equal
weights, inverting the stock selection step would lead
to pronounced differences between the upside-down
strategy and the original. The stock selection stage plays
an important role in the methodologies of commonly
used smart beta strategies (see Amenc, Goltz, and Lodh
[2012]). Therefore, inverting not only the weighting
scheme but also the stock selection methodology is necessary for a relevant analysis of upside-down strategies.
Having clarified the methodology used to create
upside-down strategies, we now turn to the results
obtained when inverting a broad range of smart beta
strategies. Exhibit 7 shows performance and risk statistics
for our test strategies as well as for the corresponding
upside-down strategies. The results in Exhibit 7 are very
homogeneous across the different strategies. Overall, the
results suggest that most upside-down strategies have
lower returns than the originals, as well as lower Sharpe
and information ratios.
Indeed, many of the original smart beta portfolios outperform their upside-down counterparts, and
they mostly do so by large margins. For example, the
mid cap and value diversified multi-strategy portfolios
outperform their upside-down portfolios by more than
3%, and this figure for the momentum diversified multistrategy is about 1.5%. Intuitively, this result can be
expected to arise if the inverse of a factor-tilted portfolio
tilts negatively toward the rewarded factor and hence
does not benefit from risk premiums. To test this explanation of the results of upside-down strategies, we will
assess their factor exposures.
In addition to looking at the explicit factor tilt
strategies, it is also interesting to see if the upside-down
version of the fundamental-weighted portfolio provides
similar performance to the original. The monkey portfolio proponents’ claim is only partly valid concerning
this strategy, as the upside-down type-1 strategy of the
fundamentally weighted portfolio indeed has similar
returns to the original strategy, but the type-2 upsidedown strategy underperforms the original.
When looking at risk-adjusted returns in the form
of Sharpe ratios, the results in Exhibit 7 show that all
explicit factor-tilted strategies in our set of test portfolios
Spring 2016
have Sharpe ratios superior to those of the inverted strategies. The information ratio and probability of outperformance of the original smart beta strategies is also
consistently higher than that of the respective inverted
portfolios. Our findings, while perfectly in line with
common sense, contradict the claims made by monkey
portfolio proponents.
To better understand the effect inverting a smart
beta strategy has on its risk profile, we employed the
multi-factor model from earlier and then looked at
how factor loadings are affected by the inversion of the
strategy. Exhibit 8 shows the seven-factor exposures of
original and inverted smart beta strategies to explain this
performance difference.
Overall, the results in Exhibit 8 suggest that the
upside-down strategies result in less rewarding factor
exposures, which is consistent with their lower performance compared to the original strategies.
The value exposure of the value diversified multistrategy is 0.40, whereas its inverse portfolios have
negative value exposure. Similarly, the low volatility
diversified multi-strategy has an exposure of 0.10 to the
BAB factor and 0.88 to the market factor, whereas its
inverse portfolios have negative BAB betas and market
betas greater than one. In fact, for each factor tilt, it is
shown that inverting a factor-tilted smart beta portfolio
will result in a portfolio with reduced exposure to the
concerned factor.
The previous f indings are similar for scoreweighted portfolios. When compared to its upsidedown type-1 portfolio, the momentum (low volatility)
score–weighted portfolio has a higher exposure to the
momentum (BAB) factor. The fact that the momentum
(low volatility) score–weighted portfolio outperforms
its type-1 inverse by 4.81% (2.43%) per annum suggests
that the momentum (BAB) factor is the main driver of
its performance.
Since this process of “artificially” inverting the
portfolios does not rely on a stock’s characteristics, such
as its SMB (small minus big) or HML (high minus low)
score, it is possible that inverting a factor-tilted portfolio
increases its SMB or HML exposure. That said, it does
not mean that the inverted portfolio will exhibit superior performance compared to the original smart
beta strategy. The case of the low investment diversified multi-strategy and high profitability diversified
multi-strategy portfolios in Exhibit 8 provides an excellent example to this effect.
Exhibit 7
Performance and Risk Analysis of Upside-Down Strategies
Notes: The probability of outperformance is the probability of obtaining positive excess returns from investing in the strategy for a period of three years at any
point during the history of the strategy. All statistics are annualized, and daily total returns from December 31, 1973, to December 31, 2013 are used for
the analysis. The CRSP S&P 500 index is used as the cap-weighted benchmark. Yield on secondary U.S. Treasury bills (3M) is used as a proxy for the
risk-free rate.
Spring 2016
Exhibit 8
Regression Analysis of Upside-Down Strategies
Notes: The market factor is the excess returns of the CRSP S&P 500 index over risk-free rate. The yield on secondary U.S. Treasury bills (3M) is used
as a proxy for the risk-free rate. The size, value, momentum, high profitability, and low investment factors were obtained from the Kenneth French data
library. The BAB factor was obtained from the Andrea Frazzini data library. The Newey–West (Newey and West [1987]) estimator is used to correct for
autocorrelation. Daily total returns from December 31, 1973 to December 31, 2013 are used for the analysis. Regression coefficients that have P-values
less than 5% are highlighted indicated in bold font. The regression coefficient that corresponds to the variable used in stock selection is indicated by a shaded
background. Alphas are annualized.
Spring 2016
We notice that inverting these two smart beta
strategies increases the portfolio’s exposure to SMB
and HML factors. However, it does not improve the
performance. Upon close inspection of the results, it is
apparent that the process of inversion also deteriorates
momentum, investment, and profitability betas. In the
case of these two strategies, the overall performance drag
resulting from these decreasing betas outweighs the performance gain that stems from SMB and HML factors,
which explains the overall performance of these smart
betas and their upside-down portfolios. It is therefore
important to examine the effect of factors in addition
to small size and value.
The factor loadings are also able to provide perspective on the behavior of the fundamentally weighted
portfolio’s inverse. We saw earlier that the fundamentally weighted upside-down strategy (at least for type-1)
shows risk and returns characteristics similar to the original strategy. In fact, this is consistent with the fact that
inversion of this portfolio reduces the value beta on the
one hand and increases the small size beta on the other.
A similar observation can be made when inspecting the
results of Arnott et al. [2013].
Our results in Exhibits 7 and 8 suggest that investment beliefs in the form of explicit factor tilts do indeed
play an important role in determining the performance
of an investment strategy. Inverting the strategy not
only turns the weights upside-down but also changes
the results, both in terms of performance and factor
exposures. This finding contradicts the monkey portfolio claim, at least for the explicit factor-tilted strategies
tested here.
CONCLUSION: ASSESSING SMART BETA
STRATEGIES IS NOT MONKEY BUSINESS
The main arguments of monkey portfolio proponents are that all smart beta strategies generate positive
value and small-cap exposure similar to that generated
by any random portfolio strategy, and the inverses of
such strategies perform similarly or better. Although
we have not attempted to conduct an exhaustive assessment of these claims across all possible strategies, our
analysis of some commonly employed smart beta strategies suggests that these statements do not hold in general. Our results show that, although some strategies
such as fundamental equity indexation may perhaps be
mostly driven by a value tilt and may generate similar
performance to their upside-down counterpart, many
smart beta strategies display exposure to additional factors, as well as pronounced differences in factor exposures across different strategies. Moreover, and perhaps
reassuringly, the inverses of these strategies generate
lower performance.
The differences with the original results underlying the monkey portfolio arguments can be explained
by the fact that we considered strategies that explicitly
tilt toward a variety of risk factors, whereas the evidence in favor of monkey portfolio claims omitted such
strategies and instead focused on a particular selection
of strategies that may better correspond to such claims.
In particular, if one selects strategies that stay relatively
close to equal-weighting, it may not be entirely surprising that the performance of such strategies is extensively driven by value and small cap exposure, as has
been documented for equal-weighted strategies. Moreover, it may not be surprising that strategies that are close
to equal-weighting can be inverted while maintaining
comparable performance benefits.
An important insight from our tests is that one
should be careful to not overgeneralize results that have
been derived from testing particular strategies. Although
the monkey portfolio arguments may apply to particular
strategies, they have been invalidated for the explicit
factor strategies we chose as our main test portfolios
here, and therefore these claims cannot be applied to
smart beta strategies in general.
Our findings of significant differences across various smart beta strategies imply that care must be taken
not to fall into the trap of oversimplification and overgeneralization. The differences in factor exposures across
smart beta strategies imply that using a particular set of
indexes corresponds to particular factor selection and
factor allocation decisions. Moreover, the different factor
tilts play an important role in shaping the risk–return
profile of smart beta strategies. Factor-tilted smart beta
strategies perform due to large positive exposure to their
respective factors, whereas their inverted counterparts
underperform the originals due to less pronounced or
negative exposure to the same factors. When considering the adoption of smart beta strategies, investors
should carefully consider which set of factor exposures is
best aligned with their investment beliefs and objectives.
Making a selection from among smart beta strategies is
not monkey business after all.
Spring 2016
A P P E N DIX
DESCRIPTION OF DIVERSIFICATION-BASED
WEIGHTING SCHEMES
The maximum deconcentration weighting scheme
is a naive diversification strategy that aims at maximizing
the effective number of stocks of an equity portfolio. The
strategy owes its popularity mainly to its robustness, and it has
been shown to deliver attractive performance despite highly
unrealistic conditions of optimality, even when compared to
sophisticated portfolio optimization strategies (DeMiguel,
Garlappi, and Uppal [2009]).
The diversified risk–weighted strategy aims to achieve
equal risk contribution from all stocks under the assumption
of identical pairwise correlation across stocks, and it is the
same as inverse volatility weighting. It aims to balance risk
exposures across portfolio constituents and can thus be seen
as a response to issues with risk concentration, which may
arise in cap-weighted equity indexes (Maillard, Roncalli, and
Teiletche [2010]).
The maximum decorrelation weighting scheme aims
at minimizing portfolio volatility under the assumption
that stock volatilities are identical, thus only exploiting the
correlation structure. In line with the idea that correlation
effects are the main driver of diversification, this approach
is an application of the diversification measure suggested by
Christoffersen et al. [2010] for equity portfolio construction.
Exhibit A1
Closed Form Solutions and Required Parameters
for Five Weighting Schemes
The efficient minimum volatility weighting scheme
attempts to minimize the portfolio volatility using the information on stock volatilities and pairwise correlations. Theoretically, it is the portfolio on the efficient frontier with the
lowest level of volatility and is the only one that does not
require estimates of expected returns (Markowitz [1952]).
The efficient maximum Sharpe ratio weighting scheme
aims to maximize the Sharpe ratio—the risk-adjusted performance. Theoretically, it is the portfolio on the efficient
frontier with the highest reward per unit of risk and is the
only portfolio of risky assets that should be of interest to a
rational investor (Tobin [1958]).
The diversified multi-strategy weighting scheme combines these five weighting schemes in equal proportions to
reduce the unrewarded or specific risks of each strategy.
ENDNOTES
1
The weight of a stock in the fundamental-weighted
portfolio is calculated using its composite fundamental score.
The composite fundamental score is the average of four
scores, each based on current book value, trailing five-year
cash f low, trailing five-year dividend, and trailing five-year
sales, respectively. When the score of certain fundamental
variables is not available for a particular stock, its composite
fundamental score is the average score across remaining
variables. For example, for the companies that did not distribute dividends (i.e., their dividend score is missing), we
omitted the dividend variable score and instead used the
average across the other three fundamental variable scores.
The fundamental-weighted portfolio is rebalanced yearly on
the third Friday in March.
2
Amenc et al. [2014] introduced diversified multistrategy weighting as an equal-weighted combination of the
following five weighting schemes: maximum deconcentration, diversified risk weighted, maximum decorrelation, efficient minimum volatility, and efficient maximum Sharpe
ratio. For a detailed description, refer to the Appendix.
3
For a given factor, each stock is assigned a factor
Z-score. The cumulative distribution function (CDF) of a
normal distribution with zero mean and unit standard deviation is used to convert Z-scores to standardized S-scores.
The S-scores range from 0 to 1.
Z
Si =
Notes: The exhibit lists the closed form solutions and the required
parameters for the five weighting schemes. N is the number of stocks, µ is
the (N × 1) vector of expected return, 1 is the (N × 1) vector of ones, σ is
the (N × 1) vector of volatilities, Ω is the (N × N) correlation matrix,
and Σ is the (N × N) covariance matrix.
Spring 2016
2
1 i − 2x
∫ e dx
2 π −∞
Score weighting is done by weighting the stocks in
proportion to their market-cap times the S-score of the
respective factor.
Wi =
S × MC i
Σ Sk × MC k
i
N
k =1
The upside-down type-1 (UD1) portfolio weights are
given by the following expression:
4
We also tested the same strategies using simple scoreweighting rather than score-adjusted market cap weighting.
The results with these portfolios do not lead to different
conclusions on the validity of the monkey portfolio arguments from our analysis of diversified tilted portfolios and
score-adjusted market cap weighted portfolios. Because of
this, and because of the fact that indexes used in practice do
not commonly employ direct score weighting, we do not
report these results here.
5
The following selection rules are applied to select
stocks for each tilt. Mid cap: bottom 50% free f loat-adjusted
market cap stocks are selected. Value: top 50% stocks are
selected by book-to-market (B/M) ratio, defined as the ratio
of the available book value of shareholders’ equity to company
market cap. High momentum: top 50% stocks are selected by
returns over the past 52 weeks, minus the last four weeks. Low
volatility: bottom 50% stocks are selected by their standard
deviation of weekly stock returns over the past 104 weeks.
Low investment: bottom 50% stocks with two-year total asset
growth rate. High profitability: top 50% stocks with highest
gross profit/total asset ratio. The score-based selection is done
twice a year ( June and December) for momentum and once
a year ( June) for the remaining five factors.
6
Rank-weighted portfolios that are long the 50% low
market beta stocks and short the 50% high market beta stocks
were constructed. Betas were estimated using the shrinkage
method of Vasicek [1973] for long and short legs separately.
Both long and short portfolios were rescaled to have a beta
of one at portfolio formation.
7
The size, value, momentum, high profitability, and
low investment factors were obtained from Kenneth French’s
data library (http://mba.tuck.dartmouth.edu/pages/faculty/
ken.french/data_library.html). The BAB factor was obtained
from Andrea Frazzini’s data library (http://www.econ.yale
.edu/~af227/data_library.htm).
8
The idea of turning a strategy upside-down is to
make counter bets (i.e., to overweight stocks that are underweighted in the smart beta strategy and vice versa). Therefore,
to turn a factor-tilted strategy upside-down would require the
remaining half of the stock universe to be selected and then
the weights in smart beta portfolios to be inverted in this subuniverse. For example, the upside-down version of the mid cap
diversified multi-strategy portfolio is the portfolio obtained
by inverting the large cap diversified multi-strategy portfolio.
Let the weight vector of a smart beta strategy be given by W:
W = (w1 , w 2 , …, w n )
w max = max(w1 , w 2 , …, w n )
 w − w1 , w max − w 2 , , w max − w n 
WUD1 =  max
n. w max − 1
 n. w max − 1 n. w max − 1
and the second inverted, or upside-down type-2 (UD2),
portfolio weights are given by the following expression:
WUD2
1
1
1


wn
w1 ,
w2 , ,


=
n
n
n
1 
 ∑ 1w ∑ 1w
∑
wk 
k =1
k =1
 k =1
k
k
Similarly, the UD1 and UD2 versions of the mid
cap score–weighted portfolio is the portfolio obtained by
inverting the size scores of large cap stocks such that larger
stocks have higher S-scores:
SiUD1 = max(S1 , S2 , …, Sn ) − Si
1
SiUD2 =
Si
These scores are then used to tilt the market capweighted portfolio of large cap stocks as follows:
Wi UD1 =
∑
SiUD1 .MC i
n
k =1
S
UD1
k
.MC k
, Wi UD2 =
∑
SiUD2 .MC i
n
k =1
S
UD2
k
.MC k
REFERENCES
Amenc, N., and F. Goltz. “Smart Beta 2.0.” The Journal of
Index Investing, Vol. 4, No. 3 (2013), pp. 15-23.
Amenc, N., F. Goltz, and A. Lodh. “Choose Your Betas:
Benchmarking Alternative Equity Index Strategies.”
The Journal of Portfolio Management, Vol. 39, No. 1 (2012),
pp. 88-111.
Amenc, N., F. Goltz, A. Lodh, and L. Martellini. “Towards
Smart Equity Factor Indices: Harvesting Risk Premia without
Taking Unrewarded Risks.” The Journal of Portfolio Management, Vol. 40, No. 4 (2014), pp. 106-122.
Ang, A., W. Goetzmann, and S. Schaefer. “Evaluation of
Active Management of the Norwegian Government Pension
Fund – Global.” Report to the Norwegian Ministry of
Finance, 2009.
Spring 2016
Arnott, R.D., J. Hsu, V. Kalesnik, and P. Tindall. “The Surprising Alpha from Malkiel’s Monkey and Upside-Down
Strategies.” The Journal of Portfolio Management, Vol. 39, No. 4
(2013), pp. 91-105.
DeMiguel, V., L. Garlappi, and R. Uppal. “Optimal Versus
Naive Diversification: How Inefficient Is the 1/N Portfolio
Strategy?” Review of Financial Studies, Vol. 22, No. 5 (2009),
pp. 1915-1953.
Asness, C.S. “Changing Equity Risk Premia and Changing
Betas over the Business Cycle and January.” Working paper,
University of Chicago, 1992.
Fama, E.F., and K.R. French. “A Five-Factor Asset Pricing
Model.” Journal of Financial Economics, Vol. 116, No. 1 (2015),
pp. 1-22.
Asness, C.S., A. Frazzini, and L.H. Pedersen. “Quality Minus
Junk.” Working paper, AQR Capital Management, 2013.
Frazzini, A., and L.H. Pedersen. “Betting Against Beta.”
Journal of Financial Economics, Vol. 111, No. 1 (2014), pp. 1-25.
Asness, C.S., T.J. Moskowitz, and L.H. Pedersen. “Value and
Momentum Everywhere.” Journal of Finance, Vol. 68, No. 3
(2013), pp. 929-985.
Hsu, J.C., V. Kalesnik, and F. Li. 2012. “An Investor’s Guide
to Smart Beta Strategies.” American Association of Individual
Investors Journal, December 2012, pp. 11-16.
Brightman, C. “What Makes Alternative Beta Smart?” Fundamentals, Research Affiliates Publication, September 2013,
pp. 1-5.
Leote de Carvalho, R., X. Lu, and P. Moulin. “Demystifying Equity Risk–Based Strategies: A Simple Alpha Plus
Beta Description.” The Journal of Portfolio Management, Vol. 38,
No. 3 (2012), pp. 56-70.
Carhart, M.M. “On Persistence in Mutual Fund Performance.” Journal of Finance, Vol. 52, No. 1 (1997), pp. 57-82.
Chow, T., J. Hsu, V. Kalesnik, and B. Little. “A Survey of
Alternative Equity Index Strategies.” Financial Analysts Journal,
Vol. 67, No. 5 (2011), pp. 37-57.
Christoffersen, P., V. Errunza, K. Jacobs, and X. Jin. “Is the
Potential for International Diversification Disappearing?”
SSRN working paper 1573345, 2010.
Clare, A.D., N.E. Motson, and S.H. Thomas. “An Evaluation
of Alternative Equity Indices – Part 1: Heuristic and Optimised Weighting Schemes.” Working paper, Cass Business
School, 2013.
Maillard, S., T. Roncalli, and J. Teiletche. “The Properties of
Equally Weighted Risk Contributions Portfolios.” The Journal
of Portfolio Management, Vol. 36, No. 4 (2010), pp. 60-70.
Markowitz, H.M. “Portfolio Selection.” Journal of Finance,
Vol. 7, No. 1 (1952), pp. 77-91.
Newey, W.K., and K.D. West. “A Simple, Positive SemiDefinite, Heteroskedasticity and Autocorrelation Consistent
Covariance Matrix.” Econometrica, Vol. 55, No. 3 (1987),
pp. 703-708.
Tobin, J. “Liquidity Preferences as Behavior towards Risk.”
Review of Economic Studies, Vol. 25, No. 2 (1958), pp. 65-68.
Clarke, R., H. de Silva, and S. Thorley. “Risk Parity, Maximum Diversification, and Minimum Variance: An Analytic
Perspective.” The Journal of Portfolio Management, Vol. 39,
No. 3 (2013), pp. 39-53.
Vasicek, O.A. “A Note on Using Cross-Sectional Information
in Bayesian Estimation of Security Betas.” Journal of Finance,
Vol. 28, No. 5 (1973), pp. 1233-1239.
Cohen, R.B., C. Polk, and T. Vuolteenaho. “The Value
Spread.” Journal of Finance, Vol. 58, No. 2 (2003), pp. 609-642.
To order reprints of this article, please contact Dewey Palmieri
at dpalmieri@ iijournals.com or 212-224-3675.
Spring 2016

Smart Beta Is Not Monkey Business

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