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Can the Whole Be More Than
the Sum of the Parts? Bottom-Up
versus Top-Down Multifactor
Portfolio Construction
IN
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JENNIFER BENDER AND TAIE WANG
TAIE WANG
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is a vice president at State
Street Global Advisors in
Hong Kong.
[email protected]
Special Issue 2016
JPM-BENDER.indd 39
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portfolio from the security level. The latter is
a better approach theoretically because each
security’s portfolio weight will depend on
how well it ranks on multiple factors simultaneously. The former approach, combining
single-factor portfolios, may miss the effects
of cross-sectional interaction between the
factors at the security level. But it may be that
the differences between the two approaches
are small—and the combination approach
has benefits such as more transparent performance attribution and greater f lexibility.
In this article, we explore this issue and
discuss the implications of both approaches.
Our conclusion is that there are, in fact,
beneficial interaction effects among factors
that are not captured by the combination
approach. Both intuition and empirical evidence favor employing the bottom-up multifactor approach.
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ultifactor rules-based portfolios—portfolios constructed
to capture multiple factor
exposures—have become
increasingly popular in recent years. The
main rationale for combining multiple factors
is that it enables investors to achieve potential diversification benefits. Historically, factors such as value, size, quality, low volatility,
and momentum have earned a long-run premium over the market, but all factors have
experienced periods of underperformance
during certain market environments. However, they have not all experienced periods
of underperformance at the same time. Some
factor pairs such as value and momentum
naturally diversify each other based on their
definitions. When a stock’s price rises, it
simultaneously becomes more momentumlike (as long as its price is rising faster than
others) and less value-like (because value is
typically defined as book-to-price, earningsto-price, or some other fundamental-to-price
ratio). Other factors tend to diversify each
other historically based on cycles of investment sentiment; quality and low volatility
stocks are favored by investors in times of
uncertainty.
Given the interest in multifactor portfolios, there has been much discussion about
the best way to build them. One often-asked
question is whether it is better to combine single-factor portfolios or to build a multifactor
A
is a managing director at
State Street Global Advisors in Boston, MA.
[email protected]
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JENNIFER BENDER
THE RISE OF RULES-BASED
INVESTING
The earliest alternative indexing strategies (also known as advanced beta or smart beta)
applied an alternative weighting scheme to
market-capitalization weighting. Examples
include GDP-weighted portfolios in the 1980s,
equal-weighted portfolios in the 1990s, and,
more recently, fundamental-weighted portfolios in the 2000s. Proponents of these strategies were typically critical of cap-weighting
The Journal of Portfolio Management
39
5/17/16 8:36:42 PM
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Single-Factor Portfolio Construction
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A number of techniques have been used to build single-factor portfolios. Equal weighting, GDP weighting,
fundamental indexation, and its close companion wealth
weighting were compelling; they were intuitive and did
not employ a black box algorithm such as optimization,
which meant security weights could be directly tied to the
securities’ observable characteristics. Subsequent factorbased approaches could also be constructed in a similar
fashion; most employ a set of rules that specify security
weights as a function of the factor characteristics.
Consider the following examples of first-­generation
rules-based portfolios—equal weighting, fundamental
indexation, and risk weighting:
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1
• Equal weights: w i =
(1)
N
Fi
(2)
• Fundamental indexation weights : w i = N
∑ Fi 1
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• Risk weights: w i =
i
σ i2
N
1
∑ σi2
i
w i = w i ,mktcap γ i
(4)
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earnings, etc.), and σ 2i is the variance of stock i. All the
methods are relatively straightforward, linking the desired
attribute (e.g., volatility or book value) to the securities’
weights.
Our preferred approach has been to adopt a
benchmark-relative framework in which multipliers are
applied to market-cap weights. For tilted factor portfolio
weights,
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where γi is a scalar applied to the market-cap weight of
each stock. The scalar γi can be specified in many ways.
It can be the result of a mapping function based on the
security’s factor characteristics. It can be nonlinear or
linear cross-sectionally, and it can be unique for each
security or unique for groups of securities. Furthermore,
we can screen out certain securities by setting the multiplier equal to zero. We have favored this approach
because it allows f lexibility and coherency for building
portfolios across different factors and at different levels
of tracking error and concentration for benchmark-sensitive investors (see Bender and Wang [2015]).
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and argued that these ­alternative weighting schemes were
superior because they were either more representative of
investment value or more diversified.
Since 2008, an alternative method of constructing
such portfolios has emerged—one that focuses on capturing factor premiums more explicitly. This approach
focuses on the “pure” factors that the portfolios are
exposed to and derive their returns from. The pure
factors, beginning with the multifactor models of Ross
[1976], are those that have been widely researched in
the academic literature, have strong theoretical foundations, and have exhibited persistence over multiple
decades. Viewing factor indexation as a means of capturing pure factors is consistent with the way academics
have viewed factors; it was most widely popularized
by Fama and French’s seminal three-factor model and
extended over the years by countless others. The most
widely discussed factors include the original Fama–
French–Carhart factors—value, (low) size, momentum—plus a handful of additional factors that have
received moderate treatment—(low) volatility, quality,
liquidity, and yield.
(3)
In Equations 1–3, wi is the weight of stock i in the
portfolio, N is the number of stocks in the universe, Fi
is the fundamental value of stock i (e.g., book value,
MULTI-FACTOR PORTFOLIO
CONSTRUCTION
Combining multiple factors with strong investment
merit can produce benefits from the potential diversification among the factors. Historically, factors such
as value, size, quality, low volatility, and momentum
have exhibited substantial diversification benefits over
short horizons such as a week or a month, but also over
longer, multiyear periods. Correlations between excess
returns are generally below 0.5 and sometimes negative
(see Bender, Brandhorst, and Wang [2014] for further
discussion).
Multifactor portfolios can be constructed in two
main ways. The simplest way is to combine single-factor
portfolios into one portfolio. Within each factor portfolio, security weights are determined according to a
specific methodology (ideally, one that is consistent
across factors). Single-factor portfolios are blended by
assigning weights to individual portfolios. This approach
is analogous to building blocks, and its benefits include
clear performance attribution and f lexibility in reallocation across factors. For the remainder of this article, we
refer to this as the combination approach.
40 Bottom-Up versus Top-Down Multifactor Portfolio ConstructionSpecial Issue 2016
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•
⎡ m1 ⎤
⎢
⎥
⎢ m2 ⎥
Multipliers for factor 1: M = ⎢
; Multipliers
. ⎥
⎢
⎥
⎡ l1 ⎤
⎢⎣ mn ⎥⎦
⎢
⎥
l
for factor 2: L = ⎢⎢ 2 ⎥⎥
.
⎢
⎥
⎢⎣ ln ⎥⎦
⎡ s1 ⎤
⎢
⎥
s2 ⎥
⎢
Starting weights: S = ⎢
; Final weights:
. ⎥
⎢
⎥
⎡ w1 ⎤
⎢⎣ sn ⎥⎦
⎢
⎥
⎢ w ⎥
W =⎢ 2 ⎥
.
⎢
⎥
⎢⎣ w n ⎥⎦
Weight in factor 1: λ
Weight in factor 2: 1 – λ
•
•
The symbol denotes element-wise product.
The resulting weight for the bottom-up, twofactor portfolio is
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Let us begin with a stylized example using 10
stocks to construct a three-factor portfolio. We consider three representative factors: dividend yield,
book-to-price ratio, and return on assets. To build
the portfolio, we rank the securities based on each
security’s factor characteristics and assign multipliers
based on their ranks. The multipliers can be assigned
to any set of starting weights; market-cap weights and
equal weights are our candidates. The new weights
(starting weight x multiplier) are then rescaled to sum
to 100%.
What are the conditions under which the bottom-up
and combination approaches will yield the same results?
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A STYLIZED EXAMPLE
where
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The second way to build the portfolios is from
the security level up, bottom up, by incorporating all the
factor characteristics simultaneously. Security weights
are assigned based on the security’s combined characteristics. Asness [1997], for example, highlighted the
effects of interaction between value and momentum.
The top-down combination approach may miss these
interaction effects.
In our example, we construct portfolios by multiplying the starting weights (e.g., cap weights) with
a multiplier, as in Equation 4. Note that if we were
employing optimization, these two approaches would
not yield the same results. (The typical objective function is a quadratic function; because it is nonlinear,
a linear combination of factor portfolios cannot be
equivalent.) We focus on non-optimized portfolios, and
therefore it is less clear if there are differences and how
large those differences are in actuality.
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• The starting weight is equal weight.
• The sums of the multipliers for each single-factor
portfolio and for the bottom-up factor portfolio
are identical.
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We show this as follows: If there are two-factor
portfolios, the resulting weights for a linear combination
of two-factor portfolios can be calculated as
M S
W =λ
+(
S ′ M
Factor 1
Special Issue 2016
JPM-BENDER.indd 41
L S
S
L ′ )
Factor
actor 2
(5)
⎡λ + (
W = ⎣
⎡⎣ λ + (
) L ⎤⎦ S
) L ⎤⎦
′
×S
(6)
For Equations 5 and 6 to be identical, the necessary
conditions are as follows:
s1 = s2 = ...sn ,
(7)
∑
(8)
∑l .
i
The derivation appears in Appendix A.
To illustrate this equivalence, we built the following two portfolios (as shown in Exhibit 1).
• Combination Portfolio: In the two left-most
panels, we sorted the securities based on raw metrics one factor at a time and ranked them. We
assigned multipliers identical to the stock’s rank
(e.g., a stock ranked fifth has a multiplier of 5).
The multipliers were applied to equal weights (e.g.,
The Journal of Portfolio Management
41
5/17/16 8:36:44 PM
Exhibit 1
each security’s weight is 10%). We then rescaled
the weights to sum to 100%. Next, we blended
the three resulting portfolios into one using equal
weights—that is, we averaged the security weights
across the three portfolios.
• Bottom-Up Portfolio: In the right-most panel,
we ranked the securities for each factor, as we did
for the combination portfolio. This time, we computed an average (equally weighted) rank across
the three factors and multiplied this combined rank
by security equal weights (10%). We then rescaled
the weights to sum to 100%.
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Scenario in Which the Combination Portfolio Is
Equivalent to the Bottom-Up Portfolio
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The bottom-up multifactor portfolio is identical
to the combination portfolio shown in Exhibit 1. Note
that the multiplier does not capture information on the
cross-sectional distribution (e.g., each security receives
one unique multiplier, no two multipliers are the same;
the sums of the multipliers are identical), and the starting
weights are equal weights.
We next replaced the starting weights with market-cap weights instead of equal weights. As summarized
in Exhibit 2, the correlation between the weights of the
combination and bottom-up portfolios is exactly 1, but
the actual weights of the two portfolios are no longer
identical. Applying the multipliers to cap weights creates
differences due to the way the “excess weight” (i.e., the
difference between the sum of the weights and the 100%
target weight) is allocated. If equal weights are used as
starting points, this excess weight is uniformly distributed
across the 10 securities. But when cap weights are used,
the excess weight is prorated based on the market-cap
weights; larger-cap, higher-ranked securities receive
more of that excess weight. Because the two approaches
­distribute the excess weight differently, the final combination and bottom-up portfolios are not the same.1 That
said, we interpret the correlation of 1 to mean that the
essence of the portfolios is comparable. The ordering of
the security weights is identical in the two portfolios.
A SECOND STYLIZED EXAMPLE
What would happen if we used scores and not
ranks?
To calculate scores for each factor, we standardize
the raw metrics for each security by subtracting the mean
and dividing by the standard deviation across securities.
Thus,
42 Bottom-Up versus Top-Down Multifactor Portfolio ConstructionSpecial Issue 2016
EXHIBIT 2
Why does the score-based approach
create a larger difference between combination and bottom-up portfolios? In
this example, the score-based approach
preserves the distributional differences in
a way that the rank-based approach does
not. Does employing a rank-based method
rule out the ability to capture these interaction effects between factors? In this stylized
example, it did, because of the way we had
specified the multipliers (i.e., one unique
multiplier for each unique rank). However,
there are portfolio construction methods
that use ranks and do capture interaction
effects (see Brandhorst [2013]).
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Summary of Rank-Based Approach
xi − x
σ
To test the impact of combination versus bottom-up portfolio construction with an actual investment
universe, we consider a four-factor portfolio capturing
value, low volatility, quality, and momentum (as commonly defined in existing literature).3 The two multifactor portfolios were constructed as follows:
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where xi is the security’s raw characteristic, x is the
average across all securities, and σ is the standard deviation of the raw characteristics across all securities. Scores
preserve the distributional characteristics of each factor
in a way that ranks do not. If a security has an extremely
high price-to-book relative to the other securities, that
“extremeness” will be captured by the score.
Once we assign scores to the securities, how do we
determine the multiplier? Because scores can be negative
(e.g., −5 to +5), we cannot just set multipliers equal to
scores, as we did for the ranks.2 Keeping it simple, we
assigned ranks based on scores and set the multipliers equal
to the ranks. Note that ranking based on scores is equivalent to ranking based on raw metrics in the combination
approach. However, in the bottom-up approach, average
rank is not the same as a rank based on average score. This
is a critical point. In this example, the multiplier applied to
each security for the bottom-up portfolio is no longer just
the average of the multipliers for the single-factor portfolios. That is, we changed the way we specify multipliers;
Equation 5 still holds, but Equation 6 does not.
Exhibit 3 summarizes the difference between the
rank-based approach and the score-based approach. When
we use scores instead of ranks, the two portfolios’ weights
are meaningfully different, and the correlations are not
1—whether we apply the multipliers to equal weights or
cap weights.
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(9)
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GLOBAL PORTFOLIO
SIMULATIONS
Special Issue 2016
JPM-BENDER.indd 43
• Combination Portfolio: For the combination
portfolio, we created single-factor portfolios for
the four factors: value, low volatility, quality, and
momentum. First, we calculated the security scores
for each factor and sorted the securities based on
those scores. Second, we grouped the securities
into 20 subportfolios, in which each subportfolio
held 5% of market-cap weight.4 We applied a fixed
set of multipliers (linearly interpolated between
0.05 and 1.95 in increments of 0.10) to the subportfolios; the multiplier was applied to the market-cap
weight of the security, depending on which subportfolio it fell in. The multipliers are shown in
Exhibit 4. Finally, we rescaled the weights so that
they summed to 100%. The combination portfolio is an equally weighted average of the four
single-factor portfolios. Portfolios are rebalanced
monthly.
• Bottom-Up Portfolio: First, we assigned scores
to securities for each factor. Second, we averaged
the scores (equally weighting the factors). Third,
we grouped the securities into 20 subportfolios,
The Journal of Portfolio Management
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Exhibit 3
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Bottom-Up vs. Combination Method: Rank vs. Score-Based Approach, December 31, 2014 (10-stock example)
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Multipliers Assigned to Subportfolios in Simulations
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Exhibit 4
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Notes: The starting weight is 5% for each subportfolio. Because each subportfolio holds 5% of market cap, this is equivalent to holding market-cap weights.
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each holding 5% of market cap, based on their
average scores. We then applied the same fixed set
of multipliers, depending on the subportfolio the
security fell in. Finally, we rescaled the weights so
that they summed to 100%. The factor definitions,
universe, and rebalancing frequency are the same
as previously.
Exhibit 5 shows the results of the simulations.
The bottom-up portfolio returns are higher than any
of the underlying component factor returns and higher
than the combinations. The difference is not insignificant—a spread of 86 basis points. Moreover, the
volatility of the bottom-up portfolio is significantly
lower, and risk-adjusted return increases from 0.73 in
the combination portfolio to 0.84 in the bottom-up
approach.
The differences between the two portfolios should
not come as a surprise to us. Consider the case in which
we group stocks into quartiles for value, momentum,
44 Bottom-Up versus Top-Down Multifactor Portfolio ConstructionSpecial Issue 2016
EXHIBIT 5
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Combination vs. Bottom-Up Approach, Four-Factor Portfolios, January 1993–March 2015 (gross USD returns)
EXHIBIT 6
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Interaction Effects Captured in Distribution of
Aggregate Scores, May 2015
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and quality for a global developed market universe.
We assigned scores based on the quartiles,5 and then
we summed the scores for two pairs: value–quality and
value–momentum.6 Stocks with a score of zero rank the
worst on both factors simultaneously; stocks with a score
of 6 rank the best on both factors simultaneously. It is
apparent in Exhibit 6 that the distribution for value–
quality is quite different from value–momentum; for
example, a greater percentage of stocks rank the highest
on both value and momentum than on value and quality.
These are the interaction effects captured by the bottom-up approach.
In our backtest, has the bottom-up approach consistently produced better performance over the combination approach in all periods? Exhibit 7 shows the
36-month excess rolling returns for the two portfolios
in Exhibit 5, as well as the rolling excess difference in
returns. The bottom-up approach underperformend
only during a short period in the early 2010s.
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Notes: Average historical turnover for the portfolios ranges from 30% to 70% (one-way annual)—the lowest for low volatility and the highest for
momentum. The bottom-up portfolio has exhibited 46% average annual one-way turnover for the period shown. Turnover in smart beta strategies is largely
driven by the rebalancing frequency. Here, our choice to deploy quarterly rebalancing (to incorporate momentum, which requires the higher rebalancing frequency to be effective) causes turnover to be higher than the 15% –25% range typically seen in smart beta strategies. Semi-annual and annual rebalancing
frequencies for non-momentum factors is more the norm.
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THE IMPACT OF STOCK SCREENING
OR SELECTION
IT
How would the results change if we were to
remove, or screen out, a subset of securities from the
starting universe? Stock screening (also sometime
referred to as stock selection) is equivalent to assigning
a multiplier of zero in our framework. In other words,
the bottom-ranked securities are not held but are merely
Special Issue 2016
JPM-BENDER.indd 45
underweighted. Intuitively, we would not expect the
impact of stock screening to change our hypothesis that
the interaction effects captured by bottom-up techniques
may have a meaningful impact on the final multifactor
portfolio.
To test this, we repeat the simulations in the previous section, but this time we employ stock screening.
The only change is to remove the bottom 5 subportfolios; the same multipliers shown in Exhibit 4 are applied
to the top 15 subportfolios. The results, shown in
Exhibit 8, confirm our expectation that the bottom-up
approach captures interaction effects in a way that the
combination does not, even when stock screening is
employed.
The Journal of Portfolio Management
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Exhibit 7
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Time Variation in the Performance of the Bottom-Up vs. Combination Approaches
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There are in fact several ways to introduce stock
screening in bottom-up multifactor portfolio construction. To understand what occurs under different forms
of screening, imagine stocks arrayed along two factor
dimensions, as shown in Exhibit 9. In our bottom-up
­simulations, we ranked stocks along each dimension and
then combined the two dimensions while simultaneously removing the bottom-ranked stocks. This is shown
in Exhibit 9, Diagram A (for illustrative purposes, we
removed the bottom half, rather than the bottom quarter,
in the simulations). In the combination approach, we
ranked stocks along each dimension, creating one factor
portfolio at a time by deleting the lowest-ranked stocks
and then combining the two-factor portfolios. This is
shown in Diagram B of Exhibit 9. The third approach,
shown in Diagram C, is to take the ­intersection of the
two ­factors. This is another valid approach to creating
a bottom-up portfolio that accounts for the interaction
effects between factors.
INTERACTION EFFECTS FOR DIFFERENT
FACTOR COMBINATIONS
In this last section, we drill down into the interaction effects by pair. We evaluate the following six pairs:
•momentum–quality
• momentum–low volatility
• quality–low volatility
•value–momentum
•value–quality
• value–low volatility
46 Bottom-Up versus Top-Down Multifactor Portfolio ConstructionSpecial Issue 2016
EXHIBIT 8
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Combination vs. Bottom-Up Approach: Four-Factor Portfolios, January 1993–March 2015 (gross USD returns)
EXHIBIT 9
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The Impact of Screening
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Exhibit 10 first plots joint distributions, similar
to Exhibit 6. We grouped stocks into quartiles for
value, momentum, and quality. We assigned scores
based on the quartiles and summed the scores for each
pair. As before, stocks with a score of zero rank the
worst on both factors simultaneously; stocks with a
score of 6 rank the best on both factors simultaneously.
There are clear distributional differences between the
pairs—the most distinct distributions being value–
quality and value–momentum. Note that these two
factors also have the lowest correlations among all
the factors.
Special Issue 2016
JPM-BENDER.indd 47
We repeated the simulations shown in the previous section, but this time for each pair at a time.
Based on the distributions shown in Exhibit 10, we
expect value–quality and value–momentum to show
the largest performance differences between combination and bottom-up approaches. Exhibit 11 shows the
backtested performance of the pairs, and it corroborates
what we expect based on the joint distributions. Moreover, there is surprising consistency across the pairs;
the bottom-up approach universally outperforms the
combination approach.
The Journal of Portfolio Management
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Exhibit 10
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Joint Distributions for Factor Characteristics, January 1993–March 2015 (gross USD returns)
48 Bottom-Up versus Top-Down Multifactor Portfolio ConstructionSpecial Issue 2016
EXHIBIT 11
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Combination vs. Bottom-Up Differences by Factor Pair
APPENDIX A
Indexed-factor portfolios have emerged in recent
years as an alternative for investors dissatisfied with market-cap weighting or as an explicit way to achieve exposure to well-known factors that have been shown to drive
stock returns. These portfolios are passively implemented
and thus retain the same benefits as traditional passive
investing—transparency, implementation efficiency, and
low costs. These innovations are changing the investment landscape, which until recently was characterized
by traditional passive investing and active management.
Portfolios that utilize multiple factors have emerged
as a way of accessing the diversification opportunities
inherent across factors. Multifactor portfolios can be
constructed either by combining individual singlefactor portfolios or by creating bottom-up portfolios in
which security weights are a function of multiple factors simultaneously. We evaluate the two approaches in
this article and show that a bottom-up approach will
produce different, and sometimes superior, results than
will a combination of individual single-factor portfolios. This is because bottom-up approaches capture nonlinear cross-sectional interaction effects between factors
that combination approaches do not. We evaluate wellknown factors such as value, quality, low volatility, and
momentum and find that the bottom-up approach results
in higher excess returns over the long run. We further
find that there are differences in the joint distributional
characteristics of various pairs. The most distinct differences arise for the value–quality and value–momentum
pairings—where the difference between bottom-up and
combination portfolios is the starkest.
EQUIVALENCE OF BOTTOM-UP AND
COMBINATION PORTFOLIOS
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CONCLUSION
M S
W =λ
+(
S ′ ×
M
Factor 1
O
D
⎡λ + (
W = ⎣
⎣⎡ λ + (
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Special Issue 2016
JPM-BENDER.indd 49
L S
S
L ′ ×
)
(A-1)
Factor 2
Note that ° denotes element-wise product.
The bottom-up two-factor portfolio is
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The combination of a two-factor portfolio is
) L ⎤⎦ S
′
) L ⎤⎦ × S
(A-2)
The following two conditions that must be satisfied for
these to be the same:
•
s1 = s2 = ...sn
(A-3)
•
∑
(A-4)
∑l
i
Here is the proof:
If s1 = s2 = …sn and
it is also true that
⎡⎣ λ
+(
)
∑
∑ l , then M′ × S = L′ × S,
i
⎤⎦′ × S = λ ′ × S + (
) L ′S
= M ′ × S = L′ × S
(A-5)
We substitute Equation A-5 in Equation A-1 as
follows:
The Journal of Portfolio Management
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5/17/16 8:36:53 PM
M S
L S
M S
λ
+ (1 − λ )
=λ
′
M
S S
′×
′ ×
L
 λM + (1 − λ ) L  × S
Factor 1
Factor 2
T
M
A
R
Y
N
Factor 2
 λM + (1 − λ ) L   S  λM + (1 − λ ) L   S
=
M′ ×S
M′ ×S
(A-8)
A
M S
M S
[ λM + (1 − λ )L ] S
λ
+ (1 − λ )
=
M
S M
S
M′ ×S
′×
′ ×
Factor 1
FO
(A-7)
IN
(A-9)
C
E
TH
It should be noted that the condition M × S = L′ × S is
all that is required for the two equations to be equivalent.
Thus, if Equation A-3 were not to hold, there could still be a
set of multipliers such that M′ × S = L′ × S; likewise, if Equation A-4 were not to hold, there could still be a set of starting
weights such that M′ × S = L′ × S.
LE
 λM + (1 − λ ) L   S
= 
′
 λM + (1 − λ ) L  × S
(A-6)
C
λ ( M  S ) + (1 − λ ) L  S
′
 λM + (1 − λ ) L  × S
R
TI
′
 λM + (1 − λ ) L  × S
A
=
L S
IS
+ (1 − λ )
12-month return minus the last month’s return. Size is measured as free-f loat market capitalization in U.S. dollars.
4
In the previous section, we applied a unique multiplier
to each security. When we implement the multiplier-based
approach to real portfolios, we apply multipliers to groupings,
or “subportfolios,” of securities, rather than at the stock level
(e.g., a unique multiplier for each security). Intuitively, this
reduces the impact of security-specific noise, because our
objective is to tilt in aggregate toward securities with the
desired factor characteristics. (Factor effects are not generally
monotonically increasing, i.e., for each incremental increase in
book-to-price, there is not an incremental increase in return.)
The number of subportfolios we decide to employ is relatively
arbitrary because there is no theoretical relationship between
the level of granularity and the level of noise. Our aim is to
balance the objective of reducing noise with the desire to sufficiently differentiate the weighting scheme across securities,
because the latter occurs as we reduce granularity.
5
The least desirable Quartile 1 receives a score of zero;
Quartile 2 receives a score of 1, etc.
6
The data are shown as of May 20, 2015. Momentum is
defined as previously discussed. Value here is approximated
by annual price-to-book ratio, and quality is approximated
by return on equity.
U
ENDNOTES
IT
IS
IL
LE
G
A
L
TO
R
EP
R
O
D
The authors would like to thank Xiaole Sun for her
many contributions to this research and Scott Conlon and Ana
Harris for their many insights. We are also grateful to Marc
Reinganum for providing the idea behind the article’s title.
1
We note that this “reweighting effect” is an artifact
of the multiplier-based framework. By determining weights
in this way in Equation 4, we introduce this effect, which is
not present in the equal-weighting or fundamental-weighting
approaches.
2
Note that the issue of transforming scores to weights
is a widespread one in heuristic rules-based portfolios. Index
vendors have proposed various mapping functions to deal
with this transformation, including linear and nonlinear functions. Probability-based functions have also been suggested.
3
Value measured as exponentially weighted five-year
averages of earnings, cash f low, sales, dividend, and book
value in the denominator and price in the numerator. The
five price-fundamental ratios are equally weighted. Low
volatility is measured as 60-month variance of returns.
Quality is measured as return on assets, debt to equity, and
five-year variability in earnings per share. These three measures are equally weighted. Momentum is measured as trailing
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Asness, C. “The Interaction of Value and Momentum Strategies.” Financial Analysts Journal, Vol. 53, No. 2 (1997).
Bender, J., E. Brandhorst, and T. Wang. “The Latest Wave
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A Coherent Framework for Advanced Beta Portfolio Construction.” The Journal of Index Investing, Vol. 6, No. 1 (2015),
pp. 51-64.
Brandhorst, E. “A Methodology for Capturing Advanced
Beta Factors.” Capital Insights, State Street Global Advisors,
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Ross, S.A. “The Arbitrage Theory of Capital Asset Pricing.”
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To order reprints of this article, please contact Dewey Palmieri
at dpalmieri@ iijournals.com or 212-224-3675.
50 Bottom-Up versus Top-Down Multifactor Portfolio ConstructionSpecial Issue 2016