Pure Quintile Portfolio

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Pure Quintile Portfolios

Ding LiuAllianceBernstein

March 30, [email protected]

Abstract

In this paper we propose a new portfolio construction framework called Pure Quintile

Portfolios. These portfolios overcome the main drawback of naïve quintile portfolios based on single sorts, namely, not having pure exposures to the target factor. Each pure

quintile portfolio has the same exposure to the target factor as its naïve counterpart, but

also has zero exposures to all other factors. Therefore pure quintile portfolios moreaccurately reflect the cross sectional distribution of true factor returns. In addition, whenwe long Q1 and short Q5 to capture factor premia as is most commonly done in research

and practice, we find that pure Q1-Q5 portfolio has lower risk and higher Sharpe ratio

than naïve Q1-Q5 portfolio for a group of widely used factors, thus providing evidencethat our new framework creates more efficient and stable factor premia than naïve

quintile portfolios.


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1) Introduction

It is well known that equity returns are driven to a large extent by factors such as Value

and Size (Fama and French 1992, 1996). A common way to evaluate the efficacy of afactor is to form quintile portfolios by sorting on that factor. It is also common to long Q1

short Q5, and use the resulting market-neutral portfolio to capture the factor return. These“naïve” quintile portfolios are intuitive and easy to construct, but they don’t represent pure exposures to the factor. For example, the top quintile portfolio by book-to-price

outperforms the bottom quintile over the long run, but high book-to-price stocks

generally also have smaller market caps and lower balance sheet accruals than low book-to-price stocks, and therefore more exposed to the small cap effect (Banz 1981) and the

accruals effect (Sloan 1996). It is not clear how much of the naïve Q1-Q5 portfolio return

is due to its book-to-price versus market cap or balance sheet accruals exposures. So,

naïve Q1-Q5 portfolio by sorting book-to-price does not reflect the efficacy of a pure book-to-price factor.

Besides masking the true factor efficacy, naïve Q1-Q5 portfolios often suffer fromoffsetting effects from unintended factor exposures that wash out the intended factor performance. For example, performance of high book-to-price stocks (i.e. Value) is hurt

by their negative exposures1 to Momentum, which has positive return over the long run.

Likewise, high Momentum stocks do not realize the full outperformance of pureMomentum because of wash-out from negative Value exposures. Therefore it should be

possible to improve the performance of both factors simultaneously by disentangling

interactions between them. There are also other benefits of pure factor returns such as

they are more predictable than naïve factor returns and are additive (Jacobs and Levy1989).

One way to disentangle factors is two-way sort (Basu 1983). For example, to createValue quintile portfolios with roughly the same Momentum exposures, first sort by

Momentum and form quintile portfolios. Next, within each Momentum quintile sort by

Value and form the next level quintile portfolios (called buckets). Then take the highest

Value bucket from each Momentum quintile and combine them as new Value quintileone, take the second highest Value bucket from each Momentum quintile and combine

them as new Value quintile two, and so on. These new Value quintiles have

monotonically lower Value exposures, but similar Momentum exposures because eachone draws one-fifth of its stocks from every Momentum quintile. However, this method

does not generalize well to multiple factors: with 5 factors it would create 3125 buckets,

way more than the number of large cap stocks in the U.S. Another limitation is that it

does not disentangle factors completely: these new Value quintiles have similar but notexactly the same Momentum exposures.

Another way to disentangle factors is to run multivariate regressions (Jacobs and Levy

1988, 1989, Back, Kapadia and Ostdiek 2013). Let X be the matrix of standardized factor

1 In this paper we standardize factor exposures across all stocks by subtracting the equally-weighted

average and dividing by the cross-sectional standard deviation, so negative exposure means below average.


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exposures (e.g. standardized book-to-price, market capitalization) and r the vector of

excess returns over all stocks in a universe such as the Russell 1000 index2.

Regressing r against all column vectors of X simultaneously gives r X X X ')'( 1 , which

can be interpreted as returns of the standardized factors. The row vectors

of ')'( 1 X X X define a set of portfolios that mimic these factors (called Factor-Mimicking

Portfolios or FMPs). Because I X X X X )')'(( 1 , each FMP has one unit of exposure to

a single factor and zero exposures to all other factors, and therefore represents pure

exposure to that factor. Alternatively, the same FMPs can be created by optimizations:

consider minimizing ww' subject to iew X ' , where ie is a k by1vector with one on

the i -th positon and zeros elsewhere. It is easy to show that the solution of this

optimization is the i -th row of ')'( 1 X X X (see Grinold and Kahn 2000).

We adapt and extend this FMP optimization framework to create pure quintile portfolios.

In previous research (for example, Melas, Suryanarayanan and Cavaglia 2010),

optimization is used to create only one FMP per factor, typically a long-short market-

neutral portfolio with one unit of standardized exposure to that factor. We adapt theoptimization in a number of ways: 1) for each target factor we run 5 optimizations to

create 5 long-only portfolios called pure quintile portfolios; 2) in each optimization we

set the number of stocks in the pure quintile portfolio to be the same as a naïve quintile portfolio; 3) in each optimization we set the pure quintile portfolio’s exposure to the

target factor to be the same as the corresponding naïve quintile portfolio; 4) in each

optimization we set the pure quintile portfolio’s exposures to all other factors to zero. Asa result, pure quintile portfolios have the same number of stocks and span the same cross

section of exposures to the target factor as naïve quintile portfolios, but also have zero

exposures to all non-targeted factors. Therefore they represent a spectrum of pure

exposures to the target factor. We are not aware of any previous studies that have done

this, which is the first contribution of this paper.

As the second contribution of this paper, we found that when pure quintile portfolios arecreated in the US large cap universe using a set of simple and commonly used factors,

pure Q1-Q5 portfolios have substantially higher Sharpe ratios than naïve Q1-Q5

portfolios across all factors. Interestingly, this is driven by both risk reduction and returnenhancement. Each pure Q1-Q5 portfolio has lower risk, and with the exception of one,

higher return than naïve Q1-Q5 portfolio. Looking at each Q1 and Q5 separately, we

found that almost every pure quintile portfolio has lower risk than its naïve counterpart

(which is also true for Q2, Q3 and Q4). The higher return of pure Q1-Q5 portfolio comesfrom both long and short sides. That is, pure Q1 has higher return than naïve Q1, and

pure Q5 has lower return than naïve Q5. This is evidence that our pure quintilemethodology creates stronger and more stable Q1-Q5 factor returns than naïve quintile

sorts. Similar evidences exist in Developed International and Emerging Market stocks,albeit weaker than in the US.

2 Assume there are n stocks and k factors, then X is a n by k matrix and r is a n by1vector.


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The rest of this paper is organized as follows. The next section describes data and

demonstrates unintended factor exposures in naïve quintile portfolios. Then, the pure

quintile portfolio framework is described and their performances are compared withnaïve quintile portfolios in the US large cap universe. Afterwards, we repeat the analysis

in Developed International and Emerging Market stocks. This is followed by some brief

concluding remarks.

2) Data and Factor Exposures of Naïve Quintile Portfolios

In this paper we focus on five factors: Value, Size, Price Momentum, Profitability andEarnings Quality. We choose these factors because they are all well-known, extensively

studied in the literature, and widely used in practice. To define these factors, we use

book-to-price for Value, the natural log of market capitalization for Size, 11 month past

price return lagged by 1 month for Price Momentum, return-on-equity for Profitability,and balance sheet accruals

3 for Earnings Quality. These factor definitions are simple and

fairly standard. There are many other ways of defining these factors4, but here we are not

interested in fine-tuning factor definitions to make the most economic sense or to realizethe best performance. This paper is focused on demonstrating the pure quintile portfolioframework and comparing performance with naïve quintile portfolios, and for that

purpose we prefer simple definitions.

Every month from January 1979 to December 2014, we collect these factors for all stocks

in the Russell 1000 index from AllianceBernstein’s internal equity research database,

which in turn gets the raw data from multiple sources including Compustat, CRSP, and

Russell. We choose the Russell 1000 universe because it is widely used by institutionalmanagers as a barometer for US large cap investments. Table A1 in the appendix shows

the number of stocks in Russell 1000, those with data on each factor, and those with data

on all factors at the beginning of each year. To create naïve and pure quintile portfolios ofeach factor, we use all stocks with data on that factor, even though some of them miss

data on other factors. We have repeated the analysis using only stocks with data on all

five factors, and the results are very similar.

Following common practice, every month for every factor we first winsorize its raw

values at 5% and 95% levels, and then standardize them by subtracting the equally-

weighted average and dividing by the cross-sectional standard deviation. We call thesestandardized factor values “exposures”. Because of standardization, exposures are

comparable across factors and across months. Throughout this paper, we use “market” to

mean the equally weighted portfolio of all stocks, which has zero exposures to all factors

because of standardization. Therefore a portfolio with exposure of one to Value, forexample, means that its weighted average book-to-price is one standard deviation above

the market. For Value, Profitability and Price Momentum, higher factor exposures have

3 Balance sheet accruals is calculated as one year change of asset accruals minus liability accruals, divided

by average total assets, where asset accruals is total assets minus cash and short term investments, and

liability accruals is total liabilities minus debt in current liabilities and total long term debt.4 For example, Value is sometimes defined as a combination of book-to-price, earnings-to-price and

dividend yield.


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higher expected returns; but for Size and Earnings Quality the opposite is true because

stocks with smaller market cap and lower balance sheet accruals tend to outperform. For

consistency, we flip the signs of Size and Earnings Quality exposures so that in all caseshigher exposures have higher expected returns. Naïve quintile portfolios are then created

every month by sorting on these factor exposures, with quintile one having the highest

exposure and quintile five having the lowest exposure, and stocks equally weightedwithin each quintile.

The left side of Table 1 shows average standardized factor exposures of naïve quintile

portfolios from 1979 to 2014. It is clear that they all pick up unintended factor exposuresto some degree on average. For example, stocks in Value Q1 have smaller market caps,

lower recent past returns, lower profits, and lower balance sheet accruals than other

stocks. These unintended exposures are not driven by some extreme correlations between

Value and the other factors during a short period of time. In fact they are generally persistent over time: Figures 1 and 2 show the rolling 12 month average factor exposures

of Value naïve Q1 and Q5.

Figure 1.

Figure 2.


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Table 1. Average Standardized Factor Exposures of Naïve and Pure Quintile Portfolios, 1979 – 2014

Value Naive Quintiles Value Pure Quintiles

Value SizePrice

MomentumProfitability

Earnings

QualityValue Size

Price


Earnings

Quality

Q1 1.58 0.21 -0.48 -0.68 0.20 Q1 1.58 0.00 0.00 0.00 0.00

2 0.46 0.07 -0.17 -0.31 0.12 2 0.46 0.00 0.00 0.00 0.003 -0.17 -0.03 -0.01 -0.06 0.03 3 -0.17 0.00 0.00 0.00 0.00

4 -0.68 -0.13 0.16 0.30 -0.11 4 -0.68 0.00 0.00 0.00 0.00

Q5 -1.19 -0.18 0.50 0.78 -0.26 Q5 -1.19 0.00 0.00 0.00 0.00

Size Naive Quintiles Size Pure Quintiles

Value SizePrice


Earnings

QualityValue Size

Price


Earnings

Quality

Q1 0.21 1.17 -0.30 -0.21 -0.04 Q1 0.00 1.17 0.00 0.00 0.00

2 0.06 0.70 0.02 -0.08 -0.03 2 0.00 0.70 0.00 0.00 0.00

3 -0.03 0.20 0.10 -0.03 -0.03 3 0.00 0.20 0.00 0.00 0.00

4 -0.03 -0.47 0.08 0.05 0.05 4 0.00 -0.47 0.00 0.00 0.00

Q5 -0.21 -1.59 0.09 0.26 0.05 Q5 0.00 -1.59 0.00 0.00 0.00

Price Momentum Naive Quintiles Price Momentum Pure Quintiles

Value SizePrice


Earnings

QualityValue Size

Price


Earnings

Quality

Q1 -0.50 -0.05 1.53 0.19 -0.10 Q1 0.00 0.00 1.53 0.00 0.00

2 -0.16 -0.15 0.42 0.13 0.04 2 0.00 0.00 0.42 0.00 0.00

3 0.04 -0.11 -0.11 0.05 0.08 3 0.00 0.00 -0.11 0.00 0.00

4 0.20 -0.01 -0.58 -0.04 0.05 4 0.00 0.00 -0.58 0.00 0.00

Q5 0.43 0.27 -1.27 -0.30 -0.06 Q5 0.00 0.00 -1.27 0.00 0.00

Profitability Naive Quintiles Profitability Pure Quintiles

Value SizePrice


Earnings

QualityValue Size

Price


Earnings

Quality

Q1 -0.86 -0.26 0.22 1.40 -0.15 Q1 0.00 0.00 0.00 1.40 0.00

2 -0.37 -0.15 0.10 0.43 -0.10 2 0.00 0.00 0.00 0.43 0.003 0.12 0.01 0.02 -0.01 -0.01 3 0.00 0.00 0.00 -0.01 0.00

4 0.59 0.07 -0.12 -0.42 0.07 4 0.00 0.00 0.00 -0.42 0.00

Q5 0.58 0.26 -0.24 -1.41 0.18 Q5 0.00 0.00 0.00 -1.41 0.00

Earnings Quality Naive Quintiles Earnings Quality Pure Quintiles

Value SizePrice


Earnings

QualityValue Size

Price


Earnings

Quality

Q1 0.14 0.06 0.00 -0.34 1.16 Q1 0.00 0.00 0.00 0.00 1.16

2 0.21 -0.10 -0.03 -0.04 0.55 2 0.00 0.00 0.00 0.00 0.55

3 0.08 -0.12 -0.03 0.08 0.19 3 0.00 0.00 0.00 0.00 0.19

4 -0.11 -0.06 -0.03 0.17 -0.29 4 0.00 0.00 0.00 0.00 -0.29

Q5 -0.29 0.11 0.05 0.12 -1.60 Q5 0.00 0.00 0.00 0.00 -1.60

3) Pure Quintile Portfolios

In each month, for each naïve quintile portfolio of each factor (called the target factor) we

run the following optimization to create a corresponding pure quintile portfolio. Here n


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is the number of stocks with exposure data to the target factor 5, exp the naïve quintile

portfolio’s exposure to the target factor,i

x the vector of stock exposures to factor i , w

the vector of weights being optimized, and e a vector of ones. All vectors have size n by

1.

Minimize ww' (3.1)Subject to:

0w (3.2)

1' we , (3.3)

expw xi ' , for the target factor i (3.4)

0' w xi , for other non-targeted factors i (3.5)

The number of stocks is the same as the naïve quintile, i.e. about 5/n (3.6)

Constraints (3.2) and (3.3) make sure the pure quintile portfolio is long only and 100%

invested. Constraint (3.4) makes its exposure to the target factor the same as the naïve

quintile portfolio, and constraint (3.5) makes sure it has zero exposures to other factors.The right side of Table 1 confirms that these constraints are satisfied. Constraint (3.6)

makes sure it has the same number of stocks as the naïve quintile portfolio. This is a

name count constraint and is supported by Axioma’s optimizer. Finally, the objectiveterm (3.1) pushes the optimized weights towards equal weights as much as possible,

which is the weighting scheme of naïve quintile portfolios.

It is important to note the similarity and difference between our optimization and those in

previous studies such as Melas et al. (2010) and Grinold and Kahn (2000). In both

situations the exposures to non-targeted factors are set to zero. In previous studies

exposure to the target factor is set to one to create a single long-short portfolio, but here

we vary that exposure to create five portfolios that span the same spectrum of exposuresas naïve quintile portfolios. The long only and name count constraints are used to make

them comparable to naïve quintile portfolios, but are also needed to avoid the following

situation: the optimizer simply returns a combination of a fixed portfolio with unitexposure to the target factor and the market for each optimization, and varies their

proportion to get different target factor exposures. Obviously that is not what we intend

to do. Instead we want to find a group of stocks with the same size as a naïve quintile portfolio and weight them, as equally as possible, to satisfy all the desired factor

exposures. It is reasonable to expect a substantial overlap between each naïve quintile

portfolio and its pure counterpart because they have the same exposure to the target factor.

On the other hand, the pure quintile portfolio also draws stocks from other naïve quintile

portfolios in order to offset other factor exposures. Later we will see that this is indeedthe case.

5 Note that n is generally different across factors; see Table A1 in the appendix.


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Table 2. Performance of Naïve and Pure Quintile Portfolios, 1979 – 2014

Value Naive Quintiles Value Pure QuintilesAvg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 2.6% 8.7% 0.30 153% Q1 3.0% 7.7% 0.39 319%

2 0.4% 4.4% 0.10 248% 2 2.2% 3.9% 0.56 612%

3 -0.8% 3.1% -0.25 255% 3 0.0% 2.9% 0.01 670%

4 -1.2% 4.0% -0.29 220% 4 -2.0% 3.9% -0.52 582%

Q5 -1.1% 8.2% -0.14 130% Q5 -2.7% 7.2% -0.37 211%

Q1-Q5 3.7% 15.7% 0.24 282% Q1-Q5 5.7% 12.3% 0.46 530%

Size Naive Quintiles Size Pure QuintilesAvg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 2.8% 9.0% 0.31 195% Q1 3.6% 6.1% 0.59 264%

2 0.2% 3.1% 0.06 225% 2 0.4% 3.7% 0.11 746%

3 -0.6% 2.8% -0.22 185% 3 -0.5% 2.4% -0.19 566%

4 -0.4% 3.3% -0.11 135% 4 -1.2% 3.3% -0.38 652%

Q5 -2.0% 5.8% -0.35 69% Q5 -2.2% 5.0% -0.44 98%

Q1-Q5 4.8% 14.1% 0.34 265% Q1-Q5 5.8% 10.3% 0.56 362%

Price Momentum Naive Quintiles Price Momentum Pure QuintilesAvg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 3.7% 10.9% 0.34 328% Q1 4.2% 10.0% 0.43 398%

2 0.7% 5.4% 0.13 604% 2 0.2% 4.6% 0.04 736%

3 -0.3% 4.3% -0.08 664% 3 -0.5% 3.3% -0.15 865%

4 -1.3% 4.6% -0.27 606% 4 -1.5% 4.5% -0.34 818%

Q5 -2.8% 13.2% -0.21 335% Q5 -3.2% 11.4% -0.28 378%

Q1-Q5 6.5% 22.3% 0.29 664% Q1-Q5 7.4% 19.4% 0.38 776%

Profitability Naive Quintiles Profitability Pure QuintilesAvg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 1.4% 4.7% 0.31 103% Q1 3.3% 5.6% 0.60 245%

2 1.0% 4.0% 0.24 150% 2 2.2% 3.7% 0.58 576%

3 0.2% 4.0% 0.06 173% 3 -0.7% 3.3% -0.21 743%

4 -0.4% 3.5% -0.11 168% 4 -1.5% 2.9% -0.52 527%

Q5 -2.3% 9.9% -0.23 132% Q5 -3.0% 6.7% -0.44 188%

Q1-Q5 3.7% 13.4% 0.28 235% Q1-Q5 6.3% 9.3% 0.68 433%

Earnings Quality Naive Quintiles Earnings Quality Pure QuintilesAvg. Annual

Excess

Return

AnnualizedRisk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

AnnualizedRisk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 3.4% 4.5% 0.76 183% Q1 3.6% 3.3% 1.09 208%

2 2.5% 4.3% 0.58 253% 2 1.4% 3.2% 0.43 610%

3 1.0% 3.4% 0.28 267% 3 0.9% 3.0% 0.30 619%

4 -1.1% 2.9% -0.39 241% 4 -1.3% 2.6% -0.52 606%

Q5 -5.8% 8.1% -0.71 169% Q5 -5.0% 5.4% -0.92 202%

Q1-Q5 9.2% 11.0% 0.84 352% Q1-Q5 8.6% 7.8% 1.11 411%


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Table 2 compares the historical performance of naïve and pure quintile portfolios side by

side for the 1979 to 2014 period. The first column shows the average annual returns inexcess of the market for Q1 through Q5, as well as the Q1 minus Q5 return. The second

column shows the annualized risks of these returns measured by standard deviation. The

third column shows the return-to-risk ratio. For Q1 through Q5 it is the Sharpe ratio oflong each quintile and short the market; for Q1-Q5 it is the Sharpe ratio of long-Q1 andshort-Q5. The last column shows annualized one-way turnover. Returns are before

transaction costs and include dividends.

Perhaps the most notable observation from Table 2 is that each factor ’s pure Q1-Q5

portfolio risk is lower by 3% or more than naïve Q1-Q5 portfolio, and each pure Q1-Q5

portfolio return is higher by about 1% or more, except for Earnings Quality. As a result,

each pure Q1-Q5 portfolio has substantially higher Sharpe ratio than naïve Q1-Q5 portfolio. Looking at Q1 and Q5 separately, we found that almost every pure quintile

portfolio has lower risk than its naïve counterpart. This is also true for quintile portfolios

Q2, Q3 and Q4. The higher return of each pure Q1-Q5 portfolio comes from both its longand short sides. For example, Value pure Q1 return is 40bps higher than Value naïve Q1,

and Value pure Q5 return is 1.6% lower than Value naïve Q5. The same pattern holds for

other factors (except Earnings Quality Q5). This is evidence that our pure quintile

methodology creates stronger and more stable Q1-Q5 factor returns than naïve quintilesorts.

Annual turnover is high for naïve Q1-Q5 portfolios, and even higher for pure Q1-Q5 portfolios. Do higher Sharpe ratios of pure Q1-Q5 portfolios survive after transaction

costs? The answer is yes. Table 3 shows net of transaction costs performance of naïve

and pure Q1-Q5 portfolios, assuming 30bps of transaction costs per trade one-way6. For

example, annual transaction costs of naïve Value Q1-Q5 is 282%*2*30bps, or about1.7%, which drags down its return from 3.7% to 2.0%. The turnover of pure Value Q1-

Q5 is almost twice as high and drags down its return more (from 5.7% to 2.5%) but still

higher than naïve Q1-Q5. The net of transaction costs Sharpe ratio of pure Value Q1-Q5is still handsomely higher than its naïve counterpart. The same is true for other factors.

Table 3. Net of Transaction Costs Performance of Naïve and Pure Q1-Q5 Portfolios, 1979 – 2014

Naive Q1-Q5

Portfolios

Gross of

T-Cost

Return

Net of

T-Cost

Return

Annualized

Risk

Net of

T-Cost

IR

Pure Q1-Q5

Portfolios

Gross of

T-Cost

Return

Net of

T-Cost

Return

Annualized

Risk

Net of

T-Cost

IR

Value 3.7% 2.0% 15.7% 0.13 Value 5.7% 2.5% 12.3% 0.21

Size 4.8% 3.2% 14.1% 0.23 Size 5.8% 3.6% 10.3% 0.35

Momentum 6.5% 2.5% 22.3% 0.11 Momentum 7.4% 2.8% 19.4% 0.14

Profitability 3.7% 2.3% 13.4% 0.17 Profitability 6.3% 3.7% 9.3% 0.40

Quality 9.2% 7.1% 11.0% 0.65 Quality 8.6% 6.1% 7.8% 0.79

6 We think this is a reasonable assumption for US large cap stocks.


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Quarterly Rebalance. We can also reduce turnover in both naïve and pure quintile portfolios by rebalancing less frequently than monthly, such as quarterly. To be exact, for

each factor and each quintile (naïve or pure) we track 3 portfolios starting from January,February and March of 1979 respectively. The only change we make is that now each

portfolio is rebalanced quarterly instead of monthly, and returns are calculated by

averaging across the 3 portfolios. Table 4 shows performance of naïve and pure quintile portfolios using quarterly rebalance. Similar to Table 2, here we have the same

observations that pure Q1-Q5 portfolios have much lower risks and higher Sharpe ratios

than naïve Q1-Q5 portfolios, and it is still mostly true that pure Q1 portfolios have higherreturns than naïve Q1 portfolios; pure Q5 portfolios have lower returns than naïve Q5

portfolios; and pure quintile portfolios generally have lower risks than their naïve

counterparts.

Comparing turnover numbers in Tables 4 and 2, we see that as expected quarterly

rebalance results in lower turnovers in all cases. The turnover savings are especially big

for Price Momentum, with naïve Q1-Q5 annual turnover saving of almost 300% (from

664% to 370%), and pure Q1-Q5 annual turnover saving of 371% (from 776% to 405%).We also observe that the turnover saving is always bigger for pure quintile and Q1-Q5

portfolios than their naïve counterparts, which means quarterly rebalance mitigates

transaction costs more for pure quintile portfolios. Table 5 shows net of transaction costs performance of naïve and pure Q1-Q5 portfolios using quarterly rebalance (again

assuming 30bps of transaction costs per trade one-way). Same results as in Table 3: pure

Q1-Q5 portfolios have substantially higher Sharpe ratios than their naïve counterpartsafter transaction costs.


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Table 4. Performance of Naïve and Pure Quintile Portfolios, 1979 – 2014, Quarterly Rebalance


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 1.7% 9.8% 0.18 89% Q1 1.8% 7.8% 0.23 156%

2 0.2% 4.8% 0.05 134% 2 1.0% 4.4% 0.23 245%

3 -0.7% 3.7% -0.18 139% 3 -0.1% 3.1% -0.04 259%

4 -0.9% 4.7% -0.19 122% 4 -1.3% 4.3% -0.31 237%

Q5 -0.4% 9.4% -0.04 78% Q5 -1.5% 8.3% -0.19 116%

Q1-Q5 2.1% 17.7% 0.12 166% Q1-Q5 3.3% 13.8% 0.24 273%


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 1.9% 9.4% 0.20 124% Q1 2.7% 6.6% 0.41 147%

2 0.4% 3.1% 0.14 125% 2 0.7% 3.8% 0.19 277%

3 -0.3% 3.0% -0.09 104% 3 -0.1% 2.4% -0.04 224%

4 -0.1% 3.4% -0.03 76% 4 -0.8% 3.6% -0.24 258%

Q5 -1.9% 6.2% -0.31 39% Q5 -2.0% 5.1% -0.40 54%

Q1-Q5 3.7% 14.8% 0.25 164% Q1-Q5 4.7% 10.8% 0.43 201%


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 3.2% 11.8% 0.27 180% Q1 3.7% 10.6% 0.35 201%

2 1.0% 5.5% 0.18 265% 2 0.8% 4.3% 0.18 286%

3 0.0% 4.4% -0.01 278% 3 -0.3% 3.3% -0.09 316%

4 -1.1% 5.2% -0.22 267% 4 -1.2% 4.9% -0.24 308%

Q5 -3.1% 13.5% -0.23 190% Q5 -3.6% 10.7% -0.34 204%

Q1-Q5 6.3% 23.2% 0.27 370% Q1-Q5 7.4% 19.1% 0.38 405%


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 1.3% 5.1% 0.26 76% Q1 2.2% 6.0% 0.37 146%

2 0.8% 4.2% 0.19 115% 2 1.2% 4.0% 0.30 247%

3 0.1% 4.3% 0.03 134% 3 0.0% 3.4% 0.00 286%

4 -0.6% 3.8% -0.15 130% 4 -1.1% 3.0% -0.37 225%

Q5 -1.7% 10.7% -0.16 100% Q5 -2.3% 7.4% -0.30 123%

Q1-Q5 3.1% 14.4% 0.21 176% Q1-Q5 4.4% 10.4% 0.43 269%


Excess

Return

AnnualizedRisk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

AnnualizedRisk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 3.5% 5.0% 0.70 148% Q1 3.3% 3.4% 0.98 155%

2 2.2% 4.1% 0.53 209% 2 1.6% 3.1% 0.53 275%

3 0.9% 3.7% 0.25 221% 3 0.9% 3.0% 0.30 284%

4 -1.0% 3.0% -0.32 196% 4 -1.4% 2.8% -0.52 268%

Q5 -5.6% 8.1% -0.69 135% Q5 -4.6% 5.2% -0.88 144%

Q1-Q5 9.0% 11.1% 0.82 283% Q1-Q5 7.9% 7.6% 1.05 299%


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12

Table 5. Net of Transaction Costs Performance of Naïve and Pure Q1-Q5 Portfolios, 1979 – 2014,

Quarterly Rebalance

Naive Q1-Q5

Portfolios

Gross of

T-Cost

Excess

Return

Net of

T-Cost

Excess

Return

Annualized

Risk

Net of

T-Cost

IR

Pure Q1-Q5

Portfolios

Gross of

T-Cost

Excess

Return

Net of

T-Cost

Excess

Return

Annualized

Risk

Net of

T-Cost

IR

Value 2.1% 1.1% 17.7% 0.06 Value 3.3% 1.7% 13.8% 0.12

Size 3.7% 2.8% 14.8% 0.19 Size 4.7% 3.5% 10.8% 0.32

Momentum 6.3% 4.1% 23.2% 0.17 Momentum 7.4% 4.9% 19.1% 0.26

Profitability 3.1% 2.0% 14.4% 0.14 Profitability 4.4% 2.8% 10.4% 0.27

Quality 9.0% 7.3% 11.1% 0.66 Quality 7.9% 6.1% 7.6% 0.81

Pure Quintile Portfolio Distributions. How the stock weights are distributed in pure quintile portfolios? To answer this question, every month we group each pure and

naïve quintile portfolio weights into 100 buckets sorted by each factor and calculate

average bucket weights across all months. Figure 3 shows average weight distributions of pure and naïve Value Q1 and Q5 in buckets sorted by Value, with the lowest Value

exposure in bucket 1 and highest Value exposure in bucket 100. Because the total numberof stocks is a little less than 1000, each bucket contains either 9 or 10 stocks depending

on rounding. By construction, naïve Value Q1 weights are evenly distributed among the

top 20 buckets with about 5% in each bucket7. Similarly, naïve Value Q5 weights are

evenly distributed among the bottom 20 buckets. On the other hand, pure Value Q1 andQ5 have wider distributions than the naïve quintiles. For example, while the majority of

pure Value Q1 weights are from the top 20 buckets, it also has a long tail of distributions

in other buckets in order to neutralize other factor exposures. It also has more weights

than naïve Q1 in the top 10 or so buckets in order to offset lower Value exposures from

the long left tail and match the Value exposure of naïve Q1. Pure Value Q5 has a similarstory.

Figure 4 shows these same portfolios in buckets sorted by Momentum, with bucket 1

having the lowest Momentum exposure and bucket 100 having the highest Momentum

exposure. Because of the negative correlation between naïve Value and Momentum, thedistributions of naïve Value Q1 are skewed towards low Momentum buckets, and naïve

Value Q5 are skewed towards high Momentum buckets. The Pure Value Q1 and Q5

distributions are flat around 1% (i.e. average bucket weight) because they are constructed

to have zero exposures to Momentum.

7 It is not exactly 5% in each bucket because of rounding.


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13

Figure 3.

Figure 4.


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14

Figure 5.

Similar observations hold for pure and naïve quintiles Q2, Q3 and Q4. Figure 5 shows

their distributions in Value buckets. By definition each naïve quintile portfolio spans a

continuous block of 20 buckets with 5% in each bucket. Each pure quintile portfoliodistribution centers on the buckets of its naïve counterpart, but is wider in order to

include more stocks to neutralize other factor exposures. We are not showing their

distributions in Momentum buckets as they are all close to the 1% line, but pure Q2 and

Q4 are still flatter than naïve Q2 and Q4, while pure Q3 and naïve Q3 are on top of each

other.

4) International Developed and Emerging Market Universe

Do the same findings exist in non-US stocks? To answer that we repeat the analysis using

Developed International and Emerging Market stocks. In the former case we use stocks in

the MSCI World ex. USA index from January 1995 to December 2014; in the latter casewe use stocks in the MSCI Emerging Market index from January 1999 to December 2014.

These time periods are chosen so that a majority of the stocks in the universe has data on

each factor (especially Earnings Quality). Raw data is sourced from WorldScope and

MSCI and processed in AllianceBernstein’s internal equity research database. To remove

currency related effects we use USD hedged returns to measure performance.

Similar to Table 2, Tables 6 and 7 show historical performance of naïve and pure quintile portfolios using Developed International and Emerging Market stocks. We observe that

in both tables pure Q1-Q5 portfolios still have lower risks than naïve Q1-Q5 portfolios,

although the differences are smaller than in the US: they range from 1% for Price

Momentum to 2.7% for Size in Emerging Markets. Unlike in the US, in DevelopedInternational and Emerging Markets pure Size Q1-Q5 portfolios have lower returns and


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15

Sharpe ratios than naïve Size Q1-Q5 portfolios. On the other hand, for Value, Price

Momentum and Profitability the return enhancements from naïve to pure Q1-Q5 are

substantially stronger than in the US. For example, while pure Value Q1-Q5 return is 2%higher than naïve Q1-Q5 return in the US (Table 2), it is 5.3% higher in Developed

International market and 9.2% higher in Emerging Market. As we have demonstrated

with US data, these higher returns survive the higher transaction costs. For these threefactors it is also true that pure Q1 return is higher than naïve Q1, and pure Q5 return islower than naïve Q5.

We conclude that, though not as consistent as in the US, overall there are still notableevidences in Developed International and Emerging Markets that pure Q1-Q5 portfolios

represent purer, stronger and more stable factor returns than naïve Q1-Q5 portfolios.


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16

Table 6. Performance of Naïve and Pure Quintile Portfolios, 1995 – 2014, International Developed

Stocks


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 3.9% 8.4% 0.47 156% Q1 6.4% 8.6% 0.75 323%

2 0.7% 3.6% 0.21 261% 2 2.1% 3.4% 0.60 590%

3 -1.3% 2.8% -0.48 265% 3 -0.8% 2.7% -0.32 665%

4 -2.0% 3.6% -0.57 212% 4 -2.9% 3.6% -0.80 491%

Q5 -1.2% 7.0% -0.17 115% Q5 -3.9% 7.1% -0.56 195%

Q1-Q5 5.1% 14.4% 0.35 271% Q1-Q5 10.4% 12.8% 0.81 518%


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 1.4% 6.0% 0.23 128% Q1 0.4% 4.7% 0.09 156%

2 -0.1% 3.1% -0.03 178% 2 0.4% 3.6% 0.12 663%

3 -0.4% 2.5% -0.17 166% 3 0.4% 3.4% 0.13 681%

4 -0.5% 2.5% -0.20 127% 4 0.0% 2.8% -0.01 640%Q5 -0.4% 4.8% -0.09 67% Q5 0.0% 4.6% -0.01 99%

Q1-Q5 1.8% 9.9% 0.18 195% Q1-Q5 0.4% 7.8% 0.06 255%


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 2.3% 8.4% 0.27 324% Q1 2.9% 7.9% 0.36 385%

2 1.4% 4.8% 0.29 612% 2 2.1% 4.4% 0.48 780%

3 0.7% 3.4% 0.20 665% 3 0.7% 2.8% 0.23 895%

4 -0.8% 3.7% -0.21 606% 4 -1.9% 3.8% -0.49 825%

Q5 -3.6% 11.7% -0.30 324% Q5 -4.1% 10.9% -0.37 383%

Q1-Q5 5.8% 19.1% 0.30 648% Q1-Q5 6.9% 17.6% 0.39 767%


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 1.9% 5.4% 0.35 93% Q1 4.5% 6.7% 0.68 240%

2 1.8% 4.3% 0.41 140% 2 2.4% 3.2% 0.76 611%

3 0.6% 3.0% 0.19 157% 3 -1.2% 4.0% -0.29 765%

4 -1.5% 4.2% -0.35 151% 4 -2.2% 4.0% -0.55 602%

Q5 -2.6% 7.7% -0.34 120% Q5 -5.0% 5.5% -0.92 179%

Q1-Q5 4.5% 12.3% 0.37 213% Q1-Q5 9.6% 10.6% 0.90 419%


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 1.6% 3.6% 0.43 134% Q1 1.4% 3.1% 0.46 159%

2 1.4% 3.5% 0.39 149% 2 1.2% 3.1% 0.38 634%

3 0.2% 3.1% 0.08 156% 3 1.4% 2.6% 0.52 530%

4 -0.3% 2.6% -0.11 149% 4 -0.5% 2.2% -0.24 566%

Q5 -2.9% 5.4% -0.54 127% Q5 -3.1% 4.4% -0.69 164%

Q1-Q5 4.5% 7.5% 0.60 261% Q1-Q5 4.5% 5.9% 0.76 323%


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Table 7. Performance of Naïve and Pure Quintile Portfolios, 1999 – 2014, Emerging Market Stocks


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 5.9% 10.1% 0.59 175% Q1 9.6% 8.5% 1.14 287%

2 1.9% 4.2% 0.46 286% 2 1.3% 4.2% 0.32 685%

3 -1.4% 3.0% -0.48 299% 3 -1.7% 4.2% -0.41 714%

4 -2.5% 4.9% -0.52 251% 4 -4.5% 5.6% -0.80 551%

Q5 -3.7% 7.9% -0.46 140% Q5 -9.2% 9.4% -0.98 231%

Q1-Q5 9.6% 16.9% 0.57 316% Q1-Q5 18.8% 14.7% 1.27 518%


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 1.9% 10.9% 0.17 155% Q1 0.9% 9.0% 0.11 187%

2 0.5% 3.8% 0.12 229% 2 -0.6% 4.7% -0.13 554%

3 0.1% 3.7% 0.02 218% 3 -1.1% 3.3% -0.33 579%

4 -0.2% 4.4% -0.04 172% 4 1.2% 4.3% 0.28 716%

Q5 -1.8% 5.3% -0.33 93% Q5 -0.3% 5.3% -0.05 98%

Q1-Q5 3.7% 15.0% 0.24 248% Q1-Q5 1.2% 12.3% 0.10 286%


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 5.2% 9.2% 0.56 319% Q1 6.6% 9.3% 0.71 356%

2 1.6% 5.9% 0.26 601% 2 1.7% 5.0% 0.33 812%

3 -0.3% 4.8% -0.07 661% 3 -1.2% 4.1% -0.29 824%

4 -3.2% 4.5% -0.71 610% 4 -3.9% 5.0% -0.79 780%

Q5 -3.2% 12.5% -0.26 340% Q5 -6.9% 11.4% -0.60 396%

Q1-Q5 8.4% 19.5% 0.43 659% Q1-Q5 13.5% 18.5% 0.73 752%


Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Avg. Annual

Excess

Return

Annualized

Risk

Return-

to-Risk

Ratio

Annual

One-Way

Turnover

Q1 1.9% 6.0% 0.32 119% Q1 2.9% 7.2% 0.40 236%

2 0.2% 4.1% 0.05 175% 2 2.9% 4.7% 0.62 601%

3 1.0% 4.0% 0.24 189% 3 1.6% 4.2% 0.38 694%

4 0.4% 3.1% 0.12 180% 4 -1.8% 3.8% -0.46 565%

Q5 -3.4% 9.3% -0.37 140% Q5 -7.1% 7.9% -0.90 182%

Q1-Q5 5.3% 14.0% 0.38 259% Q1-Q5 10.0% 12.0% 0.83 418%


ExcessReturn

Annualized

Risk

Return-

to-RiskRatio

Annual

One-WayTurnover

Avg. Annual

ExcessReturn

Annualized

Risk

Return-

to-RiskRatio

Annual

One-WayTurnover

Q1 0.3% 5.2% 0.05 140% Q1 0.0% 4.0% 0.00 164%

2 1.5% 3.9% 0.38 154% 2 -0.3% 4.3% -0.07 709%

3 1.4% 3.4% 0.41 162% 3 0.7% 3.2% 0.21 571%

4 -0.5% 3.5% -0.15 159% 4 -0.8% 4.0% -0.19 660%

Q5 -2.6% 6.3% -0.41 142% Q5 -2.3% 5.7% -0.41 178%

Q1-Q5 2.8% 10.0% 0.28 281% Q1-Q5 2.3% 8.4% 0.28 342%


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18

5) Concluding Remarks

Over the last few decades many factors have been identified as being predictive of thecross-section of stock returns. When a new factor is reported, a typical way of assessing

its predictive power is sorting by the factor to create a number of portfolios (such as

quintiles or deciles) with increasing factor scores and examine their performance.However, sorting does not control the impact of other factors and often leads to

significant other exposures. Therefore it does not help us understand the efficacies of

pure exposures to the factor in question. Existing techniques to disentangle among factors

either does not generalize to multiple factors (e.g. two-way sorts), or only revealsinformation on one point of the spectrum of pure exposures (e.g. Factor-Mimicking

Portfolios). We are not aware of any other method that solves both problems. In this

paper we extend and adapt the optimizations used to create Factor-Mimicking Portfolios,and then use them to create pure quintile portfolios that disentangle among multiplefactors and reveal the cross-sectional efficiencies of pure factor exposures.

By construction, each pure quintile portfolio has the same number of stocks and exposureto the target factor as its naïve counterpart, and therefore there is a big overlap between

their stock distributions. On the other hand, the pure quintile portfolio has a wider

distribution in order to include stocks outside of the naïve quintile to neutralize exposures

to other factors. By comparing the performance of pure and naïve quintile portfolios, wefound strong evidence in US large cap stocks that pure quintile and Q1-Q5 portfolios

have lower risks, and pure Q1-Q5 portfolios have higher returns and Sharpe ratios.

Similar but weaker evidences exist in Developed International and Emerging Marketstocks. We also note that pure quintile portfolios have higher turnovers than naïve

quintile portfolios and therefore higher transaction costs when being implemented, but

their net of transaction costs Sharpe ratios are still comfortably higher. This should come

as good news when we explore their practical applications.


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References

Back, Kerry, Nishad Kapadia and Barbara Ostdiek. “Slopes as Factors: Characteristic

Pure Plays.” http://ssrn.com/abstract= 2295993. July 2013.

Banz, Rolf. “The Relationship between Return and Market Value of Common Stocks.” Journal of Financial Economics, Vol. 9 (1981), pp. 3-18.

Basu, Sanjoy. “The Relationship between Earning’s Yield, Market Value and Return for NYSE Common Stocks: Further Evidence.” Journal of Financial Economics, Vol. 12

(1983), pp. 129-156.

Fama, Eugene F., and Kenneth R. French. “The Cross-Section of Expected StockReturns.” The Journal of Finance, Volume 47, Issue 2 (1992), pp. 427-465.

Fama, Eugene F., and Kenneth R. French. “Multifactor Explanations of Asset Pricing

Anomalies.” The Journal of Finance, Volume 51, Issue 1 (1996), pp. 55-84.

Grinold, R. and R. Kahn. Active Portfolio Management: A Quantitative Approach for Providing Superior Returns and Controlling Risk. New York: McGraw-Hill, 2

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2000.

Jacobs, I., Bruce and Kenneth N. Levy. “Disentangling Equity Return Regularities: NewInsights and Investment Opportunities.” Financial Analyst Journal , 44, May/June 1988,

pp. 18-43.

Jacobs, I., Bruce and Kenneth N. Levy. “The Complexity of the Stock Market.” The

Journal of Portfolio Management , Fall 1989, Vol. 16, No. 1, pp. 19-27.

Melas, Dimitris, Raghu Suryanarayanan and Stefano Cavaglia. “Efficient Replication ofFactor Returns: Theory and Applications.” The Journal of Portfolio Management , Winter

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Appendix

Table A1. Number of Stocks by Year and Factor, Russell 1000 Universe

R1000 Value Size ProfitabilityPrice

Momentum

Earnings

QualityAll

1979 955 900 955 900 946 839 8321980 948 894 948 892 943 837 833

1981 946 903 946 903 927 841 830

1982 955 909 955 913 936 839 830

1983 956 917 956 923 944 865 853

1984 952 914 952 919 923 907 887

1985 987 936 987 942 966 931 919

1986 985 937 985 946 970 933 920

1987 985 941 985 948 956 929 906

1988 996 948 996 950 976 934 918

1989 993 955 993 954 976 942 9261990 976 937 976 931 967 932 914

1991 986 962 986 956 969 953 927

1992 988 962 988 962 975 961 943

1993 998 973 998 970 983 968 945

1994 990 964 990 956 976 957 932

1995 981 956 981 948 963 952 928

1996 982 957 982 952 964 954 931

1997 983 958 983 954 954 948 925

1998 970 950 970 951 950 957 924

1999 956 933 956 929 932 936 900

2000 970 958 970 953 927 954 908

2001 956 948 956 940 926 948 911

2002 969 963 969 957 950 963 935

2003 989 981 989 977 974 982 965

2004 989 981 989 979 985 981 973

2005 988 952 988 954 968 955 947

2006 979 977 979 974 961 974 957

2007 984 982 984 973 957 979 953

2008 969 965 969 965 947 963 946

2009 954 950 954 950 936 949 934

2010 933 932 933 933 927 932 926

2011 948 946 948 948 936 946 934

2012 955 955 955 932 934 951 913

2013 965 965 965 939 948 962 923

2014 975 973 975 950 950 969 926

Documents

Pure Quintile Portfolio