It is well known that equity returns are

DING LIU is an SVP and senior quantitative analyst at AllianceBernstein in New York, NY. ding.liu@bernstein.com Pure Quintile Portfolios DING LIU 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 a factor is to form quintile portfolios by sorting on that factor. It is also common to go long Q1 and go 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 are more exposed to the smallcap 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 exposure versus its market cap or balance sheet accruals exposures. So, a naïve Q1 Q5 portfolio created by sorting on 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 from offsetting 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 exposures 1 to momentum, which has positive return over the long run. Likewise, high Momentum stocks do not realize the full outperformance of pure momentum 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 being more predictable than naïve factor returns and being additive (Jacobs and Levy [1989]). One way to disentangle factors is using a two-way sort (Basu [1983]). For example, to create Value 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 quintile 1, take the second-highest Value bucket from each Momentum quintile and combine them as new Value quintile 2, and so on. These new Value quintiles have monotonically lower Value exposures, but similar Momentum exposures because each one draws one-fifth of its stocks from every Momentum quintile. However, this method does not generalize well to multiple factors: with five factors it would create 3,125 buckets, way more than the number of largecap stocks in the U.S. Another limitation is THE JOURNAL OF PORTFOLIO MANAGEMENT 115

that it does not disentangle factors completely; these new Value quintiles have similar but not exactly 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 exposures (e.g., standardized book-to-price, return-on-equity, and log of market capitalization) and r the vector of returns over all stocks in a universe such as the Russell 1000 index. 2 Regressing r against all column vectors of X simultaneously gives (X X) 1 X r, which can be interpreted as returns of the standardized factors. The row vectors of (X X) 1 X define a set of portfolios that mimic these factors (called Factor-Mimicking Portfolios or FMPs). Because ((X X) 1 X ) X = I, 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 w w subject to X w = e i, where e i is a k by 1 vector with one on the i-th position and zeros elsewhere. It is easy to show that the solution of this optimization is the i -th row of (X X) 1 X (see Grinold and Kahn [2000]). We adapt and extend this FMP optimization framework to create pure quintile portfolios. In previous research (e.g., 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 the optimization in a number of ways: 1) for each target factor we run five optimizations to create five 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; and 4) in each optimization we set the pure quintile portfolio s exposures to all other factors to zero. As a 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, making this the first contribution of this article. The second contribution of this article is our finding that when pure quintile portfolios are created in the U.S. 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 return enhancement. Each pure Q1 Q5 portfolio has lower risk, and with the exception of one of them, higher return than naïve Q1 Q5 portfolio. Looking at each Q1 and Q5 portfolio 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 the pure Q1 Q5 portfolio comes from both the long and short sides. That is, the pure Q1 portfolio has a higher return than the naïve Q1 portfolio, and the pure Q5 portfolio has a lower return than the naïve Q5 portfolio. This is evidence that our pure quintile methodology is more efficient at capturing factor returns than naïve quintile sorts. Similar evidence exists in Developed International and Emerging Market stocks, although the evidence is weaker than for stocks in the U.S. The rest of this article is organized as follows: The next section describes the data used in our analysis, and demonstrates unintended factor exposures in naïve quintile portfolios. Then, the pure quintile portfolio framework is described and its performance is compared to that of naïve quintile portfolios in the U.S. large-cap universe. Afterward, we repeat the analysis for Developed International and Emerging Market stocks. This is followed by some brief concluding remarks. DATA AND FACTOR EXPOSURES OF NAÏVE QUINTILE PORTFOLIOS In this article we focus on five factors: Value, Size, Price Momentum, Profitability, and Earnings 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 factors 4, but here we are not interested in fine-tuning factor definitions to make the most economic sense or to realize 116 PURE QUINTILE PORTFOLIOS

the best performance. This article is focused on demonstrating the pure quintile portfolio framework and comparing its performance with that of naïve quintile portfolios. Simple factor definitions are sufficient for our purpose. We collected these factors for all stocks in the Russell 1000 index for every month from January 1979 to December 2014 from multiple sources including Compustat, CRSP, and Russell. We choose the Russell 1000 universe because it is widely used by institutional managers as a barometer for U.S. large-cap investments. 5 Exhibit 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 for each factor, we used all stocks with data on that factor, even though some of them are missing data on other factors. We repeated the analysis using only stocks with data on all five factors, and the results were very similar. Following common practice, every month for every factor we first winsorize its raw values at 5% and 95% levels, and then standardize the data by subtracting the equally weighted average and dividing by the crosssectional standard deviation. We call these standardized factor values exposures. Because of standardization, exposures are comparable across factors and across months. Throughout this article, we use market to mean the equally weighted portfolio of all stocks, which has zero exposure to all factors because of standardization. Therefore, a portfolio with exposure of 1 to Value, for example, means that its weighted average book-to-price is one standard deviation above the market. For Value, Profitability, and Price Momentum, a higher factor exposure has 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 cases a higher exposure has higher expected returns. Naïve quintile portfolios are then created every month by sorting on these factor exposures, with quintile 1 having the highest exposure and quintile 5 having the lowest exposure, and stocks are equally weighted within each quintile. The left side of Exhibit 3 shows the average standardized factor exposures of naïve quintile portfolios from 1979 to 2014. It is clear that they all pick up unintended factor exposures to 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: Exhibits 1 and 2 show the rolling 12-month average factor exposures of Value naïve Q1 and Q5 portfolios. PURE QUINTILE PORTFOLIOS Each month, for each naïve quintile portfolio of each factor (called the target factor) we run the following optimization to create the corresponding pure quintile portfolio. Minimize w w (1) Subject to: w 0 (2) e w = 1, (3) x i w = exp, for the target factor i (4) x i w = 0, for other non-targeted factors i (5) The number of stocks is the same as in the naïve quintile, i.e., about n/5 (6) Here n is the number of stocks with exposure data to the target factor 6, exp is the naïve quintile portfolio s exposure to the target factor, x i is the vector of stock exposures to factor i, w is the vector of weights being optimized, and e is a vector of ones. All vectors have size n by 1. Constraints (2) and (3) make sure the pure quintile portfolio is long only and 100% invested. Constraint (4) makes its exposure to the target factor the same as in the naïve quintile portfolio, and constraint (5) makes sure it has zero exposure to other factors. The right column of Exhibit 3 confirms that these constraints are satisfied. Constraint (6) makes sure it has the same number of stocks as in the naïve quintile portfolio. 7 Finally, the objective term (1) pushes the optimized weights toward equal weights as much as possible, which is the weighting scheme of naïve quintile portfolios. It is important to note the similarities and differences between our optimization and those in previous studies such as Melas, Suryanarayanan, and Cavaglia [2010] and Grinold and Kahn [2000]. In both cases, THE JOURNAL OF PORTFOLIO MANAGEMENT 117

E XHIBIT 1 Rolling 12-Month Average Factor Exposures (Value Naïve Q1) E XHIBIT 2 Rolling 12-Month Average Factor Exposures (Value Naïve Q5) the exposures to non-targeted factors are set to zero. In the previous studies, the exposure to the target factor is set to one to create a single long short portfolio; but here we match the exposure to that of each individual naïve quintile portfolio to create the corresponding pure quintile portfolio. The long-only and name count constraints are used to make pure and naïve quintile portfolios directly comparable, but are also needed to avoid a situation in which the optimizer simply returns a combination of a fixed portfolio with a unit exposure to the target factor and the market for each optimization, and varies their proportion to get different target-factor 118 PURE QUINTILE PORTFOLIOS

E XHIBIT 3 Average Standardized Factor Exposures of Naïve and Pure Quintile Portfolios, 1979 2014 Notes: This exhibit shows average standardized factor exposures of naïve and pure quintile portfolios from 1979 to 2014 using stocks in the Russell 1000 universe. High exposures to Value, Price Momentum, and Profitability correspond to high raw factor values, while high exposures to Size and Earnings Quality correspond to low raw factor values. The exposures are standardized so that exposure of one means the raw factor value is one standard deviation above the equally weighted average of all stocks. Each naïve quintile portfolio has exposure to the target factor (i.e., the factor used to create it) as well as other factors. Each pure quintile portfolio has exactly the same exposure to the target factor as the naïve quintile portfolio and zero exposure to other factors. 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 weigh the stocks as equally as possible while satisfying 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. But the pure quintile portfolio construction process can also draw on stocks from other naïve quintile portfolios to offset other factor exposures. Later we will see that the pure quintile portfolios indeed hold different stocks than the naïve quintile portfolios hold. Exhibit 4 compares the simulated historical performance of naïve and pure quintile portfolios side by side for the 1979 2014 period. Note that all quintile portfolios are created at the beginning of each month, and their THE JOURNAL OF PORTFOLIO MANAGEMENT 119

E XHIBIT 4 Performance of Naïve and Pure Quintile Portfolios, 1979 2014 Notes: This exhibit shows simulated historical performance of naïve and pure quintile portfolios from 1979 to 2014 using stocks in the Russell 1000 universe. All quintile portfolios are created at the beginning of each month, and returns are calculated through the end of the month. For Q1 through Q5 we use their returns in excess of equal-weighted market. Returns are before transaction costs and include dividends. Average returns in column one and risks in column two are annualized. Column three, the return-to-risk ratio, is the ratio between columns one and two. For Q1 through Q5 it is the Sharpe ratio of long each quintile and short the market; for Q1 Q5 it is the Sharpe ratio of long Q1 and short Q5. 120 PURE QUINTILE PORTFOLIOS

E XHIBIT 5 Net of Transaction Costs Performance of Naïve and Pure Q1 Q5 Portfolios, 1979 2014 Notes: This exhibit shows simulated gross and net of T-Cost historical performance of naïve and pure Q1 Q5 portfolios from 1979 to 2014 using stocks in the Russell 1000 universe. All quintile portfolios are created at the beginning of each month, and returns are calculated through the end of the month. Transaction costs are assumed to be 30 bps per trade one-way. Returns include dividends. Returns and risks are annualized. The last column, net of T-Cost Sharpe ratio, is the ratio between net of T-Cost return and risk. It is the net of T-Cost Sharpe ratio of long Q1 and short Q5. returns are calculated through the end of the month. The first column shows the average annual returns in excess 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 of long each quintile and short the market; for Q1 Q5, it is the Sharpe ratio of long Q1 and short Q5. The last column shows annualized one-way turnover. Returns are before transaction costs and include dividends. Perhaps the most notable observation from Exhibit 4 is that for each factor the pure Q1 Q5 portfolio risk is lower by 3% or more than the naïve Q1 Q5 portfolio risk, 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 a substantially higher Sharpe ratio than the 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 long and short sides. For example, Value pure Q1 return is 40 bps 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 is more efficient at capturing factor returns than naïve quintile sorts. 8 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? Yes. Exhibit 5 shows net of transaction costs performance of naïve and pure Q1 Q5 portfolios, assuming 30 bps of transaction costs per trade one-way. 9 For example, annual transaction costs of naïve Value Q1 Q5 is 282% 2 30 bps, or about 1.7%, 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 the net return is still higher than the net return of the naïve Q1 Q5. The net of transaction costs Sharpe ratio of pure Value Q1 Q5 is still amply higher than its naïve counterpart. The same is true for other factors. Quarterly Rebalance We can also reduce turnover in both naïve and pure quintile portfolios by rebalancing less frequently than monthly, such as quarterly. For each factor and each quintile (naïve or pure) we tracked three portfolios starting in January, February, and March of 1979, respectively. Each portfolio was rebalanced quarterly instead of monthly, and returns were calculated by averaging across the three portfolios. Exhibit 6 shows the performance of naïve and pure quintile portfolios using quarterly rebalancing. Once again the results show 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 higher returns than naïve Q1 portfolios; pure Q5 portfolios have lower returns than naïve Q5 portfolios; and pure THE JOURNAL OF PORTFOLIO MANAGEMENT 121

E XHIBIT 6 Performance of Naïve and Pure Quintile Portfolios, 1979 2014, Quarterly Rebalance Notes: Exhibit 6 is similar to Exhibit 4. The difference is that in Exhibit 6 all quintile portfolios are rebalanced quarterly instead of monthly. To be exact, each quintile portfolio return is calculated as the average of three subportfolios created at the beginning of January, February, and March of 1979, respectively. Each subportfolio is recreated every three months, and its return is calculated through the end of the three-month period. 122 PURE QUINTILE PORTFOLIOS

E XHIBIT 7 Net of Transaction Costs Performance of Naïve and Pure Q1 Q5 Portfolios, 1979 2014, Quarterly Rebalance Notes: Exhibit 7 is similar to Exhibit 5. The difference is that in Exhibit 7 all quintile portfolios are rebalanced quarterly instead of monthly. To be exact, each quintile portfolio return is calculated as the average of three subportfolios created at the beginning of January, February, and March of 1979, respectively. Each subportfolio is recreated every three months, and its return is calculated through the end of the three-month period. quintile portfolios generally have lower risks than their naïve counterparts. Comparing the turnover numbers in Exhibits 4 and 6, we see that, as expected, quarterly rebalancing results in lower turnovers in all cases. The turnover savings are especially big for Price Momentum, with naïve Q1 Q5 annual turnover declining by almost 300% (from 664% to 370%), and pure Q1 Q5 annual turnover declining by 371% (from 776% to 405%). We also observe that the turnover reduction is always bigger for pure quintile and Q1 Q5 portfolios than their naïve counterparts, which means quarterly rebalancing mitigates transaction costs more for pure quintile portfolios. Exhibit 7 shows net of transaction costs performance of naïve and pure Q1 Q5 portfolios using quarterly rebalancing (again assuming 30 bps of transaction costs per trade one-way). As in Exhibit 5: pure Q1 Q5 portfolios have substantially higher Sharpe ratios than their naïve counterparts after transaction costs. Pure Quintile Portfolio Distributions How are the stock weights distributed in pure quintile portfolios? To answer this question, we grouped every month each pure and naïve quintile portfolio weights into 100 buckets sorted by each factor, and calculated average bucket weights across all months. Exhibit 8 shows the average weight distribution of pure and naïve Value Q1 and Q5 portfolios in buckets sorted by Value, with the lowest Value exposure in bucket 1 and the highest Value exposure in bucket 100. Because the total number of 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 bucket. 10 Similarly, naïve Value Q5 weights are evenly distributed among the bottom 20 buckets. On the other hand, pure Value Q1 and Q5 have wider distributions than the naïve quintiles. For example, while the majority of pure Value Q1 weights are from the top 20 buckets, there is also a long tail of stocks in other buckets in order to neutralize other factor exposures. Pure Value Q1 portfolios also have more weights than naïve Q1 portfolios in the top 10 or so buckets in order to offset lower Value exposures from the long tail and match the Value exposure of naïve Q1. Pure Value Q5 has a similar pattern. Exhibit 9 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, the distribution of naïve Value Q1 is skewed towards low- Momentum buckets and naïve Value Q5 is skewed towards high-momentum buckets. The pure Value Q1 and Q5 distributions are flat, at around 1% (i.e., average bucket weight) because they are constructed to have zero exposure to Momentum. Similar observations hold for pure and naïve quintiles Q2, Q3, and Q4. Exhibit 10 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 portfolio distribution centers on the buckets of its naïve counterpart, but is wider in order to include more stocks to neutralize other THE JOURNAL OF PORTFOLIO MANAGEMENT 123

E XHIBIT 8 Average Portfolio Weight Distributions by Value Buckets Notes: This exhibit shows average weight distributions of pure and naïve Value Q1 and Q5 in buckets sorted by Value. Every month we create 100 buckets sorted by each factor, with bucket 1 having the lowest exposure and bucket 100 having the highest exposure. For each pure or naïve quintile portfolio we calculate the total percentage of its weights from each bucket each month, and average over all months. By construction, naïve Value Q1 (Q5) weights should be evenly distributed among the top (bottom) 20 Value buckets with 5% weight in each bucket. Distribution is not exactly 5% because of rounding; that is, some buckets have 9 stocks while others have 10 stocks. E XHIBIT 9 Average Portfolio Weight Distributions by Momentum Buckets Notes: Exhibit 9 is similar to Exhibit 8. The difference is that in Exhibit 9 the buckets are sorted by Price Momentum. 124 PURE QUINTILE PORTFOLIOS

E XHIBIT 10 Average Portfolio Weight Distributions by Value Buckets Notes: Exhibit 10 is similar to Exhibit 8. The difference is that Exhibit 10 shows pure and naïve Value Q2, Q3, and Q4. factor exposures. We are not showing their distributions in Momentum buckets because they are all close to the 1% line, but pure Q2 and Q4 are still flatter than naïve Q2 and Q4, whereas pure Q3 and naïve Q3 are on top of each other. INTERNATIONAL DEVELOPED AND EMERGING MARKET UNIVERSE Do the same findings exist in non-u.s. stocks? To answer that we repeated the analysis using Developed International and Emerging Market stocks. In the former case, we use stocks in the MSCI World ex U.S. index from January 1995 to December 2014; in the latter case, we use stocks in the MSCI Emerging Market index from January 1999 to December 2014. 11 These time periods are chosen so that a majority of the stocks in the universe had data on every factor (especially Earnings Quality). Raw data is sourced from WorldScope and MSCI. To remove currency-related effects we use USDhedged returns to measure performance. For the sake of simplicity, we don t neutralize country, region or sector exposures in either pure or naïve quintile portfolios. Similar to Exhibit 4, Exhibits 11 and 12 show the historical performance of naïve and pure quintile portfolios using Developed International and Emerging Market stocks. We observe that in both Exhibits pure Q1 Q5 portfolios again have lower risks than naïve Q1 Q5 portfolios, although the differences are smaller than in the U.S.: They range from 1% for Price Momentum to 2.7% for Size in Emerging Markets. Unlike in the U.S., in Developed International and Emerging Markets pure Size Q1 Q5 portfolios have lower returns and Sharpe ratios than naïve Size Q1 Q5 portfolios. However, for Value, Price Momentum and Profitability the return enhancements from naïve to pure Q1 Q5 are substantially stronger than in the U.S. For example, while the pure Value Q1 Q5 return is 2% higher than the naïve Q1 Q5 return in the U.S. (Exhibit 4), it is 5.3% higher in Developed International markets and 9.2% higher in Emerging Markets. As we have demonstrated with the U.S. data, these higher returns survive the higher transaction costs. For these three factors, it is also true that the pure Q1 return is higher than the naïve Q1 return, and the pure Q5 return is lower than the naïve Q5 return. Finally, we note that for Earnings Quality, in Developed International markets pure Q1 Q5 has the same return but higher Sharpe ratio than naïve Q1 Q5; and in Emerging Markets pure Q1 Q5 has the same Sharpe ratio but lower return than naïve Q1 Q5. THE JOURNAL OF PORTFOLIO MANAGEMENT 125

E XHIBIT 11 Performance of Naïve and Pure Quintile Portfolios, 1995 2014, International Developed Stocks Notes: Exhibit 11 is similar to Exhibit 4. The difference is that in Exhibit 11 the stock universe is the MSCI World ex. U.S.A index and the time period is 1995 to 2014. The time period is chosen so that a majority of the stocks in the universe has data on every factor. 126 PURE QUINTILE PORTFOLIOS

E XHIBIT 12 Performance of Naïve and Pure Quintile Portfolios, 1999 2014, Emerging Market Stocks Notes: Exhibit 12 is similar to Exhibit 4. The difference is that in Exhibit 12 the stock universe is the MSCI Emerging Market index and the time period is 1999 to 2014. The time period is chosen so that a majority of the stocks in the universe has data on every factor. THE JOURNAL OF PORTFOLIO MANAGEMENT 127

We conclude that, though not as consistent as in the U.S., overall there is still notable evidence in Developed International and Emerging Markets that pure Q1 Q5 portfolios are more efficient at capturing factor returns than naïve Q1 Q5 portfolios. CONCLUDING REMARKS Over the last few decades many factors have been identified as being predictive of the cross-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 examining their performance. However, sorting does not control for the impact of other factors, and the portfolios often have significant exposures to other factors. This approach, therefore, does not help us understand the efficacies of pure exposures to the factor in question. Existing techniques to disentangle factors either do not generalize to multiple factors (e.g., two-way sorts), or only reveal information 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 article, we extend and adapt the optimizations used to create Factor-Mimicking Portfolios, and then use them to create pure quintile portfolios that disentangle multiple factors and reveal the crosssectional efficiencies of pure factor exposures. By construction, each pure quintile portfolio has the same number of stocks and exposure to the target factor as its naïve counterpart. As a result, there is a big overlap between their stock distributions. But the pure quintile portfolio has a wider distribution of stock weights and includes stocks outside of the naïve quintile in order to neutralize exposures to other factors. By comparing the performance of pure and naïve quintile portfolios, we find strong evidence in U.S. 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 evidence exists in Developed International and Emerging Market stocks. We also note that pure quintile portfolios have higher turnover 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. A PPENDIX E XHIBIT A1 Number of Stocks by Year and Factor, Russell 1000 Universe Notes: This exhibit shows, at the beginning of each year, the number of stocks in Russell 1000 (column 1), among those the number of stocks with data on each individual factor (columns 2 6), and the number of stocks with data on all factors (column 7). 128 PURE QUINTILE PORTFOLIOS

ENDNOTES REFERENCES The author thanks Andrew Chin, Inna Okounkova, Paul Robertson, and the anonymous reviewer for their helpful comments and suggestions. The author also thanks his employer, AllianceBernstein, for providing the data and tools used in this study. The views expressed in this article are the author s own and do not necessarily reflect the views of AllianceBernstein. 1 In this article 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 raw exposure. 2 Assume there are n stocks and k factors, then X is a n by k matrix and r is a n by 1 vector. 3 Balance sheet accruals are 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 are total liabilities minus debt in current liabilities and total longterm debt. 4 For example, Value is sometimes defined as a combination of book-to-price, earnings-to-price, and dividend yield. 5 We acknowledge that using the Russell 1000 universe introduces a mid- to large-cap bias to the results. This is particularly true for our size factor, which ref lects Mid-minus- Big rather than Small-minus-Big. 6 Note that n is generally different across factors; see Exhibit A1 in the Appendix. 7 Such name count constraint is supported by Axioma s optimizer. 8 We acknowledge that there may be other factors (such as liquidity or low beta) at play here that have not been neutralized in the pure quintile portfolios. While it would be virtually impossible to consider all the factors, we think that these five well-documented factors are sufficient to demonstrate the construction of pure quintile portfolios and their higher Sharpe ratios. 9 We think this is a conservative assumption for U.S. large-cap stocks. The true costs depend on many factors but are likely to be lower, which only makes our results even stronger. 10 It is not exactly 5% in each bucket because of rounding. 11 Again, using these two MSCI indexes introduces a mid- to large-cap bias to the results because they don t capture international small-cap stocks. Back, K., N. Kapadia, and B. Ostdiek. Slopes as Factors: Characteristic Pure Plays. http://ssrn.com/abstract=2295993. July 2013. Banz, R. The Relationship between Return and Market Value of Common Stocks. Journal of Financial Economics, Vol. 9 (1981), pp. 3-18. Basu, S. 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, E.F., and K.R. French. The Cross-Section of Expected Stock Returns. The Journal of Finance, Vol. 47, Issue 2 (1992), pp. 427-465.. Multifactor Explanations of Asset Pricing Anomalies. The Journal of Finance, Vol. 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. 2nd Edition, New York: McGraw-Hill, 2000. Jacobs, B.I., and K.N. Levy. Disentangling Equity Return Regularities: New Insights and Investment Opportunities. Financial Analyst Journal, Vol. 44, No. 3 (May June 1988), pp. 18-43.. The Complexity of the Stock Market. The Journal of Portfolio Management, Vol. 16, No. 1 (Fall 1989), pp. 19-27. Melas, D., R. Suryanarayanan, and S. Cavaglia. Efficient Replication of Factor Returns: Theory and Applications. The Journal of Portfolio Management, Vol. 36, No. 2 (Winter 2010), pp. 39-51. Sloan, R.G. Do Stock Prices Fully Reflect Information in Accruals and Cash Flows about Future Earnings? The Accounting Review, Vol. 71, No. 3 (1996), pp. 289-315. To order reprints of this article, please contact Dewey Palmieri at dpalmieri@iijournals.com or 212-224-3675. THE JOURNAL OF PORTFOLIO MANAGEMENT 129