Does factor investing outperform market cap weighted indexes

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Does Factor Investing Outperform Market Cap

Weighted Indexes

By: Zhanyu Yang

Thesis Supervisor: Dr. Philippe Versijp

Master Thesis 

Master in International Finance 

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Abstract:

This article chooses six factors, including market factor, size factor, book-to-market factor, profitability factor, investment pattern factor and momentum factor, to construct factor-investing portfolios, explores three ways to improve the construction, notably using a minimum 15-year data, updating the standard deviation of data via Exponential Weighted Moving Average (EWMA) model and implementing a minimum 10% of factors allocation, and conducts performance comparison with market cap weighted indexes in terms of Sharpe ratios. The study finds that 1) factor-investing portfolios derived using more than 15-year data deliver a better Sharpe ratio, ranging from 0.36 to 0.39, while those using the most recent 5-year and 10-year data are doing worse with Sharpe ratios of 0.16 and 0.24; 2) EWMA model makes the Sharpe ratios even worse, ranging from 0.16 to 0.29; 3) a minimum 10% of factors allocation makes the Sharpe ratios better, ranging from 0.28 to 0.40; 4) Sharpe ratios of factor-investing portfolios in the six-year short-term in-sample test are not significantly different from those of market cap weighted indexes; 5) In the ten-year, fifteen-year and twenty-year long-term in-sample test, factor-investing portfolios are doing better than those in the short-term test and can sometimes deliver higher Sharpe ratios than market cap weighted indexes with a 10% confidence level.

Key words:

Factor-investing, Market, Size, Book-to-Market, Profitability, Investment Pattern, Momentum, Market cap weighted indexes

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Content

1. Introduction & Literature ... 1

Single-factor model: Sharpe-Lintner-Black ... 1

Three-factor model: Fama-French ... 2

Four-factor model: Fama-French-Carhart ... 3

Five-factor model: Updated Fama-French ... 4

2. Data ... 5

Data source ... 5

Five-by-five portfolios ... 5

Factor portfolios ... 6

3. Methodology ... 8

Selection of factors portfolios ... 8

Construction of factor-investing portfolios ... 8

Performance analysis of factor-investing portfolios ... 10

Software ... 10

4. Factors Selection ... 11

Selection of five-by-five portfolios ... 11

Selection of factor portfolios ... 13

Further analysis on six-factor portfolios ... 15

5. Portfolios Construction and Analysis ... 18

Factor investing portfolios construction ... 18

Construction improvements ... 22

Performance analysis of factor-investing portfolios ... 26

Long-term in-sample test ... 28

6. Conclusion ... 30

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1. Introduction & Literature

Asset pricing has always been a challenge for both academic researchers and practical investors. Researchers have constantly explored historical data, trying to come up with models that are able to capture all related risks, and extended the asset pricing model from single-factor one, notably the Sharpe(1964)-Lintner(1965)-Black(1972) model capturing the market-premium, to as complex as five-factor one such as the Fama and French (2015) model capturing the market-, size-, book-to-market-, profitability- and investment-pattern premiums. Simultaneously, investors also exert all their efforts to apply the asset pricing models into practices, attempting to forecast future return and make a fortune. In this article, we are going to examine the performance of factors on explaining stock returns, select a set of factors that have the strongest explanatory power to construct factor-investing portfolios and compare the performances, in terms of Sharpe ratios, of these portfolios with those of the market cap weighted indexes in short- and long-term in-sample tests. Our contribution would mainly focus on exploring a protocol to directly invest into factors, followed by the quality test of this protocol.

Single-factor model: Sharpe-Lintner-Black

Markowitz (1952) raises a model to make investment decisions by two steps, notably firstly achieving a unique optimum combination of risky assets and then applying allocations of funds between such a combination and a single riskless asset. Based on Markowitz’s theory, Sharpe (1964), Lintner (1965) and Black (1972) further raise a model that describes a simple linear relationship between the expected return and standard deviation of return for efficient combinations of risky assets. They point out that an investor may obtain a higher expected return by introducing more risks on his holdings and that the stock returns can be estimated by using risk-free rate, beta and market risk premium.

However, Fama and MacBeth (1973) find that the SLB (Sharpe-Lintner-Black) model is able to describe a relation between average stock returns and beta only during the period before 1969. Reinganum (1981), and Lakonishok and Shapiro (1986) point out that when using beta alone to explain the average stock returns, the relation between returns and beta disappears. Things become interesting when a negative relation between average stock returns and firm size is recorded by Banz (1981) and a positive relation between average stock returns and book-to-market equity is discovered by Stattman (1980) and Rosenberg, Reid, and Lanstein (1985). Moreover earning-to-price (E/P) and leverage are also discovered to have a positive relationship with average stock returns by Basu (1983) and Bhandari (1988), respectively.

According to Keim (1988), all these variables, including size, leverage, E/P and book-to-market equity, can be treated as alternative ways to reflect the information about risk and expected returns in terms of scale stock prices. However, it can be expected that some of these factors are mutually correlated since

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they are all scaled versions of price, indicating the necessity of further tests to extract the representative factors that absorb the relationship between the rest factors and average stock return.

Three-factor model: Fama-French

Fama and French (1992) have done both univariate and multivariate cross-section tests to find out the relations between average stock returns and beta, size, leverage, E/P and book-to-market equity. During univariate tests, the average stock returns are strongly related to size, leverage, E/P and book-to-market equity. However, when using the five-by-five SIZE-BETA portfolios to do multivariate tests, the simple relation between average returns and beta is flat but the relations between average stock returns and size and book-to-market equity are significantly different from zero. In further attempt, Fama and French (1993) expand previous cross-section regression on stock returns into time-series regressions. They find that the three-factor model, using market-, size- and book-to-market- factors, can do a good job during the regressions.

According to Chan, Chen and Hsieh (1985), low-grade corporate bonds have a relatively high return compared with high-grade corporate bonds due to the compensation of default risk premium. The similar principle brings an interesting explanation for the negative relation between expected returns and size factor which captures the higher default risk of small-size companies, according to Chen, Roll and Rose (1986). Moreover, Chan and Chen (1991) discover that the firms’ earning prospects, represented by book-to-market equity, capture the relative distress factor which is compensated with a corresponding return. Hence, firms that have poor earning prospects and that are signaled with relatively low stock prices, have high book-to-market equity ratios and thus are frequently compensated with higher expected returns.

Of course, three-factor model will also cause criticisms. From the survivor bias point of view, Kothari, Shanken, and Sloan (1995) question the quality of data used by Fama and French. They argue that the return distribution of BE/ME firms is positively skewed as a result of the survivorship of high BE/ME firms from distress, and thus the recorded high BE/ME returns are overstated. From the data-snooping point of view, Black (1993) and MacKinlay (1995) argue that the size and book-to-market factors are the results of heavily rummaging through the same data for the sample specific patterns to explain the anomalies and predict that average SMB and HML returns will fail to levels that are consistent with their market beta in the out-of-sample tests. Last but not least, Lakonishok, Shleifer, and Vishny (1994) and Haugen (1995), from the perspective of financial behavior, argue that the distress premium is real but irrational and that the overstated price of distress stocks and the understated price of growth stocks are driven by investors’ over-reaction.

Chan, Jegadeesh, and Lakonishok (1995) test the explanatory power of selection bias on the return difference between high and low book-to-market stocks, and find that selection bias on COMPUSTAT

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is not a severe problem. Kothari, Shanken, and Sloan (1995) themselves also concede that survivor bias cannot explain the lacking of ability of CAPM model to explain abnormal returns of high book-to-market stocks and thus it is not a major problem for value-weight portfolios. It is impossible and unrealistic to completely rule out the data-snooping bias. Fama and French (1996) argue that regardless of the economic explanation the three-factor model is applicable in capturing anomalies, which cannot be caught by CAPM model, that appeared in portfolios sorted by size, BE/ME, E/P, C/P, sales rank, and long-term past return. Moreover, Davis (1994) also finds a significant relation between average returns and BE/ME factor from 1941 to 1962 when using a sample of large firms, indicating distress premium is not special to the post-1962 COMPUSTAT data used by Fama and French (1992, 1993). Chan, Hamao, and Lakonishok (1991) and Capaul, Rowley and Sharpe (1993) also show relations between average returns and factors like size, BE/ME, E/P, and C/P in tests on international data, which can be treated as out-of-sample. As for the over-reaction skepticism, Fama and French (1995, 1996) argue that over-reaction cannot totally deny the explanation power of three-factor model otherwise the distress premium in stock returns cannot persist for more than five years after portfolio formation and that the three-factor model can do a better job in expected returns estimation if the CAPM is true but the market portfolio is unobservable.

Four-factor model: Fama-French-Carhart

DeBondt and Thaler (1985) find that in the long term, 3 to 5 years, stocks with lower past returns tend to deliver higher future returns while Jegadeesh and Titman (1993) discover that in the short term, 3 to 12 months, stocks with higher returns continue to deliver higher future returns. Even the three-factor model (1996) can to some degree explain the reversal of long-term returns as the result of distress premium delivered by stocks with lower past returns, it has a tough time to explain the continuation of short-term returns.

In the short term, 3 to 12 months, a momentum seems display in stocks that pushes the previous returns to continue, Levy (1967) claims that significant abnormal returns can be realized through buying stocks with higher current prices than their average prices over the past 27 weeks. Grinblatt and Titman (1989, 1993) shows a large number of mutual funds tend to buy stocks with increased price over the previous quarter. Hendricks, Patel and Zeckhauser (1993), Goetzmann and Ibbotson (1994) and Brown and Goetzmann (1995) also find persistence of mutual funds performance over short-term horizons, normally 1 to 3 years. Carhart (1997) constructs a four-factor model by introducing a momentum factor on the basis of Fama and French’s three-factor model and finds that funds with high returns in last year will have above-average expected returns during next year, but this momentum will disappear in years thereafter.

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Five-factor model: Updated Fama-French

Fama and French (2015) find that value factor in the three-factor model of Fama French (1993) is unable to catch depression premium all the time. Specifically, keeping the book-to-market equity value constant, a robust profitability and a conservative investment pattern will lead to a higher expected return. By introducing profitability factor and investment pattern factor, which are denoted as RMW (return of stocks with robust profitability minus return of stocks with weak profitability) and CMA (return of stocks with conservative investment pattern mine return of stocks with aggressive investment pattern), Fama and French (2015) bring in the five-factor model and find that it performs better than the three-factor model of Fama and French (1993).

Since a lot of efforts have been made to explore factors that can be used to explain the returns, this article would move a step into practice and focus on the use of these factors, specifically constructing factor-investing portfolios that might beat the market cap weighted indexes. Aiming at this target, this article firstly tests the quality of the aforementioned factors in explaining stock returns and selects a set of factors that have the strongest explanatory power. Secondly, this article assumes investors are able to apply shorting strategy and thus constructs factor-investing portfolios by allocating exposures directly to the selected factors. Lastly, this article explores three ways to improve the construction of factor-investing portfolios and tests whether these portfolios can outperform the passive investment such as market cap weighted indexes. Section 2 describes the data used in this article. Section 3 introduces the methodologies to select factors, construct and improve factor-investing portfolios, and examines portfolios performance. Section 4 focuses on testing the performance of and selecting factors. Section 5 aims at constructing, improving the construction of and compare the performance of factor-investing portfolios. Section 6 gives the conclusion of this article and raises some areas need to be explored in the future.

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2. Data

Data source

We use data downloaded from Kenneth R. French Data Library (The Library) in our analysis.

Basically The Library only uses data from nonfinancial firms since financial firms always have high leverage which is normal for financial firms but means high distress for nonfinancial firms. Two databases are used by The Library, notably (1) the return files of nonfinancial firms listed in the NYSE, AMEX and NASDAQ from the Center for Research in Security Prices (CRSP) and (2) the annual income statement and balance sheet of these firms from COMPUSTAT. We only use post-1962 data since COMPUSTAT data for pre-1962 years have a serious selection bias, tilting toward big historically successful firms.

Theoretically, there should have a 3-month gap between fiscal year-ends and the return tests since the accounting data are normally available within three months after the fiscal yearend. Practically firms are required to file 10-K reports within 90 days of their fiscal year-ends, however 19.8% of companies are unable to file reports within three months and 40% of the companies file financial report on the last day of the three months and public even later. Conservatively, to make sure the accounting data match the returns, The Library uses a 6-month gap, pairing the accounting data at fiscal year-ends in calendar year t-1 with the returns from July of year t to June of t+1. For instance, The Library uses market equity on 31 December of year t-1 to calculate earnings-price and book-to-market ratios and market equity on June of year t to calculate company size. One skepticism about this matching is that not all the firms mixed have the same fiscal year-ends, however Fama and French test smaller sample firms with December fiscal yearend and find similar results with the population firms.

Five-by-five portfolios

Five-by-five portfolios formed on every two of the factors are used as dependent variables in regressions. Taking the construction of five-by-five SIZE-BE/ME for example, seen Figure1 (a) for the procedure, in June of each year The Library uses NYSE stocks to determine five SIZE-related categories based on the quintile breakpoints of company sizes. Then each of the SIZE-related categories is further divided into five sub-categories based on the quintile breakpoints of book-to-market values. Finally all NYSE, AMEX and NASDAQ stocks are allocated into each SIZE-BE/ME sub-category. Here we further subtract one-month Treasury bill rate to get monthly excess returns.

Two-by-three portfolios

Similar to five-by-five portfolios, two-by-three portfolios are also formed on every two of the factors. Taking the construction of two-by-three SIZE-BE/ME portfolios for instance, seen Figure1 (b) for the

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procedure, in June of each year The Library uses NYSE stocks to determine two SIZE-related categories based on the median breakpoint of company size. Then each of the SIZE-related categories is further divided into three sub-categories based on the book-to-equity values, accounting for 30%, 40% and 30% by weight, respectively. Finally all NYSE, AMEX and NASDAQ stocks are allocated into each SIZE-BE/ME sub-category. The newly formed two-by-three portfolios are used to determine factor portfolios.

Factor portfolios

Factor portfolios deduced from two-by-three portfolios are used as independent variables in regressions. Taking the three-factor portfolios of Fama and French (1993) for example, which consist of three sets of single factor portfolios, notably SIZE factor denoted as SMB (return of small stocks minus return of big stocks) mimicking risk premium related to size, BE/ME factor denoted as HML (return of high book-to-market values minus return of low book-to-market values) mimicking risk premium related to book-to-market equity and MARKET factor denoted as MKT (excess market return) capturing the market premium relative to risk-free rate. Using the two-by-three SIZE-BE/ME portfolios as indicated in Figure1 (c), SMB is calculated as the difference between the equal-weighted average of monthly returns on the three small-stock sub-categories and the equal-weighted average of monthly returns on the three big-stock sub-portfolios. Similarly, HML is calculated as the difference between the equal-weighted average of monthly returns on the two high-BE/ME sub-categories and the equal-weighted average of monthly returns on the two low-BE/ME sub-categories. MKT is calculated as the difference between the value-weighted average return of all NYSE, AMEX and NASDAQ stocks and the one-month Treasury bill rate. Similarly, we can calculate profitability factor denoted as RMW (return of stocks with robust profitability minus return of stocks with weak profitability), investment pattern factor denoted as CMA (return of stocks with conservative investment pattern mine return of stocks with aggressive investment pattern) and momentum factor denoted as MOM (return of higher sum of stocks’ monthly return from t – 12 to t – 2 minus return of lower sum of stocks’ monthly return from t – 12 to t – 2).

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(a) construction of five-by-five SIZE-BE/ME portfolios

(b) construction of two-by-three SIZE-BE/ME portfolios

(c) construction of SIZE factor portfolios and BE/ME portfolios

Figure1 Construction procedure of five-by-five, two-by-three and factor portfolios

size1 size2 size3 size4 size5 NYSE size1

beme1 beme2 size1 beme3 size1 beme4 size1 beme5size1 size2

beme1 beme2 size2 beme3 size2 beme4 size2 beme5 size2 size3

beme1 beme2 size5 beme3 size5 beme4 size5 beme5 size5

AMEX

NASDAQ

step1 step2 step3

step1: allocate NYSE stocks into five categories based on the market value of firms (size). step2: divide each size category into five sub-category based on book-to-market value (beme). step3: allocate AMEX and NASDAQ stock into each sub-category. NYSE AMEX NASDAQ

step1 step2 step3

step1: allocate NYSE stocks into two categories based on the market value of ﬁrms (size). step2: divide each size category into three sub-category based on book-to-market value (beme). step3: allocate AMEX and NASDAQ stock into each sub-category. size1 size2 size1

beme1 _beme2size1 beme3siez1

size2

beme1 _beme2size2 beme3size2

size1

size2

size1

beme1 beme1size2

size1

beme2

size2

size1

beme3 beme3size2

value-weighted average return1 value-weighted

average return2 value-weighted _{average return3}

value-weighted average return4

All NYSE, AMEX and NASDAQ stocks included in the two-by-three SIZE-BE/ME porKolios SMB = average return1 – average return 2

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3. Methodology

Selection of factors portfolios

One of the most important properties that factors must have is persistence, which means factors must have been studied for some time and their existence should be long-time lasting and be independent from whether the factors are disclosed by published literatures or not (Chen et al 1986). Since investors mainly make investments decisions based on expected returns predictions using historical data, factor persistence is a demanding requirement for the successful investment. We firstly examine factor persistence by observing the stability of return trend along factors in different five-by-five portfolios and then select three sets of five-by-five portfolios that demonstrate the most obvious persistence of return trend.

Another important property that factors must have possessed is the explanatory power, which can be examined through Adjusted R2_{(Brooks, C. (2014)) and GRS statistics (Gibbons, Ross and Shanken} (1989)). Adjusted R2_{measures the percentage of variances, adjusted by the number of independent} variables, that can be explained by the related regression and GRS statistics test whether the expected intercepts values of regressions are jointly zero. The higher the adjusted R2_{and the lower the GRS} statistics, the better the regression is, namely the stronger explanatory power the factors have. We test the explanatory power of factors by regressing the selected three sets of five-by-five portfolios on different factors portfolios and examine the adjusted R2_{and GRS statistics. By saying different factors} portfolios, we refer to MKT portfolios containing only one independent variable, MKT-SMB-HML portfolios containing three independent variables, MKT-SMB-HML-MOM portfolios containing four independent variables, MKT-SMB-HML-RMW-CMA portfolios containing five independent variables and MKT-SMB-HML-RMW-CMA-MOM portfolios containing six independent variables. Then the factors portfolios that have the strongest explanatory power, namely the highest adjusted R2_{and lowest} GRS statistics, are selected for construction of factor-investing portfolios.

Construction of factor-investing portfolios

Sharpe ratio is calculated as excess return divided by standard deviation and indicates the return that investors can achieve by taking a unit of risk. Hence a higher Sharpe ratio means a better allocation of investment. We use historical data of factor portfolios to calculate the optimal factors allocation, which delivers the highest Sharpe ratio, and then apply the same factors allocation into in-sample test, hoping to achieve the new highest Sharpe ratio as well.

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1) Investors are able to exercise shorting strategy. Since all the factors except MKT are deduced by subtracting the return of one extreme sub-categories in two-by-three portfolios from the return of the other extreme sub-categories, investing directly on the factors portfolios require the ability of investors to exercise shorting strategy.

2) Constant standard deviation and correlations. We deduce optimal factors allocation from historical data by achieving the highest Sharpe ratio, which relies on the standard deviation of each factor and the correlations between every two of the factors. The same factors allocation can be used to construct factor-investing portfolios only on condition that the Sharpe ratio will remain constant, namely the standard deviation and correlations will remain unchanged. Otherwise, when applying factor-investing portfolios into in-sample test or practical investment, the achieved Sharpe ratio will be lower compared to the new highest Sharpe ratio derived from the in-sample data.

However, in reality the standard deviation and correlations will never stay constant. We propose three ways to improve the construction of factor-investing portfolios, even without unchanged standard deviation and correlation.

First way, setting minimum years of historical data used to obtain the factors allocation. Since long-term historical data turned to be more stable compared to short-term data, by enlarging the horizon of historical data we can achieve a relatively constant data.

Second way, updating standard deviation with Exponential Weighted Moving Average (EWMA) model. Since more recent data are more related to the most recently economic situation and thus provide a better prediction towards the future trend of data, that we give more weight to the more recent data to calculate standard deviation rather than use equally weights may provide a better factors allocation.

To apply EWMA mode, we firstly randomly assume a positive constant lambda (𝝀) smaller than 1. Then we allocate the most recent data with a weight of (𝟏 − 𝝀), the second most recent data with a weight of (𝟏 − 𝝀) × 𝝀𝟏_{, the third most recent data with a weight of (𝟏 − 𝝀) × 𝝀}𝟐_{and the n}th_{most recent} data with a weight of (𝟏 − 𝝀) × 𝝀𝒏−𝟏_{. By using formula (1), we can calculate the updated standard}

deviation under the assumed 𝝀. Through maximizing the likelihood indicated in formula (2) by changing 𝝀, we can achieve the updated standard deviation with the highest likelihood.

𝝈_𝒏𝟐 _{= 𝝀 × 𝝈} 𝒏−𝟏 𝟐 _{+ (𝟏 − 𝝀) × 𝒖} 𝒏−𝟏 𝟐 ₍₁₎ ∏ [ 𝟏 √𝟐𝝅𝝂𝒊 𝒎 𝒊=𝟏 𝐞𝐱𝐩 (−𝒖𝒊 𝟐 𝟐𝝂𝒊)] 𝒐𝒓 ∑ [− 𝐥𝐧(𝝂𝒊) − 𝒖_𝒊𝟐 𝝂𝒊] 𝒎 𝒊=𝟏 (2)

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Where 𝝈𝒏, 𝝈𝒏−𝟏, 𝒖𝒏−𝟏, 𝝂𝒊, 𝒖𝒊 are the forecasted standard deviation at time point of n, the forecasted

standard deviation at time point of n – 1, return at the time point of n – 1, the forecasted variance at time point of i and return at time point of i, respectively.

Third way, applying a minimum 10% allocation to each factor. It is possible that the highest Sharpe ratio will deliver an allocation that put none weight to some factors. In order to lower this risk, we set a minimum 10% allocation to each factor.

Performance analysis of factor-investing portfolios

The constructed factor-investing portfolios will be used to conduct in-sample test. We compare the performance of these portfolios by testing whether the Sharpe ratios are significantly different from those of market cap weighted indexes during the same time period such as S&P500 (GSPC), Dow Jones Industrial Average (DJI) and NASDAQ Composite (IXIC). According to Jobson and Korkie (1981) and Christoph Memmel (2003), the hypothesis that the Sharpe ratios of factor-investing portfolios equal those of market cap weighted indexes can be tested using formula (3) and (4).

𝒛 =𝝈𝒏𝝁𝒊−𝝈𝒊𝝁𝒏 √𝜽 (3) 𝜽 =𝟏 𝑻[𝟐𝝈𝒊 𝟐_𝝈 𝒏 𝟐_{− 𝟐𝝈} 𝒊𝝈𝒏𝝈𝒊𝒏+𝟏_𝟐𝝁𝒊𝟐𝝈𝒏𝟐+𝟏_𝟐𝝁𝒏𝟐𝝈𝒊𝟐−𝝁_𝝈𝒊𝝁𝒏 𝒊𝝈𝒏𝝈𝒊𝒏 𝟐 _{] (4)}

Where 𝝈_𝒊, 𝝈_𝒏, 𝝈_𝒊𝒏, 𝝁_𝒊, 𝝁_𝒏 are standard deviation of portfolio i, standard deviation of portfolio n, square root of covariance between portfolio i and portfolio n, return of portfolio i and return of portfolio n, respectively.

Software

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4. Factors Selection

Selection of five-by-five portfolios

We form different five-by-five portfolios using every two of the factors including size (SIZE), book-to-market equity (BE/ME), profitability (OP), investment patterns (INV), momentum (MOM) and market beta (BETA). Three of these portfolios will be selected, on the basis of better return trend along with each factor, as dependent variables to do further regression test

Table1 lists the descriptive statistics for eight sets of five-by-five portfolios using historical data from 1963 to 2016. Taking SIZE-BETA portfolios for example, no clear return trend is observed in the portfolios. In size1-category the return increases from 0.71 to 0.94 with beta increasing from sub-category1 to sub-category 4 but decreases to 0.74 in sub-category5. In size2-category the return increases from 0.70 to 0.93 with beta increasing from sub-category1 to sub-category3 but decreases to 0.86 in sub-category4 and 0.68 in sub-category5. In size3-category the return increases from 0.69 to 0.84 with beta increasing from sub-category1 to sub-category2 but decreases all the way to 0.71 from sub-category3 to sub-category5. Similar instability is also observed in size4-category and size5-category. However, when looking at each beta-category the return trend is pretty stable, mostly decreasing with size increasing from sub-category1 to sub-category5. Only two exceptions occurred, notably returns in size1beta3 sub-category and size2beta5 subcategory, but that does not affect the return trend along SIZE factor in the five-by-five SIZE-BETA portfolios as a whole. In summary, even the return along size factor displays a stable trend, we would not select the five-by-five SIZE-BETA portfolios due to the instable return trend along beta factor.

Similarly, we choose not to select SIZE-OP portfolios, SIZE-INV portfolios, BE/ME-INV portfolios and OP-INV portfolios due to instable return trend along one or both factors.

BE/ME-OP portfolios, highlighted in blue, are selected since the return trend along with both factors are stable and only three exceptions occurred, notably returns in beme3op2 sub-category, beme2op3 sub-category and beme5op5 sub-category which however do not affect the return trend along with two factors as a whole.

Similarly, SIZE-BE/ME portfolios and SIZE-MOM portfolios are also selected. These two portfolios are not as good as BE/ME-OP portfolios, taking into consideration that the return trends in beme1-category of SIZE-BE/ME portfolios and in mom1-category of SIZE-MOM portfolios are not stable. However, as a whole the return trend along with both factors in these two portfolios is still observable and much better than the five dropped portfolios, except that certain exceptions occurred size2beme2 sub-category in SIZE-BE/ME portfolios and size4mom4 sub-category in SIZE-MOM portfolios.

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Table1

Descriptive statistics for five-by-five portfolios constructed using different factors, July1963-June 2016 (636 months)

This table describes the average excess return, standard deviation and t-test of each sub-category in different five-by-five portfolios formed using every two of factors including size (SIZE), beta (BETA), book-to-market equity (BE/ME), profitability (OP), investment pattern (INV) and momentum (MOM). The number i in the suffix of each factor means the ith quintile of this factor, e.g. beta1 means the first quintile of five breakpoints on size factor.

SIZE-BETA portfolios beta1 beta2 beta3 beta4 beta5 beta1 beta2 beta3 beta4 beta5 beta1 beta2 beta3 beta4 beta5

size1 0.71 0.86 0.89 0.94 0.74 4.39 5.07 5.91 6.48 8.18 4.07 4.27 3.79 3.66 2.28

size2 0.70 0.85 0.93 0.86 0.68 4.29 4.75 5.44 6.18 7.89 4.13 4.49 4.31 3.51 2.16

size3 0.69 0.84 0.83 0.77 0.71 3.84 4.62 5.21 5.97 7.64 4.52 4.61 4.01 3.28 2.33

size4 0.66 0.76 0.72 0.57 0.71 3.88 4.61 5.14 5.83 7.54 4.31 4.16 3.55 2.47 2.38

size5 0.49 0.52 0.48 0.47 0.37 3.61 4.24 4.86 5.69 7.13 3.43 3.07 2.50 2.08 1.30

SIZE-BE/ME portfolios beme1 beme2 beme3 beme4 beme5 beme1 beme2 beme3 beme4 beme5 beme1 beme2 beme3 beme4 beme5

size1 0.21 0.78 0.79 0.97 1.08 7.93 6.87 5.98 5.62 6.06 0.67 2.85 3.34 4.34 4.50

size2 0.46 0.71 0.90 0.92 0.96 7.20 5.94 5.43 5.28 5.99 1.62 3.02 4.20 4.37 4.03

size3 0.49 0.75 0.77 0.85 1.02 6.61 5.47 5.02 4.92 5.50 1.88 3.48 3.86 4.35 4.69

size4 0.60 0.58 0.70 0.82 0.82 5.88 5.16 5.06 4.80 5.47 2.55 2.82 3.49 4.32 3.79

size5 0.47 0.52 0.48 0.55 0.60 4.66 4.43 4.35 4.35 5.00 2.54 2.99 2.80 3.18 3.01

SIZE-OP portfolios op1 op2 op3 op4 op5 op1 op2 op3 op4 op5 op1 op2 op3 op4 op5

size1 0.51 0.90 0.86 0.91 0.82 7.34 5.83 5.69 5.82 6.58 1.75 3.90 3.83 3.93 3.14

size2 0.57 0.75 0.81 0.77 0.94 7.02 5.66 5.38 5.51 6.24 2.04 3.35 3.81 3.55 3.82

size3 0.52 0.75 0.69 0.74 0.91 6.51 5.04 5.04 5.25 5.85 2.01 3.77 3.45 3.56 3.92

size4 0.54 0.64 0.64 0.69 0.80 6.00 4.94 4.90 4.94 5.42 2.25 3.26 3.30 3.53 3.73

size5 0.38 0.34 0.44 0.48 0.57 5.26 4.42 4.38 4.46 4.34 1.84 1.92 2.53 2.73 3.34

SIZE-INV portfolios inv1 inv2 inv3 inv4 inv5 inv1 inv2 inv3 inv4 inv5 inv1 inv2 inv3 inv4 inv5

size1 0.92 0.94 0.95 0.84 0.32 7.26 5.66 5.66 5.98 7.13 3.20 4.19 4.22 3.55 1.14

size2 0.87 0.86 0.92 0.87 0.45 6.35 5.26 5.26 5.68 6.97 3.45 4.14 4.39 3.85 1.64

size3 0.86 0.91 0.78 0.80 0.48 5.73 4.83 4.81 5.36 6.53 3.78 4.72 4.10 3.75 1.86

size4 0.77 0.72 0.72 0.74 0.52 5.40 4.77 4.68 4.95 6.33 3.59 3.82 3.89 3.79 2.08

size5 0.71 0.53 0.49 0.48 0.43 4.62 4.00 4.13 4.45 5.47 3.88 3.37 2.97 2.71 1.99

BE/ME-OP portfolios op1 op2 op3 op4 op5 op1 op2 op3 op4 op5 op1 op2 op3 op4 op5

beme1 0.12 0.28 0.44 0.46 0.55 8.23 6.97 5.66 4.95 4.61 0.36 1.01 1.97 2.34 3.01

beme2 0.28 0.47 0.43 0.64 0.70 6.36 5.13 4.85 4.67 4.85 1.11 2.32 2.22 3.45 3.66

beme3 0.36 0.44 0.58 0.71 0.90 5.36 4.54 4.59 4.66 5.60 1.69 2.42 3.17 3.83 4.07

beme4 0.50 0.62 0.76 0.85 1.03 5.09 4.41 4.51 5.15 6.62 2.50 3.58 4.26 4.16 3.92

beme5 0.73 0.81 0.91 1.13 0.82 5.37 4.94 5.75 7.14 8.47 3.41 4.13 3.98 3.98 2.44

BE/ME-INV portfolios inv1 inv2 inv3 inv4 inv5 inv1 inv2 inv3 inv4 inv5 inv1 inv2 inv3 inv4 inv5

beme1 0.54 0.52 0.53 0.51 0.50 5.58 4.70 4.53 4.75 6.14 2.42 2.76 2.93 2.69 2.03

beme2 0.78 0.75 0.48 0.50 0.44 5.12 4.55 4.60 4.91 5.57 3.84 4.15 2.63 2.55 2.01

beme3 0.68 0.63 0.67 0.54 0.48 5.05 4.53 4.56 4.75 5.60 3.41 3.52 3.72 2.85 2.17

beme4 0.76 0.63 0.70 0.83 0.66 5.19 4.38 4.47 5.18 5.40 3.70 3.65 3.95 4.06 3.09

beme5 0.93 0.80 0.71 0.94 0.60 5.74 5.17 5.26 5.92 6.27 4.10 3.91 3.39 3.99 2.41

OP-INV portfolios inv1 inv2 inv3 inv4 inv5 inv1 inv2 inv3 inv4 inv5 inv1 inv2 inv3 inv4 inv5

op1 0.72 0.47 0.58 0.49 0.12 6.28 5.24 5.44 5.77 6.49 2.90 2.26 2.68 2.15 0.45

op2 0.62 0.61 0.43 0.56 0.41 4.97 4.45 4.47 4.92 5.86 3.15 3.48 2.41 2.85 1.76

op3 0.72 0.67 0.45 0.61 0.23 5.12 4.38 4.53 4.90 5.84 3.55 3.85 2.52 3.15 1.01

op4 0.93 0.79 0.54 0.54 0.41 5.06 4.25 4.47 4.73 5.77 4.62 4.70 3.06 2.86 1.80

op5 0.91 0.62 0.65 0.58 0.68 5.01 4.61 4.36 4.68 5.89 4.58 3.40 3.77 3.12 2.90

SIZE-MOM portfolios mom1 mom2 mom3 mom4 mom5 mom1 mom2 mom3 mom4 mom5 mom1 mom2 mom3 mom4 mom5

size1 -0.02 0.65 0.90 1.05 1.35 8.00 5.86 5.41 5.47 6.75 -0.05 2.78 4.18 4.81 5.00

size2 0.11 0.64 0.80 1.00 1.21 7.85 5.86 5.25 5.38 6.71 0.35 2.76 3.82 4.65 4.51

size3 0.21 0.59 0.70 0.77 1.18 7.38 5.51 5.05 4.98 6.28 0.70 2.70 3.49 3.86 4.72

size4 0.15 0.56 0.65 0.79 1.03 7.27 5.51 4.85 4.77 5.87 0.52 2.55 3.35 4.14 4.39

size5 0.13 0.45 0.38 0.55 0.78 6.80 4.87 4.37 4.31 5.24 0.49 2.30 2.20 3.18 3.75

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13

Selection of factor portfolios

In the last part, we select five-by-five BE/ME-OP, SIZE-BE/ME and SIZE-MOM portfolios as the dependent variables due to their relatively stable return trend along with each factor compared with other five-by-five portfolios. This part we focus on the selection of factor portfolios that have the strongest explanatory power in regressions.

Table2 summarizes statics of regressions on one, three, four, five and six factors using SIZE-BE/ME, BE/ME-OP and SIZE-MOM portfolios during three time periods, including 1963 to 2016, 1963 to 2010 and 2010 to 2016.

Firstly, we confirm that five-by-five SIZE-BE/ME portfolios deliver relatively better regression performance regardless of how many factors used in the regression. Compared with regressions on BE/ME-OP portfolios, regressions on SIZE-BE/ME portfolios have a stronger explanatory power reflected in adjusted R2_{which shows the percentage of total variations that can be explained by the} regressions. Compared with regressions on SIZE-MOM portfolios, regressions SIZE-BE/ME portfolios have a stronger explanatory power reflected in GRS statistics which test whether the expected intercepts values of regressions are jointly zero. Even though regressions on SIZE-BE/ME portfolios have a relatively higher GRS statistics compared to those on BE/ME-OP portfolios and comparable adjusted R2 compared to those on SIZE-MOM portfolios, they have the lowest absolute value of intercept, indicating a better regression. Hence, we select five-by-five SIZE-BE/ME portfolios as dependent variables.

Secondly, we confirm that six-factor portfolios deliver relatively better regression performance compared with other factor portfolios. In SIZE-BE/ME portfolios, the regression using six-factor models has stronger explanatory power compared to other regressions, which is reflected on a lower absolute value of intercepts, a lower GRS statistics and a comparable adjusted R2_.

Besides, the aforementioned two confirmations can be further justified by observing regressions data in different time periods. Even though, GRS statistics and adjusted R2_{show various explanatory powers} during different time periods, the SIZE-BE/ME portfolios and the six-factor portfolios are always delivering better regression performances compared with others.

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14

Table2

Summary statics of regressions on one, three, four, five and six factors using SIZE-BE/ME, BE/ME-OP and SIZE-MOM portfolios during three time periods

This table summarizes how well regressions on one, three, four, five and six factors explain monthly excess returns using SIZE-BE/ME portfolios, BE/ME-OP portfolios and SIZE-MOM portfolios. The evaluation of regressions performance is based on (1) the average of absolute intercepts, (2) the GRS statistic testing whether the expected intercepts values of regressions are jointly zero, (3) the p-value for the GRS statistic and (4) the average of adjusted R2_{. MKT is the excess return}

on the value-weighted portfolio, RM-RF. SMB, HML, RMW, CMA and MOM are factor proxies derived from related two-by-three portfolios described in the Data section.

Abs intercept GRS p-value Adjusted R2 Abs intercept GRS p-value Adjusted R2 Abs intercept GRS p-value Adjusted R2

SIZE-BE/ME portfolios MKT 0.250 4.563 0.000 0.745 0.285 4.205 0.000 0.738 0.279 1.189 0.299 0.829 MKT SMB HML 0.100 3.737 0.000 0.915 0.105 3.286 0.000 0.914 0.111 1.035 0.448 0.933 MKT SMB HML MoM 0.093 3.192 0.000 0.916 0.095 2.770 0.000 0.914 0.106 0.939 0.557 0.934 MKT SMB HML RMW CMA 0.091 2.873 0.000 0.919 0.099 2.612 0.000 0.918 0.104 0.883 0.623 0.939 MKT SMB HML RMW CMA MoM 0.083 2.541 0.000 0.919 0.089 2.270 0.000 0.918 0.099 0.822 0.694 0.939 BE/ME-OP portfolios MKT 0.250 1.785 0.011 0.707 0.285 1.951 0.004 0.705 0.379 1.259 0.245 0.754 MKT SMB HML 0.140 1.651 0.025 0.784 0.152 1.720 0.017 0.785 0.283 1.176 0.312 0.797 MKT SMB HML MoM 0.128 1.255 0.183 0.786 0.138 1.336 0.129 0.787 0.235 1.058 0.425 0.801 MKT SMB HML RMW CMA 0.112 1.152 0.279 0.797 0.119 1.326 0.135 0.798 0.248 1.031 0.454 0.814 MKT SMB HML RMW CMA MoM 0.108 0.910 0.592 0.799 0.114 1.059 0.387 0.800 0.234 0.945 0.551 0.817 SIZE-MoM portfolios MKT 0.338 5.392 0.000 0.733 0.349 4.958 0.000 0.728 0.384 1.735 0.059 0.808 MKT SMB HML 0.328 5.279 0.000 0.847 0.333 4.846 0.000 0.844 0.346 1.829 0.045 0.896 MKT SMB HML MoM 0.133 4.011 0.000 0.914 0.134 3.749 0.000 0.913 0.217 1.619 0.090 0.937 MKT SMB HML RMW CMA 0.279 4.374 0.000 0.854 0.287 4.272 0.000 0.851 0.324 1.673 0.078 0.899 MKT SMB HML RMW CMA MoM 0.115 3.617 0.000 0.920 0.125 3.629 0.000 0.919 0.204 1.511 0.128 0.938 1963 to 2016 1963 to 2010 2010 to 2016

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15

Further analysis on six-factor portfolios

In this part, we further explore regression performance of five-by-five SIZE-BE/ME portfolios on six-factor portfolios in two aspects.

Firstly, we further compare statistics of regressions on three- and six-factors using SIZE-BE/ME portfolios during the time period between 1963 and 2016, shown in Table3. By introducing three more factors, RMW, CMA and MOM, six-factor portfolios seem to obtain more benefits than detriments compared to the three-factor model. Generally, the adjust R2_{is improved slightly by 0.01 for many} SIZE-BE/ME sub-categories, the adjusted residual standard error term declines by as much as 0.2 for certain SIZE-BE/ME sub-categories and the average value of intercepts decreases slightly as well, indicating the improvement on accuracy. However, all the t-test values on coefficients of MKT, SML and HML decrease on average but this decrease is very slight, less than 10%, and would not cause a significant difference. In addition, the introduction of RMW, CMA and MOM will compensate for the slight decrease of t-test values on coefficients in terms of explanatory power as a whole.

Besides, an interesting thing is that we find the CMA factor is highly correlated with HML factor and the correlation reaches 0.7, which implies that either the CMA factor or the HML factor to some degree is redundant. However, the correlations between other factors are relatively low, the absolute value ranging from 0.09 to 0.39.

Secondly, we compare GRS statistics and adjusted R2_{of regressions on three-, four-, five- and six-factor} portfolios in more various time horizons, since regressions on all the portfolios display a similar adjusted R2_{. Figure2 shows that no matter on every five-year basis or on accumulated five-year basis, the} adjusted R2_{of the four sets of portfolios are moving together tightly, ranging from 0.88 to 0.95, and five-} and six-factor portfolios have a 0.1 higher of adjusted R2_{. As for GRS statistics, Figure2 (a) and (b)} display a significant difference. Specifically, after 1983 GRS statistics on every five-year basis are fluctuating all the time while GRS statistics on accumulated five-year basis are increasing constantly. Moreover, GRS statistics of five- and six-factor portfolios are higher than those of three-and four-factor portfolios on every five-year basis since 2013 while lower on accumulated five-year basis since 1983. However, despite of the exception of GRS statistics on every five-year basis since 2013, the six-factor portfolios are always displaying a better performance in terms of both the highest adjusted R2_{and lowest} GRS statistic. Hence, we confirm the use of six-factor portfolios to construct factor-investing portfolios.

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16

Table3

Summary statics of regressions on three and six factors using SIZE-BE/ME portfolios, July 1963-June 2016 (636 months)

This table summarizes statistics of regressions on three and six factors using SIZE-BE/ME portfolios. RM-RF is the excess return on the value-weighted portfolio while SMB, HML, RMW, CMA and MOM are factor proxies derived from related two-by-three portfolios described in the Data section. Statistic data in each sub-category include (1) coefficient of each factor, (2) t-test value of coefficients, (3) adjusted R2

indicating the proposition of variance that can be explained by the regression, (4) the adjusted residual standard error term and (5) the correlation between each two factors on the left bottom of this table.

beme1 beme2 beme3 beme4 beme5 beme1 beme2 beme3 beme4 beme5 beme1 beme2 beme3 beme4 beme5 beme1 beme2 beme3 beme4 beme5

size1 1.06 0.96 0.91 0.87 0.97 48.22 58.61 71.78 67.02 69.57 size1 1.02 0.94 0.92 0.89 0.97 47.55 58.38 68.22 64.37 66.03 size2 1.10 1.00 0.96 0.96 1.08 73.60 76.25 76.39 74.85 78.00 size2 1.08 1.00 0.97 0.98 1.08 69.06 74.39 76.10 73.21 73.52 size3 1.08 1.03 0.99 0.99 1.06 74.74 66.87 64.60 66.50 60.62 size3 1.05 1.04 1.00 1.00 1.07 70.20 65.21 64.28 64.60 58.54 size4 1.06 1.07 1.07 1.01 1.13 71.97 65.12 63.40 64.21 60.40 size4 1.05 1.09 1.08 1.01 1.12 67.68 65.61 61.88 60.70 56.73 size5 0.97 0.99 0.97 0.98 1.04 86.16 71.05 60.49 68.34 47.61 size5 0.97 1.02 0.99 0.98 1.02 87.12 72.51 58.11 64.87 44.22 size1 1.36 1.29 1.09 1.03 1.07 44.24 56.62 61.39 56.39 55.17 size1 1.25 1.22 1.08 1.03 1.07 41.96 54.12 57.80 53.86 52.42 size2 1.00 0.88 0.78 0.73 0.86 48.01 48.48 44.33 40.55 44.45 size2 0.98 0.92 0.82 0.76 0.87 44.90 48.90 46.45 40.78 42.44 size3 0.73 0.55 0.46 0.41 0.55 36.18 25.50 21.47 19.90 22.62 size3 0.71 0.60 0.51 0.45 0.59 33.86 26.84 23.74 20.66 23.23 size4 0.38 0.24 0.20 0.23 0.26 18.64 10.31 8.30 10.43 10.02 size4 0.36 0.29 0.24 0.23 0.28 16.71 12.53 9.80 9.97 10.11 size5 -0.24 -0.20 -0.21 -0.18 -0.05 -15.38 -10.45 -9.36 -8.96 -1.78 size5 -0.19 -0.16 -0.21 -0.16 -0.09 -12.55 -7.91 -8.76 -7.54 -2.96 size1 -0.48 -0.12 0.15 0.33 0.56 -14.40 -4.81 7.96 16.88 26.90 size1 -0.42 -0.12 0.12 0.30 0.51 -9.77 -3.74 4.27 10.69 17.27 size2 -0.51 0.03 0.30 0.47 0.70 -22.60 1.30 15.75 24.55 33.91 size2 -0.47 -0.04 0.28 0.43 0.70 -14.92 -1.30 10.98 15.98 23.99 size3 -0.53 0.12 0.39 0.57 0.71 -24.37 5.30 17.16 25.67 27.24 size3 -0.44 0.10 0.36 0.52 0.66 -14.61 3.13 11.57 16.76 18.01 size4 -0.47 0.19 0.43 0.55 0.79 -21.12 7.60 16.87 23.11 27.83 size4 -0.45 0.06 0.35 0.52 0.79 -14.49 1.87 9.97 15.45 19.90 size5 -0.33 0.13 0.33 0.63 0.77 -19.56 6.13 13.56 29.33 23.33 size5 -0.31 0.03 0.26 0.61 0.84 -13.88 1.10 7.64 20.10 18.13 size1 -0.11 0.41 0.40 0.55 0.52 -1.17 5.97 7.56 10.18 9.02 size1 0.10 0.50 0.39 0.51 0.54 1.16 7.53 7.04 8.95 8.94 size2 0.22 0.37 0.52 0.48 0.36 3.59 6.86 9.89 9.03 6.17 size2 0.32 0.36 0.46 0.42 0.34 5.00 6.42 8.83 7.58 5.70 size3 0.35 0.45 0.42 0.45 0.50 5.73 7.00 6.56 7.27 6.93 size3 0.44 0.41 0.36 0.39 0.47 7.22 6.33 5.62 6.16 6.29 size4 0.53 0.31 0.36 0.47 0.31 8.60 4.59 5.14 7.13 4.01 size4 0.55 0.24 0.33 0.46 0.35 8.68 3.52 4.61 6.64 4.24 size5 0.56 0.43 0.33 0.28 0.22 11.89 7.40 4.97 4.70 2.41 size5 0.52 0.32 0.30 0.27 0.33 11.28 5.44 4.26 4.33 3.46 size1 0.92 0.94 0.95 0.94 0.94 2.27 1.68 1.31 1.34 1.44 size1 0.93 0.95 0.95 0.94 0.94 2.08 1.57 1.31 1.34 1.43 size2 0.95 0.95 0.94 0.94 0.94 1.55 1.35 1.30 1.32 1.42 size2 0.96 0.95 0.95 0.94 0.94 1.52 1.31 1.24 1.30 1.42 size3 0.95 0.91 0.90 0.90 0.89 1.49 1.59 1.58 1.53 1.80 size3 0.95 0.92 0.91 0.91 0.90 1.45 1.55 1.51 1.50 1.77 size4 0.93 0.89 0.88 0.89 0.87 1.51 1.69 1.74 1.62 1.94 size4 0.93 0.90 0.89 0.89 0.88 1.51 1.61 1.70 1.62 1.92 size5 0.94 0.89 0.85 0.88 0.80 1.16 1.44 1.66 1.48 2.26 size5 0.95 0.90 0.86 0.89 0.80 1.08 1.37 1.65 1.47 2.23 size1 -0.47 -0.31 -0.01 0.04 0.01 -10.74 -9.38 -0.48 1.48 0.20 size2 -0.12 0.15 0.21 0.15 0.05 -3.87 5.38 7.84 5.31 1.55 size3 -0.12 0.20 0.24 0.15 0.17 -3.74 6.08 7.63 4.83 4.59 size4 -0.09 0.25 0.19 0.02 0.07 -2.79 7.17 5.32 0.53 1.67 size5 0.20 0.22 0.02 0.09 -0.19 8.69 7.61 0.54 2.85 -3.93 size1 -0.14 0.02 0.08 0.09 0.09 -2.19 0.40 2.07 2.23 2.14 size2 -0.12 0.09 0.02 0.10 0.00 -2.55 2.29 0.57 2.53 -0.11 size3 -0.21 0.02 0.05 0.10 0.09 -4.89 0.42 1.01 2.22 1.67 size4 -0.03 0.23 0.12 0.06 -0.05 -0.57 4.66 2.44 1.20 -0.81

mktfr smb hml rmw cma mom size5 -0.06 0.21 0.15 0.03 -0.15 -1.88 5.10 3.01 0.65 -2.28

mktfr 1.00 0.28 -0.30 -0.21 -0.39 -0.13 smb 0.28 1.00 -0.12 -0.36 -0.11 -0.02 size1 -0.03 0.01 0.00 0.02 -0.04 -1.51 0.38 -0.04 1.28 -2.85 hml -0.30 -0.12 1.00 0.09 0.70 -0.17 size2 -0.04 -0.05 -0.02 0.00 0.00 -2.44 -4.18 -1.39 0.06 -0.20 rmw -0.21 -0.36 0.09 1.00 -0.08 0.09 size3 -0.02 -0.04 -0.03 -0.01 -0.05 -1.49 -2.38 -2.20 -0.94 -2.60 cma -0.39 -0.11 0.70 -0.08 1.00 -0.01 size4 0.01 -0.06 -0.06 -0.01 -0.05 0.53 -3.64 -3.72 -0.40 -2.62 mom -0.13 -0.02 -0.17 0.09 -0.01 1.00 size5 -0.01 0.00 0.00 -0.03 -0.02 -1.29 0.14 0.04 -1.79 -0.80

correlation between each factor

b t(b) three factors s t(s) h t(h) R(t) - RF(t) = a + b[RM(t) - RF(t)] + sSMB(t) + hHML(t) + mMOM(t) + e(t) h t(h) a six factors t(a)

Adjusted R-square s(e)

a t(a)

Adjusted R-square s(e)

b t(b)

s t(s)

R(t) - RF(t) = a + b[RM(t) - RF(t)] + sSMB(t) + hHML(t) + rRMW(t) cCMA(t) + mMOM(t) + e(t)

m t(m)

r t(r)

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17

(a) GRS statistics and adjusted R2_{on every five-year basis}

(b) GRS statistics and adjusted R2_{on accumulated five-year basis}

Figure2 GRS statistics and adjusted R2_{of regressions on three, four, five and six factors using}

SIZE-BE/ME portfolios.

These two graphs compare the regression performance of three, four, five and six factors using GRS statistics and adjusted R2_{. In graph (a) the regressions are based on every five-year basis, namely from}

July 1963 to June 1968, from July 1968 to June 1973, from July 1973 to June 1983, etc. In graph (b) the regressions are based on accumulated five-year basis, namely from July 1963 to June 1968, from July 1963 to June 1973, from July 1963 to June 1983, etc.

0 0.5 1 1.5 2 2.5 3 3.5 0.84 0.86 0.88 0.9 0.92 0.94 0.96 1968 1973 1978 1983 1988 1993 1998 2003 2008 2013 2016 G RS T ES T AD JU ST ED R2 TIME PERIOD

Adjusted R2_3-factor Adjusted R2_4-factor Adjusted R2_5-factor Adjusted R2_6-factor GRS_3-factor GRS_4-factor GRS_5-factor GRS_6-factor

0 0.5 1 1.5 2 2.5 3 3.5 4 0.88 0.89 0.9 0.91 0.92 0.93 0.94 1968 1973 1978 1983 1988 1993 1998 2003 2008 2013 2016 G RS T ES T AD JU ST ED R2 TIME PERIOD

Adjusted R2_3-factor Adjusted R2_4-factor Adjusted R2_5-factor Adjusted R2_6-factor GRS_3-factor GRS_4-factor GRS_5-factor GRS_6-factor

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18

5. Portfolios Construction and Analysis

Factor investing portfolios construction

We use historical data prior to July 2010 to explore the factors allocation, which delivers the highest Sharpe ratio of six-factor portfolios, and use the same factors allocation to conduct in-sample test using data from July 2010 to June 2016. When exploring the factors allocation, we use historical data of different time periods, including 47-year data from July 1963 to June 2010 marked as period1, 40-year data from July 1970 to June 2010 marked as period2, 30-year data from July 1980 to June 2010 marked as period3, 20-year data from July 1990 to June 2010 marked as period4, 15-year data from July 1995 to June 2010 marked as period5, 10-year data from July 2000 to June 2010 marked as period6 and 5-year data from July 2005 to June 2010 marked as period7. In addition, we also calculate the factors allocation of in-sample test during July 2010 to June 2016 marked as period8 and compare the in-sample factors allocation and Sharpe ratio with those of factor-investing portfolios.

Table4 summarizes basic statistics of historical data, factors allocation that delivers the highest Sharpe ratio, the derived highest Sharpe ratio and the in-sample test Sharpe ratio. Table5 displays correlations between each two factors during different time periods. Taking period1 for example, we firstly summarizes the mean return and standard deviation of each factor, list correlations between each factor and assume an initial allocation for each factor as 10%, 10%, 20%, 20%, 20%, 20%. Then we calculate the Sharpe ratio and through excel solver we get the highest Sharpe ratio by changing the allocation of each factor. Here, we allocate 14%, 12%, 5%, 26%, 32% and 11% to MKT, SML, HML, RMW, CMA and MOM, respectively, to get the mean return of 0.37, the standard deviation of 0.98 and the highest Sharpe ratio of 0.38. By applying the same factors allocation into in-sample test during period8, we get a mean return of 0.27, a standard deviation of 0.77 and a Sharpe ratio of 0.36. Similarly, we apply the same procedures into different periods, achieve the factors allocation and apply the same allocations to run in-sample test using data in period8.

To do further analysis conveniently, we also denote the in-sample-test Sharpe ratios as Si, where i stands for different time periods, e.g. S1 means the in-sample-test Sharpe ratio of factor-investing portfolios constructed using data from period1, S2 means the in-sample-test Sharpe ratio of factor-investing portfolios constructed using data from period2, etc.

Generally, Si varies a lot with different time periods, ranging from 0.16 in period7 to 0.47 in period8. Si becomes closer to S8 with i increasing from 1 to 5 while Si becomes far below than S8 when i equals to 6 and 7, indicating that applying the factors allocation derived from the first five periods to form factor-investing portfolios would deliver a better performance than applying the factors allocation derived from period6 and period7. As we discussed in the methodology part, Sharpe ratio is determined

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by the portfolio excess return and the portfolio standard deviation which is further determined by the standard deviation of each factor and the correlation between every two factors. Hence, allocations derived from the time period, which has similar or close patterns of return, standard deviation and correlation matrix with that of period8, would form factor-investing portfolios that have similar or close Sharpe ratio of period8. Obviously, the patterns of returns, standard deviation and correlation matrix of period8 dramatically differ from that of period6 and period7. Specifically, the signs of MKT, SML, HML and MOM in period6 and period7 are opposite to those in period8, the standard deviation of MOM ranks the highest in period6 and period7 while the standard deviation of MKT ranks the highest in period8. Moreover, the correlation matrix also changes quite a lot, for instance the correlations of HML-MKT, HML-SMB, CMA-MKT, CMA-SMB and CMA-RMW decline to almost zero from period7 to period8, the correlations of MOM-MKF and MOM-RMW declines by more than a half and the correlation of CMA-HML increases by 50%. Similar huge inconsistency is also observed between period6 and period8. Since the patterns of returns, standard deviation and correlation matrix during period6 and period7 are most on the opposite to those during period8, it is expectable that the S6 and S7 would be quite different from S8, indicating a poor in-sample test. However, the patterns of return, standard deviation and correlation matrix in other periods are to some degree similar to those of period8 and thus Sharpe ratios are relatively closer to S8.

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Table4

Construction of factor-investing portfolios using historical data in different time periods, without adjustment.

mean return sigma allocation mean return sigma Sharpe ratio mean return sigma allocation mean return sigma Sharpe ratio

MKT 0.41 4.51 0.14 0.37 0.98 0.38 MKT 0.48 4.64 0.15 0.38 1.01 0.38

SMB 0.29 3.15 0.12 SMB 0.25 3.17 0.12

HML 0.41 2.95 0.05 HML 0.40 3.08 0.03

RMW 0.26 2.19 0.26 RMW 0.31 2.28 0.27

CMA 0.33 2.08 0.32 mean return sigma Sharpe ratio CMA 0.35 2.05 0.33 mean return sigma Sharpe ratio

MOM 0.72 4.36 0.11 0.27 0.77 0.36 MOM 0.70 4.59 0.10 0.29 0.79 0.37

MKT 0.52 4.57 0.17 0.40 1.00 0.40 MKT 0.44 4.46 0.20 0.38 1.02 0.38

SMB 0.16 3.06 0.12 SMB 0.25 3.38 0.14

HML 0.36 3.21 0.00 HML 0.35 3.38 0.00

RMW 0.42 2.45 0.29 RMW 0.41 2.82 0.31

MOM 0.64 4.79 0.08 0.32 0.84 0.38 MOM 0.59 5.37 0.08 0.35 0.89 0.39

MKT 0.36 4.76 0.20 0.38 1.08 0.35 MKT -0.16 4.83 0.11 0.53 1.27 0.42

SMB 0.30 3.63 0.17 SMB 0.57 2.78 0.30

HML 0.33 3.66 0.00 HML 0.75 3.39 0.00

RMW 0.43 3.16 0.32 RMW 0.69 2.84 0.44

MOM 0.48 5.97 0.07 0.34 0.90 0.37 MOM -0.01 6.35 0.00 0.20 0.82 0.24

MKT -0.10 4.93 0.05 0.27 0.95 0.28 MKT 1.21 3.73 0.28 0.49 1.04 0.47

SMB 0.25 2.53 0.24 SMB -0.06 2.18 0.00

HML 0.07 2.87 0.00 HML -0.17 1.81 0.00

RMW 0.30 1.50 0.70 RMW 0.14 1.50 0.37

MOM -0.39 6.35 0.00 0.15 0.90 0.16 MOM 0.44 3.09 0.17 0.49 1.04 0.47

2005 to 2010 2010 to 2016

In-sample test (2010-2016) In-sample test (2010-2016)

statistics factor-investing portfolios

1995 to 2010 2000 to 2010

statistics factor-investing portfolios statistics factor-investing portfolios

1980 to 2010 1990 to 2010

In-sample test (2010-2016)

1963 to 2010 1970 to 2010

In-sample test (2010-2016)

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Table5

Correlations between each factor during different time periods

MKT SMB HML RMW CMA MOM MKT SMB HML RMW CMA MOM MKT 1.00 0.27 -0.31 -0.19 -0.42 -0.12 MKT 1.00 0.24 -0.33 -0.21 -0.41 -0.15 SMB 0.27 1.00 -0.12 -0.35 -0.12 -0.02 SMB 0.24 1.00 -0.13 -0.39 -0.08 -0.06 HML -0.31 -0.12 1.00 0.10 0.70 -0.16 HML -0.33 -0.13 1.00 0.16 0.70 -0.15 RMW -0.19 -0.35 0.10 1.00 -0.09 0.09 RMW -0.21 -0.39 0.16 1.00 -0.02 0.08 CMA -0.42 -0.12 0.70 -0.09 1.00 0.00 CMA -0.41 -0.08 0.70 -0.02 1.00 0.04 MOM -0.12 -0.02 -0.16 0.09 0.00 1.00 MOM -0.15 -0.06 -0.15 0.08 0.04 1.00 MKT SMB HML RMW CMA MOM MKT SMB HML RMW CMA MOM MKT 1.00 0.18 -0.37 -0.27 -0.45 -0.14 MKT 1.00 0.20 -0.27 -0.41 -0.44 -0.27 SMB 0.18 1.00 -0.23 -0.44 -0.09 0.02 SMB 0.20 1.00 -0.21 -0.50 -0.07 0.02 HML -0.37 -0.23 1.00 0.30 0.70 -0.18 HML -0.27 -0.21 1.00 0.48 0.67 -0.14 RMW -0.27 -0.44 0.30 1.00 0.09 0.10 RMW -0.41 -0.50 0.48 1.00 0.20 0.05 CMA -0.45 -0.09 0.70 0.09 1.00 0.01 CMA -0.44 -0.07 0.67 0.20 1.00 0.08 MOM -0.14 0.02 -0.18 0.10 0.01 1.00 MOM -0.27 0.02 -0.14 0.05 0.08 1.00 MKT SMB HML RMW CMA MOM MKT SMB HML RMW CMA MOM MKT 1.00 0.19 -0.26 -0.47 -0.42 -0.30 MKT 1.00 0.31 -0.09 -0.66 -0.26 -0.51 SMB 0.19 1.00 -0.22 -0.53 -0.05 0.05 SMB 0.31 1.00 0.14 -0.32 0.05 -0.18 HML -0.26 -0.22 1.00 0.57 0.65 -0.15 HML -0.09 0.14 1.00 0.45 0.64 0.11 RMW -0.47 -0.53 0.57 1.00 0.27 0.06 RMW -0.66 -0.32 0.45 1.00 0.34 0.43 CMA -0.42 -0.05 0.65 0.27 1.00 0.07 CMA -0.26 0.05 0.64 0.34 1.00 0.24 MOM -0.30 0.05 -0.15 0.06 0.07 1.00 MOM -0.51 -0.18 0.11 0.43 0.24 1.00 MKT SMB HML RMW CMA MOM MKT SMB HML RMW CMA MOM MKT 1.00 0.46 0.48 -0.57 -0.11 -0.44 MKT 1.00 0.43 -0.02 -0.38 0.02 -0.23 SMB 0.46 1.00 0.43 -0.38 0.11 -0.27 SMB 0.43 1.00 0.00 -0.50 -0.01 -0.03 HML 0.48 0.43 1.00 -0.22 0.43 -0.42 HML -0.02 0.00 1.00 -0.16 0.68 -0.41 RMW -0.57 -0.38 -0.22 1.00 0.10 0.31 RMW -0.38 -0.50 -0.16 1.00 0.01 0.09 CMA -0.11 0.11 0.43 0.10 1.00 -0.04 CMA 0.02 -0.01 0.68 0.01 1.00 -0.13 MOM -0.44 -0.27 -0.42 0.31 -0.04 1.00 MOM -0.23 -0.03 -0.41 0.09 -0.13 1.00 2005 to 2010 2010 to 2016 1963 to 2010 1970 to 2010 1980 to 2010 1990 to 2010 1995 to 2010 2000 to 2010

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Construction improvements

To improve the in-sample-test Sharpe ratio of factor-investing portfolios derived from different periods, we need to modify the factors allocation by introducing certain restrictions. Here we explore three ways to optimize the construction of factor-investing portfolios, namely setting minimum years of historical data used to obtain the factors allocation, updating standard deviation with Exponential Weighted Moving Average (EWMA) model and applying a minimum 10% allocation to each factor.

Table4 reveals that factors allocation derived from periods with more than 15 years can deliver higher Sharpe ratios than those derived from periods with less than 15 years. Hence we set a minimum use of 15-year data to derive factors allocation.

When calculating factors allocation through maximizing the Sharpe ratio, we assume the standard deviation as a constant. However, the standard deviation has been changing all the time and thus we introduce EWMA model to update the standard deviation so that the most recent variance would have more weight in the process of deducing the standard deviation. According to the procedure described in the methodology section and applying formula (1) and (2), we can achieve an updated forecasted standard deviation. By applying these new standard deviations, we can generate table5.

In general, table6 seems to display worse results compared with table4. S6 and S7 are not improved at all while S1 to S5 become even lower. However, this abnormal phenomenon can be explained. We have discussed previously that patterns of return, standard deviation and correlation matrix in period6 and period7 are quite opposite to those in period8 and thus if the factors allocation derived from period6 and period7 are used to do in-sample test using data in period8, the Sharpe ratio will be well below S8. Since EWMA model puts more weight on the most recent data to update standard deviation, especially data in period6 and period7, the new factors allocation would tilt towards those derived from data in period6 and period7. Therefore, the in-sample-test Sharpe ratio would be moving towards S6 and S7, and thus become lower. In our case, updated standard deviation is not applicable.

Moreover, we notice that in the first five periods of table4 each factor is allocated certain weight while in the last three periods at least two factors are not allocated any weight, specifically there is no allocation of HML and MOM in period6, no allocation of HML, CMA and MOM in period7 and no allocation of SMB and HML in period8. This sort of allocation may achieve the highest Sharpe ratio when deriving the factors allocation but at the same time takes great risks during in-sample test due to low level of diversification. Here we set a minimum 10% allocation to each factor in order to lower the risk. Considering the high correlation between CMA and HML, reaching 0.7, we set a minimum 10% criteria for the sum of allocations in these two factors. Hence, 50% of weight are fixed while the other 50% are flexible.

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Table6

Construction of factor-investing portfolios using historical data in different time periods, with standard deviation adjustment using EWMA.

mean return sigma allocation mean return sigma Sharpe ratio mean return sigma allocation mean return sigma Sharpe ratio

MKT 0.41 5.73 0.07 0.32 0.72 0.45 MKT 0.48 5.72 0.08 0.35 0.74 0.47

SMB 0.29 3.05 0.10 SMB 0.25 3.04 0.09

HML 0.41 3.39 0.00 HML 0.40 3.38 0.00

RMW 0.26 1.43 0.36 RMW 0.31 1.40 0.39

CMA 0.33 1.43 0.44 mean return sigma Sharpe ratio CMA 0.35 1.46 0.40 mean return sigma Sharpe ratio

MOM 0.72 5.95 0.03 0.20 0.77 0.26 MOM 0.70 6.07 0.03 0.21 0.76 0.27

MKT 0.52 5.72 0.08 0.39 0.71 0.54 MKT 0.44 5.60 0.09 0.38 0.66 0.57

SMB 0.16 3.06 0.10 SMB 0.25 3.09 0.11

HML 0.36 3.32 0.00 HML 0.35 3.27 0.00

RMW 0.42 1.37 0.45 RMW 0.41 1.29 0.52

MOM 0.64 6.13 0.02 0.21 0.76 0.28 MOM 0.59 5.59 0.03 0.22 0.76 0.29

MKT 0.36 5.57 0.09 0.40 0.62 0.63 MKT -0.16 5.47 0.09 0.58 0.75 0.77

SMB 0.30 3.09 0.12 SMB 0.57 3.02 0.10

HML 0.33 3.28 0.00 HML 0.75 3.28 0.00

RMW 0.43 1.21 0.57 RMW 0.69 1.31 0.67

MOM 0.48 4.91 0.03 0.21 0.79 0.27 MOM -0.01 4.11 0.00 0.22 0.90 0.24

MKT -0.10 5.49 0.05 0.27 0.88 0.31 MKT 1.21 3.73 0.30 0.49 1.10 0.45

SMB 0.25 2.53 0.21 SMB -0.06 2.18 0.00

HML 0.07 3.29 0.00 HML -0.17 1.81 0.00

RMW 0.30 1.35 0.74 RMW 0.14 1.50 0.40

MOM -0.39 3.23 0.00 0.15 0.94 0.16 MOM 0.44 4.28 0.11 0.49 1.07 0.46

1963 to 2010 1970 to 2010

1980 to 2010 1990 to 2010

1995 to 2010 2000 to 2010

2005 to 2010 2010 to 2016