Factor Factory Series 2: Anomaly Report
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1. Introduction
In the stock market, one company’s stock yields higher returns than the other’s. If the common characteristics of an overachieving entity can be identified, it could develop a theory that could explain complex stock market movements on this basis. In addition, investing in companies with good characteristics can also yield stable returns. The beginning of the study on the characteristics associated with time-series and cross-sectional variation in stock return is the Capital Asset Pricing Model (CAPM) proposed by Sharp (1964). The CAPM explains the relationship between the systematic risk and the expected return of an individual asset, explaining the phenomenon that an asset with a high systematic risk has a higher return than an asset that is not. Starting with this, Banz (1981), Statman (1980) and Rosenberg, Reid, and Lanstein (1985), Fama and French (1992, 1993, 2015), Novy Marx (2013), which explains the relationship between the firm’s unique characteristics and stock returns has been the most active subjects of study in the financial field for about 50 years.
A number of prior studies have reported firm-specific characteristics related to cross-sectional variations of returns on common stock. Banz (1981) found the size effect in which firms’ size are closely associated with the cross-sectional variation of the return on stocks. Stattman (1980) and Rosenberg, Reid, and Lanstein (1985) showed that the book-to-market ratio(B/M, book value divided by the market price of the firm) was related to the stock returns. Fama and French (1992) demonstrated that among the various unique firm-specific characteristics, the size(ME) and the book-to-market ratio(B/M) are the main components that explain the cross-sectional variation of the return on the stock through Fama-Macbeth(1973) regression. Fama-French(1993) formed two factor portfolios, SMB and HML, based on the size and book-to-market ratio of firms which were found to be closely related to the cross-section of stock returns in Fama-French(1992). The SMB and HML explained most of the time-series and cross-sectional variation in the excess return of 25 portfolios. Carhart (1997) showed that the momentum effect exists on mutual funds’ performance. French (2015) proposed two additional factor portfolios RMW, CMA, to the Fama-French three-factor model. The Fama-French five-factor model improved the explanatory power of the cross-sectional variation of the stock returns.
2. Methodology
2.1. Data Configuration
First, common stocks listed on NYSE and NASDAQ were used only to construct a portfolio. ClassA filters were applied and data from June 30, 1990 to June 30, 2020 were used. To eliminate the survivorship bias, all the delisted stocks were included. In addition, all items from the financial statement were delayed for six months to eliminate look-ahead bias due to the disclosure time difference. Therefore, data published in December of year t are accessible after June of year t+1. In the case of small-cap stocks with small market capitalization small-cap transactions, high-slippage not only makes it practically difficult to invest but also creates a huge gap between backtest performance and actual return on investment. Therefore, stocks that were on the top 30% of the market capitalization of the previous month before rebalancing based on the total number of stocks listed in NYSE and NASDAQ were used for the portfolio composition.
After applying the market cap filter, only those stocks whose calculated factor scores are above the top 20% of NYSE criteria were included in the portfolio. The portfolio is rebalanced in June every year and is composed of a value-weight method. For the entire period, portfolios with fewer than 30 items included in the portfolio were excluded. FactorGym’s depth constraint is set between 2 and 15 so that various depth factors can be explored. The analysis results related to the depth parameter are discussed in detail in Chapter 3.2. Additionally, a factor portfolio that has a time-series length of less than 120 months was excluded.
2.2 Evaluation Metric
Factor Factory includes an evaluation process to assess the explored factor in accordance with various constraints. The evaluation process includes several MetricFn such as Regression Alpha, RankIC, Shape, Sortino, MDD, etc., in order to assess the portfolio from various perspectives. In this paper, the Regression Alpha was used as the most important evaluation indicator. In the regression analysis, the independent variables were selected from the 5-factor asset-pricing model proposed by Fama-French (2015) and the QMJ factor proposed by Asness, Frazini, and Pedersen (2014), the BAB factor proposed by Frazzini (2013), and the MOM (large) factor provided by AQR.
The following figure shows the correlations coefficients between independent variables. The closer the color red, the higher the correlation coefficient. As shown in the figure, there is a high correlation in some of the above-mentioned independent variables. In this case, the problem is that multicollinearity makes it difficult to trust the estimated coefficients in the regression model. Therefore, the HML, RMW, and MOM (large) factors were finally excluded from the independent variables after verification processes such as the spanning test.
[Figure 01. Correlation Table of Independent Variables]
3. Result
3.1. Decreasing Alpha
The analysis was conducted on 1,851 factor portfolios among the sample 1,915 portfolios explored by the Factor Factory after removing deduplicated data. The main assessment of the factor portfolio was based on the Region Alpha (which is subsequently referred to as Alpha). Figure 03 shows the distribution of factor portfolio alpha estimated on a 10-year basis(Period 1(1990~1999), Period 2(2000~2009), Period 3(2010~2019)). As you can see in the figure, alpha distribution tends to be less distributed and higher in addition. The average for Period 1, Period 2, and Period 3 was 0.43%, 0.44%, and 0.13%, respectively, in accordance with the alpha shrinking trend in recent years. As the performance of the factor is being downgraded, it is becoming more difficult to achieve high returns by relying on some of the “super factors” that are as good as in the past.
[Figure 02. 10-year basis alpha distribution of all factors ]
3.2. Robustness of Alpha
Among the factors that the Factor Factory has found, the factor that has performed well in the past tends to maintain good performance afterwards. The analysis was performed by dividing the factor portfolio into sub-periods to verify the robustness of the factor performance. The following figure shows a scatterplot of the training and test period alpha on the x- and y-axes, respectively. The training period is from July 1991 to April 2015, and the test period is from May 2015 to April 2020. A red dot represents a portfolio that has performed better in the test period, while a blue dot represents a portfolio that has performed better in the training period. Due to the overall downward leveling of alpha, most factors have shown relatively poor performance in the test period. However, there is a distinct linear relationship between the performance of the training period and that of the test period, and the factor that has shown good performance during the training period can confirm the tendency to show good performance even during the test period. The RankIC calculated by the alpha of the training and test periods is 0.67, which shows the robust performance persistence of the factor portfolio.
[ Figure 03. Scatter plot of training(x) and test period(y) alpha ]
3.3 Factor Performance by Depth
The complexity of the portfolio explored through the Factor Factory can be seen through the depth index. Depth means the depth of the factor tree and is determined by the number of NormalizerNodes, OperatorNodes, and DataNodes that make up the factor tree and how they are combined. In general, the higher the depth, the greater the complexity. The following figure visualizes the tree structure of a relatively simple Factor Tree (A) to a complex Factor Tree (C) corresponding to Depth 7.
The Expression Tree of the Factor Factory is characterized by its very difficult interpretation. However, one interesting fact is that some of the anomalies that the Factor Factory found inductively are consistent with the results found in prior research. Factor A is a market anomaly that has performed well in the past by incorporating stocks with momentum effect, which subtracts from the past 12 months’ return to the last six months’ return, into the portfolio. In other words, the 7–12month return contributes more to the momentum effect than the latest six-month return. This is similar to the result of R. Novy-Marx (2012).
[ Figure 04. The structure of factor trees by its depth ]
In general, the complexity of the factors reported in the preceding study to generate an excess return is about 3 to 4 levels. Factors with a depth of 5 or higher have multi-factor-like characteristics and are difficult to interpret. And they are characterized by an exponential increase in the number of cases in which the factor can be constructed as the depth increases. The complexity of the factor that researchers can identify through deductive research has some limitations. However, unlike humans, the factor factory can explore a variety of factors because it efficiently explores countless numbers of cases by utilizing computational optimization and search algorithms.
As one of the common preconceptions, there is concern that a complex factor may be a case of over-fitting which results in good performance only for given datasets. However, in practice, complex factors not only achieve as good performance as simple factors, but also show no noticeable difference in indicators such as average alpha, variance, and RankIC. Thus, by utilizing the complex factors discovered by Factor Factory, it is able to expand the universe of selectable factor assets and maximize the dispersion effect.
[ Table 01. Factor portfolio performances by depth ]
3.4. Examples of Best Factors
Next, two of the factors(Factor F and Factor G) found through the factor factory were selected and basic analysis was performed. The portfolio selected stocks in the same way as mentioned in the Configuration chapter. Rebalancing was carried out in June every year and transaction costs arising from rebalancing were not taken into account. The S&P 500 index was used as a benchmark index. The following figure visualizes the tree structure of Factor F and G. The interpretation of the factor tree was omitted as it deviates from the main points covered in the paper.
[ Figure 05. Factor tree structure of Factor F and G ]
The table below shows the basic statistics of the performance of Factor F and G. The analysis was performed by separating the entire period into the training period (1992.01–2015.04) and the test period (2015.05–2020.04). Both Factor F and Factor G performed well against the benchmark in both sub-periods. The Sharp ratio, which measures the return on risk increased significantly, and the MDD, a risk indicator, also has been improved.
[ Table 02. Summary statistics of Factor F and G returns ]
The following figure illustrates the cumulative return on Factor F and G from January 1992 to May 2020 and the cumulative excess return relative to BM. The log scale was used for smooth comparison. The blue line represents the cumulative return of the factor portfolio, and the red line represents the cumulative return of the S&P 500 index used as a benchmark. The areas indicated in orange represent cumulative excess return relative to the BM. Both factors consistently outperformed the benchmark index, and their cumulative performance was also steadily upward.
[ Figure 06. Cumulative return of portfolio. Factor F(up), Factor G(down) ]
The performance of the above portfolios is more pronounced when performance is measured yearly. The following figure shows the annual return on the factor portfolio and benchmark index. The yellow and blue bars, respectively, refer to the annual return of the factor portfolio and benchmark index. Factor F outperformed the benchmark in 27 years of the total 28 years, while Factor G outperformed the benchmark index in all years. Factors F and G achieve excess return relative to benchmarks at 96.5% and 100%. Since Factors F and G were selected to have shown excellent performance during the training period (in-samples), and it is somewhat natural that they have performed well before 2015. However, it is noteworthy that the two factors have shown robust performance over benchmarks for all years since 2015.
[ Figure 07. Annual returns of factor portfolios. Factor F(above), Factor G(below) ]
Next, the performance of the factor portfolios was decomposed using the five-factor model proposed by Fama-French (2015). Factors explored by the Factor Factory have exposure to other risk factors together, so it is desirable to eliminate these effects for rigorous performance analysis. Regression analysis using the Fama-French 5 factor model provides insight into the portfolio’s exposure to each risk factor and whether there is a unique performance(alpha) that cannot be explained by these risk factors. In the regression analysis, Newey-West correction was applied to compensate for the heteroskedasticity and autocorrelation present in the dependent variables.
[ Eq 01. Fama-French(2015) 5-factor model ]
The table below shows the regression results for the two-factor portfolios. Both factors showed a high coefficient of determination in both sections of Panel A and Panel B. Factors F and G were found to have significant alpha after removing linear exposure to the five risk factors proposed by Fama-French(2015) during the training period. In particular, Factor F showed significant alpha up to the test period. Therefore, it can be seen that the excess return of Factor F during the test period is due to not only the appropriate exposure to other risk factors but also to the unique alpha that Factor F has.
All factor portfolios consist of a market capitalization weighted approach, with some of the NYSE and NASDAQ listed, excluding small-cap stocks. Therefore, it can be seen that all estimated beta has a value close to 1. In both factors, the exposure to the SMB, which is a size factor, changes from positive(+) to negative(-) values over time, so it is believed that the portion of the stock has shifted to large-cap stocks over the recent period. The regression coefficient h of HML, corresponding to the value factor, has a value of significant negative. Therefore, factor F and G are expected to be portfolios composed mainly of growth stocks. Factor F has a small positive exposure to the profit factor RMW and significant negative exposure to the CMA factor during the learning period, but it has become non-significant during the test period. Unlike Factor F, Factor G has significant negative exposure to the RMW factor during the learning period, and during the test period, it has become insignificant.
As such, exposure to risk factors not only varies from factor to factor but also changes dynamically, so it is not easy to leave only pure alpha by hedging them. However, it is possible to offset exposure to the risk factors of individual factors and to leave pure alpha by forming a portfolio of different factors. This is discussed in detail in the Chapter 3.6.
[ Table 03. Regression coefficients for Factor Portfolios ]
3.5. Best/Worst Factors Out of Sample Test
Next, a performance analysis was conducted on the best/worst factor of the training period(in-sample). The following figure illustrates the cumulative performance of the test period(out-of-sample) of the best factor with the highest alpha in the training period and the worst factor with the smallest alpha. The Best/Worst factors were selected for 300 each. The orange line, blue line and red line represent the cumulative performance of the Best factor, Worst factor and benchmark index (S&P 500), respectively. The cumulative performance was adjusted to log scale for smooth comparison. As shown in the figure, the best factor(orange) performed better than the benchmark index (red) and the worst factor (blue) during the test period. 97% of the best factors outperformed the BM. On the other hand, only 42.3% of the Worst Factor achieved excess return.
[ Figure 08. Best/Worst Portfolios of Train Period in Test Period ]
3.6 Monotonous Performance of Factors
In the process of factor verification, it is important to confirm the monotony of the factor scores and portfolio returns. A robust factor is monotonous in terms of portfolio returns organized according to factor score, and the long-short performance is also stable. The following figure illustrates the cumulative return on a quartile portfolio consisting of any factor. The return was expressed in log-scale. The Q1 and Q4 portfolios, respectively, refer to a portfolio consisting of the cross-sectional upper/lower 25% of the NYSE factor-score on the rebalancing month.On the graph, the Q1 portfolio showed steady excess return compared to other portfolios. On the other hand, Q4 showed the lowest performance.
[ Figure 09. Cumulative Return of Quantile Factor Portfolios ]
The figure below shows the number of times each portfolio showed the best return in a particular year compared to other quantile portfolios. The Q1 portfolio showed the best performance compared to other portfolios in the 17-year (56.7 percent). The long-short (Q1-Q4) portfolio also shows a steady cumulative performance.
[ Figure 10. Yearly Best Portfolio Counts(Up), Cumulative Return of Long-Short Portfolio(Down) ]
3.7. A Portfolio of Different Factors
The purpose of the factor factory is to compose an optimal subset to compose a security-level asset pricing model, but here, if you use the factors found through the factor factory for verification, we will construct a portfolio in various ways. The discussion of dynamic factor allocation is not covered in detail since it is outside the scope of this paper. In this paper, the analysis was conducted on an equally-weighted(EW) portfolio. The EW portfolio consisted of the best factors(factor with the largest alpha in the learning period). mentioned in Chapter 3.4
The following figure illustrates the cumulative performance over the test period of the EW portfolio. The blue and red lines represent the cumulative performance of the EW portfolio and benchmark index (S&P 500), respectively. The orange box area represents the cumulative excess return of the EW portfolio relative to BM index. The EW portfolio consistently outperformed the BM during the test period. In addition, the Sharp ratio, Hit Ratio and MDD indicators all improved significantly against the benchmark. The EW portfolio outperformed BM index for all years of the test period.
[Figure 11. Cumulative return of EW portfolio(above), Summary Table(center), annual return(below)]
3.8. Distribution of Factors After applying PCA
PCA(Principal Component Analysis) refers to a methodology for locating high-dimensional data on a low-dimensional basis, which maximizes dispersion. PCA is widely used for feature engineering and visualization as a feature that can reduce the dimensions of a given data while preserving the maximum amount of explanatory power for the data.
The figure below is a three-dimensional graph of the application of PCA to the excess return of 1,851 factors found by Factor Factory. The data were used from July 2000 to April 2020 when returns of all factor portfolios exist. The best factors represented by orange dots mean 300 factors with the highest alpha in the training period, and the Worst factors marked by blue dots represent 300 factors with the smallest alpha. The graph shows the best/worst factors in the training period creating clusters not only during the training period but also during the test period.
[Figure 12. excess return of facters(PCA). Training Period(up), Test Period(down)]
4. Conclusion and Limitations
Factor Factory can easily and quickly find a factor that corresponds to the desired conditions. It is a great advantage of the Factor Factory that you can choose the various variables used to construct the factor, such as the stock universe, data period, and portfolio composition methodology, with a simple configuration modification. In addition, by adding a new MetricFn, the explored factor can be evaluated from various perspectives. Although this paper used the regression alpha, it is also possible to assess a portfolio from a risk perspective by using a variety of indicators (ex, Sharp, MDD) that take into account variance in portfolio returns.
Alpha found by Factor Factory is very robust and shows excellent performance. Even when the entire period was divided into sub-periods, the performance of the best and worst factors of the in-sample period was consistent in the out-of-sample period, and the robustness index (RankIC, etc.) was found to have high values. Furthermore, by forming a portfolio of individual factors found by the Factor Factory, it is possible to create a broad concept of factor portfolios (ex, Pure Alpha, Value Enforce).
However, the Factor Factory still needs improvement. This paper used fixed linear factor models as independent variables in regression analyses. In this case, because of the misspecification, the assessment bias may be present in the main evaluation index of this report, the estimated alpha. Minimizing these bias and estimation errors, increasing reliability and creating a more stringent factor evaluation process are the top priorities facing the Factor Factory.
In addition, the factor found by the Factorial Factory does not have an industrial neutralization option. Often, as a rebuke of factor investment, words like “factor investment is actually sector investment.” are mentioned. For example, the Low-Vol factor and the Value factor are the factors that invest in stocks with low return/residual volatility and low market value relative to book value, respectively. However, because these values are closely related to the industry in which the firms are included, the portion of investment in the portfolio is often excessively focused on specific industries. To solve these problems, it is called industrial neutralization by calculating the proportion of investments in stocks evenly by industry through intra-industry comparisons. Industrial neutralization proves the robustness of the factor and contributes greatly to the resolution of the unbalanced investment weight by industry in the composition of the multi-factor portfolio. Therefore, it aims to apply industry neutralization as a future task of Factorial Factory.
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