Multifactor rules-based portfolios portfolios

JENNIFER BENDER is a managing director at State Street Global Advisors in Boston, MA. jennifer_bender@ssga.com TAIE WANG is a vice president at State Street Global Advisors in Hong Kong. taie_wang@ssga.com Can the Whole Be More Than the Sum of the Parts? Bottom-Up versus Top-Down Multifactor Portfolio Construction JENNIFER BENDER AND TAIE WANG Multifactor rules-based portfolios portfolios constructed to capture multiple factor exposures have become increasingly popular in recent years. The main rationale for combining multiple factors is that it enables investors to achieve potential diversification benefits. Historically, factors such as value, size, quality, low volatility, and momentum have earned a long-run premium over the market, but all factors have experienced periods of underperformance during certain market environments. However, they have not all experienced periods of underperformance at the same time. Some factor pairs such as value and momentum naturally diversify each other based on their definitions. When a stock s price rises, it simultaneously becomes more momentumlike (as long as its price is rising faster than others) and less value-like (because value is typically defined as book-to-price, earningsto-price, or some other fundamental-to-price ratio). Other factors tend to diversify each other historically based on cycles of investment sentiment; quality and low volatility stocks are favored by investors in times of uncertainty. Given the interest in multifactor portfolios, there has been much discussion about the best way to build them. One often-asked question is whether it is better to combine single-factor portfolios or to build a multifactor portfolio from the security level. The latter is a better approach theoretically because each security s portfolio weight will depend on how well it ranks on multiple factors simultaneously. The former approach, combining single-factor portfolios, may miss the effects of cross-sectional interaction between the factors at the security level. But it may be that the differences between the two approaches are small and the combination approach has benefits such as more transparent performance attribution and greater f lexibility. In this article, we explore this issue and discuss the implications of both approaches. Our conclusion is that there are, in fact, beneficial interaction effects among factors that are not captured by the combination approach. Both intuition and empirical evidence favor employing the bottom-up multifactor approach. THE RISE OF RULES-BASED INVESTING The earliest alternative indexing strategies (also known as advanced beta or smart beta) applied an alternative weighting scheme to market-capitalization weighting. Examples include GDP-weighted portfolios in the 980s, equal-weighted portfolios in the 990s, and, more recently, fundamental-weighted portfolios in the 2000s. Proponents of these strategies were typically critical of cap-weighting Special Issue 206 The Journal of Portfolio Management 39

and argued that these alternative weighting schemes were superior because they were either more representative of investment value or more diversified. Since 2008, an alternative method of constructing such portfolios has emerged one that focuses on capturing factor premiums more explicitly. This approach focuses on the pure factors that the portfolios are exposed to and derive their returns from. The pure factors, beginning with the multifactor models of Ross [976], are those that have been widely researched in the academic literature, have strong theoretical foundations, and have exhibited persistence over multiple decades. Viewing factor indexation as a means of capturing pure factors is consistent with the way academics have viewed factors; it was most widely popularized by Fama and French s seminal three-factor model and extended over the years by countless others. The most widely discussed factors include the original Fama French Carhart factors value, (low) size, momentum plus a handful of additional factors that have received moderate treatment (low) volatility, quality, liquidity, and yield. Single-Factor Portfolio Construction A number of techniques have been used to build single-factor portfolios. Equal weighting, GDP weighting, fundamental indexation, and its close companion wealth weighting were compelling; they were intuitive and did not employ a black box algorithm such as optimization, which meant security weights could be directly tied to the securities observable characteristics. Subsequent factorbased approaches could also be constructed in a similar fashion; most employ a set of rules that specify security weights as a function of the factor characteristics. Consider the following examples of first- generation rules-based portfolios equal weighting, fundamental indexation, and risk weighting: Equal weights: wi = () N Fi Fundamental indexationweights : w i = N (2) Fi 2 σ i Risk weights: w i = N (3) 2 i σi In Equations 3, w i is the weight of stock i in the portfolio, N is the number of stocks in the universe, F i is the fundamental value of stock i (e.g., book value, i earnings, etc.), and σ i 2 is the variance of stock i. All the methods are relatively straightforward, linking the desired attribute (e.g., volatility or book value) to the securities weights. Our preferred approach has been to adopt a benchmark-relative framework in which multipliers are applied to market-cap weights. For tilted factor portfolio weights, w = w γ i i, mktcap i (4) where γ i is a scalar applied to the market-cap weight of each stock. The scalar γ i can be specified in many ways. It can be the result of a mapping function based on the security s factor characteristics. It can be nonlinear or linear cross-sectionally, and it can be unique for each security or unique for groups of securities. Furthermore, we can screen out certain securities by setting the multiplier equal to zero. We have favored this approach because it allows flexibility and coherency for building portfolios across different factors and at different levels of tracking error and concentration for benchmark-sensitive investors (see Bender and Wang [205]). MULTI-FACTOR PORTFOLIO CONSTRUCTION Combining multiple factors with strong investment merit can produce benefits from the potential diversification among the factors. Historically, factors such as value, size, quality, low volatility, and momentum have exhibited substantial diversification benefits over short horizons such as a week or a month, but also over longer, multiyear periods. Correlations between excess returns are generally below 0.5 and sometimes negative (see Bender, Brandhorst, and Wang [204] for further discussion). Multifactor portfolios can be constructed in two main ways. The simplest way is to combine single-factor portfolios into one portfolio. Within each factor portfolio, security weights are determined according to a specific methodology (ideally, one that is consistent across factors). Single-factor portfolios are blended by assigning weights to individual portfolios. This approach is analogous to building blocks, and its benefits include clear performance attribution and f lexibility in reallocation across factors. For the remainder of this article, we refer to this as the combination approach. 40 Bottom-Up versus Top-Down Multifactor Portfolio Construction special Issue 206

The second way to build the portfolios is from the security level up, bottom up, by incorporating all the factor characteristics simultaneously. Security weights are assigned based on the security s combined characteristics. Asness [997], for example, highlighted the effects of interaction between value and momentum. The top-down combination approach may miss these interaction effects. In our example, we construct portfolios by multiplying the starting weights (e.g., cap weights) with a multiplier, as in Equation 4. Note that if we were employing optimization, these two approaches would not yield the same results. (The typical objective function is a quadratic function; because it is nonlinear, a linear combination of factor portfolios cannot be equivalent.) We focus on non-optimized portfolios, and therefore it is less clear if there are differences and how large those differences are in actuality. A STYLIZED EXAMPLE Let us begin with a stylized example using 0 stocks to construct a three-factor portfolio. We consider three representative factors: dividend yield, book-to-price ratio, and return on assets. To build the portfolio, we rank the securities based on each security s factor characteristics and assign multipliers based on their ranks. The multipliers can be assigned to any set of starting weights; market-cap weights and equal weights are our candidates. The new weights (starting weight x multiplier) are then rescaled to sum to 00%. What are the conditions under which the bottom-up and combination approaches will yield the same results? The starting weight is equal weight. The sums of the multipliers for each single-factor portfolio and for the bottom-up factor portfolio are identical. We show this as follows: If there are two-factor portfolios, the resulting weights for a linear combination of two-factor portfolios can be calculated as W M S L = λ + ( ) M S L Factor Factor 2 S S (5) where m m2 Multipliers for factor : M = ; Multipliers. l m n for factor 2: l2 L =. l n s Starting weights: s2 S = ; Final weights:. w s n w2 W =. w n Weight in factor : λ Weight in factor 2: λ The symbol denotes element-wise product. The resulting weight for the bottom-up, twofactor portfolio is λ + ( )L = S W λ + ( )L S (6) For Equations 5 and 6 to be identical, the necessary conditions are as follows: s = s 2 =... s n, (7) l li. (8) The derivation appears in Appendix A. To illustrate this equivalence, we built the following two portfolios (as shown in Exhibit ). Combination Portfolio: In the two left-most panels, we sorted the securities based on raw metrics one factor at a time and ranked them. We assigned multipliers identical to the stock s rank (e.g., a stock ranked fifth has a multiplier of 5). The multipliers were applied to equal weights (e.g., Special Issue 206 The Journal of Portfolio Management 4

E x h i b i t Scenario in Which the Combination Portfolio Is Equivalent to the Bottom-Up Portfolio each security s weight is 0%). We then rescaled the weights to sum to 00%. Next, we blended the three resulting portfolios into one using equal weights that is, we averaged the security weights across the three portfolios. Bottom-Up Portfolio: In the right-most panel, we ranked the securities for each factor, as we did for the combination portfolio. This time, we computed an average (equally weighted) rank across the three factors and multiplied this combined rank by security equal weights (0%). We then rescaled the weights to sum to 00%. The bottom-up multifactor portfolio is identical to the combination portfolio shown in Exhibit. Note that the multiplier does not capture information on the cross-sectional distribution (e.g., each security receives one unique multiplier, no two multipliers are the same; the sums of the multipliers are identical), and the starting weights are equal weights. We next replaced the starting weights with market-cap weights instead of equal weights. As summarized in Exhibit 2, the correlation between the weights of the combination and bottom-up portfolios is exactly, but the actual weights of the two portfolios are no longer identical. Applying the multipliers to cap weights creates differences due to the way the excess weight (i.e., the difference between the sum of the weights and the 00% target weight) is allocated. If equal weights are used as starting points, this excess weight is uniformly distributed across the 0 securities. But when cap weights are used, the excess weight is prorated based on the market-cap weights; larger-cap, higher-ranked securities receive more of that excess weight. Because the two approaches distribute the excess weight differently, the final combination and bottom-up portfolios are not the same. That said, we interpret the correlation of to mean that the essence of the portfolios is comparable. The ordering of the security weights is identical in the two portfolios. A SECOND STYLIZED EXAMPLE What would happen if we used scores and not ranks? To calculate scores for each factor, we standardize the raw metrics for each security by subtracting the mean and dividing by the standard deviation across securities. Thus, 42 Bottom-Up versus Top-Down Multifactor Portfolio Construction special Issue 206

E XHIBIT 2 Summary of Rank-Based Approach xi x (9) σ where x i is the security s raw characteristic, x is the average across all securities, and σ is the standard deviation of the raw characteristics across all securities. Scores preserve the distributional characteristics of each factor in a way that ranks do not. If a security has an extremely high price-to-book relative to the other securities, that extremeness will be captured by the score. Once we assign scores to the securities, how do we determine the multiplier? Because scores can be negative (e.g., 5 to +5), we cannot just set multipliers equal to scores, as we did for the ranks. 2 Keeping it simple, we assigned ranks based on scores and set the multipliers equal to the ranks. Note that ranking based on scores is equivalent to ranking based on raw metrics in the combination approach. However, in the bottom-up approach, average rank is not the same as a rank based on average score. This is a critical point. In this example, the multiplier applied to each security for the bottom-up portfolio is no longer just the average of the multipliers for the single-factor portfolios. That is, we changed the way we specify multipliers; Equation 5 still holds, but Equation 6 does not. Exhibit 3 summarizes the difference between the rank-based approach and the score-based approach. When we use scores instead of ranks, the two portfolios weights are meaningfully different, and the correlations are not whether we apply the multipliers to equal weights or cap weights. Why does the score-based approach create a larger difference between combination and bottom-up portfolios? In this example, the score-based approach preserves the distributional differences in a way that the rank-based approach does not. Does employing a rank-based method rule out the ability to capture these interaction effects between factors? In this stylized example, it did, because of the way we had specified the multipliers (i.e., one unique multiplier for each unique rank). However, there are portfolio construction methods that use ranks and do capture interaction effects (see Brandhorst [203]). GLOBAL PORTFOLIO SIMULATIONS To test the impact of combination versus bottom-up portfolio construction with an actual investment universe, we consider a four-factor portfolio capturing value, low volatility, quality, and momentum (as commonly defined in existing literature). 3 The two multifactor portfolios were constructed as follows: Combination Portfolio: For the combination portfolio, we created single-factor portfolios for the four factors: value, low volatility, quality, and momentum. First, we calculated the security scores for each factor and sorted the securities based on those scores. Second, we grouped the securities into 20 subportfolios, in which each subportfolio held 5% of market-cap weight. 4 We applied a fixed set of multipliers (linearly interpolated between 0.05 and.95 in increments of 0.0) to the subportfolios; the multiplier was applied to the market-cap weight of the security, depending on which subportfolio it fell in. The multipliers are shown in Exhibit 4. Finally, we rescaled the weights so that they summed to 00%. The combination portfolio is an equally weighted average of the four single-factor portfolios. Portfolios are rebalanced monthly. Bottom-Up Portfolio: First, we assigned scores to securities for each factor. Second, we averaged the scores (equally weighting the factors). Third, we grouped the securities into 20 subportfolios, Special Issue 206 The Journal of Portfolio Management 43

E x h i b i t 3 Bottom-Up vs. Combination Method: Rank vs. Score-Based Approach, December 3, 204 (0-stock example) E x h i b i t 4 Multipliers Assigned to Subportfolios in Simulations Notes: The starting weight is 5% for each subportfolio. Because each subportfolio holds 5% of market cap, this is equivalent to holding market-cap weights. each holding 5% of market cap, based on their average scores. We then applied the same fixed set of multipliers, depending on the subportfolio the security fell in. Finally, we rescaled the weights so that they summed to 00%. The factor definitions, universe, and rebalancing frequency are the same as previously. Exhibit 5 shows the results of the simulations. The bottom-up portfolio returns are higher than any of the underlying component factor returns and higher than the combinations. The difference is not insignificant a spread of 86 basis points. Moreover, the volatility of the bottom-up portfolio is significantly lower, and risk-adjusted return increases from 0.73 in the combination portfolio to 0.84 in the bottom-up approach. The differences between the two portfolios should not come as a surprise to us. Consider the case in which we group stocks into quartiles for value, momentum, 44 Bottom-Up versus Top-Down Multifactor Portfolio Construction special Issue 206

E XHIBIT 5 Combination vs. Bottom-Up Approach, Four-Factor Portfolios, January 993 March 205 (gross USD returns) Notes: Average historical turnover for the portfolios ranges from 30% to 70% (one-way annual) the lowest for low volatility and the highest for momentum. The bottom-up portfolio has exhibited 46% average annual one-way turnover for the period shown. Turnover in smart beta strategies is largely driven by the rebalancing frequency. Here, our choice to deploy quarterly rebalancing (to incorporate momentum, which requires the higher rebalancing frequency to be effective) causes turnover to be higher than the 5% 25% range typically seen in smart beta strategies. Semi-annual and annual rebalancing frequencies for non-momentum factors is more the norm. and quality for a global developed market universe. We assigned scores based on the quartiles, 5 and then we summed the scores for two pairs: value quality and value momentum. 6 Stocks with a score of zero rank the worst on both factors simultaneously; stocks with a score of 6 rank the best on both factors simultaneously. It is apparent in Exhibit 6 that the distribution for value quality is quite different from value momentum; for example, a greater percentage of stocks rank the highest on both value and momentum than on value and quality. These are the interaction effects captured by the bottom-up approach. In our backtest, has the bottom-up approach consistently produced better performance over the combination approach in all periods? Exhibit 7 shows the 36-month excess rolling returns for the two portfolios in Exhibit 5, as well as the rolling excess difference in returns. The bottom-up approach underperformend only during a short period in the early 200s. THE IMPACT OF STOCK SCREENING OR SELECTION How would the results change if we were to remove, or screen out, a subset of securities from the starting universe? Stock screening (also sometime referred to as stock selection) is equivalent to assigning a multiplier of zero in our framework. In other words, the bottom-ranked securities are not held but are merely E XHIBIT 6 Interaction Effects Captured in Distribution of Aggregate Scores, May 205 underweighted. Intuitively, we would not expect the impact of stock screening to change our hypothesis that the interaction effects captured by bottom-up techniques may have a meaningful impact on the final multifactor portfolio. To test this, we repeat the simulations in the previous section, but this time we employ stock screening. The only change is to remove the bottom 5 subportfolios; the same multipliers shown in Exhibit 4 are applied to the top 5 subportfolios. The results, shown in Exhibit 8, confirm our expectation that the bottom-up approach captures interaction effects in a way that the combination does not, even when stock screening is employed. Special Issue 206 The Journal of Portfolio Management 45

E x h i b i t 7 Time Variation in the Performance of the Bottom-Up vs. Combination Approaches There are in fact several ways to introduce stock screening in bottom-up multifactor portfolio construction. To understand what occurs under different forms of screening, imagine stocks arrayed along two factor dimensions, as shown in Exhibit 9. In our bottom-up simulations, we ranked stocks along each dimension and then combined the two dimensions while simultaneously removing the bottom-ranked stocks. This is shown in Exhibit 9, Diagram A (for illustrative purposes, we removed the bottom half, rather than the bottom quarter, in the simulations). In the combination approach, we ranked stocks along each dimension, creating one factor portfolio at a time by deleting the lowest-ranked stocks and then combining the two-factor portfolios. This is shown in Diagram B of Exhibit 9. The third approach, shown in Diagram C, is to take the intersection of the two factors. This is another valid approach to creating a bottom-up portfolio that accounts for the interaction effects between factors. INTERACTION EFFECTS FOR DIFFERENT FACTOR COMBINATIONS In this last section, we drill down into the interaction effects by pair. We evaluate the following six pairs: momentum quality momentum low volatility quality low volatility value momentum value quality value low volatility 46 Bottom-Up versus Top-Down Multifactor Portfolio Construction special Issue 206

E XHIBIT 8 Combination vs. Bottom-Up Approach: Four-Factor Portfolios, January 993 March 205 (gross USD returns) E XHIBIT 9 The Impact of Screening Exhibit 0 first plots joint distributions, similar to Exhibit 6. We grouped stocks into quartiles for value, momentum, and quality. We assigned scores based on the quartiles and summed the scores for each pair. As before, stocks with a score of zero rank the worst on both factors simultaneously; stocks with a score of 6 rank the best on both factors simultaneously. There are clear distributional differences between the pairs the most distinct distributions being value quality and value momentum. Note that these two factors also have the lowest correlations among all the factors. We repeated the simulations shown in the previous section, but this time for each pair at a time. Based on the distributions shown in Exhibit 0, we expect value quality and value momentum to show the largest performance differences between combination and bottom-up approaches. Exhibit shows the backtested performance of the pairs, and it corroborates what we expect based on the joint distributions. Moreover, there is surprising consistency across the pairs; the bottom-up approach universally outperforms the combination approach. Special Issue 206 The Journal of Portfolio Management 47

E x h i b i t 0 Joint Distributions for Factor Characteristics, January 993 March 205 (gross USD returns) 48 Bottom-Up versus Top-Down Multifactor Portfolio Construction special Issue 206

E XHIBIT Combination vs. Bottom-Up Differences by Factor Pair CONCLUSION Indexed-factor portfolios have emerged in recent years as an alternative for investors dissatisfied with market-cap weighting or as an explicit way to achieve exposure to well-known factors that have been shown to drive stock returns. These portfolios are passively implemented and thus retain the same benefits as traditional passive investing transparency, implementation efficiency, and low costs. These innovations are changing the investment landscape, which until recently was characterized by traditional passive investing and active management. Portfolios that utilize multiple factors have emerged as a way of accessing the diversification opportunities inherent across factors. Multifactor portfolios can be constructed either by combining individual singlefactor portfolios or by creating bottom-up portfolios in which security weights are a function of multiple factors simultaneously. We evaluate the two approaches in this article and show that a bottom-up approach will produce different, and sometimes superior, results than will a combination of individual single-factor portfolios. This is because bottom-up approaches capture nonlinear cross-sectional interaction effects between factors that combination approaches do not. We evaluate wellknown factors such as value, quality, low volatility, and momentum and find that the bottom-up approach results in higher excess returns over the long run. We further find that there are differences in the joint distributional characteristics of various pairs. The most distinct differences arise for the value quality and value momentum pairings where the difference between bottom-up and combination portfolios is the starkest. A PPENDIX A EQUIVALENCE OF BOTTOM-UP AND COMBINATION PORTFOLIOS The combination of a two-factor portfolio is M S L S W =λ + ( ) M S L S Factor Factor 2 Note that denotes element-wise product. The bottom-up two-factor portfolio is λ + ( )L = S W λ + ( )L S (A-) (A-2) The following two conditions that must be satisfied for these to be the same: s = s2 =... s n (A-3) l li (A-4) Here is the proof: If s = s 2 = s n and it is also true that i ll, then M S = L S, ( ) λ ( ) λ + ) S = λ S+ L S = M S = L S (A-5) We substitute Equation A-5 in Equation A- as follows: Special Issue 206 The Journal of Portfolio Management 49

M S λ ( ) + λ L S =λ M S M S L S λ M + λ L S Factor Factor 2 ( ) + λ ( ) ( ) ( ) = λ M S + λ L S λ M + λ L S ( ) ( ) λ + λ = M L S λ M + λ L S ( ) LS ( ) M S [ ] λ + λ M S = λ M ( ) + ( λ ) L S M S M S M S Factor Factor 2 λ M + λ L S λ M + ( λ) L S ( ) = λ M + λ L S M S M S (A-6) (A-7) (A-8) (A-9) It should be noted that the condition M S = L S is all that is required for the two equations to be equivalent. Thus, if Equation A-3 were not to hold, there could still be a set of multipliers such that M S = L S; likewise, if Equation A-4 were not to hold, there could still be a set of starting weights such that M S = L S. ENDNOTES The authors would like to thank Xiaole Sun for her many contributions to this research and Scott Conlon and Ana Harris for their many insights. We are also grateful to Marc Reinganum for providing the idea behind the article s title. We note that this reweighting effect is an artifact of the multiplier-based framework. By determining weights in this way in Equation 4, we introduce this effect, which is not present in the equal-weighting or fundamental-weighting approaches. 2 Note that the issue of transforming scores to weights is a widespread one in heuristic rules-based portfolios. Index vendors have proposed various mapping functions to deal with this transformation, including linear and nonlinear functions. Probability-based functions have also been suggested. 3 Value measured as exponentially weighted five-year averages of earnings, cash flow, sales, dividend, and book value in the denominator and price in the numerator. The five price-fundamental ratios are equally weighted. Low volatility is measured as 60-month variance of returns. Quality is measured as return on assets, debt to equity, and five-year variability in earnings per share. These three measures are equally weighted. Momentum is measured as trailing 2-month return minus the last month s return. Size is measured as free-f loat market capitalization in U.S. dollars. 4 In the previous section, we applied a unique multiplier to each security. When we implement the multiplier-based approach to real portfolios, we apply multipliers to groupings, or subportfolios, of securities, rather than at the stock level (e.g., a unique multiplier for each security). Intuitively, this reduces the impact of security-specific noise, because our objective is to tilt in aggregate toward securities with the desired factor characteristics. (Factor effects are not generally monotonically increasing, i.e., for each incremental increase in book-to-price, there is not an incremental increase in return.) The number of subportfolios we decide to employ is relatively arbitrary because there is no theoretical relationship between the level of granularity and the level of noise. Our aim is to balance the objective of reducing noise with the desire to sufficiently differentiate the weighting scheme across securities, because the latter occurs as we reduce granularity. 5 The least desirable Quartile receives a score of zero; Quartile 2 receives a score of, etc. 6 The data are shown as of May 20, 205. Momentum is defined as previously discussed. Value here is approximated by annual price-to-book ratio, and quality is approximated by return on equity. REFERENCES Asness, C. The Interaction of Value and Momentum Strategies. Financial Analysts Journal, Vol. 53, No. 2 (997). Bender, J., E. Brandhorst, and T. Wang. The Latest Wave of Advanced Beta. The Journal of Index Investing, Vol. 5, No. (204), pp. 67-76. Bender, J., and T. Wang. Tilted and Anti-Tilted Portfolios: A Coherent Framework for Advanced Beta Portfolio Construction. The Journal of Index Investing, Vol. 6, No. (205), pp. 5-64. Brandhorst, E. A Methodology for Capturing Advanced Beta Factors. Capital Insights, State Street Global Advisors, 203. Ross, S.A. The Arbitrage Theory of Capital Asset Pricing. Journal of Economic Theory, 3 (976), pp. 34-360. To order reprints of this article, please contact Dewey Palmieri at dpalmieri@iijournals.com or 22-224-3675. 50 Bottom-Up versus Top-Down Multifactor Portfolio Construction special Issue 206