如何消除股票因素中不受欢迎的偏见

1、The quest for pure equity factor exposureHow to eliminate the unwanted biases in equity factors?In the Equity Risk Premia Strategies primer by Kolanovic (2014) various factor neutralization methods were discussed beta-neutral styles as well as sector/country normalization methods have been analyzed.

2、 We note that removing any potential overlap among styles calls for additional techniques.Pure equity factor is an equity factor portfolio that solely contains an exposure to one particular with no impact from the market, country sector exposures on the portfolioreturn.We propose and compare various

3、 methods for constructing pure equity factors. Namely, we put forward the factor-mimicking, minimum torsion and numerical optimization approaches and compare their performance and characteristics with those of the widely accepted ranking-based (unconstrained or factors.The difference between the pur

4、e factors and the unconstrained factors is most noticeable for the Quality and Sizefactors.The correlations among the pure factors are closer to zero, the pure factors are diversified, and the risk can be better Neutralizing additional factors is easy to achieve and we show an example with neutraliz

5、ing ratesexposure.We determine that the top-down and the bottom-up approaches for constructing multi-factor portfolios are almost equivalent in the context of pure equityfactors.While the multi-factor portfolio based on the pure factor approach strongly outperforms the unconstrained one, it has also

6、 recently faced some performanceheadwinds.Global Cross-Asset Risk Premia Strategy28 November 2019Global Quantitative and Derivatives StrategyDobromir Tzotchev, PhD AC(44-20) 7134-5331 HYPERLINK mailto:dobromir.tzotchev dobromir.tzotchevJ.P. Morgan Securities plcAda Lau(852) 2800-7618 HYPERLINK mailt

7、o:ada.lau ada.lauJ.P. Morgan Securities (Asia Pacific) Limited/J.P. Morgan Broking (Hong Kong) LimitedRahul Dalmia(44 20) 7134-5883 HYPERLINK mailto:rahul.dalmia rahul.dalmiaJ.P. Morgan Securities plcMarko Kolanovic, PhD(1-212) 622-3677 HYPERLINK mailto:marko.kolanovic marko.kolanovicJ.P. Morgan Sec

8、urities LLCDubravko Lakos-Bujas(1-212) 622-3601 HYPERLINK mailto:dubravko.lakos-bujas dubravko.lakos- HYPERLINK mailto:bujas bujasJ.P. Morgan Securities LLCKhuram Chaudhry(44-20) 7134-6297 HYPERLINK mailto:khuram.chaudhry khuram.chaudhryJ.P. Morgan Securities plcRobert Smith, PhD(852) 2800 8569 HYPE

9、RLINK mailto:robert.z.smith robert.z.smithJ.P. Morgan Securities (Asia Pacific) LimitedComparison between the pure and unconstrained multi-factor approachesPerformance statisticsBottom-up Numerical Optimization Position LimitAnnualizedUnconstrained/ Raw Multi- factor ModelSource: J.P. Morgan Quantit

10、ative and Derivatives StrategyReturn6.5%5.0%AnnualizedVolatility5.0%5.0%Sharpe1.290.99Maximum Drawdown9.2%13.9%Source: J.P. Morgan Quantitative and Derivatives StrategySee page 41 for analyst certification and important disclosures, including non-US analyst disclosures.J.P.Morgandoesandseekstodobusi

11、nesswithcompaniescoveredinitsresearchreports.Asaresult,investorsshouldbeawarethatmay a of the of as only a HYPERLINK / Table of Contents HYPERLINK l _bookmark0 Unconstrained (Raw)EquityFactors3 HYPERLINK l _bookmark1 FactorDefinitions3 HYPERLINK l _bookmark2 Construction of the Unconstrained (Raw)Eq

12、uityFactors3 HYPERLINK l _bookmark3 PureEquityFactors3 HYPERLINK l _bookmark4 Constructing PureEquityFactors4 HYPERLINK l _bookmark16 EmpiricalResults7 HYPERLINK l _bookmark19 Combined EquityFactorsPortfolios16 HYPERLINK l _bookmark20 Improving top-down andmulti-factorapproaches21 HYPERLINK l _bookm

13、ark25 References:34 HYPERLINK l _bookmark26 Appendix35 HYPERLINK l _bookmark27 Factor-mimicking portfolios as the mostdiversifiedportfolios35 HYPERLINK l _bookmark28 Factor-mimicking portfolios as the minimumvarianceportfolios35 HYPERLINK l _bookmark29 Sharpe optimization for the bottom-up factor ap

14、proach under market, sector andregionconstraints36 HYPERLINK l _bookmark30 Incorporating the stocks rates sensitivities in thefactor-mimickingregression37 HYPERLINK l _bookmark31 Factor-mimicking portfolio based on a regressionwithoutintercept39Unconstrained (Raw) Equity FactorsFactor DefinitionsThe

15、 goal of the research note is to illustrate and compare various ways of eliminating undesired biases and overlaps in equity styles. Therefore, we rely on a standard set of factor definitions as presented in Equity Risk Premia Strategies primer by Kolanovic (2014) and shown below.Figure 1: Classifica

16、tion of Equity Risk Factor Styles, and our selection of Prototype FactorsSource: J.P. Morgan Quantitative and Derivatives StrategyConstruction of the Unconstrained (Raw) Equity FactorsEach month-end all the stocks listed in our universe are ranked. The universe of stocks is the applicable MSCI regio

17、nal index (MSCI World), excluding stocks that are undergoing corporate actions. Stocks are ranked according to the measure (or measures) used to construct the Risk Factor converted to a Z-score. Note that the Z- scores are additionally sector-normalized when we construct the unconstrained factors (b

18、ut the non-normalized scores are used as an input for the pure factors).For instance, the Value Risk Factor is constructed based on Earnings Yield (inverse of P/E). The factor is designed by selecting the top 200 names as our Long basket and selecting the bottom 200 names as our Short basket. At eve

19、ry point in time we target an annualized volatility of the factor of 5%.Pure Equity FactorsThe common definition of a pure equity factor is an equity factor portfolios that solely contains an exposure to one particular style with no any overlap with other styles and no impact of country or sector ex

20、posures on the portfolio return.In the Equity Risk Premia Strategies primer by Kolanovic (2014) various factor neutralization methods have discussed. In particular, various ways to construct beta-neutral styles as well sector/country normalization methods have been analyzed. Removing any potential o

21、verlap among styles calls for additional techniques and processes.The early attempts to arrive at a pure factor exposure involved variants of multiple sorts. For in Fama and French (1996), the Size (SMB) portfolio return is obtained after a secondary sort on the Value metrics and in turn Value (HML)

22、 portfolio return is obtained after a secondary sort the Size metrics. While the framework is tractable with two factors, it becomes unmanageable with a bigger number of factors. For example, if there are 5 factors and we use quintiles, we need to create 3125buckets.Nowadays obtaining pure equity st

23、yle exposure almost certainly entails a statistical/numerical procedure.Below we discuss three alternative implementation approaches. The first two have an academic and statistical flavor, while the latter one is a real-life practitioners approach based on numerical optimization. Later we compare th

24、e empirical results for the various approaches and we discuss the advantages of the pure approach in light of constructing multi-factor portfolios from a top-down point of view, i.e. combining the individual factor portfolios into a global portfolio.Constructing Pure Equity FactorsPure Equity Factor

25、s Based on Fama-MacBeth RegressionsLets assume that the cross-section of n stock returns is driven by a m factor model:i = in +1Scori,1 + 2Scori,2 + +nScori,n + i,rei s e rn f k i, n s e at rtur, i s e at bta f k i,j s e rn r jd Scori,j se/xposure fk i htor HYPERLINK l _bookmark5 1In our case the fa

26、ctors are the various equity styles Quality, Value, Momentum, Size and Low Volatility and also the respective region and sector affiliations. Similarly, the scores of the individual stocks are either Z-scores for the equity factors or 0/1 dummy variables for the region/sector affiliation.If the samp

27、le for the regression above includes all the stocks in the universe, the average score per equity factor is zero. HYPERLINK l _bookmark6 2 Furthermore, if average excess return across all sectors and the average excess return across all regions returns are zero (see next paragraph), it that estimate

28、 of the market return is the equally-weighted average stock return. HYPERLINK l _bookmark7 3The setup above, while flexible, poses some statistical challenges. There exists multicollinearity among the dummy variables for the regions and the sectors in the regression setup above. For a certain stock

29、all the region scores and all the sector scores sum to 1 (as a stock belongs to one region and sector). To avoid multicollinearity we need to introduce additional assumptions. A reasonable assumption is to assume that the average country return is zero and the average sector return is zero. Hence, t

30、he excess return for one of the regions can be1 The average (non-weighted) market return will be estimated in the regression rather than imposed. As we will see later this assures that the factor-mimicking portfolios will also be beta-neutral.2 The statement is equivalent to saying that the equally-

31、weighted market has no exposure to any style.3 Note that in a regression without an intercept the average residual might differ from zero. In the Appendix we have provided a derivation of the factor-mimicking portfolio results that assure that the average residual is zero. We have not found signific

32、ant impact of on our empirical results when we force the average residual to be equal to zero.expressed as a linear combination of the remaining ones and a similar logic can be applied for one of the sectors (one of the regions and one of the sectors are redundant variables in the regression above).

33、 HYPERLINK l _bookmark8 4Using the known scores of the individual stocks we can estimate the factor returns with a standard regression technique. HYPERLINK l _bookmark9 5 The approach of obtaining the factor return via a regression using the factor exposures is also known in the academic world as Fa

34、ma-MacBeth regression (1973). HYPERLINK l _bookmark10 6s e by Scoree m atrix that s e s f the s: Score = Scor1 . . Scorn d Q e e r f k btas h rt o e y d arket . n e atrix Score y n s e s ) o a . tB = Q Scor1 . . Scorn. t s that e s f e at rndrs j rj = ,.,n areny q = n 1 2 n = 1u.Beyond the estimates

35、 of the factor returns, the setup of the Fama-MacBeth regression provides a convenient numerical way to construct factor-mimicking portfolios, i.e. portfolios that have a unit exposure to the factor of interest and zero exposure to the remaining ones.The weights/positions in the individual stocks in

36、 the mimicking portfolios are provided in the 2,.,m+1 columns of the matrix W:W = v12.n = 1is W = I reI seyxtst uScori = 1 iuScorj = 0 ri j. r,ergse aneutral uw= 0. HYPERLINK l _bookmark11 7iiIt can be shown that the factor mimicking-portfolios are the ones that minimize the sum of squared weights (

37、positions) and hence can be considered most diversified one. HYPERLINK l _bookmark12 8 Therefore, even though there is no explicit control of the position size in this approach (in contrast to the numerical portfolio optimization approach discussed later) it is expected that those portfolios should

38、not contain many outsized positions.Pure Equity Factors Based on Minimum Torsion BetsMeuci et al. (2013) introduced the minimum torsion bets as the linear transformation of the initial set of factors that results in a new set of de-correlated factors with the smallest tracking error with respect to

39、the original set factors, i.e. the minimum torsion transformation. In order words, minimum torsion is the linear combination that disrupts the least the original factors to achieve orthogonality. The main advantage of the minimum torsion4Forexample,letsassumewehavenregions.Wecanchoosethefactorreturn

40、forn-thregiontoberecalculatedfromthereturnsremainingn-1regions,i.e. = q1RFR1q2RFR2 qn1RFRn1,whereisthereturnforregioniqi is the number of stocks in region i. The relationship can be enforced if the scores of 1 for region n are replaced with scores of -qi /qn with respect to regions i=1,2,.,n-1. Simi

41、lar approach has been used by Heston and Rouwenhorst (1994).5 An alternative is to attempt to appropriately model the volatility of the noise term in the regression. Typically it is assumed that thevolatility of noise term is inversely proportional to market capitalization and regression methods tha

42、t handle heteroscedasticity are employed.6 Note that the returns for the sector/region that have been expressed as a function of the remaining ones due to the multicollinearity will have to be recalculated.7 An additional constraints imposing that the factor mimicking portfolios are also cash-neutra

43、l can be added by including a constant in the Fama-MacBeth regression. Note that such constraint can be deemed inappropriate in the case of the Low Volatility style.8 For a proof see the Appendix.approach in comparison to the more widely used PCA analysis is that the new set of factors are easily in

44、terpretable as they are close to the time-series dynamics of the original set of factors.We apply minimum torsion to the various unconstrained equity style factors. The correlation matrix is a critical input for the analysis, and we can use either the historical correlation matrix of the factors or

45、the implied correlation matrix based on the actual composition of the equity factors a point We have opted to use the implied correlation matrix, as such an approach will be reactive. HYPERLINK l _bookmark13 9 The empirical results can be easily replicated using the historical correlation matrix of

46、the equity factors aswell.Pure Equity Factors Based on Numerical OptimizationAnother alternative is to adopt a completely numerical optimization approach. The pure equity styles are constructed via maximization of the style exposure of interest, while simultaneously keeping neutral exposure to the o

47、ther remaining styles and complying with a portfolio volatility constraint HYPERLINK l _bookmark14 10. The main advantage of the approach is its flexibility to incorporate real-life constraints that convert the pure factors an actual tradable solution. Additional portfolio construction/risk manageme

48、nt constraints, neutral overall country/sector exposures, deviation of long/short legs composition from the benchmark one, position sizes, individual and portfolio-level risk can be added. In addition, tradability constraints like turnover constraints, constraints on the borrowing costs are easilyin

49、corporated.Note that, in the absence of the additional trading constraints, it can be shown that the optimal solution is just the factor-mimicking portfolio for the respective style (please see the Appendix). As the trading constraints can be mandate specific on our side we have imposed only one sup

50、plementary constraint on the maximum size of the position in addition to the standard constraints of neutrality to the remaining styles and the broad market, countries and sectors HYPERLINK l _bookmark15 11.Comparison among the different approachesThere are substantial differences between the minimu

51、m torsion approach and the Fama-MacBeth one. The minimum torsion approach assures (statistical) orthogonality among the individual styles, but does not strictly impose neutrality versus additional factors. In simple terms in the minimum torsion approach we redistribute some of the positions among th

52、e styles so that we achieve uncorrelated factors. The factor-mimicking portfolios are additionally neutral to region and sector as well as being beta and cash neutral.The factor-mimicking portfolios framework is quite flexible as it allows easily to incorporate additional factors. For example, if la

53、tent duration exposure is thought to be present in any of the styles, we can add the sensitivity to moves in the interest rates as an additional factor.Both the minimum torsion and the factor-mimicking portfolio approaches their merits. The minimum torsion is easy to implement, the factor-mimicking

54、portfolio methodology is quite flexible and analytical. Nevertheless, both the approaches can easily miss on constraints that might be important for the realistic trading. Incorporating those constraints will eventually bring us to some form ofoptimization.9 We simulate backwards the returns of the

55、equity styles based on the current stock composition and then calculate the correlation matrix of those implied factor returns.10 Please see the section on the numerical optimization in the multi-factor approach for more details. The pure equity factors based on numerical optimization are just a spe

56、cial case of the numerical optimization for the multi-factor approach.11 In our case the maximum absolute size of the position cannot exceed 50bps.2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019Dobromir Tzotchev, PhD (44-20) 7134-5331 HYPERLINK mailto:dobromir.tzotchev dobromir

57、.tzotchevGlobal Cross-Asset Risk Premia Strategy28 November 2019Table 1: Comparison among the different methods for constructing pure equity factorsInputVariablesQuantitative ApproachCovariance matrix based on theAdvantagesDisadvantagesDoes not assure neutrality to additional factors. Real-lifeMinim

58、umTorsioncurrentstyle compositionNumericalEasy and quick toapplyFlexible to incorporateimplementation requires additional optimization/constraints.Factor-mimicking portfoliosFactorscores,region/sectoraffiliation, stockbeta.additional factors. Provides estimates of pure factor returns unit factor exp

59、osure.Real-life implementation requires additional optimization/constraints.NumericalOptimizationFactor scores,region/sector affiliation, stock beta,riskNumerical OptimizationCanimplementreal-lifeMost complex toimplementconstraints managementconstraintsSource: J.P. Morgan Quantitative and Derivative

60、s StrategyEmpirical ResultsBelow we compare the minimum torsion approach, the factor-mimicking portfolios and the numerical optimization approaches to the unconstrained factors presented in the earlier Both the Fama-MacBeth and the numerical optimization approaches use the factor scores while minimu