Interpretable Proximate Factors For Large Dimensions
Interpretable Proximate Factors for LargeDimensionsMarkus Pelger1Ruoxuan Xiong1 StanfordUniversity2 StanfordUniversityFebruary 1, 2018Risk Management SeminarUC Berkeley2
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixMotivationMotivation: What are the factors?Statistical Factor AnalysisFactor models are widely used in big data settingsReduce data dimensionalityFactors are traded extensivelyProblem: Which factors should be used?Statistical (latent) factors perform wellFactors estimated from principle component analysis (PCA)Weighted averages of all features/assetsProblem: Hard to interpretGoals of this paper:Create interpretable proximate factorsShrink most assets’ weights to zero to get proximate factors More interpretable Significantly lower transaction costs when trading factors1
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixMotivationContribution of this paperContributionThis Paper: Estimation of interpretable proximate factorsKey elements of estimator:1Statistical factors instead of pre-specified (and potentiallymiss-specified) factors2Uses information from large panel data sets: Many assets withmany time observations3Proximate factors approximate latent factors very well with afew assets without sparse structure in population factors4Only 5-10% of the cross-sectional observations with the largestexposure are needed for proximate factors2
IntroIllustrationModelSimulationEmpirical oretical ResultsAsymptotic probabilistic lower bound for generalized correlations ofproximate factors with population factorsGuidance on how to construct proximate factorsEmpirical ResultsVery good approximation to population factors with 5-10%portfolios, measured by generalized correlation, variance explained,pricing error and Sharpe-ratioInterpret statistical latent factors forDouble-sorted portfolio data370 single-sorted anomaly portfoliosHigh-frequency returns of S&P 500 companies3
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixMotivationLiterature (partial list)Large-dimensional factor models with PCABai (2003): Distribution theoryFan et al. (2013): Sparse matrices in factor modelingFan et al. (2016): Projected PCA for time-varying loadingsPelger (2016), Aı̈t-Sahalia and Xiu (2015): High-frequencyLarge-dimensional factor models with penalty termBai and Ng (2017): Robust PCA with ridge shrinkageLettau and Pelger (2017): Risk-Premium PCA with pricingpenaltyZhou et al. (2006): Sparse PCA4
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixIllustrationEmpirical example: Double-sorted portfoliosDaily data of 25 double-sorted Fama-French portfolios(a) Size and Book-to-Market(b) Size and InvestmentFigure: Sum of generalized correlation ρ̂ between estimated 3 PCAfactors and 3 proximate factorsProblem in interpreting factors: Factors only identified up toinvertible linear transformations.Generalized correlation measures how many factors two sets have incommon.5
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixIllustrationEmpirical Application: Size and Book-to-market Portfolios25 portfolios formed on size and book-to-market(07/1963-10/2017, 3 factors, daily data)(a) Generalized correlation(b) Variance explained(c) RMS pricing error(d) Max Sharpe Ratio6
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixIllustrationEmpirical Application: Size and Book-to-market PortfoliosFigure: Portfolio weights of 1. statistical factor Equally weighted market factor7
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixIllustrationEmpirical Application: Size and Book-to-market PortfoliosFigure: Portfolio weights of 2. statistical factor Small-minus-big size factor Proximate factor with 4 largest weights correlation 0.88 with sizefactor8
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixIllustrationEmpirical Application: Size and Book-to-market PortfoliosFigure: Portfolio weights of 3. statistical factor High-minus-low value factor Proximate factor with 4 largest weights correlation 0.91 with valuefactor9
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixIllustrationEmpirical Application: Size and Investment Portfolios25 portfolios formed on size and investment(07/1963-10/2017, 3 factors, daily data)(a) Generalized correlation(b) Variance explained(c) RMS pricing error(d) Max Sharpe Ratio10
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixIllustrationEmpirical Application: Size and Investment PortfoliosFigure: Portfolio weights of 1. statistical factor Equally weighted market factor11
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixIllustrationEmpirical Application: Size and Investment PortfoliosFigure: Portfolio weights of 2. statistical factor Small-minus-big size factor Proximate factor with 4 largest weights correlation 0.97 with sizefactor12
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixIllustrationEmpirical Application: Size and Investment PortfoliosFigure: Portfolio weights of 3. statistical factor High-minus-low value factor Proximate factor with 4 largest weights correlation 0.79 withinvestment factor13
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixModelThe ModelApproximate Factor ModelObserve excess returns of N assets over T time periods:Xt,i Ft Λi et,i {z}1 KK 1 {z } {z} idiosyncratici 1, ., N t 1, ., Tfactors loadingsMatrix notationX {z}F {z}Λ {z}e {z}T NT K K NT NN assets (large)T time-series observation (large)K systematic factors (fixed)F , Λ and e are unknown14
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixModelThe ModelApproximate Factor ModelSystematic and non-systematic risk (F and e uncorrelated):Var (X ) ΛVar (F )Λ {z}systematicVar (e) {z }non systematic Systematic factors should explain a large portion of thevariance Idiosyncratic risk can be weakly correlatedEstimation: PCA (Principal component analysis)Apply PCA to the sample covariance matrix:X̄ sample mean of asset excess returns1TX X X̄ X̄ withEigenvectors of largest eigenvalues estimate loadings Λ̂.F̂ estimator for factors: F̂ 1N X Λ̂ X Λ̂ (Λ̂ Λ̂) 1 .15
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixModelThe ModelProximate FactorsSparse loadings Λ̃ are obtained fromSelect finitely many mN loadings with largest absolute valuefrom Λ̂kShrink estimated loadings Λ̂ to 0 except for mN largest valuesDivide by column norms, i.e. λ̃ k λ̃k 1Proximate factors F̃ X T Λ̃16
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixModelThe ModelCloseness measureFor 1-factor model: Correlation between F̃ and F .Problem for multiple factors: Factors are only identified up toinvertible linear transformations Need measure for closenessbetween span of two vector spacesFor multi-factor model: The ”closeness” between F̃ and F ismeasured by generalized correlation:Total generalizedcorrelation measure: Tρ trace (F F /T ) 1 (F T F̃ /T )(F̃ T F̃ /T ) 1 (F̃ T F /T )ρ 0: F̃ and F are orthogonalρ K : F̃ and F are span the same spaceAlternative measure: Element-wise generalized correlations areeigenvalues instead of trace of above matrixElement-wise generalized correlations close to 1 measure howmany factors are well approximated17
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixIntuitionIntuition: Why picking largest elements in Λ̂ works?Consider one factor and one nonzero element in Λ̃:F [f1t ] RT 1 , Λ [λ1,i ] RN 1Λ̃ [λ̃1,i ] is sparse. Assume nonzero element in λ̃1,i is λ̃1,1 .F̃ X T Λ̃ F ΛT Λ̃ e T Λ̃ f1 λ1,1 e1Assumef1,t (0, σf2 ),f1T f1 σf2 ,TDefine signal-to-noise ratio s iide1,t (0, σe2 )e1T e1 σe2Tσfσe18
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixIntuitionIntuition: Why pick the largest elements in Λ̂?ρ tr (F T F /T ) 1 (F T F̃ /T )(F̃ T F̃ /T ) 1 (F̃ T F /T ) 2f1T (f1 λ1,1 e1 )/T(f1T f1 /T )1/2 ((f1 λ1,1 e1 )T (f1 λ1,1 e1 )/T )1/2λ21,1λ21,1 1/s 2(Generalized) correlation increases in size of loading λ1,1 .(Generalized) correlation increases in signal-to-noise ratio s.No sparsity in population loadings assumed!19
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixAsymptotic resultsAsymptotic resultsProximate factors F̃ are in general not consistent.F̃ X T Λ̃ F ΛT Λ̃ e T Λ̃Idiosyncratic component not diversified awayiidAssume ei,l (0, σe2·,l ), then each element in e T Λ̃ hasVarmNXi 1!λ̃j,ji eji ,l mNXλ̃2j,ji σe2·,l σe2·,l 6 0i 1Instead we provide probabilistic lower bound for (generalized)correlation ρ given a target correlation level ρ0 :P(ρ ρ0 )20
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixAsymptotic resultsAssumptionsAssumptions1Factors: Uncorrelated and demeaned factors:E [F ] 0F F ΣF diag (σf21 , σf22 , · · · , σf2r )T2Loadings: Random variables λi,j Op (1) and Λ Λ ΣΛ3Systematic factors: Eigenvalues of ΣΛ ΣF bounded away from 0.4Residuals: Weak DependencyE [ei,l ] 0 and Var (ei,l ) σe2 i, le independent from F and Λ 1 e T e(k) Op (1) i, k and i 6 kT (i)5Consistent estimator: 1fˆj Hfj Op Nλ̂i H 1 λi Op 1 T N, T Sufficient conditions in Bai (2003) and Bai and Ng (2002)21
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixOne Factor CaseOne factor caseTheoremAssume K 1 factor and population loadings λ1,i are i.i.d for all i.For any ρ0 we have for N, T P(ρ ρ0 ) 1 mXN 1 j 0 N(1 F λ1,i (ymN ))j F λ1,i (ymN )N j (1)jwheresymN F λ1,i 1 σe2 ρ0mN σf21 1 ρ0(y ) P( λ1,i y )22
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixOne Factor CaseOne factor caseDenote theprobabilitybound for P(ρ ρ0 ) by PmlowerjN jN 1 Np 1 j 0j (1 F λ1,i (ymN )) F λ1,i (ymN )It holds, p 0 F λ1,i (ymN )p is decreasing inF λ1,i (ymN ). Hence p isdecreasing in ρ0increasing in s σf1 /σeincreasing in mNincreasing in the dispersion of the distribution of λ1,i 23
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixOne Factor CaseMultiple FactorsMultiple Factor: Simple CaseDenote by {j1 , j2 , · · · , jmN } indexes of nonzero entries in λ̃j (i.e.largest mN entries in λ̂j in absolute value).Let U be the “sparse” rotated population loadings ΛH R N k withnon-zero entries {j1 , j2 , · · · , jmN }.Assume U columns do not overlapLet vj,(mN ) min( uj,j1 , uj,j2 , · · · , uj,jmN ) to be the mN -th orderstatistic of uj For any threshold ρ0 and for N, T we have kX1m(K ρ)N0 P(ρ ρ0 ) P 22σsj vj,(meN)j 124
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixOne Factor CaseMultiple FactorsMultiple Factor: Threshold and then rotateDenote by {j1 , j2 , · · · , jmN } indices of nonzero entries in λ̃jLet Λ̆ be the “sparse” population loadings ΛH with non-zero entries{j1 , j2 , · · · , jmN }.Assume there exists orthonormal matrix P s.t. Λ̆P columns do not overlapSignal matrix S is diagonal matrix of the eigenvalues of ΣΛ ΣF indecreasing orderPPPDefine [wM,1, wM,2, · · · , wM,k] as normalized elements of Λ̆S 1/2 PPPPPLet wj,(m min( wj,j , wj,j , · · · , wj,j ) to be the mN -th orderm12N)Nstatistic of wjP For any threshold ρ0 and for N, T we haveP(ρ ρ0 ) PKXj 1mN (1 γ)(K ρ0 ) P2σe2 (1 )4(wj,(m)N)1with known constants c and and γ.!25
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixOne Factor CaseMultiple FactorsMultiple Factor: Rotate and thresholdSimilar to previous theorem, but first find a rotation of the data and thenthreshold such that columns of sparse loadings to not overlapFor any threshold ρ0 and for N, T we haveP(ρ ρ0 ) PKXj 1mN (1 γ)(K ρ0 ) P2σe2(wj,(m)N)1!with known constants c and and γ.26
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixOne Factor CaseMultiple FactorsDenote the lower probability bound for P(ρ ρ0 ) by pIt holds (very similar to the one factor case) that p isdecreasing in ρ0increasing in s σf1 /σeincreasing in mNincreasing in the dispersion of the distribution of λ1,i 27
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixLassoRelationship with LassoAlternative approach with Lasso:12Estimate factors by PCA, i.e X T X F̂ F̂ V with V matrix ofeigenvalues.Estimate loadings by X ΛF̂ T2F α kΛk1 . Divide theminimizer by its column norm (standardize each loading) toobtain Λ̄3Proximate factors from Lasso approach are F̄ X T Λ̄(Λ̄T Λ̄) 1 Same selection of non-zero elements (for one factor case) butdifferent weighting Under certain conditions worse performance than thresholdingapproachTuning parameter less transparent28
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixSimulationSimulation: One Factor (σe 1, λ N(0, 1), 500 MCs)(a) N 50(b) N 100(c) N 200(d) N 500Figure: σf 1.5, ρ0 0.9529
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixSimulationSimulation: One Factor (σe 1, λ N(0, 1), 500 MCs)(a) N 50(b) N 100(c) N 200(d) N 500Figure: σf 1.0, ρ0 0.9530
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixSimulationSimulation: One Factor (σe 1, λ N(0, 1), 500 MCs)(a) N 250(b) N 500(c) N 750(d) N 1000Figure: σf 0.5, ρ0 0.9531
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixSimulationSimulation: Two Factors (σe 1, λ N(0, 1), 500 MCs)(a) N 50(b) N 100(c) N 200(d) N 500Figure: σf 2.0, ρ0 1.832
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixSimulationSimulation: Two Factors (σe 1, λ N(0, 1), 500 MCs)(a) N 100(b) N 200(c) N 300(d) N 500Figure: σf 1.5, ρ0 1.733
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixSimulationSimulation: Two Factors (σe 1, λ N(0, 1), 500 MCs)(a) N 100(b) N 200(c) N 300(d) N 500Figure: σf 1.0, ρ0 1.634
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsExtreme deciles of single-sorted portfoliosPortfolio DataKozak, Nagel and Santosh (2017) data: 370 decile portfolios sortedaccording to 37 anomaliesMonthly return data from 07/1963 to 12/2016 (T 638)First only lowest and highest decile portfolio for each anomaly(N 74).Risk-Premium PCA (RP-PCA) from Lettau and Pelger (2017)applies PCA to T1 X X γ X̄ X̄ penalty for pricing errorFactors:1234RP-PCA: K 6 and γ 100.PCA: K 6Fama-French 5: The five factor model of Fama-French(market, size, value, investment and operating profitability).Proxy factors: RP-PCA and PCA factors approximated with 8largest positions.35
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsExtreme DecilesSRRP-PCAPCARP-PCA ProxyPCA ProxyFama-French 50.640.350.620.370.32In-sampleRMS α Idio. 0.280.480.3150.31Out-of-sampleRMS α Idio. : First and last decile of 37 single-sorted portfolios from 07/1963 to12/2016 (N 74 and T 638): Maximal Sharpe-ratios,root-mean-squared pricing errors and unexplained idiosyncratic variation.K 6 statistical factors.Proximate factors approximate latent factors very wellResults hold out-of-sample.36
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsInterpreting factors: Generalized correlations with 880.721.001.000.990.970.950.89Table: Generalized correlations of statistical factors with proxy factors(portfolios of 8 assets).Generalized correlations close to 1 measure of how many factors twosets have in common.Total generalized correlation ρ sum of element-wise generalizedcorrelations Proxy factors approximate statistical factors well.37
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsExtreme Deciles: Maximal Sharpe-ratioSR (In-sample)0.7SR (Out-of-sample)3 factors4 factors5 factors6 factors7 PPCFigure: Maximal Sharpe-ratios. Spike in Sharpe-ratio for 6 factors Proximate factors capture similar Sharpe-ratio pattern38
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsExtreme Deciles: Pricing 53 factors4 factors5 factors6 factors7 Figure: Root-mean-squared pricing errors. RP-PCA has smaller out-of-sample pricing errors Proximate factors have similar pricing errors39
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsExtreme Deciles: Idiosyncratic VariationIdiosyncratic Variation (In-sample)7Idiosyncratic Variation (Out-of-sample)7665544333 factors4 factors5 factors6 factors7 yox-PRPyAroxPCAPPCFigure: Unexplained idiosyncratic variation. Unexplained variation similar for RP-PCA and PCA Proximate factors explain the same variation40
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsInterpreting factors: Composition of proxies2. Proxy 0.422. Proxy .033. Proxy .26-0.40-0.41-0.673. Proxy luem10.460.380.370.360.320.310.31-0.294. Proxy e1mom1210.280.260.25-0.24-0.30-0.44-0.45-0.494. Proxy 1mom121mom10.360.350.340.33-0.31-0.35-0.39-0.395. Proxy 5. Proxy .30-0.25-0.27-0.28-0.29-0.37-0.38-0.576. Proxy 6. Proxy able: Portfolio-composition of proxy factors for first and last decile of 37single-sorted portfolios: First proxy factors is an equally-weightedportfolio.41
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsInterpreting factors: Cumulative absolute proxy weightsRP-PCA ProxyMomentum (12m)Idiosyncratic VolatilityIndustry Rel. Rev. (L.V.)Momentum (6m)PriceIndustry Mom. ReversalsValue (M)Industry Rel. ReversalsShare VolumeNet Operating AssetsValue-Momentum-Prof.SizeReturn on Book Equity (A)Industry MomentumValue-MomentumLong Run ReversalsEarnings/PricePCA .420.320.290.270.240.22Idiosyncratic VolatilityMomentum (12m)Momentum (6m)PriceAsset TurnoverIndustry Rel. Rev. (L.V.)Value (M)Industry MomentumIndustry Mom. ReversalsDividend/PriceGross ProfitabilitySales/PriceValue-ProfitabilityNet Operating AssetsValue (A)Value-Momentum-Prof.Cash 70.640.580.580.370.360.330.3142
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsSingle-sorted portfoliosPortfolio DataMonthly return data from 07/1963 to 12/2016 (T 638) forN 370 portfoliosKozak, Nagel and Santosh (2017) data: 370 decile portfolios sortedaccording to 37 anomaliesRisk-Premium PCA (RP-PCA) from Lettau and Pelger (2017)applies PCA to T1 X X γ X̄ X̄ penalty for pricing errorFactors:1234RP-PCA: K 6 and γ 100.PCA: K 6Fama-French 5: The five factor model of Fama-French(market, size, value, investment and operating profitability, allfrom Kenneth French’s website).Proxy factors: RP-PCA and PCA factors approximated with5% of largest position.43
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsSingle-sorted portfoliosSRRP-PCAPCAFama-French 5RP-PCA Proxy 6PCA Proxy 60.660.280.320.570.34In-sampleRMS α Idio. of-sampleRMS α Idio. 194.623.153.12Table: Deciles of 37 single-sorted portfolios from 07/1963 to 12/2016(N 370 and T 638): Maximal Sharpe-ratios, root-mean-squaredpricing errors and unexplained idiosyncratic variation. K 6 statisticalfactors.Proximate factors approximate latent factors very wellResults hold out-of-sample.44
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsInterpreting factors: Generalized correlations with 920.781.001.000.990.990.940.89Table: Generalized correlations of statistical factors with proxy factors(portfolios of 5% of assets).Generalized correlations close to 1 measure of how many factors twosets have in common.Total generalized correlation ρ sum of element-wise generalizedcorrelations Proxy factors approximate statistical factors well.45
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsSingle-sorted portfolios: Maximal Sharpe-ratioSR (In-sample)0.7SR (Out-of-sample)3 factors4 factors5 factors6 factors7 PPCFigure: Maximal Sharpe-ratios. Spike in Sharpe-ratio for 6 factors Proximate factors capture similar Sharpe-ratio pattern46
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsSingle-sorted portfolios: Pricing t-of-sample)0.13 factors4 factors5 factors6 factors7 CAPryox-PRPyAroxPCAPPCFigure: Root-mean-squared pricing errors. RP-PCA has smaller out-of-sample pricing errors Proximate factors have similar pricing errors47
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsSingle-sorted portfolios: Idiosyncratic VariationIdiosyncratic Variation (In-sample)5Idiosyncratic Variation (Out-of-sample)54433223 factors4 factors5 factors6 factors7 x-PRPyAroxPCAPPCFigure: Unexplained idiosyncratic variation. Unexplained variation similar for RP-PCA and PCA Proximate factors explain the same variation48
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsInterpreting factors: 6th proxy factor6. Proxy RP-PCAMomentum (6m) 1Momentum (6m) 2Value (M) 10Value-Momentum 1Industry Momentum 1Industry Reversals 9Industry Momentum 2Momentum (6m) 3Idiosyncratic Volatility 2Industry Mom. ReversalsValue-Momentum 8Momentum (6m) 10Value-Momentum 9Value-Momentum 10Short-Term Reversals 1Industry-Momentum 10Industry Rel. Reversals 1Idiosyncratic Volatility -0.20-0.21-0.23-0.23-0.24-0.24-0.28-0.386. Proxy PCALeverage 10Asset Turnover 10Value-Profitability 10Profitability 10Asset Turnover 9Sales/Price 10Sales/Price 9Size 10Value-Momentum-Profitability 1Profitability 2Value-Profitability 1Profitability 4Value-Profitability 2Profitability 1Idiosyncratic Volatility 1Profitability 3Asset Turnover 2Asset Turnover -0.20-0.20-0.20-0.23-0.24-0.25-0.28-0.3549
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsInterpreting factors: Cumulative absolute proxy weightsRP-PCA ProxyIdiosyncratic VolatilityMomentum (12m)Industry Mom. ReversalsIndustry Rel. Reversals (L.V.)PriceMomentum (6m)Value-MomentumSizeIndustry MomentumNet Operating AssetsIndustry Rel. ReversalsValue (M)Value-Momentum-Prof.Share VolumeInvestment/CapitalEarnings/PriceShort-Term ReversalsPCA .750.510.460.410.400.40Idiosyncratic VolatilityMomentum (12m)Asset TurnoverGross ProfitabilityIndustry Rel. Rev. (L.V.)SizeIndustry Mom. ReversalsNet Operating AssetsMomentum mentum-Prof.Industry MomentumValue 041.011.000.990.920.860.820.800.730.670.560.4550
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsHigh-Frequency price dataDataHigh-frequency factor analysis from Pelger (2017)Time period: 2003 to 2012Xi (t) is the log-return from the TAQ databaseN between 500 and 600 firms from the S&P 5005-min sampling: on average 250 days with 77 increments eachEstimator for number of factors indicate 4 latent factorsCreate factors for continuous (normal) movements and for jumps(rare large) movementsQuestion: What are the factors?51
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsIdentification of factorsInterpretation of continuous factorsApproach: Rotate and thresholdNon-zero elements are almost all in specific industries4 economic candidate factors:Market (equally weighted)Oil and gas (40 equally weighted assets)Banking and Insurance (60 equally weighted assets)Electricity (24 equally weighted assets)52
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsMain result: Interpretation of factors4 continuous factors with industry continuous factors1.00 0.98 0.95 0.804 jump factors with industry jump factors0.99 0.75 0.29 0.054 continuous factors with Fama-French Carhart Factors0.95 0.74 0.60 0.00Table: Generalized correlations of first four largest statistical factors for2007-2012 with economic factorsElement-wise generalized correlations close to 1 measure of howmany factors two sets have in commonEconomic industry factors: Market, oil, finance, electricity Jump structure different from continuous structure Size, value, momentum do not explain factors53
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsInterpretation of continuous 76Generalized correlation of market, oil, finance and energy factors withfirst four largest statistical factors for 2007-2012 Stable continuous factor structure Proximate factors approximate latent factors well54
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixEmpirical ResultsInterpretation of continuous .830.78Generalized correlation of market, oil, finance and energy factors withfirst four largest statistical factors for 2003-2012 Finance factor disappears in 2003-200655
IntroIllustrationModelSimulationEmpirical dologyProximate factors (portfolios of a few assets) for latent populationfactors (portfolios of all assets)Simple thresholding estimator based on largest loadingsProximate factors approximate population factors well withoutsparsity assumptionAsymptotic probabilistic lower bound for (generalized) correlationFuture work: Sharpen bounds based on extreme value theory Few observations summarize most of the informationEmpirical ResultsGood approximation to population factors with 5-10% portfoliosInterpretation of RP-PCA and high-frequency PCA factors56
IntroIllustrationModelSimulationEmpirical ResultsConclusionAppendixExtreme e-ratioAccruals - accrualAsset Turnover - aturnoverCash Flows/Price - cfpComposite Issuance - cissDividend/Price - divpEarnings/Price - epGross Margins - gmarginsAsset Growth - growthInvestment Growth - igrowthIndustry Momentum - indmomIndustry Mom. Reversals - indmomrevIndustry Rel. Reversals - indrrevIndustry Rel. Rev. (L.V.) - indrrevlvInvestment/Assets - invInvestment/Capital - invcapIdiosyncratic Volatility - ivolLeverage - levLong Run Reversals - lrrevMomentum (6m) - 240.440.160.030.080.050.090.06Momentum (12m) - mom12Momentum-Reversals - momrevNet Operating Assets - noaPrice - priceGross Profitability - profReturn on Assets (A) - roaaReturn on Book Equity (A) - roeaSeasonality - seasonSales Growth - sgrowthShare Volume - shvolSize - sizeSales/Price spShort-Term Reversals - strevValue-Momentum - valmomValue-Momentum-Prof. - valmomprofValue-Profitability - valprofValue (A) - valueValue (M) - 070.100.170.200.110.07Table: Long-Short Portfolios of extreme deciles of 37 single-sortedportfolios from 07/1963 to 12/2016: Mean, standard deviation andSharpe-ratio.
(a)Size and Book-to-Market (b)Size and Investment Figure:Sum of generalized correlation ˆbetween estimated 3 PCA factors and 3 proximate factors Problem in interpreting factors: Factors only identi ed up to invertible linear transformations. Generalized correlation measures how many factors two sets have in common.
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