Estimating Latent Asset-Pricing Factors

Transcription

1 Estimating Latent Asset-Pricing Factors Markus Pelger Martin Lettau Stanford University UC Berkeley September 4th 8 IEOR-DRO Seminar Columbia University

2 Motivation Motivation: Asset Pricing with Risk Factors The Challenge of Asset Pricing Most important question in finance: Why are prices different for different assets? Fundamental insight: Arbitrage Pricing Theory: Prices of financial assets should be explained by systematic risk factors. Problem: Chaos in asset pricing factors: Over 3 potential asset pricing factors published! Fundamental question: Which factors are really important in explaining expected returns? Which are subsumed by others? Goals of this paper: Bring order into factor chaos Summarize the pricing information of a large number of assets with a small number of factors

3 Motivation Why is it important? Importance of factors for investing Optimal portfolio construction Only factors are compensated for systematic risk Optimal portfolio with highest Sharpe-ratio must be based on factor portfolios (Sharpe-ratio=expected excess return/standard deviation) Smart beta investments = exposure to risk factors Arbitrage opportunities Find underpriced assets and earn alpha 3 Risk management Factors explain risk-return trade-off Factors allow to manage systematic risk exposure

4 Motivation Contribution of this paper Contribution This Paper: Estimation approach for finding risk factors Key elements of estimator: Statistical factors instead of pre-specified (and potentially miss-specified) factors Uses information from large panel data sets: Many assets with many time observations 3 Searches for factors explaining asset prices (explain differences in expected returns) not only co-movement in the data 4 Allows time-variation in factor structure 3

5 Motivation Contribution of this paper Results Asymptotic distribution theory for weak and strong factors No blackbox approach Estimator discovers weak factors with high Sharpe-ratios high Sharpe-ratio factors important for asset pricing and investment Estimator strongly dominates conventional approach (Principal Component Analysis (PCA)) PCA does not find all high Sharpe-ratio factors Empirical results: New factors much smaller pricing errors in- and out-of sample than benchmark (PCA, 5 Fama-French factors, etc.) times higher Sharpe-ratio then benchmark factors (PCA) 4

6 Motivation Literature (partial list) Large-dimensional factor models with strong factors Bai (3): Distribution theory Bai and Ng (7): Robust PCA Fan et al. (6): Projected PCA for time-varying loadings Kelly et al. (7): Instrumented PCA for time-varying loadings Pelger (6), Aït-Sahalia and Xiu (5): High-frequency Large-dimensional factor models with weak factors (based on random matrix theory) Onatski (): Phase transition phenomena Benauch-Georges and Nadakuditi (): Perturbation of large random matrices Asset-pricing factors Feng, Giglio and Xiu (7): Factor selection with double-selection LASSO Kozak, Nagel and Santosh (7): Bayesian shrinkage 5

7 Model The Model Approximate Factor Model Observe excess returns of N assets over T time periods: X t,i = F t K }{{} factors Λ i K }{{} loadings + e t,i }{{} idiosyncratic i =,..., N t =,..., T Matrix notation X }{{} T N = }{{} F }{{} Λ T K K N + e }{{} T N N assets (large) T time-series observation (large) K systematic factors (fixed) F, Λ and e are unknown 6

8 Model The Model Approximate Factor Model Systematic and non-systematic risk (F and e uncorrelated): Var(X ) = ΛVar(F )Λ + }{{} Var(e) }{{} systematic non systematic Systematic factors should explain a large portion of the variance Idiosyncratic risk can be weakly correlated Arbitrage-Pricing Theory (APT): The expected excess return is explained by the risk-premium of the factors: E[X i ] = E[F ]Λ i Systematic factors should explain the cross-section of expected returns 7

9 Model The Model: Estimation of Latent Factors Conventional approach: PCA (Principal component analysis) Apply PCA to the sample covariance matrix: T X X X X with X = sample mean of asset excess returns Eigenvectors of largest eigenvalues estimate loadings ˆΛ. Much better approach: Risk-Premium PCA (RP-PCA) Apply PCA to a covariance matrix with overweighted mean T X X + γ X X γ = risk-premium weight Eigenvectors of largest eigenvalues estimate loadings ˆΛ. ˆF estimator for factors: ˆF = N X ˆΛ = X (ˆΛ ˆΛ) ˆΛ. 8

10 Model The Model: Objective Function Conventional PCA: Objective Function Minimize the unexplained variance: min Λ,F NT N T (X ti F t Λ i ) i= t= RP-PCA (Risk-Premium PCA): Objective Function Minimize jointly the unexplained variance and pricing error min Λ,F N T (X ti F t Λ i ) +γ N ( X i F Λ ) i NT N i= t= i= }{{}}{{} unexplained variation pricing error with X i = T T t= X t,i and F = T T t= F t and risk-premium weight γ 9

11 Model The Model: Interpretation Interpretation of Risk-Premium-PCA (RP-PCA): Combines variation and pricing error criterion functions: Select factors with small cross-sectional pricing errors (alpha s). Protects against spurious factor with vanishing loadings as it requires the time-series errors to be small as well. Penalized PCA: Search for factors explaining the time-series but penalizes low Sharpe-ratios. 3 Information interpretation: (GMM interpretation) PCA of a covariance matrix uses only the second moment but ignores first moment Using more information leads to more efficient estimates. RP-PCA combines first and second moments efficiently.

12 Model The Model: Interpretation Interpretation of Risk-Premium-PCA (RP-PCA): continued 4 Signal-strengthening: Intuitively the matrix T X X + γ X X converges to Λ ( Σ F + ( + γ)µ F µ ) F Λ + Var(e) with Σ F = Var(F ) and µ F = E[F ]. The signal of weak factors with a small variance can be pushed up by their mean with the right γ.

13 Illustration Illustration (Size and accrual) Illustration: Anomaly-sorted portfolios (Size and accrual) Factors PCA: Estimation based on PCA of correlation matrix, K = 3 RP-PCA: K = 3 and γ = 3 FF-long/short: market, size and accrual (based on extreme quantiles, same construction as Fama-French factors) Data Double-sorted portfolios according to size and accrual (from Kenneth French s website) Monthly return data from 7/963 to /7 (T = 65) for N = 5 portfolios Out-of-sample: Rolling window of years (T=4) Stochastic Discount Factor (SDF): maximum Sharpe-ratio portfolio R opt = F Σ F µ F

14 Illustration Illustration (Size and accrual) In-sample Out-of-sample SR RMS α Idio. Var. SR RMS α Idio. Var. RP-PCA PCA FF-long/short Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and unexplained idiosyncratic variation. K = 3 factors and γ =. SR: Maximum Sharpe-ratio of linear combination of factors Cross-sectional pricing errors α: Pricing error α i = E[X i ] E[F ]Λ i N RMS α: Root-mean-squared pricing errors N i= α i N T Idiosyncratic Variation: NT i= t= (X t,i Ft Λ i ) RP-PCA significantly better than PCA and quantile-sorted factors. 3

15 Illustration Loadings for statistical factors (Size and Accrual).5. PCA factor.5. PCA factor.5 3. PCA factor Loadings Loadings Loadings Portfolio.5. RP-PCA factor Portfolio.5. RP-PCA factor Portfolio.5 3. RP-PCA factor Loadings Loadings Loadings Portfolio Portfolio Portfolio RP-PCA detects accrual factor while 3rd PCA factor is noise. 4

16 Illustration Maximal Sharpe ratio (Size and accrual) =- = = = =5 = SR (In-sample) SR (Out-of-sample) factor factors 3 factors factor factors 3 factors Figure: Maximal Sharpe-ratio by adding factors incrementally. st and nd PCA and RP-PCA factors the same. RP-PCA detects 3rd factor (accrual) for γ >. 5

17 Illustration Effect of Risk-Premium Weight γ.4 SR (In-sample).4 SR (Out-of-sample) factor SR. factors 3 factors SR RMS (In-sample) RMS (Out-of-sample) Idiosyncratic Variation (In-sample) 4 Idiosyncratic Variation (Out-of-sample) Variation Variation Figure: Maximal Sharpe-ratios, root-mean-squared pricing errors and unexplained idiosyncratic variation. RP-PCA detects 3rd factor (accrual) for γ >. 6

18 Weak vs. Strong Factors The Model Strong vs. weak factor models Strong factor model ( N Λ Λ bounded) Interpretation: strong factors affect most assets (proportional to N), e.g. market factor Strong factors lead to exploding eigenvalues RP-PCA always more efficient than PCA optimal γ relatively small Weak factor model (Λ Λ bounded) Interpretation: weak factors affect a smaller fraction of assets Weak factors lead to large but bounded eigenvalues RP-PCA detects weak factors which cannot be detected by PCA optimal γ relatively large 7

19 Weak Factor Model Weak Factor Model Weak Factor Model Weak factors either have a small variance or affect a smaller fraction of assets: Λ Λ bounded (after normalizing factor variances) Statistical model: Spiked covariance models from random matrix theory Eigenvalues of sample covariance matrix separate into two areas: The bulk, majority of eigenvalues The extremes, a few large outliers Bulk spectrum converges to generalized Marchenko-Pastur distribution (under certain conditions) 8

20 Weak Factor Model Weak Factor Model Weak Factor Model Large eigenvalues converge either to A biased value characterized by the Stieltjes transform of the bulk spectrum To the bulk of the spectrum if the true eigenvalue is below some critical threshold Phase transition phenomena: estimated eigenvectors orthogonal to true eigenvectors if eigenvalues too small Onatski (): Weak factor model with phase transition phenomena Problem: All models in the literature assume that random processes have mean zero RP-PCA implicitly uses non-zero means of random variables New tools necessary! 9

21 Weak Factor Model Weak Factor Model Assumption : Weak Factor Model Rate: Assume that N T c with < c <. Factors: F are uncorrelated among each other and are independent of e and Λ and have bounded first two moments. ˆµ F := T p F t µf ˆΣF := σ F T T FtF t p Σ F = t= σf K 3 Loadings: The column vectors of the loadings Λ are orthogonally invariant and independent of ɛ and F (e.g. Λ i,k N(, N ) and Λ Λ = I K 4 Residuals: e = ɛσ with ɛ t,i N(, ). The empirical eigenvalue distribution function of Σ converges to a non-random spectral distribution function with compact support and supremum of support b. Largest eigenvalues of Σ converge to b.

22 Weak Factor Model Weak Factor Model Definition: Weak Factor Model Average idiosyncratic noise σ e := trace(σ)/n Denote by λ λ... λ N the ordered eigenvalues of T e e. The Cauchy transform (also called Stieltjes transform) of the eigenvalues is the almost sure limit: G(z) := a.s. lim T N B-function N i= c B(z) :=a.s. lim T N z λ i N i= c =a.s. lim T N trace ( = a.s. lim T N trace (zi N ) T e e) λ i (z λ i ) ( ( (zi N T e e) ) ( ) ) T e e

23 Weak Factor Model Weak Factor Model Intuition: Weak Factor Model Signal matrix for PCA of covariance matrix (γ = ): Σ F + cσ e I K K largest eigenvalues θ PCA,..., θk PCA measure strength of signal Signal matrix for RP-PCA: ( ) Σ F + cσe Σ / F µ F ( + γ) µ F Σ/ F ( + γ) ( + γ)(µ F µ F + cσe ) ( + γ) = + γ K largest eigenvalues θ RP-PCA,..., θk RP-PCA measure strength of signal RP-PCA signal matrix is close to Σ F + ( + γ)µ F µ F + cσ e I K

24 Weak Factor Model Weak Factor Model Theorem : Risk-Premium PCA under weak factor model Assumption ( holds. ) The first K largest eigenvalues ˆθ i i =,..., K of X I T T + γ X satisfy T { ( ) p G ˆθ i θ i if θ i > θ crit = lim z b G(z) b otherwise The correlation of the estimated with the true factors converges to ρ Ĉorr(F, ˆF ) p ρ }{{} Ũ }{{} Ṽ rotation rotation ρ K with ρ i p { +θ i B( ˆθ i )) if θ i > θ crit otherwise 3

25 Weak Factor Model Weak Factor Model Optimal choice or risk premium weight γ Critical value θ crit and function B(.) depend only on the noise distribution and are known in closed-form If µ F and γ > then RP-PCA signals are always larger than PCA signals: θ RP-PCA i > θ PCA i RP-PCA can detect factors that cannot be detected with PCA For θ i > θ crit correlation ρ i is strictly increasing in θ i. The rotation matrices satisfy Ũ Ũ I K and Ṽ Ṽ I K. Ĉorr(F, ˆF ) is not necessarily an increasing function in θ. Based on closed-form expression choose optimal RP-weight γ 4

26 Strong Factor Model Strong Factor Model Strong Factor Model Strong factors affect most assets: e.g. market factor N Λ Λ bounded (after normalizing factor variances) Statistical model: Bai and Ng () and Bai (3) framework Factors and loadings can be estimated consistently and are asymptotically normal distributed RP-PCA provides a more efficient estimator of the loadings Assumptions essentially identical to Bai (3) 5

27 Strong Factor Model Strong Factor Model Asymptotic Distribution (up to rotation) PCA under assumptions of Bai (3): (up to rotation) Asymptotically ˆΛ behaves like OLS regression of F on X. Asymptotically ˆF behaves like OLS regression of Λ on X. RP-PCA under slightly stronger assumptions as in Bai (3): Asymptotically ( ˆΛ behaves ) like OLS regression of FW on XW with W = I T + γ T and is a T vector of s. Asymptotically ˆF behaves like OLS regression of Λ on X. Asymptotic Efficiency Choose RP-weight γ to obtain smallest asymptotic variance of estimators RP-PCA (i.e. γ > ) always more efficient than PCA Optimal γ typically smaller than optimal value from weak factor model RP-PCA and PCA are both consistent 6

28 Strong Factor Model Simplified Strong Factor Model Assumption : Simplified Strong Factor Model Rate: Same as in Assumption Factors: Same as in Assumption 3 Loadings: Λ Λ/N p I K and all loadings are bounded. 4 Residuals: e = ɛσ with ɛ t,i N(, ). All elements and all row sums of Σ are bounded. 7

29 Strong Factor Model Simplified Strong Factor Model Proposition: Simplified Strong Factor Model Assumption holds. Then: The factors and loadings can be estimated consistently. The asymptotic distribution of the factors is not affected by γ. 3 The asymptotic distribution of the loadings is given by T ( H ˆΛ i Λ i ) D N(, Ωi ) Ω i =σe i (Σ F + ( + γ)µ F µ F ( Σ F + ( + γ)µ F µ F ) E[e t,i] =σ e i, H full rank matrix ) ( ) Σ F + ( + γ) µ F µ F 4 γ = is optimal choice for smallest asymptotic variance. γ =, i.e. the covariance matrix, is not efficient. 8

30 Time-varying loadings Time-varying loadings Model with time-varying loadings Observe panel of excess returns and L covariates Z i,t,l : X t,i = F t K g K (Z i,t,,..., Z i,t,l ) + e t,i Loadings are function of L covariates Z i,t,l with l =,..., L e.g. characteristics like size, book-to-market ratio, past returns,... Factors and loading function are latent Idea: Similar to Projected PCA (Fan, Liao and Wang (6)) and Instrumented PCA (Kelly, Pruitt, Su (7)), but we include the pricing error penalty allow for general interactions between covariates 9

31 Time-varying loadings Time-varying loadings Projected RP-PCA (work in progress) Approximate nonlinear function g k (.) by basis functions φ m (.): g k (Z i,t ) = M m= b m,k φ m (Z i,t ) g(z t ) = }{{}}{{} B Φ(Z t ) }{{} K N K M M N Apply RP-PCA to projected data X t = X t Φ(Z t ) X t = F t B Φ(Z t )Φ(Z t ) + e t Φ(Z t ) = F t B + ẽt Special case: φ m = {Zt I m} X characteristics sorted portfolios Obtain arbitrary interactions and break curse of dimensionality by conditional tree sorting projection Intuition: Projection creates M portfolios sorted on any functional form and interaction of covariates Z t. 3

32 Empirical Results Single-sorted portfolios Portfolio Data Monthly return data from 7/963 to /7 (T = 65) for N = 37 portfolios Kozak, Nagel and Santosh (7) data: 37 decile portfolios sorted according to 37 anomalies Factors: RP-PCA: K = 5 and γ =. PCA: K = 5 3 Fama-French 5: The five factor model of Fama-French (market, size, value, investment and operating profitability, all from Kenneth French s website). 4 Proxy factors: RP-PCA and PCA factors approximated with 5% of largest position. 3

33 Empirical Results Single-sorted portfolios In-sample Out-of-sample SR RMS α Idio. Var. SR RMS α Idio. Var. RP-PCA %.45..7% PCA % % Fama-French % % Table: Deciles of 37 single-sorted portfolios from 7/963 to /7 (N = 37 and T = 65): Maximal Sharpe-ratios, root-mean-squared pricing errors and unexplained idiosyncratic variation. K = 5 statistical factors. RP-PCA strongly dominates PCA and Fama-French 5 factors Results hold out-of-sample. 3

34 Empirical Results Single-sorted portfolios: Maximal Sharpe-ratio SR (In-sample) SR (Out-of-sample).7 factor factors 3 factors.6 4 factors 5 factors 6 factors RP-PCA RP-PCA Proxy PCA PCA Proxy RP-PCA RP-PCA Proxy PCA PCA Proxy Figure: Maximal Sharpe-ratios. Spike in Sharpe-ratio for 5 factors 33

35 Empirical Results Single-sorted portfolios: Pricing error.5 RMS (In-sample).5 RMS (Out-of-sample) factor factors.5 3 factors 4 factors.5 5 factors 6 factors RP-PCA RP-PCA Proxy PCA PCA Proxy RP-PCA RP-PCA Proxy PCA PCA Proxy Figure: Root-mean-squared pricing errors. RP-PCA has smaller out-of-sample pricing errors 34

36 Empirical Results Single-sorted portfolios: Idiosyncratic Variation Idiosyncratic Variation (In-sample) Idiosyncratic Variation (Out-of-sample).5 factor factors.5 3 factors 4 factors. 5 factors 6 factors RP-PCA RP-PCA Proxy PCA PCA Proxy RP-PCA RP-PCA Proxy PCA PCA Proxy Figure: Unexplained idiosyncratic variation. Unexplained variation similar for RP-PCA and PCA 35

37 Empirical Results Choice of γ: Maximal Sharpe-ratio SR (In-sample) SR (Out-of-sample) factor factors 3 factors 4 factors 5 factors 6 factors.5.5 SR.4 SR Figure: Maximal Sharpe-ratios for different RP-weights γ and number of factors K Strong increase in Sharpe-ratios for γ. 36

38 Empirical Results Signal of factors: Existence of weak factors PCA RP-PCA (γ = ) FF5 σ σ.7.7. σ3...7 σ σ σ Table: Variance signal for different factors Largest eigenvalues of N ΛΣ F Λ normalized by the average idiosyncratic variance σ e = N N i= σ e,i Higher factors are weak. 37

39 Empirical Results Signal of factors: Existence of weak factors Eigenvalues Eigenvalues Normalized Eigenvalues =- = = =5 = = Normalized Eigenvalues = = =5 = = Number Number Figure: Largest eigenvalues of the matrix N ( T X X + γ X X ). LEFT: Eigenvalues are normalized by division through the average idiosyncratic variance σ e = N N i= σ e,i. RIGHT: Eigenvalues are normalized by the corresponding PCA (γ = ) eigenvalues. Higher factors have weak variance but high mean signal. 38

40 Eigenvalue Difference Empirical Results Number of factors Onatski (): Eigenvalue-ratio test Eigenvalue Differences =- = =5 = = Critical value Number RP-PCA: 5 factors PCA: 4 factors 39

41 Empirical Results Interpreting factors: Generalized correlations with proxies RP-PCA PCA. Gen. Corr.... Gen. Corr Gen. Corr Gen. Corr Gen. Corr Table: Generalized correlations of statistical factors with proxy factors (portfolios of 5% of assets). Problem in interpreting factors: Factors only identified up to invertible linear transformations. Generalized correlations close to measure of how many factors two sets have in common. Proxy factors approximate statistical factors. 4

42 Empirical Results Interpreting factors: 5th proxy factor 5. Proxy RP-PCA Weights 5. Proxy PCA Weights Industry Rel. Reversals (LV). Leverage.6 Industry Rel. Reversals (LV) 9.98 Value-Profitability.4 Value-Momentum-Prof..95 Asset Turnover. Profitability.94 Profitability.99 Industry Mom. Reversals.9 Asset Turnover 9.9 Profitability -.86 Size.89 Profitability Long Run Reversals.85 Industry Mom. Reversals -.9 Sales/Price.84 Industry Rel. Reversals -.9 Size 9.8 Asset Turnover -.95 Value-Momentum-Prof Net Operating Assets -.97 Value-Profitability -.8 Seasonality -. Profitability -.8 Value-Profitability -. Profitability -.89 Short-Term Reversals -. Profitability Industry Rel. Reversals (LV) -.4 Value-Profitability -.94 Industry Rel. Reversals -.5 Profitability Idiosyncratic Volatility -.8 Asset Turnover -.7 Momentum (m) -.8 Asset Turnover

43 Stochastic Discount Factor Single-sorted portfolios: Maximal Sharpe-ratio SR (In-sample) SR (Out-of-sample) factor factors 3 factors 4 factors 5 factors 6 factors RP-PCA (N=74) PCA (N=74) RP-PCA (N=37) PCA (N=37) RP-PCA (N=74) PCA (N=74) RP-PCA (N=37) PCA (N=37) Figure: Maximal RMS Sharpe-ratios (In-sample) for extreme (N = 74) RMS and (Out-of-sample) all (N = 37) deciles Extreme deciles are lowest and highest. decile portfolio for each anomaly (N = 74). RP-PCA (N=74) PCA (N=74) RP-PCA (N=37) PCA (N=37) RP-PCA (N=74) PCA (N=74) RP-PCA (N=37) PCA (N=37) Extreme deciles capture most of the pricing information..5 3 Idiosyncratic Variation (In-sample) 3 Idiosyncratic Variation (Out-of-sample) 4

44 Stochastic Discount Factor All 37 portfolios: PCA..5 Model RP-PCA PCA..5 Weight Decile Loading weights within deciles for all characteristics. Almost all weights on extreme deciles. 43

45 Stochastic Discount Factor Optimal Portfolio with RP-PCA Composition of Stochast Discount Factor (RP-PCA) Industry Rel. Reversals Ind. Rel. Rev. (L.V.) Short-Term Reversals Gross Profitability High Decile Low Decile Asset Growth Seasonality Value-Profitability Asset Turnover Industry Mom. Rev. Momentum (m) Investment/Capital Value (M) Idiosyncratic Volatility Net Operating Assets Value-Mom-Prof. Return on Assets (A) Gross Margins Leverage Momentum-Reversals Return on Book Equity (A) Dividend/Price Value-Momentum Composite Issuance Size Long Run Reversals Sales Growth Sales/Price Cash Flows/Price Earnings/Price Industry Momentum Investment/Assets Price Value (A) Share Volume Accrual Investment Growth Momentum (6m) Figure: Portfolio composition of highest Sharpe ratio portfolio (Stochastic Discount Factor) based on 5 RP-PCA factors. 44

46 Stochastic Discount Factor Optimal Portfolio with RP-PCA (largest positions) Composition of Stochast Discount Factor (RP-PCA).3 High Decile Low Decile Industry Rel. Reversals Ind. Rel. Rev. (L.V.) Short-Term Reversals Gross Profitability Seasonality Value-Profitability Asset Turnover Industry Mom. Rev. Momentum (m) Investment/Capital Value (M) Idiosyncratic Volatility Net Operating Assets Value-Mom-Prof. Return on Assets (A) Figure: Largest portfolios in highest Sharpe ratio portfolio (Stochastic Discount Factor) based on 5 RP-PCA factors. 45

47 Stochastic Discount Factor Optimal Portfolio with PCA Composition of Stochast Discount Factor (PCA) Asset Turnover Value-Profitability Gross Profitability Value-Mom-Prof. High Decile Low Decile Gross Margins Earnings/Price Sales/Price Leverage Idiosyncratic Volatility Return on Assets (A) Return on Book Equity (A) Dividend/Price Momentum (m) Momentum (6m) Long Run Reversals Size Industry Momentum Cash Flows/Price Ind. Rel. Rev. (L.V.) Value-Momentum Momentum-Reversals Value (A) Accrual Share Volume Industry Mom. Rev. Sales Growth Industry Rel. Reversals Value (M) Net Operating Assets Composite Issuance Investment Growth Short-Term Reversals Investment/Capital Investment/Assets Price Asset Growth Seasonality Figure: Portfolio composition of highest Sharpe ratio portfolio (Stochastic Discount Factor) based on 5 PCA factors. 46

48 Stochastic Discount Factor Optimal Portfolio with PCA (largest positions) Composition of Stochast Discount Factor (PCA) Asset Turnover Value-Profitability High Decile Low Decile Gross Profitability Value-Mom-Prof. Earnings/Price Sales/Price Leverage Idiosyncratic Volatility Return on Assets (A) Return on Book Equity (A) Dividend/Price Momentum (m) Momentum (6m) Long Run Reversals Size Figure: Largest portfolios in highest Sharpe ratio portfolio (Stochastic Discount Factor) based on 5 PCA factors. 47

49 Stochastic Discount Factor Optimal Portfolio (SDF) Order portfolios by SR! (top RP-PCA, bottom PCA) indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep accruals value prof aturnover valmom cfp growth lrrev momrev ivol indmom valuem strev size mom roaa lev divp noa invcap roea sgrowth gmargins price shvol Decile Decile indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep accruals value prof aturnover valmom cfp growth lrrev momrev ivol indmom valuem strev size mom roaa lev divp noa invcap roea sgrowth gmargins price shvol Decile Decile 48

50 Time-stability Time-stability of loadings Figure: Time-varying rotated loadings for the first six factors. Loadings are estimated on a rolling window with 4 months. 49

51 Time-stability Time-stability: Generalized correlations RP-PCA (total vs. time-varying) Generalized Correlation st GC nd GC 3rd GC 4th GC 5th GC Time PCA (total vs. time-varying) Generalized Correlation st GC nd GC 3rd GC 4th GC 5th GC Time Figure: Generalized correlations between loadings estimated on the whole time horizon T = 65 and a rolling window with 4. 5

52 Individual Stocks Individual stocks SR (In-sample) SR (Out-of-sample).6.4 factor factors 3 factors 4 factors 5 factors 6 factors RP-PCA PCA RP-PCA PCA Figure: Stock RMS price (In-sample) data (N = 7 and T = 5): RMS Maximal (Out-of-sample) Sharpe-ratios for different number of factors. RP-weight γ = Stock price data from /97 to /6.3 (N = 7 and T = 5).. Out-of-sample performance collapses Constant loading model inappropriate.. 5

53 Individual Stocks Time-stability of loadings of individual stocks Figure: Stock price data: Generalized correlations between loadings estimated on the whole time horizon and a rolling window 5

54 Individual Stocks Time-stability of loadings of individual stocks Generalized Correlation Generalized Correlation RP-PCA (total vs. time-varying) st GC nd GC.5 3rd GC 4th GC 5th GC 6th GC Time PCA (total vs. time-varying) Time Figure: Stock price data (N = 7 and T = 5): Generalized correlations between loadings estimated on the whole time horizon and a rolling window with 4 months. 53

55 Conclusion Conclusion Methodology Estimator for estimating priced latent factors from large data sets Combines variation and pricing criterion function Asymptotic theory under weak and strong factor model assumption Detects weak factors with high Sharpe-ratio More efficient than conventional PCA Empirical Results Strongly dominates PCA of the covariance matrix. Potential to provide benchmark factors for horse races. 54

56 RMS of TS α s: N = 37 indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep Accrual value prof Aturnover RMS valmom In-Sample.6 RP-PCA PCA indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep Accrual value prof Aturnover RMS valmom cfp momrev growth lrrev indmom Out-Of-Sample ivol valuem strev size mom roaa cfp momrev growth lrrev indmom ivol valuem strev size mom roaa lev divp lev divp noa invcap roea gmargins shvol price sgrowth RP-PCA PCA noa invcap roea gmargins shvol price sgrowth A

57 RMS of TS α s: N = indrrevlv indmomrev indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep accrual value prof aturnover indrrev season valprof mom valmomprof inv ciss igrowth sp ep accrual value prof aturnover RMS valmom RMS valmom In-Sample cfp momrev growth lrrev indmom Out-Of-Sample ivol valuem strev size mom roaa cfp momrev growth lrrev indmom ivol valuem strev size mom roaa lev divp lev divp RP-PCA PCA noa invcap roea gmargins shvol price sgrowth.3 RP-PCA PCA noa invcap roea gmargins shvol price sgrowth A

58 Generalized Correlations Generalized Correlations Single-sorted portfolios: Interpreting factors RP-PCA LS-Factors Correlations PCA LS-Factors Correlations GC. GC. 3. GC 4. GC 5. GC 3 4 Number of LS-factors. GC. GC. 3. GC 4. GC 5. GC Number of LS-factors Figure: Generalized correlations of statistical factors with increasing number of long- short anomaly factors. First LS-factor is the market factor and LS-factors added incrementally based on the largest accumulative absolute loading. Long-Short Factors do not span statistical factors. A 3

59 Factors : Long-only ( Mkt ) Factor (sorted by Category) RP-PCA PCA value valuem divp ep cfp sp valmom valmomprof valprof mom mom indmom lrrev strev momrev indmomrev indrrev indrrevlv inv invcap igrowth growth sgrowth prof roaa roea noa gmargins aturnover size ivol accruals ciss lev price season shvol RP-PCA PCA Factor : Long in (almost) all portfolios A 4

60 Factor : Value and value-interaction Factor (sorted by Category) RP-PCA PCA value valuem divp ep cfp sp valmom valmomprof valprof mom mom indmom lrrev strev momrev indmomrev indrrev indrrevlv inv invcap igrowth growth sgrowth prof roaa roea noa gmargins aturnover size ivol accruals ciss lev price season shvol RP-PCA PCA RP-PCA: Long/short in value and value-interaction portfolios PCA: Mostly value portfolios A 5

61 Factor 3: Momentum Factor 3 (sorted by Category) RP-PCA PCA value valuem divp ep cfp sp valmom valmomprof valprof mom mom indmom lrrev strev momrev indmomrev indrrev indrrevlv inv invcap igrowth growth sgrowth prof roaa roea noa gmargins aturnover size ivol accruals ciss lev price season shvol RP-PCA PCA RP-PCA: Momentum-related portfolios PCA: No clear pattern A 6

62 Factor 4: Momentum-Interaction Factor 4 (sorted by Category) RP-PCA PCA value valuem divp ep cfp sp valmom valmomprof valprof mom mom indmom lrrev strev momrev indmomrev indrrev indrrevlv inv invcap igrowth growth sgrowth prof roaa roea noa gmargins aturnover size ivol accruals ciss lev price season shvol RP-PCA PCA RP-PCA and PCA: Momentum and momentum-interaction portfolios A 7

63 Factor 5: High SR Note: Order portfolios by SR instead of categories! Factor 5 (sorted by SR) RP-PCA PCA indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep value prof aturnover valmom cfp accruals momrev growth lrrev indmom ivol strev size mom roaa lev divp noa invcap roea sgrowth PCA valuem price shvol gmargins RP-PCA RP-PCA: Long in highest SR portfolios PCA: Asset Turnover and Profitability A 8

64 Interpretation of factors Factors RP-PCA PCA, long long 3 value & value interactions value 4 momentum? 5 momentum-interaction momentum-interaction 6 high SR asset turnover and profitability Note: Factors are comprised mostly of classic anomaly portfolios A 9

65 All 37 portfolios: RP-PCA indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep accruals value prof aturnover valmom cfp gmargins price shvol momrev growth lrrev indmom ivol valuem strev size mom roaa lev divp noa invcap roea sgrowth A

66 All 37 portfolios: PCA indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep accruals value prof aturnover valmom cfp gmargins price shvol momrev growth lrrev indmom ivol valuem strev size mom roaa lev divp noa invcap roea sgrowth A

67 The Model: Objective function Variation objective function: Minimize the unexplained variation: min Λ,F NT N i= t= T (X ti F t Λ i ) = min Λ NT trace ( (XM Λ ) (XM Λ ) ) s.t. F = X (Λ Λ) Λ Projection matrix M Λ = I N Λ(Λ Λ) Λ Error (non-systematic risk): e = X F Λ = XM Λ Λ proportional to eigenvectors of the first K largest eigenvalues of NT X X minimizes time-series objective function Motivation for PCA A

68 The Model: Objective function Pricing objective function: Minimize cross-sectional expected pricing error: N = N N (Ê[Xi ] Ê[F ]Λ i i= ) N ( T X i T F Λ i i= ) = N trace ( ( T XM Λ ) ( T XM Λ ) ) s.t. F = X (Λ Λ) Λ is vector T of s and thus F T estimates factor mean Why not estimate factors with cross-sectional objective function? Factors not identified Spurious factor detection (Bryzgalova (6)) A 3

69 The Model: Objective function Combined objective function: Risk-Premium-PCA min Λ,F ( (( )) ( NT trace (XM Λ ) (XM Λ + γ ) ( ) ) N trace T XM Λ T XM Λ ( ( = min Λ NT trace M Λ X I + γ ) T ) XM Λ s.t. F = X (Λ Λ) Λ The objective function is minimized by the eigenvectors of the largest eigenvalues of NT X ( I T + γ T ) X. ˆΛ estimator for loadings: proportional to eigenvectors of the first K eigenvalues of NT X ( I T + γ T ) X ˆF estimator for factors: N X ˆΛ = X (ˆΛ ˆΛ) ˆΛ. Estimator for the common component C = F Λ is Ĉ = ˆF ˆΛ A 4

70 Simulation Simulation parameters Parameters as in the empirical application N = 37 and T = 65. Factors: K = 4 or K = Factors F t N(µ F, Σ F ) Σ F = diag(5,.3,., σf ) with σ F {.3,.5,.} SR F = (.,.,.3, sr) with sr {.8,.5,.3,.} Loadings: Λ i N(, I K ) Residuals: e t ɛ t Σ with empirical correlation matrix and σ e =. A 5

71 Simulation 5. Factor. Factor 5 True factor RP-PCA = RP-PCA = RP-PCA = PCA Time Time 5 3. Factor 5 4. Factor Time Time Figure: Sample paths of the cumulative returns of the first four factors and the estimated factor processes.the fourth factor has a variance σ F =.3 and Sharpe-ratio sr =.5. A 6

72 Simulation: Multifactor Model. Factor Corr. (IS) for F =.3. Factor Corr. (OOS) for F =.3. Factor Corr. (IS) for F =.. Factor Corr. (OOS) for F = Corr.6.4 Corr.6.4 Corr.6.4 Corr Factor Corr. (IS) for F =.3. Factor Corr. (OOS) for F =.3. Factor Corr. (IS) for F =.. Factor Corr. (OOS) for F = Corr.6.4 Corr.6.4 Corr.6.4 Corr Factor Corr. (IS) for F =.3 3. Factor Corr. (OOS) for F =.3 3. Factor Corr. (IS) for F =. 3. Factor Corr. (OOS) for F = Corr.6.4 Corr.6.4 Corr.6.4 Corr Factor Corr. (IS) for F =.3 4. Factor Corr. (OOS) for F =.3 4. Factor Corr. (IS) for F =. 4. Factor Corr. (OOS) for F =. Corr Corr Corr SR=.8 SR=.5 SR=.3 SR=. Corr Figure: Correlation of estimated with true factor. A 7

73 Simulation: Multifactor Model SR Factor SR (IS) for F =.3 SR Factor SR (OOS) for F =.3 SR Factor SR (IS) for F =. SR=.8 SR=.5 SR=.3 SR=. SR Factor SR (OOS) for F = Factor SR (IS) for F =.3. Factor SR (OOS) for F =.3. Factor SR (IS) for F =.. Factor SR (OOS) for F = SR.4 SR.4 SR.4 SR Factor SR (IS) for F =.3 3. Factor SR (OOS) for F =.3 3. Factor SR (IS) for F =. 3. Factor SR (OOS) for F = SR.4 SR.4 SR.4 SR Factor SR (IS) for F =.3 4. Factor SR (OOS) for F =.3 4. Factor SR (IS) for F =. 4. Factor SR (OOS) for F = SR.4 SR.4 SR.4 SR Figure: Maximal Sharpe-ratio of factors. A 8

74 Corr Corr Simulation: Weak factor model prediction Statistical Model.5 PCA ( =-) RP-PCA ( =) RP-PCA ( =) RP-PCA ( =5).5..5 F Monte-Carlo Simulation F Correlations between estimated and true factor based on the weak factor model prediction and Monte-Carlo simulations. The Sharpe-ratio of the factor is.8. The normalized variance of the factors corresponds to σ F N. A 9

75 Weak Factor Model: Dependent residuals dependent residuals i.i.d residuals signal Figure: Model-implied values of ρ i ( +θ i B(ˆθ i )) if θ i > σ crit and otherwise) for different signals θ i. The average noise level is normalized in both cases to σ e =. A

76 Simulation: Weak factor model prediction Statistical Model F =.3 Statistical Model F =.5 Statistical Model F =. Statistical Mode Corr.5 SR=.8 SR=.5 SR=.3 SR=. Corr.5 Corr.5 Corr Monte-Carlo Simulation F =.3 Monte-Carlo Simulation F =.5 Monte-Carlo Simulation F =. Monte-Carlo Simula Corr.5 Corr.5 Corr.5 Corr Monte-Carlo Simulation OOS F =.3 Monte-Carlo Simulation OOS F =.5 Monte-Carlo Simulation OOS F =. Monte-Carlo Simulatio Corr.5 Corr.5 Corr.5 Corr Correlation of estimated with true factors for different variances and Sharpe-ratios of the factor and for different RP-weights γ SR Statistical Model F =.3 SR Statistical Model F =.5 SR Statistical Model F =. SR Statistical Mode 5 A

77 Monte-Carlo Simulation OOS F =.3 Monte-Carlo Simulation OOS F =.5 Monte-Carlo Simulation OOS F =. Monte-Carlo Simulatio Simulation: Weak factor model prediction Corr Corr Corr 5 5 Corr.5 5 SR Statistical Model F = SR Statistical Model F = SR Statistical Model F =. 5 5 SR Statistical Mode 5 Monte-Carlo Simulation F =.3 Monte-Carlo Simulation F =.5 Monte-Carlo Simulation F =. Monte-Carlo Simula SR.6.4 SR.6.4 SR.6.4 SR Monte-Carlo Simulation OOS F =.3 Monte-Carlo Simulation OOS F =.5 Monte-Carlo Simulation OOS F =. Monte-Carlo Simulatio SR.6.4 SR.6.4 SR.6.4 SR Sharpe-ratio for different variances and Sharpe-ratios of the factor and for different RP-weights γ. The residuals have the empirical residual correlation matrix. A

78 The Model: Objective function Weighted Combined objective function: Straightforward extension to weighted objective function: min Λ,F NT trace(q (X F Λ ) (X F Λ )Q) + γ ) ( N trace (X F Λ )QQ (X F Λ ) = min trace (M ( Λ Q X I + γ ) Λ T ) XQM Λ s.t. F = X (Λ Λ) Λ Cross-sectional weighting matrix Q Factors and loadings can be estimated by applying PCA to Q X ( I + γ T ) XQ. Today: Only Q equal to inverse of a diagonal matrix of standard deviations. For γ = corresponds to PCA of a correlation matrix. Optimal choice of Q: GLS type argument A 3

79 Weak Factor Model Corollary: Covariance PCA for i.i.d. errors Assumption holds, c and e t,i i.i.d. N(, σ e ). The largest K eigenvalues of S have the following limiting values: ˆλ i p { σ Fi + σ e σ F i (c + + σ e ) σ e ( + c) if σ F i + cσ e > σ crit σ F > cσ e otherwise The correlation between the estimated and true factors converges to ϱ Ĉorr(F, ˆF ) p ϱ K with ϱ i p cσ4 e σ 4 F i + cσ e σ F i + σ4 e σ 4 F i (c c) if σ F i + cσ e > σ crit otherwise A 4

80 Weak Factor Model Example: One-factor model Assume that there is only one factor, i.e. K =. The signal matrix M RP simplifies to ( ) σ M RP = F + cσe σ F µ( + γ) µσ F ( + γ) (µ + cσe )( + γ) and has the eigenvalues: θ, = σ F + cσe + (µ + cσe )( + γ) ± (σf + cσ e + (µ + cσe )( + γ)) 4( + γ)cσe (σf + µ + cσe ) The eigenvector of first eigenvalue θ has the components Ũ, = Ũ, = µσ F ( + γ) (θ (σf + cσ e )) + µ σf ( + γ) θ σ F + cσ e (θ (σ F + cσ e )) + µ σ F ( + γ) A 5

81 Weak Factor Model Corollary: One-factor model The correlation between the estimated and true factor has the following limit: Ĉorr(F, ˆF ) p ρ ρ + ( ρ ) (θ (σ F +cσ e )) + µ σ F (+γ) A 6

82 Strong Factor Model Strong Factor Model Strong factors affect most assets: e.g. market factor N Λ Λ bounded (after normalizing factor variances) Statistical model: Bai and Ng () and Bai (3) framework Factors and loadings can be estimated consistently and are asymptotically normal distributed RP-PCA provides a more efficient estimator of the loadings Assumptions essentially identical to Bai (3) A 7

83 Strong Factor Model Asymptotic Distribution (up to rotation) PCA under assumptions of Bai (3): Asymptotically ˆΛ behaves like OLS regression of F on X. Asymptotically ˆF behaves like OLS regression of Λ on X. RP-PCA under slightly stronger assumptions as in Bai (3): Asymptotically ( ˆΛ behaves ) like OLS regression of FW on XW with W = I T + γ T. Asymptotically ˆF behaves like OLS regression of Λ on X. Asymptotic Expansion Asymptotic expansions (under slightly stronger assumptions as in Bai (3)): ) T (H ˆΛi Λ i = ( F W F ) ( T ) T T F W e i + O p + o N p() ) N (H ˆFt F t = ( N Λ Λ ) ( N ) N Λ et + O p + o T p() with known rotation matrix H. A 8

84 Strong Factor Model Assumption : Strong Factor Model Assume the same assumptions as in Bai (3) (Assumption A-G) hold and in addition ( T T t= Fte ) ( ) t,i D Ω, Ω, T T t= e N(, Ω) Ω = t,i Ω, Ω, A 9

85 Strong Factor Model Theorem : Strong Factor Model Assumption holds and γ [, ). Then: For any choice of γ the factors, loadings and common components can be estimated consistently pointwise. then ( ) T H ˆΛi D Λ i N(, Φ) If T N ( ) ) Φ = Σ F + (γ + )µ F µ F (Ω, + γµ F Ω, + γω,µ F + γ µ F Ω,µ F ( ) Σ F + (γ + )µ F µ F For γ = this simplifies to the conventional case Σ F Ω,Σ F. If N T then the asymptotic distribution of the factors is not affected by the choice of γ. The asymptotic distribution of the common component depends on γ if and only if N T does not go to zero. For T N T (Ĉt,i C t,i ) D N (, F t ΦF t ) A 3

86 Extreme deciles of single-sorted portfolios Portfolio Data Monthly return data from 7/963 to /7 (T = 65) for N = 74 portfolios Kozak, Nagel and Santosh (7) data: 37 decile portfolios sorted according to 37 anomalies Here we take only the lowest and highest decile portfolio for each anomaly (N = 74). Factors: RP-PCA: K = 5 and γ =. PCA: K = 5 3 Fama-French 5: The five factor model of Fama-French (market, size, value, investment and operating profitability, all from Kenneth French s website). 4 Proxy factors: RP-PCA and PCA factors approximated with 8 largest positions. A 3

87 Extreme Deciles Anomaly Mean SD Sharpe-ratio Anomaly Mean SD Sharpe-ratio Accruals - accrual Momentum (m) - mom Asset Turnover - aturnover Momentum-Reversals - momrev Cash Flows/Price - cfp Net Operating Assets - noa Composite Issuance - ciss Price - price Dividend/Price - divp Gross Profitability - prof Earnings/Price - ep Return on Assets (A) - roaa Gross Margins - gmargins Return on Book Equity (A) - roea Asset Growth - growth Seasonality - season Investment Growth - igrowth Sales Growth - sgrowth Industry Momentum - indmom Share Volume - shvol. 6.. Industry Mom. Reversals - indmomrev Size - size Industry Rel. Reversals - indrrev Sales/Price sp Industry Rel. Rev. (L.V.) - indrrevlv Short-Term Reversals - strev Investment/Assets - inv Value-Momentum - valmom Investment/Capital - invcap Value-Momentum-Prof. - valmomprof Idiosyncratic Volatility - ivol Value-Profitability - valprof Leverage - lev Value (A) - value Long Run Reversals - lrrev Value (M) - valuem Momentum (6m) - mom Table: Long-Short Portfolios of extreme deciles of 37 single-sorted portfolios from 7/963 to /7: Mean, standard deviation and Sharpe-ratio. A 3

88 Extreme Deciles In-sample Out-of-sample SR RMS α Idio. Var. SR RMS α Idio. Var. RP-PCA %.5.5.6% PCA.3..3%.4..98% RP-PCA Proxy % % PCA Proxy.33..9%.7.8.% Fama-French % % Table: First and last decile of 37 single-sorted portfolios from 7/963 to /7 (N = 74 and T = 65): Maximal Sharpe-ratios, root-mean-squared pricing errors and unexplained idiosyncratic variation. K = 6 statistical factors. RP-PCA strongly dominates PCA and Fama-French 5 factors Results hold out-of-sample. A 33

89 Eigenvalue Difference Extreme Deciles: Number of factors Onatski (): Eigenvalue-ratio test..8.6 Eigenvalue Differences =- = =5 = = Critical value Number RP-PCA: 5 factors PCA: 4 factors A 34

90 Extreme Deciles: Maximal Sharpe-ratio SR (In-sample) SR (Out-of-sample).7 factor factors 3 factors.6 4 factors 5 factors 6 factors RP-PCA RP-PCA Proxy PCA PCA Proxy RP-PCA RP-PCA Proxy PCA PCA Proxy Figure: Maximal Sharpe-ratios. Spike in Sharpe-ratio for 5 factors A 35

91 Extreme Deciles: Pricing error RMS (In-sample) RMS (Out-of-sample) factor factors. 3 factors.5 4 factors 5 factors.5 6 factors RP-PCA RP-PCA Proxy PCA PCA Proxy RP-PCA RP-PCA Proxy PCA PCA Proxy Figure: Root-mean-squared pricing errors. RP-PCA has smaller out-of-sample pricing errors A 36

92 Extreme Deciles: Idiosyncratic Variation.3 Idiosyncratic Variation (In-sample) factor.3 Idiosyncratic Variation (Out-of-sample) factors.5 3 factors 4 factors.5 5 factors. 6 factors RP-PCA RP-PCA Proxy PCA PCA Proxy RP-PCA RP-PCA Proxy PCA PCA Proxy Figure: Unexplained idiosyncratic variation. Unexplained variation similar for RP-PCA and PCA A 37

93 Extreme Deciles: Maximal Sharpe-ratio SR (In-sample) SR (Out-of-sample) factor factors 3 factors 4 factors 5 factors 6 factors.5.5 SR.4 SR Figure: Maximal Sharpe-ratios for different RP-weights γ and number of factors K A 38

94 Interpreting factors: Generalized correlations with proxies RP-PCA PCA. Gen. Corr.... Gen. Corr Gen. Corr Gen. Corr Gen. Corr Table: Generalized correlations of statistical factors with proxy factors (portfolios of 8 assets). Problem in interpreting factors: Factors only identified up to invertible linear transformations. Generalized correlations close to measure of how many factors two sets have in common. Proxy factors approximate statistical factors. A 39

95 Interpreting factors: 5th proxy factor 5. Proxy RP-PCA Weights 5. Proxy PCA Weights Value.93 Value-Profitability.5 Industry Rel. Reversal.39 Asset Turnover.5 Price.3 Profitability.95 Industry Rel. Reversal (LV).6 Sales/Price.95 Long Run Reversals.5 Long Run Reversals.86 Short Run Reversals -. Value-Profitability -.98 Industry Rel. Reversal (LV) -.34 Profitability -.5 Industry Rel. Reversal -.37 Asset Turnover -.89 A 4

96 Interpreting factors: Composition of proxies RP-PCA divp.53 mom.4 size.4 valuem.93 growth -.46 mom.99 ivol.3 indrrev.39 igrowth -.5 indmomrev.9 valmomprof.89 price.3 ep -.53 mom -.9 mom.84 indrrevlv.6 invcap -.69 valuem -.3 mom.8 lrrev.5 shvol -.7 ivol -.93 price.69 strev -. mom -.3 price -3.5 shvol.65 indrrevlv -.34 ivol -.48 mom -4. indmomrev -.57 indrrev -.37 PCA valuem.9 divp.74 indmom.4 valprof.5 price.5 ivol.69 mom.39 Aturnover.5 divp.6 roea -.64 valmom.8 prof.95 value.4 mom -.65 mom. sp.95 lrrev.6 size -.8 valmomprof. lrrev.86 sp.98 shvol -.9 indmom -.38 valprof -.98 cfp.9 ivol -3.6 mom -.7 prof -.5 mom.88 price -3. mom -.7 Aturnover -.89 Table: Portfolio-composition of proxy factors for first and last decile of 37 single-sorted portfolios: First proxy factors is an equally-weighted portfolio. A 4

97 Extreme Deciles: Time-stability of loadings Figure: Time-varying rotated loadings for the first six factors. Loadings are estimated on a rolling window with 4 months. A 4