Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Illustration Loadings for statistical factors (Size and Accrual) 1. PCA factor 2. PCA factor 3. PCA factor 0.5 0.5 0.5 Loadings Loadings Loadings 0 0 0 -0.5 -0.5 -0.5 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Portfolio Portfolio Portfolio 1. RP-PCA factor 2. RP-PCA factor 3. RP-PCA factor 0.5 0.5 0.5 Loadings Loadings Loadings 0 0 0 -0.5 -0.5 -0.5 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Portfolio Portfolio Portfolio ⇒ RP-PCA detects accrual factor while 3rd PCA factor is noise. 14
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Illustration Maximal Sharpe ratio (Size and accrual) SR (In-sample) SR (Out-of-sample) 0.35 0.35 =-1 =0 0.3 0.3 =1 =10 =15 0.25 0.25 =20 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 1 factor 2 factors 3 factors 1 factor 2 factors 3 factors Figure: Maximal Sharpe-ratio by adding factors incrementally. ⇒ 1st and 2nd PCA and RP-PCA factors the same. ⇒ RP-PCA detects 3rd factor (accrual) for γ > 10. 15
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Illustration Effect of Risk-Premium Weight γ SR (In-sample) SR (Out-of-sample) 0.4 0.4 1 factor 2 factors SR SR 0.2 3 factors 0.2 0 0 0 5 10 15 20 0 5 10 15 20 RMS (In-sample) RMS (Out-of-sample) 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 5 10 15 20 0 5 10 15 20 Idiosyncratic Variation (In-sample) Idiosyncratic Variation (Out-of-sample) 4 4 Variation Variation 2 2 0 0 0 5 10 15 20 0 5 10 15 20 Figure: Maximal Sharpe-ratios, root-mean-squared pricing errors and unexplained idiosyncratic variation. 16 ⇒ RP-PCA detects 3rd factor (accrual) for γ > 10.
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Weak vs. Strong Factors The Model Strong vs. weak factor models Strong factor model ( 1 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 17
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix 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) 18
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix 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 (2012): 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! 19
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Weak Factor Model Weak Factor Model Assumption 1: Weak Factor Model Rate: Assume that N T → c with 0 < c < ∞ . 1 Factors: F are uncorrelated among each other and are independent of e 2 and Λ and have bounded first two moments. σ 2 · · · 0 F 1 � T µ F := 1 := 1 . . ... p ˆ p T F t F ⊤ . . ˆ F t → µ F Σ F → Σ F = . . t T t =1 σ 2 0 · · · F K Loadings: The column vectors of the loadings Λ are orthogonally 3 invariant and independent of ǫ and F (e.g. Λ i , k ∼ N (0 , 1 N ) and Λ ⊤ Λ = I K Residuals: e = ǫ Σ with ǫ t , i ∼ N (0 , 1). The empirical eigenvalue 4 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 . 20
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Weak Factor Model Weak Factor Model Definition: Weak Factor Model Average idiosyncratic noise σ 2 e := trace (Σ) / N T e ⊤ e . The 1 Denote by λ 1 ≥ λ 2 ≥ ... ≥ λ N the ordered eigenvalues of Cauchy transform (also called Stieltjes transform) of the eigenvalues is the almost sure limit: � � − 1 � N 1 1 1 ( zI N − 1 T e ⊤ e ) G ( z ) := a . s . lim z − λ i = a . s . lim N trace N T →∞ T →∞ i =1 B -function � N c λ i B ( z ) := a . s . lim N ( z − λ i ) 2 T →∞ i =1 �� �� � − 2 � 1 c ( zI N − 1 T e ⊤ e ) T e ⊤ e = a . s . lim N trace T →∞ 21
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Weak Factor Model Weak Factor Model Intuition: Weak Factor Model “Signal” matrix for PCA of covariance matrix ( γ = − 1): Σ F + c σ 2 e I K K largest eigenvalues θ PCA , ..., θ PCA measure strength of signal 1 K “Signal” matrix for RP-PCA: � � Σ 1 / 2 Σ F + c σ 2 F µ F (1 + ˜ γ ) γ ) 2 = 1 + γ e (1 + ˜ F Σ 1 / 2 µ ⊤ (1 + γ )( µ ⊤ F µ F + c σ 2 F (1 + ˜ γ ) e ) K largest eigenvalues θ RP-PCA , ..., θ RP-PCA measure strength of signal 1 K RP-PCA signal matrix is “close” to Σ F + (1 + γ ) µ F µ ⊤ F + c σ 2 e I K 22
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Weak Factor Model Weak Factor Model Theorem 1: Risk-Premium PCA under weak factor model Assumption 1 holds. The first K largest eigenvalues ˆ θ i i = 1 , ..., K of T X ⊤ � � I T + γ ✶✶ ⊤ 1 X satisfy T � G − 1 � � 1 1 if θ i > θ crit = lim z ↓ b ˆ p G ( z ) θ i → θ i b otherwise The correlation of the estimated with the true factors converges to ρ 1 0 0 · · · 0 ρ 2 0 · · · Corr ( F , ˆ � p ˜ ˜ F ) → U . V ... ���� ���� . 0 0 . rotation rotation 0 0 ρ K · · · with � 1 if θ i > θ crit ρ 2 p 1+ θ i B (ˆ θ i )) → i 0 otherwise 23
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix 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 � = 0 and γ > − 1 then RP-PCA signals are always larger than PCA signals: θ RP-PCA > θ PCA i i ⇒ RP-PCA can detect factors that cannot be detected with PCA For θ i > θ crit correlation ρ 2 i is strictly increasing in θ i . U ⊤ ˜ V ⊤ ˜ The rotation matrices satisfy ˜ U ≤ I K and ˜ V ≤ I K . ⇒ � Corr ( F , ˆ F ) is not necessarily an increasing function in θ . ⇒ Based on closed-form expression choose optimal RP-weight γ 24
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix 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) 1 Statistical model: Bai and Ng (2002) and Bai (2003) 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 (2003) 25
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Strong Factor Model Strong Factor Model Asymptotic Distribution (up to rotation) PCA under assumptions of Bai (2003): (up to rotation) Asymptotically ˆ Λ behaves like OLS regression of F on X . Asymptotically ˆ F behaves like OLS regression of Λ on X ⊤ . RP-PCA under slightly stronger assumptions as in Bai (2003): Asymptotically ˆ Λ behaves like OLS regression of FW on XW � � with W 2 = I T + γ ✶✶ ⊤ and ✶ is a T × 1 vector of 1’s . T Asymptotically ˆ F 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 > − 1) always more efficient than PCA Optimal γ typically smaller than optimal value from weak factor model RP-PCA and PCA are both consistent 26
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Strong Factor Model Simplified Strong Factor Model Assumption 2: Simplified Strong Factor Model Rate: Same as in Assumption 1 1 Factors: Same as in Assumption 1 2 p Loadings: Λ ⊤ Λ / N → I K and all loadings are bounded. 3 Residuals: e = ǫ Σ with ǫ t , i ∼ N (0 , 1). All elements and all row sums of 4 Σ are bounded. 27
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Strong Factor Model Simplified Strong Factor Model Proposition: Simplified Strong Factor Model Assumption 2 holds. Then: The factors and loadings can be estimated consistently. 1 The asymptotic distribution of the factors is not affected by γ . 2 The asymptotic distribution of the loadings is given by 3 � � D √ H ˆ T Λ i − Λ i → N (0 , Ω i ) � � − 1 � � Ω i = σ 2 Σ F + (1 + γ ) µ F µ ⊤ Σ F + (1 + γ ) 2 µ F µ ⊤ e i F F � � − 1 Σ F + (1 + γ ) µ F µ ⊤ F E [ e 2 t , i ] = σ 2 e i , H full rank matrix γ = 0 is optimal choice for smallest asymptotic variance. 4 γ = − 1, i.e. the covariance matrix, is not efficient. 28
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Time-varying loadings Time-varying loadings Model with time-varying loadings Observe panel of excess returns and L covariates Z i , t − 1 , l : ⊤ g X t , i = F t ( Z i , t − 1 , 1 , ..., Z i , t − 1 , L ) + e t , i 1 × K K × 1 Loadings are function of L covariates Z i , t − 1 , l with l = 1 , ..., 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 (2016)) and Instrumented PCA (Kelly, Pruitt, Su (2017)), but we include the pricing error penalty allow for general interactions between covariates 29
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Time-varying loadings Time-varying loadings Projected RP-PCA (work in progress) Approximate nonlinear function g k ( . ) by basis functions φ m ( . ): M � = B ⊤ g k ( Z i , t − 1 ) = b m , k φ m ( Z i , t − 1 ) g ( Z t − 1 ) Φ( Z t − 1 ) ���� � �� � � �� � m =1 K × M K × N M × N Apply RP-PCA to projected data ˜ X t = X t Φ( Z t − 1 ) ⊤ X t = F t B ⊤ Φ( Z t − 1 )Φ( Z t − 1 ) ⊤ + e t Φ( Z t − 1 ) ⊤ = F t ˜ ˜ B + ˜ e t Special case: φ m = ✶ { Z t − 1 ∈ 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 30 form and interaction of covariates Z t − 1 .
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Empirical Results Single-sorted portfolios Portfolio Data Monthly return data from 07/1963 to 12/2017 ( T = 650) for N = 370 portfolios Kozak, Nagel and Santosh (2017) data: 370 decile portfolios sorted according to 37 anomalies Factors: RP-PCA : K = 5 and γ = 10. 1 PCA : K = 5 2 Fama-French 5: The five factor model of Fama-French 3 (market, size, value, investment and operating profitability, all from Kenneth French’s website). Proxy factors : RP-PCA and PCA factors approximated with 4 5% of largest position. 31
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Empirical Results Single-sorted portfolios In-sample Out-of-sample SR RMS α Idio. Var. SR RMS α Idio. Var. RP-PCA 0.53 0.14 10.76% 0.45 0.12 12.70% PCA 0.24 0.14 10.66% 0.17 0.14 12.56% Fama-French 5 0.32 0.23 13.56% 0.31 0.21 13.66% Table: Deciles of 37 single-sorted portfolios from 07/1963 to 12/2017 ( N = 370 and T = 650): 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. 32
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Empirical Results Single-sorted portfolios: Maximal Sharpe-ratio SR (In-sample) SR (Out-of-sample) 0.7 1 factor 0.7 2 factors 3 factors 0.6 0.6 4 factors 5 factors 6 factors 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 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
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Empirical Results Single-sorted portfolios: Pricing error RMS (In-sample) RMS (Out-of-sample) 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 1 factor 2 factors 3 factors 0.05 0.05 4 factors 5 factors 6 factors 0 0 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
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Empirical Results Single-sorted portfolios: Idiosyncratic Variation Idiosyncratic Variation (In-sample) Idiosyncratic Variation (Out-of-sample) 1 factor 0.25 0.25 2 factors 3 factors 4 factors 5 factors 0.2 0.2 6 factors 0.15 0.15 0.1 0.1 0.05 0.05 0 0 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
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Empirical Results Choice of γ : Maximal Sharpe-ratio SR (In-sample) SR (Out-of-sample) 0.9 0.9 1 factor 0.8 0.8 2 factors 3 factors 4 factors 0.7 0.7 5 factors 6 factors 0.6 0.6 0.5 0.5 SR SR 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 5 10 15 20 0 5 10 15 20 Figure: Maximal Sharpe-ratios for different RP-weights γ and number of factors K ⇒ Strong increase in Sharpe-ratios for γ ≥ 10. 36
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Empirical Results Signal of factors: Existence of weak factors PCA RP-PCA ( γ = 10) FF5 σ 2 8.05 8.05 8.00 1 σ 2 0.27 0.27 0.21 2 σ 2 0.21 0.21 0.17 3 σ 2 0.14 0.14 0.03 4 σ 2 0.05 0.05 0.02 5 σ 2 0.03 0.03 0.00 6 Table: Variance signal for different factors N ΛΣ F Λ ⊤ normalized by the average Largest eigenvalues of 1 � N idiosyncratic variance σ 2 e = 1 i =1 σ 2 e , i N ⇒ Higher factors are weak. 37
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Empirical Results Signal of factors: Existence of weak factors Eigenvalues Eigenvalues 1.5 =-1 =0 0.25 =0 =1 =1 =5 Normalized Eigenvalues Normalized Eigenvalues 1.4 =5 =10 0.2 =10 =20 =20 1.3 0.15 1.2 0.1 1.1 0.05 0 1 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Number Number � 1 X ⊤ � T X ⊤ X + γ ¯ X ¯ Figure: Largest eigenvalues of the matrix 1 . N LEFT: Eigenvalues are normalized by division through the average � N idiosyncratic variance σ 2 e = 1 i =1 σ 2 e , i . N RIGHT: Eigenvalues are normalized by the corresponding PCA ( γ = − 1) eigenvalues. ⇒ Higher factors have weak variance but high mean signal. 38
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Empirical Results Number of factors Onatski (2010): Eigenvalue-ratio test Eigenvalue Differences 3 =-1 2.5 =0 Eigenvalue Difference =5 =10 2 =20 Critical value 1.5 1 0.5 0 2 4 6 8 10 12 14 16 Number RP-PCA: 5 factors PCA: 4 factors 39
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Empirical Results Interpreting factors: Generalized correlations with proxies RP-PCA PCA 1. Gen. Corr. 1.00 1.00 2. Gen. Corr. 0.99 0.99 3. Gen. Corr. 0.98 0.99 4. Gen. Corr. 0.94 0.94 5. Gen. Corr. 0.77 0.89 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 1 measure of how many factors two sets have in common. ⇒ Proxy factors approximate statistical factors. 40
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Empirical Results Interpreting factors: 5th proxy factor 5. Proxy RP-PCA Weights 5. Proxy PCA Weights Industry Rel. Reversals (LV) 10 1.12 Leverage 10 1.61 Industry Rel. Reversals (LV) 9 0.98 Value-Profitability 10 1.04 Value-Momentum-Prof. 10 0.95 Asset Turnover 10 1.02 Profitability 10 0.94 Profitability 10 0.99 Industry Mom. Reversals 10 0.91 Asset Turnover 9 0.92 Profitability 2 -0.86 Size 10 0.89 Profitability 3 -0.88 Long Run Reversals 10 0.85 Industry Mom. Reversals 1 -0.90 Sales/Price 10 0.84 Industry Rel. Reversals 2 -0.91 Size 9 0.82 Asset Turnover 1 -0.95 Value-Momentum-Prof. 1 -0.79 Net Operating Assets 1 -0.97 Value-Profitability 1 -0.81 Seasonality 1 -1.00 Profitability 2 -0.81 Value-Profitability 1 -1.12 Profitability 1 -0.89 Short-Term Reversals 1 -1.21 Profitability 4 -0.91 Industry Rel. Reversals (LV) 1 -1.24 Value-Profitability 2 -0.94 Industry Rel. Reversals 1 -1.52 Profitability 3 -1.04 Idiosyncratic Volatility 1 -1.81 Asset Turnover 2 -1.17 Momentum (12m) 1 -1.81 Asset Turnover 1 -1.35 41
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Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Stochastic Discount Factor All 370 portfolios: PCA Model 2.0 RP-PCA PCA 1.5 1.0 0.5 Weight 0.0 0.5 1.0 1.5 2.0 1 2 3 4 5 6 7 8 9 10 Decile Loading weights within deciles for all characteristics. ⇒ Almost all weights on extreme deciles. 43
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Stochastic Discount Factor Optimal Portfolio with RP-PCA Composition of Stochast Discount Factor (RP-PCA) High Decile 0.3 Low Decile 0.2 0.1 0 -0.1 -0.2 -0.3 ) s s r ) ) ) ) e s e e ) e . y y y . l M y . A s e s e m e h e e m s A a l h ) h l V l t e v m a s f A z l c m m a a t i t t o n g a l c c a t c c t c ( u t t . i l i v e t ( i t ( ( u i w i i u e i w w s L s i l a i l 2 i l e r g i a i n S s r i r r u 6 b b o R p e i P s s y r t o P r P t s P e r o r ( r n 1 t s r r P n a r P n l c o ( e e a a n a u a t r e t e s u o . o . ( s - e a e i / e u r / / / A c r m r v v v t t r m C o l A m v u d v G s s s e l V A G G i s i u m a l s M v m s a e e e f f e e q n e e w g m / u o a o T / V V o s s t V R R R o u t g L E e o R s l n n e t t t r e r n M A s R I e a o o n n e P t M t c n d M i e r P S e n e s - e S l n M a e e s . . m i i - n m k n a l F m e l l - s e m t t e o o v i - t r h m m s e s e y a a o e i u S a A r s m u r u o i s h y t S R R e s u r t r r G D u o R E s t o A t s c e l n B s r s T o l s o a t l p t e y a e n p r n a g a s e M . - r u M V u m v r d G V v y O e n V n C u v t t d n s n r n s t m o o d n o n e o I u I I o t L n I h I e R o n C d d i I S N M r n I u I t e R Figure: Portfolio composition of highest Sharpe ratio portfolio (Stochastic Discount Factor) based on 5 RP-PCA factors. 44
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Stochastic Discount Factor Optimal Portfolio with RP-PCA (largest positions) Composition of Stochast Discount Factor (RP-PCA) High Decile 0.3 Low Decile 0.2 0.1 0 -0.1 -0.2 -0.3 ) s . s y r ) ) ) y y . l M y . A l V l e v m a s f a a t t t t o . i i i v e t t ( l l l 2 ( i e s L s i a i o R i i l r b b p e t P s r ( r n 1 s e e a a n a a t o ( u s - e v . r . C l m v v t t u m l o A s i s i m a e e e f f / V o a o T o V o s R R R u t g r e r n M A t M t c n P S P e . m e n i l . m t i - n e l - s y e a t e e s e a o r s m u R R e s u r t r r A t s c e o s l n T l o e n a y . r a u p r d - M V u r G V v y O t t d n r n s t s o n e I I o u I t R h i e d d S N n I I Figure: Largest portfolios in highest Sharpe ratio portfolio (Stochastic Discount Factor) based on 5 RP-PCA factors. 45
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Stochastic Discount Factor Optimal Portfolio with PCA Composition of Stochast Discount Factor (PCA) High Decile 0.3 Low Decile 0.2 0.1 0 -0.1 -0.2 -0.3 ) r e e e ) ) ) ) s e . ) l e s ) s e y e y y . y A A e m m e V m s A a . h M e h l s h s t f c c g m l z l m v l s l a c t n i t o t c a c a ( u t a t c t a t t i v l l i i i a i ( ( i u . u e w ( w t e i w l i o i i r r r l r i 2 6 s S r i L s e r u s e n s i r a g b b P P P r t i s y t t R o e p s P o n a P 1 ( r n P ( n r c l r s a o r s n r a a - e t t e e u c o r e u s e a r r / / l e i / ( m e / . e . u r A o a t t m s s v o u d v s v v a l A V m G v l A G v C G u i i g s m m m a s s M f f e e V q n u e w e e V e e / / T o o o n s o s V s t t t a a l L A E e u t R o R o R e R g I t R n n e r r M n o M e n e s t P P n i S c d t M M r n e e s s e k n e n F l - a l . e e S - r t i n i . m a l i t m m s o s - e a o v e m u l - h y e t m m e s a o e e S a s i A r s u o i m R y h u S r R r t u s E r D o r R u r o t e t s G A o l c n B s t t e s s l a o M t l n s p T e e a n r g s a . a u y p e r V u M u d e m - v V G y n n C V d r O v t v s t n m t r n n e o o d n s o n o I o I I R L n o u t C I i n I e h d I d r M N S I u n t I e R Figure: Portfolio composition of highest Sharpe ratio portfolio (Stochastic Discount Factor) based on 5 PCA factors. 46
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Stochastic Discount Factor Optimal Portfolio with PCA (largest positions) Composition of Stochast Discount Factor (PCA) High Decile 0.3 Low Decile 0.2 0.1 0 -0.1 -0.2 -0.3 ) ) s e r y y . e e e e ) ) e f y A A m m l t t o c c g a z v i t ( c l l i i i a i ( i o i i r r r l i 2 6 s S b b P P i s y r P r t P 1 ( r n e a t t e a a - / / e ( r m s v l i / m t t s o u d v u i i g s m f f e e q n u e T o o o V s n l L E u R M a A e t r r i n t P P n S c d t e k n e n - r i n i s - e t o v e m e s a a o u s s u E r o D i m R u o A o l c n B l a o M a n r g r V M V G y u n n s t o e o o R L n i d r I u t e R Figure: Largest portfolios in highest Sharpe ratio portfolio (Stochastic Discount Factor) based on 5 PCA factors. 47
Optimal Portfolio (SDF) Stochastic Discount Factor Intro Model Illustration 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.3 0.2 0.1 0.0 0.1 0.2 Order portfolios by SR! (top RP-PCA, bottom PCA) indrrevlv indrrevlv indmomrev indmomrev indrrev indrrev Weak Factor season season valprof valprof mom12 mom12 valmomprof valmomprof inv inv ciss ciss igrowth igrowth Strong Factor sp sp ep ep accruals accruals value value prof prof aturnover aturnover valmom valmom Time-Var. cfp cfp momrev momrev growth growth lrrev lrrev indmom indmom Portfolios ivol ivol valuem valuem strev strev size size mom mom roaa roaa lev lev divp divp noa noa invcap invcap roea roea sgrowth sgrowth gmargins gmargins price price Stocks shvol shvol Decile 1 Decile 10 Decile 1 Decile 10 Conclusion Appendix 48
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Time-stability Time-stability of loadings Figure: Time-varying rotated loadings for the first six factors. Loadings are estimated on a rolling window with 240 months. 49
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Time-stability Time-stability: Generalized correlations RP-PCA (total vs. time-varying) 1 Generalized Correlation 0.8 0.6 1st GC 0.4 2nd GC 3rd GC 0.2 4th GC 5th GC 0 0 50 100 150 200 250 300 350 400 450 Time PCA (total vs. time-varying) 1 Generalized Correlation 0.8 0.6 1st GC 0.4 2nd GC 3rd GC 0.2 4th GC 5th GC 0 0 50 100 150 200 250 300 350 400 450 Time Figure: Generalized correlations between loadings estimated on the whole time horizon T = 650 and a rolling window with 240. 50
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Individual Stocks Individual stocks SR (In-sample) SR (Out-of-sample) 1 factor 0.6 0.6 2 factors 3 factors 4 factors 0.4 0.4 5 factors 6 factors 0.2 0.2 0 0 RP-PCA PCA RP-PCA PCA Figure: Stock price data ( N = 270 and T = 500): Maximal Sharpe-ratios for different number of factors. RP-weight γ = 10. Stock price data from 01/1972 to 12/2016 ( N = 270 and T = 500) ⇒ Out-of-sample performance collapses ⇒ Constant loading model inappropriate 51
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix 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 52
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Individual Stocks Time-stability of loadings of individual stocks RP-PCA (total vs. time-varying) Generalized Correlation 1 1st GC 2nd GC 0.5 3rd GC 4th GC 5th GC 6th GC 0 0 50 100 150 200 250 300 Time PCA (total vs. time-varying) Generalized Correlation 1 0.5 0 0 50 100 150 200 250 300 Time Figure: Stock price data ( N = 270 and T = 500): Generalized correlations between loadings estimated on the whole time horizon and a rolling window with 240 months. 53
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix 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
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix RMS of TS α ’s: N = 370 RMS In-Sample 0.6 RP-PCA PCA 0.4 0.2 0 v v n f 2 v s p p l e f r m v v l m v e m a v a p a s l v o o f h a o e p h m o p o e h v l e e o 1 n s t s e u f e t e e z a e v o a e n c t r r r i i w u l r v o c r w r o v e r i o o l n o i v w e r r s p m p c r a p o m m r i u t s m d i c g h r i r m d a l o c o l m s r v r p o r a o m v n l n r s d o n e m r c r a l o r d a a r m i s v o g A u m g n v i m g n i v s i m t i d A g n l a i v RMS Out-Of-Sample 0.5 RP-PCA 0.4 PCA 0.3 0.2 0.1 0 a a v v v n f 2 f v s h p p a l e o f r m p v h v m o l m v e m v p a p s o l e h l e e o o 1 o n s s e u e e e e z a e o a e n c v r r w t u r v o c f w t o v e o v v w t e r r s p m p i c i r a l p r r i r s i o l i n c o g i h r i m r a o c o m m o r m u s t m r d v r p o r d a l o m v n l l r s d r o n e r c l o r d a n a r s v m o g A r a g v i m g n m i u v m n m i t i s i d A g l n a i v A 1
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix RMS of TS α ’s: N = 74 RMS In-Sample 0.4 RP-PCA PCA 0.3 0.2 0.1 0 v v v n f 2 v s p p l e f r m p v v m l m v e m a v p a p a s l e l o o f n h a o e h o e o h v e e o 1 s t s e u u f e t e v e z a v o a e n v c t r r s r m r i i w l r v o c r w r o e r i o o l i n c o i i w e m r p p c r a p o m m r m i u t s m d g h r r d a l m o c v o l s r v r r s p o r e a o c n a l n a d o n v m g r r a l o r d r m i s o a u m g n v i m g n m i v s i t i g d a n a l i v RMS Out-Of-Sample 0.3 RP-PCA PCA 0.2 0.1 0 v v 2 v p e v v e a v a a l v n o f f s h p a l o f e r m p v h m o l m m p p s o e h v l e e o 1 o n s t s e u f e t e e z a e v o a e n c t r r r i i w u l r v o c w r o v e r i o o l o v w e r r s p m p c r a p o r r i u t s m d i n c g i h r i r m d a l o c m m o l m s r v r p o r a o m v n l r s d o n e r c l o r d a n a r i s v m o g a r a m g v i m g n m i u v n s i m t i d a g l n a i v A 2
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Single-sorted portfolios: Interpreting factors RP-PCA LS-Factors Correlations PCA LS-Factors Correlations 1 1 0.8 0.8 Generalized Correlations Generalized Correlations 0.6 0.6 0.4 0.4 1. GC 1. GC 2. GC 2. GC 0.2 0.2 3. GC 3. GC 4. GC 4. GC 5. GC 5. GC 0 0 0 10 20 30 40 0 5 10 15 20 25 30 Number of LS-factors 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
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Factors 1: Long-only (“Mkt”) Factor 1 (sorted by Category) 5.0 5.7 3.5 4.9 5.0 5.6 4.8 5.4 5.3 5.4 5.8 4.7 5.8 6.3 6.0 5.0 7.1 5.6 5.2 4.1 5.5 5.0 4.7 4.4 4.3 4.4 5.1 4.2 4.7 5.4 3.1 5.4 4.0 5.1 3.9 5.9 3.1 RP-PCA10 PCA10 4.6 5.5 3.1 4.5 4.7 5.2 4.4 4.7 4.8 5.1 5.3 4.4 5.6 6.5 5.8 4.1 6.8 4.8 4.9 3.9 5.5 4.9 4.8 4.1 4.2 4.3 5.1 4.2 4.5 5.3 2.9 5.4 3.6 5.0 3.9 5.5 2.9 RP-PCA1 4.5 4.6 5.0 5.6 4.8 4.4 4.9 5.0 4.6 6.0 5.5 5.1 5.2 4.4 5.2 3.5 4.6 3.4 5.1 5.7 5.4 5.4 5.5 3.7 5.6 6.0 4.7 4.9 3.6 3.9 6.5 5.3 5.0 4.5 6.6 4.6 6.5 PCA1 4.6 4.6 5.2 6.1 5.0 4.6 5.2 5.4 5.0 6.2 6.6 5.4 5.4 4.8 5.5 4.0 5.2 4.0 5.4 6.0 5.8 5.7 5.7 3.8 6.1 6.4 5.1 5.0 3.8 3.9 7.6 5.7 5.4 4.6 7.0 5.0 7.0 e m p p p p m f f m 2 m v v v v v v v p h h h f a a a s r e l s s v e n l o o o e o o u v e f s 1 e e e e e l n a t t t a e o n z l s e c o e c o r r o o v w w w r v v a i v l i m r r r r r i c o o n i i c l i s a u d m p p m m r t m m r e p g o s i u r h v o o o r r p a v l m l o l s d r r n r s a l a d r n r r r c e a m o o n a v o v n d i g g g r c s v m m i m u m n i s a i t d i g l a n a i v Factor 1: Long in (almost) all portfolios A 4
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Factor 2: Value and value-interaction Factor 2 (sorted by Category) RP-PCA10 1.3 0.5 1.5 1.2 1.1 1.0 1.2 1.0 1.0 0.3 0.3 0.6 0.6 -0.8 0.2 1.3 -0.3 1.1 0.4 0.7 -0.3 0.2 -0.1 -0.2 -0.7 -0.5 -0.8 -0.8 0.0 0.1 0.4 -0.7 0.9 0.8 -0.4 -0.1 0.4 PCA10 2.2 2.9 2.3 1.5 1.9 2.0 0.8 0.1 1.4 -1.0 -1.1 -0.5 2.1 0.7 1.3 -0.1 0.6 0.5 0.6 1.0 0.3 0.9 0.9 -1.2 -1.3 -1.0 -1.1 -1.4 -0.2 1.5 -0.1 -0.6 0.7 1.7 -1.1 -0.5 0.4 -1.2 -1.0 -1.0 -1.5 -1.3 -1.2 -1.4 -1.3 -1.1 -1.2 -2.3 -1.2 -1.2 -0.9 -1.2 -0.8 -1.4 -0.8 -1.1 -1.7 -1.5 -1.5 -1.2 -0.1 -1.2 -0.9 -1.2 -0.2 -0.5 -0.3 -2.5 -1.4 -0.9 -1.2 -1.1 -0.8 -1.7 RP-PCA1 PCA1 -1.7 -1.8 -1.0 -0.4 -1.6 -1.5 -0.3 0.2 -0.5 1.2 1.9 0.9 -1.4 -0.3 -1.1 0.5 -0.2 -0.3 -0.6 -1.6 -0.7 -1.0 -1.1 0.4 0.3 0.7 -0.1 0.5 0.1 -0.6 0.8 -0.5 0.1 -1.7 2.5 0.5 -0.7 e m p p p p m f f m 2 m v v v v v v v p h h h f a a a s r e l s s v e n l o o o e o o u v e f s 1 e e e e e l n a t t t a e o n z l s e c o e c o r r o o v r v a v l i m r r r r r i c w w w o o n i v i i l i s a u d m p p m m e p g s i u c r h r t m m r v o o o r r o p a v l m l o l s d r r r s a l a d n n e o o n r r r r a c a o v m d i g g g r c s v n m m i m u v m n i s a i d g t i a l n a v i RP-PCA: Long/short in value and value-interaction portfolios PCA: Mostly value portfolios A 5
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Factor 3: Momentum Factor 3 (sorted by Category) RP-PCA10 -0.5 -2.3 -0.3 0.2 -0.4 -0.6 1.0 1.5 -0.0 2.0 2.0 1.8 -1.2 -1.2 -0.8 1.9 -0.6 1.0 0.3 0.2 -0.2 -0.2 -0.6 1.5 1.1 1.1 0.8 1.1 0.7 -1.1 1.0 0.3 0.6 -0.4 1.2 0.8 0.4 PCA10 0.4 -0.8 1.7 1.1 0.8 0.1 0.5 -0.2 -0.1 0.1 -0.2 0.5 -1.1 -0.9 -0.9 1.3 -1.3 0.8 0.1 1.0 -0.8 0.0 -0.0 0.6 0.4 0.7 -0.3 0.2 0.5 -1.8 1.7 -0.6 1.2 1.4 0.8 -0.2 1.3 1.0 1.4 0.6 -0.7 0.7 0.8 -0.8 -1.2 -0.1 -2.3 -4.0 -1.9 0.7 -0.1 0.4 -0.8 -0.6 0.1 0.1 0.3 -0.4 0.1 0.4 -0.1 -1.1 -1.2 -0.5 -0.2 -0.2 0.7 -2.9 -0.5 -0.4 1.0 -3.5 -0.8 -0.5 RP-PCA1 PCA1 0.2 0.2 0.3 -1.3 0.0 0.1 -0.3 -0.2 0.3 -1.1 -1.7 -0.5 0.0 -0.3 -0.4 0.9 -0.7 1.1 -0.1 -1.5 -1.0 -0.6 -0.8 0.7 -1.5 -1.6 -0.2 -0.3 0.5 1.0 -3.2 -0.8 0.1 0.0 -3.2 -0.3 -1.9 e m p p p p m f f m 2 m v v v v v v v p h h h f a a a s r e l s s v e n l o o o e o o u v e f s 1 e e e e e l n a t t t a e o n z l s e c o e c o r r o o v w w w r v v a i v l i m r r r r r i c o o n i i l i s a u d m p p m m r t m m r e p g o s i u c r h v o o o r r p a v l m l o l s d r r r s a l a d r n r r r n c e a m o o n a v o v d i g g g r c s v n m m i m u m n i s a i t d i g a l n a i v RP-PCA: Momentum-related portfolios PCA: No clear pattern A 6
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Factor 4: Momentum-Interaction Factor 4 (sorted by Category) RP-PCA10 0.2 -0.7 -1.1 -0.3 -0.1 0.3 1.4 1.9 0.6 1.8 1.8 1.6 1.2 -0.2 0.8 -0.2 0.6 -0.6 0.4 -0.5 0.9 0.4 0.1 -0.5 -0.5 -0.7 0.2 -0.5 -0.4 2.1 -1.2 0.5 -0.7 -1.2 -0.4 0.3 -1.1 0.5 -1.7 -0.0 0.4 0.4 0.3 2.2 2.1 0.5 2.4 2.1 2.4 0.5 -1.1 0.2 0.7 -0.5 -0.3 0.5 0.2 0.6 0.4 0.1 -0.2 -0.4 -0.4 -0.0 -0.4 -0.2 1.2 -0.3 0.1 -0.0 -0.5 0.2 0.1 -0.4 PCA10 RP-PCA1 -0.6 -0.1 -0.5 1.0 -0.4 -0.5 -1.4 -1.4 -0.6 -1.4 -1.2 -1.5 -0.2 0.8 0.5 -1.6 1.0 -0.6 -0.1 0.8 0.6 0.2 0.6 -0.6 1.1 1.5 -0.2 0.8 -0.6 -1.0 2.1 0.4 -0.4 -0.4 1.7 0.4 1.7 PCA1 -0.6 -0.0 -0.5 0.3 -0.5 -0.5 -2.1 -1.9 -0.4 -2.7 -2.7 -2.4 -0.2 0.9 0.4 -1.2 0.9 0.3 -0.1 -0.2 0.1 -0.2 0.1 -0.1 0.3 0.7 -0.4 0.8 -0.4 -0.5 0.3 -0.1 -0.3 -0.5 -0.5 0.4 0.7 e m p p p p m f f m 2 m v v v v v v v p h h h f a a a s r e l s s v e n l o o o o o u v e f s 1 e e e e e l n a t t t a e o n e z l s e c o e o o o v v a v l i c r r m r r r r r i c w w w r o o n i v i i l i s a u d m p p m m e p g s i u c r h r t m m r v r r o a m l l s d r o o o r r p v l a o n n e s a l d o o n r r r r a c a v m d i g g g r s v o n m m m c v i u a m i n i s d g t i a l a n v i RP-PCA and PCA: Momentum and momentum-interaction portfolios A 7
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Factor 5: High SR Note: Order portfolios by SR instead of categories! Factor 5 (sorted by SR) RP-PCA10 1.3 0.6 1.4 0.7 1.2 -0.0 0.6 0.4 0.4 0.3 1.0 0.5 0.2 0.8 0.6 0.6 -0.1 0.5 1.0 0.2 1.3 -0.6 -0.1 1.9 1.0 0.9 -0.5 0.3 0.4 0.3 0.2 0.0 0.2 0.1 0.1 -0.4 0.0 PCA10 0.2 -0.2 -0.0 -0.1 1.2 -0.0 0.6 0.0 -0.1 -0.1 0.9 0.4 -0.4 0.0 1.0 1.1 -0.1 0.1 0.4 -0.1 0.9 -0.1 -0.1 -0.1 0.2 0.9 -0.1 0.5 0.7 0.0 0.1 -0.5 0.6 -0.5 0.0 -0.2 -0.0 -1.3 -0.4 -1.4 -0.6 -0.8 0.3 0.1 -0.3 -0.4 -0.3 -0.1 -0.4 -0.1 -0.0 -0.5 -0.7 0.5 -0.1 -0.6 -0.3 -0.3 0.8 -0.5 -0.1 -1.2 -0.2 1.2 -0.2 0.1 -0.1 -0.4 0.3 -0.0 -0.1 -0.2 1.3 -0.3 RP-PCA1 PCA1 -0.0 0.3 -0.5 0.1 -1.0 -0.1 -0.7 0.1 0.0 -0.3 -0.0 -0.3 0.5 0.4 -1.5 -1.9 0.2 0.3 -0.3 -0.0 0.1 0.1 -0.5 0.3 -0.5 -0.2 -0.0 -0.7 0.4 0.5 -0.5 0.1 -0.6 -0.2 -0.3 0.3 -0.1 v v v n f 2 f v s h p p s e f r m p v h v m l m v e m a v p a p a h s e l o o o o o l e e o 1 n s t s e l u e f e t e e z a e v o a e t n c v r r a r o o v e o v r r s m i i w l v c r w r r i o l i n c o w i i e p p c u a p m m i u s m d g r h m r a o m r t r v r r d l m o r v o l s o r p s e a o n l n r o n r c l o r d a r a d s v m o g r a g i g m c m n v m n i i a u v s m i d t g i a n l a i v RP-PCA: Long in highest SR portfolios PCA: Asset Turnover and Profitability A 8
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Interpretation of factors Factors RP-PCA PCA 1, 2 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
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix All 370 portfolios: RP-PCA 1.7 1.2 1.3 1.0 1.5 0.9 1.4 0.3 0.3 0.1 0.8 0.3 0.1 0.4 1.6 1.1 -0.0 0.0 0.8 -0.0 1.1 -0.0 0.1 0.9 0.5 0.7 0.3 0.9 0.2 -0.0 0.7 -0.4 0.7 0.6 -0.0 -0.1 -0.6 10 1.5 0.7 1.4 0.4 1.1 0.0 0.9 0.3 0.2 0.2 0.5 0.2 0.6 -0.1 1.1 1.0 -0.5 0.2 0.3 0.2 0.4 0.2 0.1 0.7 1.0 0.5 -0.2 0.2 -0.0 -0.2 0.3 -0.3 0.5 -0.1 0.1 -0.1 -0.1 9 1.0 0.4 0.9 0.1 1.0 0.1 0.4 0.2 -0.0 0.1 0.5 0.2 -0.1 -0.3 0.7 0.1 -0.5 -0.2 0.4 -0.1 0.4 -0.2 0.1 -0.2 0.8 0.5 -0.4 0.0 -0.2 -0.3 0.3 -0.1 0.2 0.1 0.0 -0.1 -0.1 8 0.5 0.2 0.5 0.1 0.7 -0.2 0.2 0.2 -0.0 0.3 -0.0 0.0 -0.0 -0.3 0.0 0.2 -0.3 -0.6 0.1 -0.1 0.1 0.0 -0.0 -0.2 0.4 0.4 -0.2 -0.5 -0.3 0.0 0.1 -0.3 -0.3 -0.1 0.1 -0.2 -0.3 7 0.1 0.2 0.2 -0.1 0.7 -0.2 -0.3 -0.0 0.2 0.0 -0.1 -0.2 -0.1 -0.6 -0.2 0.3 0.0 -0.3 0.4 -0.1 0.1 0.6 0.1 -0.7 0.2 0.3 0.0 -0.5 -0.6 0.1 0.0 -0.1 -0.5 -0.5 -0.2 0.2 0.3 6 -0.1 0.1 -0.1 0.0 0.7 -0.1 0.0 0.2 0.3 -0.0 -0.1 -0.3 0.1 -0.6 -0.5 0.3 0.2 -0.2 -0.0 -0.1 -0.0 0.2 0.2 -0.5 0.0 0.2 0.1 -0.5 -0.7 -0.2 0.1 0.0 -0.6 -0.3 0.1 0.2 0.5 5 -0.2 -0.3 -0.4 -0.4 -0.0 -0.0 -0.0 -0.1 -0.0 -0.2 -0.1 -0.1 0.1 -0.8 -1.0 0.2 0.3 -0.1 -0.2 0.1 -0.0 -0.1 -0.1 -0.7 -0.3 0.1 0.3 -0.5 -0.2 -0.2 -0.7 0.3 -0.5 -0.4 -0.2 0.5 0.1 4 -0.6 -0.1 -0.8 -0.3 -0.7 0.0 -0.0 0.2 -0.1 0.1 -0.4 0.0 0.1 0.0 -1.2 -0.2 0.3 -0.2 -0.0 0.3 -0.1 0.2 0.2 -0.4 -0.3 -0.0 0.4 -0.5 -0.5 0.1 -0.0 -0.0 -0.6 -0.2 -0.3 0.3 0.3 3 -0.8 -0.3 -1.1 -0.6 -0.8 -0.4 0.1 -0.2 -0.3 0.3 -0.2 -0.3 0.0 0.3 -1.2 -0.8 0.5 0.5 -0.2 0.4 -0.5 0.1 -0.3 0.0 -1.0 -0.2 0.5 -0.4 0.1 -0.1 -0.4 0.2 -0.6 -0.5 -0.1 0.3 0.0 2 -1.7 -1.1 -1.9 -1.1 -1.4 -1.7 -0.7 -0.7 -0.8 -0.6 0.2 -0.7 -0.3 0.5 -0.9 -1.2 0.2 0.3 -0.7 -0.4 -0.2 -0.1 -1.7 0.5 -1.5 -0.2 0.2 -0.6 0.6 0.0 -1.1 0.7 -0.6 -0.6 -0.4 -0.1 -0.0 1 v v v n f 2 f v s h p p s e f r m p v h v m l m v e m a v p a p a h s e l o o o e o o l e e o 1 n s t s e l u f e t e e z a e v o a e t n c v r r i w a r v o c w o v e o w v r r s m i l r r r i o l i n c o i i e m r p p c u a p o m m r m i u t s m d g r h a o o r v r o p r d l o m r v l l s r s r e a r c n l r d a n r o n m a o a d s v o g c r g v i g m i u v m n m n m i a s t i i d g a n l a i v A 10
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix All 370 portfolios: PCA 0.5 0.5 -0.5 -0.2 1.4 0.9 1.4 0.3 0.2 -0.1 1.2 1.3 -0.7 0.7 0.8 0.9 1.2 0.7 0.4 0.3 1.0 1.1 0.4 -1.0 -0.3 1.1 1.0 -0.0 1.9 1.1 -0.3 0.1 0.3 -0.5 0.4 -0.1 -0.3 10 0.5 0.8 -0.0 -0.0 0.9 0.8 1.0 0.4 0.1 -0.2 1.3 0.9 -0.8 0.4 0.4 1.0 0.7 0.4 0.4 0.1 0.6 1.0 0.2 -0.6 -0.0 1.0 0.6 -0.2 0.9 0.5 -0.7 0.1 0.1 -0.6 0.3 -0.0 0.0 9 0.2 0.7 -0.1 -0.1 0.8 0.7 0.9 0.2 0.5 0.1 1.1 0.7 -0.7 0.3 -0.2 0.0 0.4 0.4 0.4 0.1 0.7 1.0 0.4 -0.3 0.2 0.9 0.7 -0.1 0.5 0.3 0.1 -0.0 0.3 -0.6 0.1 0.1 0.1 8 0.5 0.4 0.1 0.3 0.7 0.4 0.5 -0.4 0.2 -0.3 0.8 0.4 -0.2 0.1 -0.6 0.2 0.1 0.2 0.3 -0.3 0.6 0.7 0.4 -0.5 0.2 0.9 0.6 -0.1 0.2 0.1 -0.1 0.1 0.2 -0.1 0.4 -0.0 0.0 7 0.2 0.5 0.1 0.2 0.4 0.3 0.3 -0.1 0.0 -0.4 0.8 0.4 -0.4 -0.2 -0.4 -0.1 -0.0 0.4 0.2 -0.1 0.5 0.6 0.5 -0.5 0.2 0.7 0.2 -0.3 0.2 0.3 -0.4 -0.1 -0.1 -0.4 0.4 0.1 -0.0 6 0.5 0.4 -0.0 0.2 0.2 0.2 -0.0 -0.4 -0.1 -0.2 0.5 -0.0 0.0 -0.4 -0.5 0.3 0.1 0.2 0.4 0.1 0.3 0.4 0.1 -0.6 0.2 0.5 -0.1 -0.2 -0.1 0.6 0.1 0.3 0.1 -0.2 0.4 0.0 -0.0 5 0.4 0.2 0.1 0.3 0.0 -0.2 -0.4 -0.4 -0.0 0.0 0.3 0.1 0.4 -0.2 -0.8 -0.3 0.1 0.4 0.2 0.0 0.2 -0.0 -0.2 -0.4 0.1 0.4 -0.2 0.0 -0.2 0.4 -0.4 0.2 -0.4 0.1 0.3 -0.3 -0.1 4 0.4 0.1 -0.0 0.4 -0.9 -0.3 -0.6 -0.6 -0.1 -0.2 0.1 0.0 0.1 0.0 -0.9 -0.8 0.0 0.1 0.2 -0.2 0.2 0.0 -0.2 -0.3 -0.0 0.1 -0.3 0.7 -0.2 0.3 -0.3 -0.3 -0.4 0.0 0.2 -0.1 0.0 3 0.1 -0.1 -0.1 0.2 -1.2 -1.0 -0.9 -0.6 0.0 -0.2 -0.2 -1.0 0.1 -0.1 -0.5 -1.2 -0.3 0.3 -0.1 -0.4 -0.0 -0.5 -0.4 -0.1 -0.1 0.1 -0.9 0.8 -0.0 0.1 -0.4 -0.4 -0.3 0.5 0.2 -0.2 -0.7 2 0.5 -0.1 -0.1 0.2 -1.0 -1.7 -1.7 -0.3 0.2 -0.7 -0.6 -0.5 -0.2 -0.3 -0.7 -1.4 -1.2 -0.2 -0.5 -0.6 -0.5 -1.3 -0.8 0.0 -0.0 -0.3 -1.6 -0.5 -0.3 0.0 -0.8 -0.9 -0.4 0.2 -0.3 -0.8 -0.7 1 v v v n f 2 f v s h p p s e f r m p v h v m l m v e m a v p a p a h s e l o o o e o o l e e o 1 n s t s e l u f e t e e z a e v o a e t n c v r r i w a r v o c w o v e o w v r r s m i l r r r i o l i n c o i i e m r p p c u a p o m m r m i u t s m d g r h a o o r v r o p r d l o m r v l l s r s r e a r c n l r d a n r o n m a o a d s v o g c r g v i g m i u v m n m n m i a s t i i d g a n l a i v A 11
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix The Model: Objective function Variation objective function: Minimize the unexplained variation: � N � T 1 ( X ti − F t Λ ⊤ i ) 2 min NT Λ , F i =1 t =1 � � 1 ( XM Λ ) ⊤ ( XM Λ ) s.t. F = X (Λ ⊤ Λ) − 1 Λ ⊤ = min NT trace Λ Projection matrix M Λ = I N − Λ(Λ ⊤ Λ) − 1 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 1 ⇒ Motivation for PCA A 12
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix The Model: Objective function Pricing objective function: Minimize cross-sectional expected pricing error: � � 2 � N 1 E [ X i ] − ˆ ˆ E [ F ]Λ ⊤ i N i =1 � 1 � 2 � N = 1 i ✶ − 1 T X ⊤ T ✶ ⊤ F Λ ⊤ i N i =1 �� 1 � ⊤ � � � 1 = 1 T ✶ ⊤ XM Λ T ✶ ⊤ XM Λ s.t. F = X (Λ ⊤ Λ) − 1 Λ ⊤ N trace ✶ is vector T × 1 of 1’s and thus F ⊤ ✶ estimates factor mean T Why not estimate factors with cross-sectional objective function? Factors not identified Spurious factor detection (Bryzgalova (2016)) A 13
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix The Model: Objective function Combined objective function: Risk-Premium-PCA �� 1 � ⊤ � � � 1 �� �� 1 + γ 1 ( XM Λ ) ⊤ ( XM Λ T ✶ ⊤ XM Λ T ✶ ⊤ XM Λ min NT trace N trace Λ , F � M Λ X ⊤ � T ✶✶ ⊤ � � 1 I + γ s.t. F = X (Λ ⊤ Λ) − 1 Λ ⊤ = min NT trace XM Λ Λ The objective function is minimized by the eigenvectors of the NT X ⊤ � T ✶✶ ⊤ � I T + γ 1 largest eigenvalues of X . ˆ Λ estimator for loadings: proportional to eigenvectors of the first K NT X ⊤ � T ✶✶ ⊤ � 1 I T + γ eigenvalues of X ˆ N X ˆ Λ = X (ˆ Λ ⊤ ˆ Λ) − 1 ˆ 1 Λ ⊤ . F estimator for factors: Estimator for the common component C = F Λ is ˆ C = ˆ F ˆ Λ ⊤ A 14
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Simulation Simulation parameters Parameters as in the empirical application N = 370 and T = 650. Factors: K = 4 or K = 1 Factors F t ∼ N ( µ F , Σ F ) Σ F = diag (5 , 0 . 3 , 0 . 1 , σ 2 F ) with σ 2 F ∈ { 0 . 03 , 0 . 05 , 0 . 1 } SR F = (0 . 12 , 0 . 1 , 0 . 3 , sr ) with sr ∈ { 0 . 8 , 0 . 5 , 0 . 3 , 0 . 2 } Loadings: Λ i ∼ N (0 , I K ) Residuals: e t ∼ ǫ t Σ with empirical correlation matrix and σ 2 e = 1. A 15
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Simulation 1. Factor 2. Factor 150 20 True factor RP-PCA =0 15 100 RP-PCA =10 RP-PCA =20 10 PCA 50 5 0 0 -50 -5 0 50 100 150 200 250 0 50 100 150 200 250 Time Time 3. Factor 4. Factor 25 25 20 20 15 15 10 10 5 5 0 0 -5 -5 0 50 100 150 200 250 0 50 100 150 200 250 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 2 F = 0 . 03 and Sharpe-ratio sr = 0 . 5. A 16
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Simulation: Multifactor Model 2 2 2 2 1. Factor Corr. (IS) for F =0.03 1. Factor Corr. (OOS) for F =0.03 1. Factor Corr. (IS) for F =0.1 1. Factor Corr. (OOS) for F =0.1 0.8 0.8 0.8 0.8 Corr Corr Corr Corr 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 2 2 2 2 2. Factor Corr. (IS) for F =0.03 2. Factor Corr. (OOS) for F =0.03 2. Factor Corr. (IS) for F =0.1 2. Factor Corr. (OOS) for F =0.1 0.8 0.8 0.8 0.8 Corr Corr Corr Corr 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 2 2 2 2 3. Factor Corr. (IS) for F =0.03 3. Factor Corr. (OOS) for F =0.03 3. Factor Corr. (IS) for F =0.1 3. Factor Corr. (OOS) for F =0.1 0.8 0.8 0.8 0.8 Corr Corr Corr Corr 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 2 2 2 2 4. Factor Corr. (IS) for F =0.03 4. Factor Corr. (OOS) for F =0.03 4. Factor Corr. (IS) for F =0.1 4. Factor Corr. (OOS) for F =0.1 0.8 0.8 0.8 0.8 SR=0.8 Corr Corr Corr Corr 0.6 0.6 0.6 0.6 SR=0.5 SR=0.3 0.4 0.4 0.4 0.4 SR=0.2 0.2 0.2 0.2 0.2 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Figure: Correlation of estimated with true factor. A 17
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Simulation: Multifactor Model 2 1. Factor SR (OOS) for 2 2 1. Factor SR (OOS) for 2 1. Factor SR (IS) for F =0.03 F =0.03 1. Factor SR (IS) for F =0.1 F =0.1 0.8 0.8 0.8 0.8 SR=0.8 0.6 0.6 0.6 0.6 SR=0.5 SR SR SR SR 0.4 0.4 0.4 0.4 SR=0.3 SR=0.2 0.2 0.2 0.2 0.2 0 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 2 2. Factor SR (OOS) for 2 2 2. Factor SR (OOS) for 2 2. Factor SR (IS) for F =0.03 F =0.03 2. Factor SR (IS) for F =0.1 F =0.1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 SR SR SR SR 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 2 3. Factor SR (OOS) for 2 2 3. Factor SR (OOS) for 2 3. Factor SR (IS) for F =0.03 F =0.03 3. Factor SR (IS) for F =0.1 F =0.1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 SR SR SR SR 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 2 4. Factor SR (OOS) for 2 2 4. Factor SR (OOS) for 2 4. Factor SR (IS) for F =0.03 F =0.03 4. Factor SR (IS) for F =0.1 F =0.1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 SR SR SR SR 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Figure: Maximal Sharpe-ratio of factors. A 18
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Simulation: Weak factor model prediction Statistical Model 1 PCA ( =-1) Corr 0.5 RP-PCA ( =0) RP-PCA ( =10) RP-PCA ( =50) 0 0 0.05 0.1 0.15 2 F Monte-Carlo Simulation 1 Corr 0.5 0 0 0.05 0.1 0.15 2 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 0.8. The normalized variance of the factors corresponds to σ 2 F · N . A 19
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Weak Factor Model: Dependent residuals 1 0.8 0.6 2 0.4 0.2 dependent residuals i.i.d residuals 0 0 10 20 30 40 50 signal Figure: Model-implied values of ρ 2 1 θ i )) if θ i > σ 2 i ( crit and 0 1+ θ i B (ˆ otherwise) for different signals θ i . The average noise level is normalized in both cases to σ 2 e = 1. A 20
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Simulation: Weak factor model prediction 2 2 2 Statistical Model F =0.03 Statistical Model F =0.05 Statistical Model F =0.1 1 1 1 SR=0.8 Corr Corr Corr Corr SR=0.5 0.5 0.5 0.5 SR=0.3 SR=0.2 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 2 2 2 Monte-Carlo Simulation F =0.03 Monte-Carlo Simulation F =0.05 Monte-Carlo Simulation F =0.1 1 1 1 Corr Corr Corr Corr 0.5 0.5 0.5 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 2 2 2 Monte-Carlo Simulation OOS F =0.03 Monte-Carlo Simulation OOS F =0.05 Monte-Carlo Simulation OOS F =0.1 1 1 1 Corr Corr Corr Corr 0.5 0.5 0.5 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Correlation of estimated with true factors for different variances and Sharpe-ratios of the factor and for different RP-weights γ . A 21
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Simulation: Weak factor model prediction 2 2 2 Statistical Model F =0.03 Statistical Model F =0.05 Statistical Model F =0.1 0.8 0.8 0.8 0.6 0.6 0.6 SR SR SR 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 2 2 2 Monte-Carlo Simulation F =0.03 Monte-Carlo Simulation F =0.05 Monte-Carlo Simulation F =0.1 0.8 0.8 0.8 0.6 0.6 0.6 SR SR SR 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 2 2 2 Monte-Carlo Simulation OOS F =0.03 Monte-Carlo Simulation OOS F =0.05 Monte-Carlo Simulation OOS F =0.1 0.8 0.8 0.8 0.6 0.6 0.6 SR SR SR 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 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 22
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix The Model: Objective function Weighted Combined objective function: Straightforward extension to weighted objective function: 1 NT trace ( Q ⊤ ( X − F Λ ⊤ ) ⊤ ( X − F Λ ⊤ ) Q ) min Λ , F � � + γ 1 ✶ ⊤ ( X − F Λ ⊤ ) QQ ⊤ ( X − F Λ ⊤ ) ⊤ ✶ N trace � M Λ Q ⊤ X ⊤ � T ✶✶ ⊤ � � I + γ s.t. F = X (Λ ⊤ Λ) − 1 Λ ⊤ = min Λ trace XQM Λ Cross-sectional weighting matrix Q Factors and loadings can be estimated by applying PCA to Q ⊤ X ⊤ � T ✶✶ ⊤ � I + γ XQ . Today: Only Q equal to inverse of a diagonal matrix of standard deviations. For γ = − 1 corresponds to PCA of a correlation matrix. Optimal choice of Q : GLS type argument A 23
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Weak Factor Model Corollary: Covariance PCA for i.i.d. errors Assumption 1 holds, c ≥ 1 and e t , i i.i.d. N (0 , σ 2 e ). The largest K eigenvalues of S − 1 have the following limiting values: � F > √ c σ 2 σ 2 σ 2 ( c + 1 + σ 2 if σ 2 F i + c σ 2 e > σ 2 crit ⇔ σ 2 F i + e e ) p ˆ σ 2 e λ i → e (1 + √ c ) 2 Fi σ 2 otherwise The correlation between the estimated and true factors converges to ̺ 1 0 · · · . . ... � p Corr ( F , ˆ . . F ) → . . 0 ̺ K · · · with 1 − c σ 4 e σ 4 Fi if σ 2 F i + c σ 2 e > σ 2 p ̺ 2 crit 1+ c σ 2 + σ 4 → e e ( c 2 − c ) i σ 2 σ 4 Fi Fi 0 otherwise A 24
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Weak Factor Model Example: One-factor model Assume that there is only one factor, i.e. K = 1. The “signal matrix” M RP simplifies to � σ 2 � F + c σ 2 σ F µ (1 + ˜ γ ) e M RP = ( µ 2 + c σ 2 µσ F (1 + ˜ γ ) e )(1 + γ ) and has the eigenvalues: θ 1 , 2 =1 e + ( µ 2 + c σ 2 2 σ 2 F + c σ 2 e )(1 + γ ) � ± 1 e + ( µ 2 + c σ 2 F + µ 2 + c σ 2 e )(1 + γ )) 2 − 4(1 + γ ) c σ 2 ( σ 2 F + c σ 2 e ( σ 2 e ) 2 The eigenvector of first eigenvalue θ 1 has the components µσ F (1 + ˜ γ ) ˜ � U 1 , 1 = e )) 2 + µ 2 σ 2 ( θ 1 − ( σ 2 F + c σ 2 F (1 + γ ) θ 1 − σ 2 F + c σ 2 ˜ e U 1 , 2 = � e )) 2 + µ 2 σ 2 ( θ 1 − ( σ 2 F + c σ 2 F (1 + γ ) A 25
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Weak Factor Model Corollary: One-factor model The correlation between the estimated and true factor has the following limit: ρ 1 � p Corr ( F , ˆ F ) → � ( θ 1 − ( σ 2 F + c σ 2 e )) 2 +1 ρ 2 1 + (1 − ρ 2 1 ) µ 2 σ 2 F (1+ γ ) A 26
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Strong Factor Model Strong Factor Model Strong factors affect most assets: e.g. market factor N Λ ⊤ Λ bounded (after normalizing factor variances) 1 Statistical model: Bai and Ng (2002) and Bai (2003) 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 (2003) A 27
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Strong Factor Model Asymptotic Distribution (up to rotation) PCA under assumptions of Bai (2003): Asymptotically ˆ Λ behaves like OLS regression of F on X . Asymptotically ˆ F behaves like OLS regression of Λ on X . RP-PCA under slightly stronger assumptions as in Bai (2003): Asymptotically ˆ Λ behaves like OLS regression of FW on XW � � with W 2 = I T + γ ✶✶ ⊤ . T Asymptotically ˆ F behaves like OLS regression of Λ on X . Asymptotic Expansion Asymptotic expansions (under slightly stronger assumptions as in Bai (2003)): � � � √ � √ � 1 � − 1 H ⊤ ˆ T F ⊤ W 2 F 1 T F ⊤ W 2 e i + O p T T Λ i − Λ i = + o p (1) 1 √ N � � � √ � √ � 1 � − 1 H ⊤− 1 ˆ 1 N Λ ⊤ Λ N Λ ⊤ e ⊤ N N F t − F t = t + O p + o p (1) 2 √ T with known rotation matrix H . A 28
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Strong Factor Model Assumption 2: Strong Factor Model Assume the same assumptions as in Bai (2003) (Assumption A-G) hold and in addition � � � T � Ω 1 , 1 � 1 t =1 F t e t , i √ Ω 1 , 2 D T � T → N (0 , Ω) Ω = 1 Ω 2 , 1 Ω 2 , 2 t =1 e t , i √ T A 29
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Strong Factor Model Theorem 2: Strong Factor Model Assumption 2 holds and γ ∈ [ − 1 , ∞ ). Then: For any choice of γ the factors, loadings and common components can be estimated consistently pointwise. � � √ √ H ⊤ ˆ D T If → 0 then Λ i − Λ i → N (0 , Φ) T N � � − 1 � � Σ F + ( γ + 1) µ F µ ⊤ Ω 1 , 1 + γµ F Ω 2 , 1 + γ Ω 1 , 2 µ F + γ 2 µ F Ω 2 , 2 µ F Φ = F � � − 1 Σ F + ( γ + 1) µ F µ ⊤ · F For γ = − 1 this simplifies to the conventional case Σ − 1 F Ω 1 , 1 Σ − 1 F . √ N If → 0 then the asymptotic distribution of the factors is not affected T 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 → 0 � � √ � � ˆ D 0 , F ⊤ t Φ F t T C t , i − C t , i → N A 30
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Extreme deciles of single-sorted portfolios Portfolio Data Monthly return data from 07/1963 to 12/2017 ( T = 650) for N = 74 portfolios Kozak, Nagel and Santosh (2017) data: 370 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 = 10. 1 PCA : K = 5 2 Fama-French 5: The five factor model of Fama-French 3 (market, size, value, investment and operating profitability, all from Kenneth French’s website). Proxy factors : RP-PCA and PCA factors approximated with 8 4 largest positions. A 31
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Extreme Deciles Anomaly Mean SD Sharpe-ratio Anomaly Mean SD Sharpe-ratio Accruals - accrual 0.37 3.20 0.12 Momentum (12m) - mom12 1.28 6.91 0.19 Asset Turnover - aturnover 0.40 3.84 0.10 Momentum-Reversals - momrev 0.47 4.82 0.10 Cash Flows/Price - cfp 0.44 4.38 0.10 Net Operating Assets - noa 0.15 5.44 0.03 Composite Issuance - ciss 0.46 3.31 0.14 Price - price 0.03 6.82 0.00 Dividend/Price - divp 0.2 5.11 0.04 Gross Profitability - prof 0.36 3.41 0.11 Earnings/Price - ep 0.57 4.76 0.12 Return on Assets (A) - roaa 0.21 4.07 0.05 Gross Margins - gmargins 0.02 3.34 0.01 Return on Book Equity (A) - roea 0.08 4.40 0.02 Asset Growth - growth 0.33 3.46 0.10 Seasonality - season 0.81 3.94 0.21 Investment Growth - igrowth 0.37 2.69 0.14 Sales Growth - sgrowth 0.05 3.59 0.01 Industry Momentum - indmom 0.49 6.17 0.08 Share Volume - shvol 0.00 6.00 0.00 Industry Mom. Reversals - indmomrev 1.18 3.48 0.34 Size - size 0.29 4.81 0.06 Industry Rel. Reversals - indrrev 1.00 4.11 0.24 Sales/Price sp 0.53 4.26 0.13 Industry Rel. Rev. (L.V.) - indrrevlv 1.34 3.01 0.44 Short-Term Reversals - strev 0.36 5.27 0.07 Investment/Assets - inv 0.49 3.09 0.16 Value-Momentum - valmom 0.51 5.05 0.10 Investment/Capital - invcap 0.13 5.02 0.03 Value-Momentum-Prof. - valmomprof 0.84 4.85 0.17 Idiosyncratic Volatility - ivol 0.56 7.22 0.08 Value-Profitability - valprof 0.76 3.84 0.20 Leverage - lev 0.24 4.58 0.05 Value (A) - value 0.50 4.57 0.11 Long Run Reversals - lrrev 0.46 5.02 0.09 Value (M) - valuem 0.43 5.89 0.07 Momentum (6m) - mom 0.35 6.27 0.06 Table: Long-Short Portfolios of extreme deciles of 37 single-sorted portfolios from 07/1963 to 12/2017: Mean, standard deviation and Sharpe-ratio. A 32
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Extreme Deciles In-sample Out-of-sample SR RMS α Idio. Var. SR RMS α Idio. Var. RP-PCA 0.57 0.17 10.40% 0.50 0.15 12.06% PCA 0.30 0.22 10.30% 0.24 0.20 11.98% RP-PCA Proxy 0.58 0.17 10.40% 0.50 0.15 11.97% PCA Proxy 0.33 0.22 11.09% 0.27 0.18 12.10% Fama-French 5 0.32 0.30 13.56% 0.31 0.26 13.66% Table: First and last decile of 37 single-sorted portfolios from 07/1963 to 12/2017 ( N = 74 and T = 650): 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
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Extreme Deciles: Number of factors Onatski (2010): Eigenvalue-ratio test Eigenvalue Differences 1.2 =-1 1 =0 Eigenvalue Difference =5 =10 0.8 =20 Critical value 0.6 0.4 0.2 0 2 4 6 8 10 12 14 16 Number RP-PCA: 5 factors PCA: 4 factors A 34
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Extreme Deciles: Maximal Sharpe-ratio SR (In-sample) SR (Out-of-sample) 0.7 1 factor 0.7 2 factors 3 factors 0.6 0.6 4 factors 5 factors 6 factors 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 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
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Extreme Deciles: Pricing error RMS (In-sample) RMS (Out-of-sample) 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 1 factor 0.1 0.1 2 factors 3 factors 4 factors 0.05 0.05 5 factors 6 factors 0 0 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
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Extreme Deciles: Idiosyncratic Variation Idiosyncratic Variation (In-sample) Idiosyncratic Variation (Out-of-sample) 0.3 0.3 1 factor 2 factors 3 factors 0.25 0.25 4 factors 5 factors 6 factors 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 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
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Extreme Deciles: Maximal Sharpe-ratio SR (In-sample) SR (Out-of-sample) 0.9 0.9 1 factor 0.8 0.8 2 factors 3 factors 4 factors 0.7 0.7 5 factors 6 factors 0.6 0.6 0.5 0.5 SR SR 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 5 10 15 20 0 5 10 15 20 Figure: Maximal Sharpe-ratios for different RP-weights γ and number of factors K A 38
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Interpreting factors: Generalized correlations with proxies RP-PCA PCA 1. Gen. Corr. 1.00 1.00 2. Gen. Corr. 0.99 0.99 3. Gen. Corr. 0.95 0.97 4. Gen. Corr. 0.95 0.94 5. Gen. Corr. 0.71 0.86 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 1 measure of how many factors two sets have in common. ⇒ Proxy factors approximate statistical factors. A 39
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Interpreting factors: 5th proxy factor 5. Proxy RP-PCA Weights 5. Proxy PCA Weights Value 10 1.93 Value-Profitability 10 1.25 Industry Rel. Reversal 10 1.39 Asset Turnover 10 1.15 Price 1 1.31 Profitability 10 0.95 Industry Rel. Reversal (LV) 10 1.26 Sales/Price 10 0.95 Long Run Reversals 10 1.25 Long Run Reversals 10 0.86 Short Run Reversals 1 -1.22 Value-Profitability 1 -0.98 Industry Rel. Reversal (LV) 1 -1.34 Profitability 1 -1.51 Industry Rel. Reversal 1 -1.37 Asset Turnover 1 -1.89 A 40
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Interpreting factors: Composition of proxies RP-PCA divp 10 1.53 mom12 10 2.04 size 10 2.14 valuem10 1.93 growth 1 -1.46 mom 10 1.99 ivol 1 2.13 indrrev 10 1.39 igrowth 1 -1.51 indmomrev 10 1.90 valmomprof 10 1.89 price 1 1.31 ep 1 -1.53 mom 1 -2.29 mom12 10 1.84 indrrevlv 10 1.26 invcap 1 -1.69 valuem 10 -2.32 mom 10 1.82 lrrev 10 1.25 shvol 1 -1.72 ivol 1 -2.93 price 1 1.69 strev 1 -1.22 mom12 1 -2.32 price 1 -3.51 shvol 1 1.65 indrrevlv 1 -1.34 ivol 1 -2.48 mom12 1 -4.00 indmomrev 1 -1.57 indrrev 1 -1.37 PCA valuem 10 2.91 divp 10 1.74 indmom 10 2.42 valprof 10 1.25 price 1 2.52 ivol 10 1.69 mom 10 2.39 Aturnover 10 1.15 divp 10 2.26 roea 1 -1.64 valmom 10 2.18 prof 10 0.95 value 10 2.24 mom12 1 -1.65 mom12 10 2.12 sp 10 0.95 lrrev 10 2.06 size 10 -1.82 valmomprof 10 2.12 lrrev 10 0.86 sp 10 1.98 shvol 1 -1.90 indmom 1 -2.38 valprof 1 -0.98 cfp 10 1.92 ivol 1 -3.16 mom12 1 -2.70 prof 1 -1.51 mom12 1 1.88 price 1 -3.21 mom 1 -2.71 Aturnover 1 -1.89 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 41
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Extreme Deciles: Time-stability of loadings Figure: Time-varying rotated loadings for the first six factors. Loadings are estimated on a rolling window with 240 months. A 42
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Extreme Deciles: Time-stability: Generalized correlations RP-PCA (total vs. time-varying) 1 Generalized Correlation 0.8 0.6 1st GC 0.4 2nd GC 3rd GC 0.2 4th GC 5th GC 0 0 50 100 150 200 250 300 350 400 450 Time PCA (total vs. time-varying) 1 Generalized Correlation 0.8 0.6 1st GC 0.4 2nd GC 3rd GC 0.2 4th GC 5th GC 0 0 50 100 150 200 250 300 350 400 450 Time Figure: Generalized correlations between loadings estimated on the whole time horizon T = 650 and a rolling window with 240. A 43
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix Portfolio categories Category Portfolios value 1-6 value interaction 7-9 momentum 10-13 reversal 14-18 investment 19-23 profitability 24-26 other 27- 37 A 44
Intro Model Illustration Weak Factor Strong Factor Time-Var. Portfolios Stocks Conclusion Appendix 10 highest SR 10 lowest SR Portfolio Mean SR Portfolio Mean SR Ind. Rel. Rev. (L.V.) 1.33 0.44 Return on Assets (A) 0.21 0.05 Industry Mom. Rev. 1.18 0.33 Leverage 0.23 0.05 Industry Rel. Reversals 1.00 0.24 Dividend/Price 0.20 0.03 Seasonality 0.81 0.20 Net Operating Assets 0.15 0.02 Value-Profitability 0.75 0.19 Investment/Capital 0.12 0.02 Momentum (12m) 1.28 0.18 Return on Book Equity (A) 0.08 0.01 Value-Mom-Prof. 0.84 0.17 Gross Margins 0.01 0.00 Investment/Assets 0.48 0.15 Share Volume 0.00 0.00 Composite Issuance 0.45 0.13 Price 0.02 0.00 Investment Growth 0.37 0.13 Sales Growth 0.04 0.00 A 45
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