Estimating Asset Pricing Factors from Large-Dimensional Panel Data

Transcription

1 Estimating Asset Pricing Factors from Large-Dimensional Panel Data Markus Pelger Martin Lettau 2 Stanford University 2 UC Berkeley January 24th, 27 UC Berkeley Risk Seminar

2 Motivation Motivation: Cochrane (2, Presidential Address) The Challenge of Cross-Sectional Asset Pricing Fundamental insight: Arbitrage Pricing Theory: Expected return of assets should be explained by systematic risk factors. Problem: Chaos in asset pricing factors: Over 33 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: Estimate priced factors: Search for priced factors and separate them from unpriced factors Bring order into factor chaos Summarize the pricing information in a small number of factors Illustrate and explain the flaws of statistical factors based on PCA

3 Motivation Contribution of this paper Contribution New estimator for estimating priced latent factors (that can explain expected returns) from large panel data sets Estimation theory: Asymptotic distribution theory for weak and strong factors Weak assumptions: Approximate factor model and arbitrage-pricing theory Estimator discovers weak factors with high Sharpe-ratios Strongly dominates PCA Empirical results: New factors explain correlation structure and cross-sectional expected returns at the same time New factors have in and out-of sample smaller pricing errors and larger Sharpe-ratios than benchmark factors 2

4 Literature Literature (partial list) Large-dimensional factor models with strong factors Bai (23): Distribution theory Ahn and Horenstein (23), Onatski (2), Bai and Ng (22): Determining the number of factors Fan et al. (23): Sparse matrices in factor modeling Pelger (26), Aït-Sahalia and Xiu (25): High-frequency Large-dimensional factor models with weak factors (based on random matrix theory) Onatski (22): Phase transition phenomena Benauch-Georges and Nadakuditi (2): Perturbation of large random matrices Asset-pricing factors Harvey and Liu (25): Lucky factors Clarke (25): Level, slope and curvature for stocks Kozak, Nagel and Santosh (25): PCA based factors Bryzgalova (26): Spurious factors 3

5 Literature Agenda Introduction ( ) 2 Factor model setup and illustration 3 Statistical model Weak factor model 2 Strong factor model 4 Simulation 5 Empirical results 6 Conclusion 4

6 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 5

7 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 6

8 Model The Model Time-series objective function: Minimize the unexplained variance: min Λ,F NT N i= t= T (X ti F t Λ i ) 2 = 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 7

9 Model The Model Cross-sectional objective function: Minimize cross-sectional expected pricing error: N = N N (Ê[Xi ] Ê[F ]Λ i i= ) 2 N ( T X i T F Λ i i= ) 2 = 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 (26)) 8

10 Model The Model Combined objective function: 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 9

11 Model The Model 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

12 Model The Model Interpretation of Risk-Premium-PCA (RP-PCA): Time- and cross-sectional regression: Combines the time- and cross-sectional criteria functions. Select factors with small cross-sectional alpha s. Protects against spurious factor with vanishing loadings as it requires the time-series errors to be small as well. 2 High Sharpe ratio factors: 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.

13 Model The Model Interpretation of Risk-Premium-PCA (RP-PCA): continued 4 Signal-strengthening: Intuitively the matrix T X ( I T + γ T ) 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. 2

14 Model 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 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, e.g. value factor RP-PCA detects weak factors which cannot be detected by PCA There exists a critical variance level, such that factors with σ 2 F < σ 2 crit cannot be estimated at all with PCA, but can reliably be estimated with RP-PCA. optimal γ relatively large 3

15 Model The Model Strong vs. weak factor models Consequences for eigenvalues of T X X : Strong factors lead to exploding eigenvalues Weak factors lead to large but bounded eigenvalues Empirical evidence (equity data): Strong and weak factors: st eigenvalue typically substantially larger than rest of spectrum (usually x larger than the 2nd) 2nd and 3rd eigenvalues typically stand out, but similar magnitudes as the rest of the spectrum 4

16 Illustration Illustration Illustration: Anomaly-sorted portfolios (Size and accrual) Factors PCA: Estimation based on PCA of correlation matrix, K = 3 2 RP-PCA: Estimation based on PCA of X ( I + γ T ) X (normalized standard deviation of X ), K = 3 and γ = 3 Fama-French 5 factor model: market, size, value, profitability and investment 4 Specific factors: market, size and accrual Data Double-sorted portfolios according to size and accrual (from Kenneth French s website) Monthly return data from July 963 to December 23 (T = 66) for N = 25 portfolios 5

17 Illustration Comparison among estimators Goodness-of-fit-measures: SR: Sharpe ratio of the stochastic discount factor: ( ) µ F Σ F µ F. Cross-sectional pricing error α: Time-series estimator: Intercept of regression: X i = α i + F Λ i + e i Cross-sectional estimator: Regression of E[X ] = E[F ]Λ + α Results the same. This presentation: Time-series regression α. N N i= α i 2 RMS α: Root-mean-squared pricing errors Out-of-sample estimation: Rolling window of years (T=2) to estimate loadings for next month: ( ) ˆα t,i = X t,i Ĉt,i with Ĉt = X t (Λ t Λ t Λ t Λ t ). Fama-MacBeth test-statistic (weighted sum of squared α s, with χ 2 N K distribution under H ). 6

18 Illustration Portfolio Data: In-sample (Size and accrual) SR RMS α Fama-MacBeth RP-PCA PCA Fama-French Specific Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and Fama-MacBeth test statistics. K = 3 statistical factors and risk-premium weight γ =. RP-PCA significantly better than PCA and quantile-sorted factors. 7

19 Illustration Cross-sectional α s for sorted portfolios (Size and Accrual) Pricing Errors Size and Accrual PCA RP-PCA Fama-French 5 Specific RP-PCA avoids large pricing errors due to penalty term. 8

20 Loadings Loadings Loadings Loadings Loadings Loadings Illustration Loadings for statistical factors (Size and Accrual) Loadings of. PCA factor.3 Loadings of 2. PCA factor.4 Loadings of 3. PCA factor Portfolio Loadings of. RP-PCA factor Portfolio Loadings of 2. RP-PCA factor Portfolio Loadings of 3. RP-PCA factor Portfolio Portfolio Portfolio RP-PCA detects accrual factor while 3rd PCA factor is noise. 9

21 Illustration Maximal Incremental Sharpe Ratio PCA RP-PCA Factor Factors Factors Table: Maximal Sharpe-ratio by adding factors incrementally. For the statistical factor estimators we use K = 3 factors and γ =. st and 2nd PCA and RP-PCA factors the same. Better performance of RP-PCA because of third accrual factor. 2

22 Illustration Portfolio Data: Objective function (Size and Accrual) PCA TS RP-PCA TS PCA XS RP-PCA XS Factor Factors Factors Table: Time-series and cross-sectional objective functions. RP-PCA and PCA explain the same amount of variation. PR-PCA explains cross-sectional pricing much better. Motivation for risk-premium weight γ =. 2

23 Illustration Portfolio Data: Out-of-sample (Size and Accrual) Out-of-sample In-sample RP-PCA.97.9 PCA Fama-French 5..2 Specific Table: Root-mean-squared pricing errors. Out-of-sample factors are estimated with a rolling window. K = 3 statistical factors and risk-premium weight γ =. RP-PCA performs better in- and out-of-sample. 22

24 Alpha Illustration Cross-sectional α s out-of-sample (Size and Accrual).35.3 Out-of-sample Pricing Errors Size and Accrual PCA RP-PCA Fama-French 5 Specific Portfolio RP-PCA avoids large pricing errors due to penalty term. 23

25 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) 24

26 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 (22): 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! 25

27 Weak Factor Model Weak Factor Model Assumption : Weak Factor Model Residual matrix can be represented as 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. The supremum of the support is b. 2 The 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 ˆΣ F := σ 2 F T T FtF t p Σ F = t= σf 2 K 3 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 Assume that N T c with < c <. 26

28 Weak Factor Model Weak Factor Model Definition: Weak Factor Model Average idiosyncratic noise σ 2 e := trace(σ)/n Denote by λ λ 2... 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 ) 2 ( ( (zi N T e e) ) 2 ( ) ) T e e 27

29 Weak Factor Model Weak Factor Model Estimator Risk-premium ( PCA (RP-PCA): ) Apply PCA estimation to S γ := X I T T + γ X T PCA : Apply PCA ( to estimated ) covariance matrix S := X I T T X, i.e. γ =. PCA special case of RP-PCA Signal Matrix for Covariance PCA T σf 2 + cσ 2 e M Var = Σ F + cσe 2 I K = σf 2 K + cσe 2 Intuition: Largest K true eigenvalues of S. 28

30 Weak Factor Model Weak Factor Model Lemma: Covariance PCA Assumption holds. Define the critical value σcrit 2 = lim z b. The first K G(z) largest eigenvalues ˆλ i of S satisfy for i =,..., K ( ) p G ˆλ if σ i σ F 2 +cσ 2 F 2 i e i + cσe 2 > σcrit 2 b otherwise The correlation between the estimated and true factors converges to ϱ Ĉorr(F, ˆF ) p ϱ K with ϱ 2 i p { +(σ 2 F i +cσ 2 e )B(ˆλ i )) if σ 2 F i + cσ 2 e > σ 2 crit otherwise 29

31 Weak Factor Model Weak Factor Model Corollary: Covariance PCA for i.i.d. errors Assumption holds, c and e t,i i.i.d. N(, σ 2 e ). The largest K eigenvalues of S have the following limiting values: ˆλ i p { σ 2 Fi + σ2 e σ 2 F i (c + + σ 2 e ) σ 2 e ( + c) 2 if σ 2 F i + cσ 2 e > σ 2 crit σ 2 F > cσ 2 e otherwise The correlation between the estimated and true factors converges to ϱ Ĉorr(F, ˆF ) p ϱ K with ϱ 2 i p cσ4 e σ 4 F i + cσ2 e σ 2 F i + σ4 e σ 4 F i (c 2 c) if σ 2 F i + cσ 2 e > σ 2 crit otherwise 3

32 Weak Factor Model Weak Factor Model Signal Matrix for RP-PCA Signal Matrix for RP-PCA ( ) Σ M RP = F + cσe 2 Σ /2 F µ F ( + γ) µ F Σ /2 F ( + γ) ( + γ)(µ F µ F + cσ2) 2 Define γ = γ + and note that ( + γ) 2 = + γ. Projection on K demeaned factors and on mean operator. Denote by θ... θ K+ the eigenvalues of the signal matrix M RP and by Ũ the corresponding orthonormal eigenvectors : θ Ũ M RP Ũ = θ K+ Intuition: θ,..., θ K+ largest K + true eigenvalues of S γ. 3

33 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 S γ satisfy { ( ) p G ˆθ i θ i if θ i > σcrit 2 = lim z b G(z) b otherwise The correlation of the estimated with the true factors converges to ρ Ĉorr(F, ˆF ) p ( I K ) ρ 2 Ũ. }{{}..... D /2 /2 K ˆΣ ˆF rotation ρ K }{{} rotation with ρ 2 i p { +θ i B( ˆθ i )) if θ i > σcrit 2 otherwise 32

34 Weak Factor Model Weak Factor Model Theorem : continued ˆΣ ˆF =D/2 K ρ ( ρ K ρ 2 ) ρ 2 K D K =diag ((ˆθ )) ˆθK ( ) Ũ IK Ũ D /2 K ρ ρ K 33

35 Weak Factor Model Weak Factor Model Lemma: Detection of weak factors If γ > and µ F, then the first K eigenvalues of M RP are strictly larger than the first K eigenvalues of M Var, i.e. For θ i > σ 2 crit it holds that ˆθ i θ i > θ i > σ 2 F i + cσ 2 e Thus, if γ > and µ F, then ρ i > ϱ i. ρ i θ i > i =,..., K For µ F RP-PCA always better than PCA. 34

36 Weak Factor Model 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 2 M RP = F + cσe 2 σ F µ( + γ) µσ F ( + γ) (µ 2 + cσe 2 )( + γ) and has the eigenvalues: θ,2 = 2 σ2 F + cσe 2 + (µ 2 + cσe 2 )( + γ) ± (σf cσ2 e + (µ 2 + cσe 2 )( + γ)) 2 4( + γ)cσe 2 (σf 2 + µ2 + cσe 2 ) The eigenvector of first eigenvalue θ has the components Ũ, = Ũ,2 = µσ F ( + γ) (θ (σf 2 + cσ2 e )) 2 + µ 2 σf 2 ( + γ) θ σ 2 F + cσ 2 e (θ (σ 2 F + cσ2 e )) 2 + µ 2 σ 2 F ( + γ) 35

37 Weak Factor Model 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 2 + ( ρ2 ) (θ (σ 2 F +cσ2 e ))2 + µ 2 σ 2 F (+γ) 36

38 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 (22) and Bai (23) 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 (23) 37

39 Strong Factor Model Strong Factor Model Asymptotic Distribution (up to rotation) PCA under assumptions of Bai (23): 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 (23): Asymptotically ( ˆΛ behaves ) like OLS regression of FW on XW with W 2 = 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 (23)): ) T (H ˆΛi Λ i = ( F W 2 F ) ( T ) T T F W 2 e i + O p + o N p() ) 2 N (H ˆFt F t = ( N Λ Λ ) ( N ) N Λ et + O p + o T p() with known rotation matrix H. 38

40 Strong Factor Model Strong Factor Model Assumption 2: Strong Factor Model Assume the same assumptions as in Bai (23) (Assumption A-G) hold and in addition ( T T t= Fte ) ( ) t,i D Ω, Ω,2 T T t= e N(, Ω) Ω = t,i Ω 2, Ω 2,2 39

41 Strong Factor Model Strong Factor Model Theorem 2: Strong Factor Model Assumption 2 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 N T ( ) ) Φ = Σ F + (γ + )µ F µ F (Ω, + γµ F Ω 2, + γω,2µ F + γ 2 µ F Ω 2,2µ F ( ) Σ F + (γ + )µ F µ F For γ = this simplifies to the conventional case Σ F Ω,Σ F. 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 ) 4

42 Strong Factor Model Strong Factor Model Example 2: Toy model with i.i.d. residuals and K = i.i.d. Assume K = and e t,i (, σe 2 ). If Assumption 2 holds and T, then N ) D T (ˆΛi Λ i N(, Ω) with ( σ 2 F + µ 2 F ( + γ) 2) Ω = σ 2 e (σf 2 + µ2 F ( + γ))2 Optimal choice minimizing the asymptotic variance is risk-premium weight γ =. Choosing γ =, i.e. the covariance matrix for factor estimation, is not efficient. 4

43 Simulation Simulation Simulation parameters N = 25 and T = 35. Factors: K = 4. Factor represent the market with N(.2, 9): Sharpe-ratio of.4 2. Factor represents an industry factors following N(., ): Sharpe-ratio of.. 3. Factor follows N(.4, ): Sharpe-ratio of Factor has a small variance but high Sharpe-ratio. It follows N(.4,.6): Sharpe-ratio of. Loadings normalized such that N Λ Λ. Λ i, = and Λ i,k N(, ) for k = 2, 3, 4. Errors: Cross-sectional and time-series correlation and heteroskedasticity in the residuals. Half of the variation due to non-systematic risk. 42

44 Simulation Simulation 4. Factor 4 2. Factor 8 3. Factor 4 4. Factor True factor PCA Var PCA Corr RP-PCA RP-PCA Corr Figure: Sample path of the first four factors and the estimated factor processes. γ = 5. 43

45 Simulation Simulation PCA Var PCA Corr RP-PCA RP-PCA Corr. Factor Factor Factor Factor Table: Average root-mean-squared (RMS) errors of estimated factors relative to the true factor processes. γ = 5. 44

46 Simulation Simulation Statistical Model PCA RP-PCA (.=) RP-PCA (.=) RP-PCA (.=5) Monte-Carlo Simulation PCA RP-PCA (.=) RP-PCA (.=) RP-PCA (.=5) Squared correlations between estimated and true factor based on the weak factor model prediction and Monte-Carlo simulations for different variances of the factor. The Sharpe-ratio of the factor is, i.e. the mean equals µ F = σ F. The normalized variance of the factors is σ 2 F N. 45

47 Simulation Weak Factor Model: Dependent residuals dependent residuals i.i.d residuals Figure: Values of ρ 2 i ( if θ +θ i B(ˆθ i )) i > σcrit 2 and otherwise) for different signals θ i. The average noise level is normalized in both cases to σe 2 = and c =. For the correlated residuals we assume that Σ /2 is a Toeplitz matrix with β, β, β, β 2 on the right four off-diagonals with β =.7. 46

48 Empirical Results Portfolio Data Portfolio Data Data Factors Monthly return data from July 963 to December 23 (T = 66) 3 double sorted portfolios (consisting of 25 portfolios) from Kenneth French s website and 49 industry portfolios PCA: K = 3 2 RP-PCA: K = 3 and γ = 3 Fama-French 5 factor model: market, size, value, profitability and investment 4 Specific factors: market + two specific anomaly long-short factors 47

49 Empirical Results Pricing errors α (in-sample) RP-PCA PCA FF 5 Specific Size and BM BM and Investment BM and Operating Profits Size and Accrual Size and Beta Size and Investment Size and Operating Profits Size and Short-Term Reversal Size and Long-Term Reversal Size and Res. Var Size and Total Var Operating Profits and Investment Size and Net Share Iss Industries

50 Empirical Results Pricing errors α (out-of-sample) RP-PCA PCA FF 5 Specific Size and BM BM and Investment BM and Operating Profits Size and Accrual Size and Beta Size and Investment Size and Operating Profits Size and Short-Term Reversal Size and Long-Term Reversal Size and Res. Var Size and Total Var Operating Profits and Investment Size and Net Share Iss Industries

51 Empirical Results Maximum Sharpe-Ratios RP-PCA PCA Specific Size and BM BM and Investment BM and Operating Profits Size and Accrual Size and Beta Size and Investment Size and Operating Profits Size and Short-Term Reversal Size and Long-Term Reversal Size and Res. Var Size and Total Var Operating Profits and Investment Size and Net Share Iss Industries

52 Empirical Results Portfolio Data Portfolio Data Monthly return data from July 963 to December 23 (T = 66) for N = 99 portfolios Novy-Marx and Velikov (24) data: 5 portfolios sorted according to 5 anomolies (same data as in Kozak, Nagel and Santosh (25)) 49 industry portfolios from Kenneth French s website Fama-French 5: The five factor model of Fama-French (market, size, value, investment and operating profitability, all from Kenneth French s website). 2 Specific: Market, value, value-momementum-profitibility and volatility factors. Number of statistical factors K = 4 and γ =. 5

53 Empirical Results Portfolio Data I: 5 Novy-Marx factors and portfolios Size Gross Profitability Value Value Prof Accruals Net Issuance Asset Growth Investment Piotrotski F-Score ValMomProf ValMom Idiosyncratic Vol Momentum Long Run Reversal Beta Arbitrage. 52

54 Empirical Results Portfolio Data: In-sample SR RMS α Fama-MacBeth RP-PCA PCA Fama-French Specific Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and Fama-MacBeth test statistics. K = 4 statistical factors and risk-premium weight γ =. RP-PCA strongly dominates PCA and Fama-French 5 factors Specific factors (Market, Value, Value-Momementum-Profitibility and Volatility) perform similar to RP-PCA. 53

55 Empirical Results Portfolio Data: Out-of-sample Out-of-sample In-sample RP-PCA PCA Fama-French Specific Table: Root-mean-squared pricing errors. Out-of-sample factors are estimated with a rolling window. K = 4 statistical factors and risk-premium weight γ =. RP-PCA performs well in- and out-of-sample. 54

56 Empirical Results Portfolio Data: Interpreting factors PCA RP-PCA. Gen. Corr Gen. Corr Gen. Corr Gen. Corr Table: Generalized Correlations between specific factors and statistical factors. 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. Specific factors: Market, Value, Value-Momementum-Profitability and Volatility factors. Specific factors approximate RP-PCA factors. 55

57 Empirical Results Maximal Incremental Sharpe Ratio PCA RP-PCA Factor Factors Factors Table: Maximal Sharpe-ratio by adding factors incrementally. K = 4 statistical factors and risk-premium weight γ =. 56

58 Empirical Results Portfolio Data: Objective function PCA TS RP-PCA TS PCA XS RP-PCA XS Factor Factors Factors Table: Time-series and cross-sectional objective functions. RP-PCA and PCA explain the same amount of variation. PR-PCA explains cross-sectional pricing much better. Motivation for risk-premium weight γ =. 57

59 Cumulative return Cumulative return Empirical Results Cumulative returns of factors Mkt-Rf RP-PCA RP-PCA 2 RP-PCA 3 RP-PCA Year Mkt-Rf PCA PCA 2 PCA 3 PCA Year 58

60 Extension Extension: 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 Literature (partial list) Projected PCA: Fan, Liao and Wang (26) Dynamic semiparametric factor model: Park, Mammen, Härdle and Borak (29) Nonparametric regression model: Connor and Linton (27) 59

61 Extension Extension: Time-varying loadings Projected RP-PCA (work in progress) Assume additive nonparametric loading model: g k (Z i,t ) = L g k,l (Z i,t,l ) l= Each additive component of g k is estimated by the sieve method. Choose appropriate basis functions φ (.),..., φ D (.) (e.g. splines, polynomial series, kernels, etc.) Define projection P t as regression on L D N matrix φ(z t ) with elements φ d (Z i,t,l ), i =,..., N, l =,..., L, d =,..., D. Apply RP-PCA to projected data X t = X t P t. Empirical results promising: We recover size, value, momentum and volatility factors from individual stock price data 6

62 Conclusion Conclusion Methodology New estimator for estimating priced latent factors from large data sets Combines time-series and cross-sectional 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 estimation based on PCA of the covariance matrix Potential to provide benchmark factors for horse races. Promising empirical results. 6

63 Expected excess return Expected excess return Expected excess return Expected excess return Predicted excess return in-sample (Size and Accrual) In-sample RP-PCA Predicted excess return In-sample PCA Predicted excess return In-sample 5 Fama-French factors Predicted excess return In-sample Specific factors Predicted excess return A

64 Expected excess return Expected excess return Expected excess return Expected excess return Predicted excess return out-of-sample (Size and Accrual) OLS out-of-sample RP-PCA Predicted excess return OLS out-of-sample PCA Predicted excess return OLS out-of-sample 5 Fama-French factors Predicted excess return OLS out-of-sample Specific factors Predicted excess return A 2

65 Strong Factor Model Asymptotic Expansion Asymptotic expansions (under slightly stronger assumptions as in Bai (23)): ) T (H ˆΛi Λ i = ( F W 2 F ) ( T ) T T F W 2 e i + O p + o N p() ) 2 N (H ˆFt F t = ( N Λ Λ ) ( N ) N Λ et + O p + o T p() ) 3 δ (Ĉt,i C t,i = δ ( T Ft F W 2 F ) T T F W 2 e i + ( δ N Λ i N Λ Λ ) N Λ et + o p() ) with H = ( F W 2 F ) ( Λ ˆΛ T N ( ) W 2 = I T + γ. T V TN, δ = min(n, T ) and A 3

66 Simulation parameters Errors Residuals are modeled as e = σ e D T A T ɛa N D N : ɛ is a T N matrix and follows a multivariate standard normal distribution Time-series correlation in errors: A T creates an AR() model with parameter ρ =. Cross-sectional correlation in errors: A N is a Toeplitz-matrix with (β, β, β, β 2 ) on the right four off-diagonals with β =.7 Cross-sectional heteroskedasticity: D N is a diagonal matrix with independent elements following N(,.2) Time-series heteroskedasticity: D T is a diagonal matrix with independent elements following N(,.2) Signal-to-noise ratio: σ 2 e = Parameters produce eigenvalues that are consistent with the data. A 4

67 Simulation True Factors PCA Var PCA Corr RP-PCA PR-PCA Corr SR Table: Maximal Sharpe Ratio with K = 4 factors. γ = 5. True PCA Var PCA Corr RP-PCA RP-PCA Corr. Factor Factor Factor Factor Table: Estimated mean of factors. γ = 5. A 5

68 Simulation True PCA Var PCA Corr RP-PCA RP-PCA Corr. Factor Factor Factor Factor Table: Estimated variance of factors. γ = 5. A 6

69 Fama-MacBeth Test-Statistics: χ 2 22 : 34(95 %) RP-PCA PCA FF 5 Specific Size and BM BM and Investment BM and Operating Profits Size and Accrual Size and Beta Size and Investment Size and Operating Profits Size and Short-Term Reversal Size and Long-Term Reversal Size and Res. Var Size and Total Var Operating Profits and Investment Size and Net Share Iss Industries A 7

70 Expected excess return Expected excess return Expected excess return Expected excess return Predicted excess return in-sample.5.5 In-SampleRP-PCA Predicted excess return In-SamplePCA Predicted excess return In-Sample5 Fama-French factors Predicted excess return In-SampleSpecific factors Predicted excess return A 8

71 Expected excess return Expected excess return Expected excess return Expected excess return Predicted excess return out-of-sample.5.5 OLS out-of-sample RP-PCA Predicted excess return OLS out-of-sample PCA Predicted excess return OLS out-of-sample 5 Fama-French factors Predicted excess return OLS out-of-sample Specific factors Predicted excess return A 9

72 Cumulative return Cumulative return Cumulative returns of optimal portfolios 3 2 Factor 2 Factors 3 Factors 4 Factors RP-PCA Optimal Portfolios Year Factor 2 Factors 3 Factors 4 Factors PCA Optimal Portfolios Year A

73 Portfolio Data: In-sample (BM and Investment) SR RMS α Fama-MacBeth RP-PCA PCA Fama-French Specific Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and Fama-MacBeth test statistics for different set of factors. For the statistical factor estimators we use K = 3 factors and γ =. A

74 Cross-sectional α s for sorted portfolios (BM and Investment).3.25 Pricing Errors BM and Investment PCA RP-PCA Fama-French 5 Specific A 2

75 Loadings Loadings Loadings Loadings Loadings Loadings Loadings for statistical factors (BM and Investment) Loadings of. PCA factor.25 Loadings of 2. PCA factor.6.4 Loadings of 3. PCA factor Portfolio Loadings of. RP-PCA factor Portfolio Loadings of 2. RP-PCA factor Portfolio Loadings of 3. RP-PCA factor Portfolio Portfolio Portfolio A 3

76 Maximal Incremental Sharpe Ratio (BM and Investment) PCA RP-PCA Factor Factors Factors Table: Maximal Sharpe-ratio by adding factors incrementally. For the statistical factor estimators we use K = 3 factors and γ =. A 4

77 Portfolio Data: Objective function (BM and Investment) PCA TS RP-PCA TS PCA XS RP-PCA XS Factor Factors Factors Table: Time-series and cross-sectional objective functions. A 5

78 Portfolio Data: Out-of-sample (BM and Investment) Out-of-sample In-sample RP-PCA PCA Fama-French 5..3 Specific Table: Root-mean-squared pricing errors for different set of factors. Out-of-sample factors are estimated with a rolling window. For the statistical factor estimators we use K = 3 factors and γ =. A 6

79 Alpha Cross-sectional α s out-of-sample (BM and Investment) Out-of-sample Pricing Errors BM and Investment PCA RP-PCA Fama-French 5 Specific Portfolio A 7

80 Expected excess return Expected excess return Expected excess return Expected excess return Predicted excess return in-sample (BM and Investment) In-sample RP-PCA Predicted excess return In-sample PCA Predicted excess return In-sample 5 Fama-French factors Predicted excess return In-sample Specific factors Predicted excess return A 8

81 Expected excess return Expected excess return Expected excess return Expected excess return Predicted excess return out-of-sample (BM and Invest.) OLS out-of-sample RP-PCA Predicted excess return OLS out-of-sample PCA Predicted excess return OLS out-of-sample 5 Fama-French factors Predicted excess return OLS out-of-sample Specific factors Predicted excess return A 9

82 Portfolio Data: In-sample (Size and BM) SR RMS α Fama-MacBeth RP-PCA PCA Fama-French Specific Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and Fama-MacBeth test statistics for different set of factors. For the statistical factor estimators we use K = 3 factors and γ =. A 2

83 Cross-sectional α s for sorted portfolios (Size and BM).6.5 Pricing Errors Size and BM PCA RP-PCA Fama-French 5 Specific A 2

84 Loadings Loadings Loadings Loadings Loadings Loadings Loadings for statistical factors (Size and BM) Loadings of. PCA factor Loadings of 2. PCA factor.6.4 Loadings of 3. PCA factor Portfolio Loadings of. RP-PCA factor Portfolio Loadings of 2. RP-PCA factor Portfolio Loadings of 3. RP-PCA factor Portfolio Portfolio Portfolio A 22

85 Maximal Incremental Sharpe Ratio PCA RP-PCA Factor Factors Factors Table: Maximal Sharpe-ratio by adding factors incrementally. For the statistical factor estimators we use K = 3 factors and γ =. A 23

86 Portfolio Data: Objective function (Size and BM) PCA TS RP-PCA TS PCA XS RP-PCA XS Factor Factors Factors Table: Time-series and cross-sectional objective functions. A 24

87 Portfolio Data: Out-of-sample (Size and BM) Out-of-sample In-sample RP-PCA.7.6 PCA.87.8 Fama-French Specific Table: Root-mean-squared pricing errors for different set of factors. Out-of-sample factors are estimated with a rolling window. For the statistical factor estimators we use K = 3 factors and γ =. A 25

88 Alpha Cross-sectional α s out-of-sample (Size and BM).7.6 Out-of-sample Pricing Errors Size and BM PCA RP-PCA Fama-French 5 Specific Portfolio A 26

89 Generalized Correlation Generalized Correlation Generalized correlations for time-varying loadings (Size and BM) st GC 2nd GC 3rd GC RP-PCA Year PCA st GC 2nd GC 3rd GC Year A 27

90 Generalized Correlation Generalized Correlation Generalized correlations for time-varying loadings (Size and BM).99 st GC 2nd GC 3rd GC RP-PCA Year PCA.995 st GC 2nd GC 3rd GC Year A 28

91 Loadings Loadings Loadings Loadings Loadings Loadings Time-varying loadings (Size and BM) Loadings of. RP-PCA factor.5 Loadings of 2. RP-PCA factor 2 Loadings of 3. RP-PCA factor Portfolio Loadings of. PCA factor Portfolio Portfolio Loadings of 2. PCA factor Portfolio Portfolio Loadings of 3. PCA factor Portfolio A 29

92 Expected excess return Expected excess return Expected excess return Expected excess return Predicted excess return in-sample (Size and BM).5 In-sample RP-PCA Predicted excess return In-sample PCA Predicted excess return In-sample 5 Fama-French factors Predicted excess return In-sample Specific factors Predicted excess return A 3

93 Expected excess return Expected excess return Expected excess return Expected excess return Predicted excess return out-of-sample (Size and BM).5 OLS out-of-sample RP-PCA Predicted excess return OLS out-of-sample PCA Predicted excess return OLS out-of-sample 5 Fama-French factors Predicted excess return OLS out-of-sample Specific factors Predicted excess return A 3

94 Portfolio Data: In-sample (Size and Momentum) SR RMS α Fama-MacBeth RP-PCA PCA Fama-French Specific Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and Fama-MacBeth test statistics for different set of factors. For the statistical factor estimators we use K = 3 factors and γ =. A 32

95 Cross-sectional α s for sorted portfolios (Size and Momentum).7.6 Pricing Errors Size and Short-Term Reversal PCA RP-PCA Fama-French 5 Specific A 33

96 Loadings Loadings Loadings Loadings Loadings Loadings Loadings for statistical factors (Size and Momentum) Loadings of. PCA factor Loadings of 2. PCA factor.4.2 Loadings of 3. PCA factor Portfolio Loadings of. RP-PCA factor Portfolio Loadings of 2. RP-PCA factor Portfolio Loadings of 3. RP-PCA factor Portfolio Portfolio Portfolio A 34

97 Portfolio Data: Out-of-sample (Size and Momentum) Out-of-sample In-sample RP-PCA.7.48 PCA Fama-French Specific.8.2 Table: Root-mean-squared pricing errors for different set of factors. Out-of-sample factors are estimated with a rolling window. For the statistical factor estimators we use K = 3 factors and γ =. A 35

98 Alpha Cross-sectional α s out-of-sample (Size and Momentum) Out-of-sample Pricing Errors Size and Short-Term Reversal PCA RP-PCA Fama-French 5 Specific Portfolio A 36

99 Expected excess return Expected excess return Expected excess return Expected excess return Predicted excess return in-sample (Size and Momentum).5 In-sample RP-PCA Predicted excess return In-sample PCA Predicted excess return In-sample 5 Fama-French factors Predicted excess return In-sample Specific factors Predicted excess return A 37

100 Expected excess return Expected excess return Expected excess return Expected excess return Predicted excess return out-of-sample (Size and Moment.).5 OLS out-of-sample RP-PCA Predicted excess return OLS out-of-sample PCA Predicted excess return OLS out-of-sample 5 Fama-French factors Predicted excess return OLS out-of-sample Specific factors Predicted excess return A 38

101 Portfolio Data: Objective function (Size and Moment.) PCA TS RP-PCA TS PCA XS RP-PCA XS Factor Factors Factors Table: Time-series and cross-sectional objective functions. A 39

102 Maximal Incremental Sharpe Ratio (Size and Moment.) PCA RP-PCA Factor Factors Factors Table: Maximal Sharpe-ratio by adding factors incrementally. For the statistical factor estimators we use K = 3 factors and γ =. A 4

103 Portfolio Data: In-sample (Size and Net Share Iss.) SR RMS α Fama-MacBeth RP-PCA PCA Fama-French Specific Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and Fama-MacBeth test statistics for different set of factors. For the statistical factor estimators we use K = 3 factors and γ =. A 4

104 Cross-sectional α s for sorted portfolios (Size and Net Share Iss.).6.5 Pricing Errors Size and Net Share Iss. PCA RP-PCA Fama-French 5 Specific A 42

105 Loadings Loadings Loadings Loadings Loadings Loadings Loadings for statistical factors (Size and Net Share Iss.) Loadings of. PCA factor -.5 Loadings of 2. PCA factor.4 Loadings of 3. PCA factor Portfolio Loadings of. RP-PCA factor Portfolio Loadings of 2. RP-PCA factor Portfolio Loadings of 3. RP-PCA factor Portfolio Portfolio Portfolio A 43

106 Alpha Cross-sectional α s out-of-sample (Size and Net Share Iss.).6.5 Out-of-sample Pricing Errors Size and Net Share Iss. PCA RP-PCA Fama-French 5 Specific Portfolio A 44

107 Portfolio Data: Out-of-sample (Size and Net Share Iss.) Out-of-sample In-sample RP-PCA.42.5 PCA Fama-French Specific Table: Root-mean-squared pricing errors for different set of factors. Out-of-sample factors are estimated with a rolling window. For the statistical factor estimators we use K = 3 factors and γ =. A 45

108 Expected excess return Expected excess return Expected excess return Expected excess return Predicted excess return in-sample (Size and Shares).5 In-sample RP-PCA Predicted excess return In-sample PCA Predicted excess return In-sample 5 Fama-French factors Predicted excess return In-sample Specific factors Predicted excess return A 46

109 Expected excess return Expected excess return Expected excess return Expected excess return Predicted excess return out-of-sample (Size and Shares).5 OLS out-of-sample RP-PCA Predicted excess return OLS out-of-sample PCA Predicted excess return OLS out-of-sample 5 Fama-French factors Predicted excess return OLS out-of-sample Specific factors Predicted excess return A 47

110 Portfolio Data: Objective function (Size and Shares) PCA TS PCA XS RP-PCA TS RP-PCA XS Factor Factors Factors Table: Time-series and cross-sectional objective functions. A 48

111 Maximal Incremental Sharpe Ratio (Size and Shares) PCA RP-PCA Factor Factors Factors Table: Maximal Sharpe-ratio by adding factors incrementally. For the statistical factor estimators we use K = 3 factors and γ =. A 49