Factors that Fit the Time Series and Cross-Section of Stock Returns

Transcription

1 Factors that Fit the Time Series and Cross-Section of Stock Returns Martin Lettau Markus Pelger UC Berkeley Stanford University November 3th 8 NBER Asset Pricing Meeting

2 Motivation Motivation Fundamental question: What are risk factors and how are they priced? Current state of the literature: Factor zoo with 3+ potential asset pricing factors! Goal of this paper: Bring order into factor chaos Summarize the pricing information of a large number of assets with a small number of factors Conventional methods (Principal Component Analysis PCA): Only time-series co-movement, ignores risk premia Ross Arbitrage Pricing Theory (APT) links time series and cross section: Risk premia depend on exposure to non-diversifiable factors Our method: PCA + APT = Risk Premium PCA (RP-PCA) RP-PCA estimates factors that explain the time-series variation explain the cross-section of risk premia 3 have high Sharpe-ratios

3 Model A factor model of asset returns Observe excess returns X nt of N assets over T time periods: X nt = F t B n + e nt n =,..., N t =,..., T X }{{} T N = }{{} F T K B }{{} K N + e }{{} T N T : time-series observation (large) N: test assets (large) K: systematic factors (fixed) F, B and e are unknown

4 Model A statistical model of asset returns: Systematic factors Systematic and non-systematic risk: Var(X) = B Var(F) B + }{{} Var(e) }{{} systematic non systematic Systematic factors explain large portion of variance Idiosyncratic risk only weakly correlated Estimation via standard PCA: Minimize the unexplained variance min B,F NT N n= t= T (X nt F t B n ) Easy to estimate: Eigenvectors/values of covariance matrix of X PCA factors capture common co-movements... but usually do not fit cross-sectional mean returns and low SR Example: Industry factors 3

5 Model A statistical model of asset returns: Risk premia Arbitrage-Pricing Theory (APT, Ross 976): The expected excess return is explained by the risk-premium of the factors: E[X n] = B n E[F] Systematic factors should explain the cross-section of expected returns Estimation: Minimize cross-sectional pricing error min B,F N N ( Xn F B i n= ) with X n = T T t= Xnt 4

6 Model Risk-Premium PCA (RP-PCA) estimator Risk-Premium PCA: PCA + APT=RP-PCA: min B,F N T (X nt F t B n ) + γ N ( Xn NT N F ) B n n= t= n= }{{}}{{} unexplained TS variation XS pricing error γ is the weight on the APT mean restriction Estimation: Apply standard PCA to the matrix with overweighted mean T X X + γ X X. Special case: γ = : (Standard) PCA on covariance matrix 5

7 Model Interpretation of RP-PCA Combines variation and pricing error criterion functions: Protects against spurious factor with vanishing loadings as it requires the time-series errors to be small as well. Penalized PCA: Search for factors explaining the time-series but penalizes low Sharpe-ratios (consistent with APT) 3 Information interpretation: (GMM interpretation) PCA of a covariance matrix uses only the second moment. RP-PCA combines first and second moments efficiently. 4 Signal-strengthening: Intuitively the matrix X X + γ X X converges to T ( B Var(F) + ( + γ)e[f]e[f] ) B + Var(e) The signal of weak factors with a small variance can be pushed up by their mean with the right γ. 6

8 Model Theoretical results Types of factors: Strong factors affect all assets: MKT Weak factors affect either some assets strongly or all asset weakly : Long-short factors Theory: Companion paper (Lettau and Pelger (8)) Paper: Strong factors: RP-PCA more efficient RP-PCA with γ = is optimal but PCA works as well Weak factors: RP-PCA dominates PCA RP-PCA is able to detect factors that are missed by PCA Optimal γ Novel results based on random matrix theory for non-zero means Criterion to select number of factors Extensive simulation study, in-sample and out-of-sample 7

9 Illustration Illustration (Size and accrual) Model Out-of-sample SR RMS α Idio.Var. RP-PCA PCA. 6.7 FF-long/sort.. 7. Table: K = 3 factors and γ =. Monthly returns of 5 double-sorted portfolios SR = Sharpe-ratio of mean-variance efficient portfolio RMSα = Root Mean Squared XS pricing error RP-PCA factors have higher SRs than PCA factors and FF factors RP-PCA model has lower pricing errors than PCA and FF factors RP-PCA has < % higher idiosyncratic TS variance than PCA 8

10 Illustration Loadings for statistical factors: Size and accrual portfolios RP-PCA: SR =. SR =.3 SR 3 =.3 PCA: SR =. SR =. SR 3 =. 9

11 Empirical Results Large Cross-section of Single-Sorted Portfolios Kozak, Nagel and Santosh (7) data: 37 decile portfolios sorted according to 37 anomalies Monthly return data from 7/963 to /7 (T = 65) for N = 37 portfolios Two cases: N = 74 extreme / decile portfolios All N = 37 portfolios Factors: RP-PCA: K = 5 and γ =. PCA: K = 5 3 Fama-French 5

12 Empirical Results Single-sorted portfolios Model Out-of-sample SR RMS α Idio.Var. RP-PCA 5..7% PCA.7.56% Fama-French % Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and unexplained idiosyncratic variation. K = 5 factors and γ =. RP-PCA strongly dominates PCA and Fama-French 5 factors RP-PCA has larger SR and smaller pricing errors RP-PCA explains (almost) the same variation as PCA Results hold out-of-sample.

13 Empirical Results Maximal Sharpe-ratio Figure: Maximal Sharpe-ratios for extreme (N = 74) and all (N = 37) deciles. RP-PCA: 5th factor (green) has big effect on SR PCA: higher order factors contribute less to SR Extreme deciles capture most of the pricing information

14 Empirical Results Choice of γ: Maximal Sharpe-ratio Figure: Maximal Sharpe-ratios for different γ and K for N = 37. Strong increase in Sharpe-ratios for γ and K = 5. 3

15 Empirical Results Unexplained Variation Figure: Idiosyncratic variation for extreme (N = 74) and all (N = 37) deciles. RP-PCA captures (almost) as much common variation as PCA factors 4

16 Empirical Results RMS of Pricing Errors α s: N = indrrevlv indmomrev indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep accrual value prof aturnover indrrev season valprof mom valmomprof inv ciss igrowth sp ep accrual value prof aturnover RMS valmom RMS valmom In-Sample cfp momrev growth lrrev indmom Out-Of-Sample ivol valuem strev size mom roaa cfp momrev growth lrrev indmom ivol valuem strev size mom roaa lev divp lev divp RP-PCA PCA noa invcap roea gmargins shvol price sgrowth.3 RP-PCA PCA noa invcap roea gmargins shvol price sgrowth Characteristics sorted according to Sharpe-ratios of LS portfolios High SR characteristics significantly better priced by RP-PCA 5

17 Empirical Results Composition of factors..5 Model RP-PCA PCA..5 Weight Decile Loading weights within deciles for all characteristics. Almost all weights on extreme deciles. 6

18 Empirical Results Stochastic Discount Factors for N = 74 Order portfolios by SR! (top RP-PCA, bottom PCA) indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep accruals value prof aturnover valmom cfp growth lrrev momrev ivol indmom valuem strev size mom roaa lev divp noa invcap roea sgrowth gmargins price shvol Decile Decile indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep accruals value prof aturnover valmom cfp growth lrrev momrev ivol indmom valuem strev size mom roaa lev divp noa invcap roea sgrowth gmargins price shvol Decile Decile 7

19 Empirical Results RP-PCA vs. PCA Factors RP-PCA PCA Mean Variance SR Mean Variance SR. Factor Factor Factor Factor Factor Normalize loadings B B = I K. Fifth RP-PCA factor weak and high SR 8

20 Empirical Results Interpretation of factors RP-PCA PCA. Factor long-only market proxy long-only market proxy. Factor value/growth + interactions value/growth 3. Factor momentum+profitability? 4. Factor momentum+momentum interaction momentum 5. Factor high SR long/short portfolios? Note: Factors are comprised mostly of classic anomaly portfolios 9

21 Generalized Correlations Generalized Correlations Empirical Results Generalized Correlation with Ad-Hoc Long/Short Portfolios RP-PCA LS-Factors Correlations PCA LS-Factors Correlations GC. GC. 3. GC 4. GC 5. GC 3 4 Number of LS-factors. GC. GC. 3. GC 4. GC 5. GC Number of LS-factors Figure: Generalized correlations of statistical factors with increasing number of long- short anomaly factors. First LS-factor is the market factor and LS-factors added incrementally based on the largest accumulative absolute loading. Ad-hoc long-short factors do not span statistical factors.

22 Time-stability Time-stability of loadings Figure: Time-varying rotated loadings for the first six factors. Loadings are estimated on a rolling window with 4 months. RP-PCA stable over time

23 Extension Next steps So far: Pre-sorted portfolios, constant factor-weights and loadings Goal: Construct factors using individual stocks without any pre-sorts Obstacle: Factor loadings of individual stocks are likely to vary over time Example: Momentum exposure Current work: Time-varying loadings General idea behind all time-varying factor models Create portfolios = projection based on characteristics Estimate constant loading model on projected data 3 Invert model to obtain loadings that vary with characteristics Our approach More general portfolios (projection) based on decision trees RP-PCA to extract factors on tree portfolios Invert model with second dimension reduction

24 Conclusion Conclusion Methodology PCA-based estimator with cross-sectional mean information Easy to implement: Conventional PCA on nd-moment matrix plus mean term RP-PCA can detect weak factors that are missed by standard PCA More efficient than conventional PCA Empirical Results RP-PCA dominates standard PCA RP-PCA higher-order factors with high SRs RP-PCA loadings put more weight on high-sr characteristics Robust in out-of-sample estimation 5 economically meaningful asset pricing factors Redundancy in characteristic information 3

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

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

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

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

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

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

31 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 A 7

32 Time-stability of loadings of individual stocks Generalized Correlation Generalized Correlation RP-PCA (total vs. time-varying) st GC nd GC.5 3rd GC 4th GC 5th GC 6th GC Time PCA (total vs. time-varying) Time Figure: Stock price data (N = 7 and T = 5): Generalized correlations between loadings estimated on the whole time horizon and a rolling window with 4 months. A 8

33 Portfolio data: Average Pricing Errors Figure: Maximal Sharpe-ratios for extreme (N = 74) and all (N = 37) deciles. RP-PCA has smaller pricing errors. A 9

34 Single-sorted portfolios In-sample Out-of-sample SR RMS α Idio. Var. SR RMS α Idio. Var. RP-PCA.53.76% 5..7% PCA.66%.7.56% Fama-French % % Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and unexplained idiosyncratic variation. K = 5 factors and γ =. RP-PCA strongly dominates PCA and Fama-French 5 factors Results hold out-of-sample. A

35 factors Intro.6 Model Illustration 3 factors Empirical.6 Results Conclusion Appendix 4 factors.5 5 factors.5 6 factors.3.3 Portfolio data: Average Pricing Errors... RP-PCA (N=74) PCA (N=74) RP-PCA (N=37) PCA (N=37). RP-PCA (N=74) PCA (N=74) RP-PCA (N=37) PCA (N=37) RMS (In-sample) RMS (Out-of-sample) RP-PCA (N=74) PCA (N=74) RP-PCA (N=37) PCA (N=37) RP-PCA (N=74) PCA (N=74) RP-PCA (N=37) PCA (N=37) Idiosyncratic Variation (In-sample) Idiosyncratic Variation (Out-of-sample) Figure: Maximal Sharpe-ratios for extreme (N = 74) and all (N = 37) deciles RP-PCA has smaller pricing errors out-of-sample RP-PCA (N=74) PCA (N=74) RP-PCA (N=37) PCA (N=37) RP-PCA (N=74) PCA (N=74) RP-PCA (N=37) PCA (N=37) A

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

37 All 37 portfolios: RP-PCA indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep accruals value prof aturnover valmom cfp gmargins price shvol momrev growth lrrev indmom ivol valuem strev size mom roaa lev divp noa invcap roea sgrowth A 3

38 All 37 portfolios: PCA indrrevlv indmomrev indrrev season valprof mom valmomprof inv ciss igrowth sp ep accruals value prof aturnover valmom cfp gmargins price shvol momrev growth lrrev indmom ivol valuem strev size mom roaa lev divp noa invcap roea sgrowth A 4

39 Eigenvalue Difference Eigenvalue Difference Number of factors Onatski (): Eigenvalue-ratio test. N=74 N=37 Eigenvalue Differences Eigenvalue Differences =- = =5 = = Critical value.5.5 =- = =5 = = Critical value Number Number RP-PCA: 5 factors PCA: 4 factors A 5

40 Time-stability: Generalized correlations RP-PCA (total vs. time-varying) Generalized Correlation Generalized Correlation st GC nd GC 3rd GC 4th GC 5th GC Time st GC nd GC 3rd GC 4th GC 5th GC PCA (total vs. time-varying) Time Figure: Generalized correlations between loadings estimated on the whole time horizon T = 65 and a rolling window with 4. A 6

41 Signal of factors: Existence of weak factors PCA RP-PCA (γ = ) FF5 σ σ.7.7. σ3...7 σ4.3 σ σ Table: Variance signal for different factors Largest eigenvalues of N ΛΣ F Λ normalized by the average idiosyncratic variance σ e = N N i= σ e,i Higher factors are weak. A 7

42 Signal of factors: Existence of weak factors Eigenvalues Eigenvalues Normalized Eigenvalues =- = = =5 = = Normalized Eigenvalues = = =5 = = Number Number ( Figure: Largest eigenvalues of the matrix X X + γ X X ). N T LEFT: Eigenvalues are normalized by division through the average idiosyncratic variance σe = N N i= σ e,i. RIGHT: Eigenvalues are normalized by the corresponding PCA (γ = ) eigenvalues. Higher factors have weak variance but high mean signal. A 8

43 Interpreting factors: Generalized correlations with proxies RP-PCA PCA. Gen. Corr.... Gen. Corr Gen. Corr Gen. Corr Gen. Corr Table: Generalized correlations of statistical factors with proxy factors (portfolios of 5% of assets). Problem in interpreting factors: Factors only identified up to invertible linear transformations. Generalized correlations close to measure of how many factors two sets have in common. Proxy factors approximate statistical factors. A 9

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

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

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

47 Interpreting factors: 5th proxy factor 5. Proxy RP-PCA Weights 5. Proxy PCA Weights Industry Rel. Reversals (LV). Leverage.6 Industry Rel. Reversals (LV) 9.98 Value-Profitability.4 Value-Momentum-Prof..95 Asset Turnover. Profitability.94 Profitability.99 Industry Mom. Reversals.9 Asset Turnover 9.9 Profitability -.86 Size.89 Profitability Long Run Reversals.85 Industry Mom. Reversals -.9 Sales/Price.84 Industry Rel. Reversals -.9 Size 9.8 Asset Turnover -.95 Value-Momentum-Prof Net Operating Assets -.97 Value-Profitability -.8 Seasonality -. Profitability -.8 Value-Profitability -. Profitability -.89 Short-Term Reversals -. Profitability Industry Rel. Reversals (LV) -.4 Value-Profitability -.94 Industry Rel. Reversals -.5 Profitability Idiosyncratic Volatility -.8 Asset Turnover -.7 Momentum (m) -.8 Asset Turnover -.35 A 3

48 Extreme Deciles Anomaly Mean SD Sharpe-ratio Anomaly Mean SD Sharpe-ratio Accruals - accrual Momentum (m) - mom Asset Turnover - aturnover Momentum-Reversals - momrev Cash Flows/Price - cfp Net Operating Assets - noa Composite Issuance - ciss Price - price Dividend/Price - divp Gross Profitability - prof Earnings/Price - ep Return on Assets (A) - roaa Gross Margins - gmargins Return on Book Equity (A) - roea Asset Growth - growth Seasonality - season Investment Growth - igrowth Sales Growth - sgrowth Industry Momentum - indmom Share Volume - shvol. 6.. Industry Mom. Reversals - indmomrev Size - size Industry Rel. Reversals - indrrev. 4. Sales/Price sp Industry Rel. Rev. (L.V.) - indrrevlv Short-Term Reversals - strev Investment/Assets - inv Value-Momentum - valmom Investment/Capital - invcap Value-Momentum-Prof. - valmomprof Idiosyncratic Volatility - ivol Value-Profitability - valprof Leverage - lev Value (A) - value Long Run Reversals - lrrev Value (M) - valuem Momentum (6m) - mom Table: Long-Short Portfolios of extreme deciles of 37 single-sorted portfolios from 7/963 to /7: Mean, standard deviation and Sharpe-ratio. A 4

49 highest SR lowest SR Portfolio Mean SR Portfolio Mean SR Ind. Rel. Rev. (L.V.).33 4 Return on Assets (A)..5 Industry Mom. Rev Leverage.3.5 Industry Rel. Reversals. Dividend/Price..3 Seasonality.8. Net Operating Assets.5. Value-Profitability.75.9 Investment/Capital.. Momentum (m).8.8 Return on Book Equity (A).8. Value-Mom-Prof Gross Margins.. Investment/Assets 8.5 Share Volume.. Composite Issuance 5.3 Price.. Investment Growth.37.3 Sales Growth.4. A 5

50 Extreme Deciles In-sample Out-of-sample SR RMS α Idio. Var. SR RMS α Idio. Var. RP-PCA.57.7 %.5.5.6% PCA.3..3%..98% RP-PCA Proxy.58.7 % % PCA Proxy.33..9%.7.8.% Fama-French % % Table: First and last decile of 37 single-sorted portfolios from 7/963 to /7 (N = 74 and T = 65): Maximal Sharpe-ratios, root-mean-squared pricing errors and unexplained idiosyncratic variation. K = 6 statistical factors. RP-PCA strongly dominates PCA and Fama-French 5 factors Results hold out-of-sample. A 6

51 Interpreting factors: Generalized correlations with proxies RP-PCA PCA. Gen. Corr.... Gen. Corr Gen. Corr Gen. Corr Gen. Corr Table: Generalized correlations of statistical factors with proxy factors (portfolios of 8 assets). Problem in interpreting factors: Factors only identified up to invertible linear transformations. Generalized correlations close to measure of how many factors two sets have in common. Proxy factors approximate statistical factors. A 7

52 Extreme Deciles: Maximal Sharpe-ratio SR (In-sample) SR (Out-of-sample).7 factor factors 3 factors.6 4 factors 5 factors 6 factors RP-PCA RP-PCA Proxy PCA PCA Proxy RP-PCA RP-PCA Proxy PCA PCA Proxy Figure: Maximal Sharpe-ratios. Spike in Sharpe-ratio for 5 factors A 8

53 Extreme Deciles: Pricing error RMS (In-sample) RMS (Out-of-sample) factor factors. 3 factors.5 4 factors 5 factors.5 6 factors RP-PCA RP-PCA Proxy PCA PCA Proxy RP-PCA RP-PCA Proxy PCA PCA Proxy Figure: Root-mean-squared pricing errors. RP-PCA has smaller out-of-sample pricing errors A 9

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

55 Extreme Deciles: Maximal Sharpe-ratio.9 SR (In-sample).9 SR (Out-of-sample) factor factors 3 factors 4 factors 5 factors 6 factors.5.5 SR SR Figure: Maximal Sharpe-ratios for different RP-weights γ and number of factors K A 3

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

57 Interpreting factors: Composition of proxies RP-PCA divp.53 mom.4 size.4 valuem.93 growth -.46 mom.99 ivol.3 indrrev.39 igrowth -.5 indmomrev.9 valmomprof.89 price.3 ep -.53 mom -.9 mom.84 indrrevlv.6 invcap -.69 valuem -.3 mom.8 lrrev.5 shvol -.7 ivol -.93 price.69 strev -. mom -.3 price -3.5 shvol.65 indrrevlv -.34 ivol -.48 mom -4. indmomrev -.57 indrrev -.37 PCA valuem.9 divp.74 indmom.4 valprof.5 price.5 ivol.69 mom.39 Aturnover.5 divp.6 roea -.64 valmom.8 prof.95 value.4 mom -.65 mom. sp.95 lrrev.6 size -.8 valmomprof. lrrev.86 sp.98 shvol -.9 indmom -.38 valprof -.98 cfp.9 ivol -3.6 mom -.7 prof -.5 mom.88 price -3. mom -.7 Aturnover -.89 Table: Portfolio-composition of proxy factors for first and last decile of 37 single-sorted portfolios: First proxy factors is an equally-weighted portfolio. A 33

58 Extreme Deciles: Time-stability of loadings Figure: Time-varying rotated loadings for the first six factors. Loadings are estimated on a rolling window with 4 months. A 34

59 Extreme Deciles: Time-stability: Generalized correlations RP-PCA (total vs. time-varying) Generalized Correlation Generalized Correlation st GC nd GC 3rd GC 4th GC 5th GC Time st GC nd GC 3rd GC 4th GC 5th GC PCA (total vs. time-varying) Time Figure: Generalized correlations between loadings estimated on the whole time horizon T = 65 and a rolling window with 4. A 35

60 Double-sorted portfolios Data Monthly return data from 7/963 to /7 (T = 65) 3 double sorted portfolios (consisting of 5 portfolios) from Kenneth French s website Factors PCA: K = 3 RP-PCA: K = 3 and γ = 3 FF-Long/Short factors: market + two specific anomaly long-short factors A 36

61 Sharpe-ratios and pricing errors (in-sample) Sharpe-Ratio α RPCA PCA FF-L/S RPCA PCA FF-L/S Size and BM BM and Investment BM and Profits Size and Accrual.3... Size and Beta Size and Investment Size and Profits Size and Momentum Size and ST-Reversal Size and Idio. Vol Size and Total Vol Profits and Invest Size and LT-Reversal A 37

62 Sharpe-ratios and pricing errors (out-of-sample) Sharpe-Ratio α RPCA PCA FF-L/S RPCA PCA FF-L/S Size and BM BM and Investment BM and Profits Size and Accrual.9... Size and Beta Size and Investment Size and Profits Size and Momentum Size and ST-Reversal Size and Idio. Vol Size and Total Vol Profits and Invest Size and LT-Reversal...4 A 38

63 Maximal Sharpe ratio (Size and accrual) =- = = = =5 = SR (In-sample) SR (Out-of-sample) factor factors 3 factors factor factors 3 factors Figure: Maximal Sharpe-ratio by adding factors incrementally. st and nd PCA and RP-PCA factors the same. RP-PCA detects 3rd factor (accrual) for γ >. A 39

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

65 Literature (partial list) Large-dimensional factor models with strong factors Bai (3): Distribution theory Bai and Ng (7): Robust PCA Fan et al. (6): Projected PCA for time-varying loadings Kelly et al. (7): Instrumented PCA for time-varying loadings Pelger (6), Aït-Sahalia and Xiu (5): High-frequency Large-dimensional factor models with weak factors (based on random matrix theory) Onatski (): Phase transition phenomena Benauch-Georges and Nadakuditi (): Perturbation of large random matrices Asset-pricing factors Feng, Giglio and Xiu (7): Factor selection with double-selection LASSO Kozak, Nagel and Santosh (7): Bayesian shrinkage A 4

66 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 A 4

67 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 A 43

68 The Model: Estimation of Latent Factors Conventional approach: PCA (Principal component analysis) Apply PCA to the sample covariance matrix: T X X X X with X = sample mean of asset excess returns Eigenvectors of largest eigenvalues estimate loadings ˆΛ. Much better approach: Risk-Premium PCA (RP-PCA) Apply PCA to a covariance matrix with overweighted mean T X X + γ X X γ = risk-premium weight Eigenvectors of largest eigenvalues estimate loadings ˆΛ. ˆF estimator for factors: ˆF = N X ˆΛ = X (ˆΛ ˆΛ) ˆΛ. A 44

69 The Model: Objective Function Conventional PCA: Objective Function Minimize the unexplained variance: min Λ,F NT N T (X ti F tλ i ) i= t= RP-PCA (Risk-Premium PCA): Objective Function Minimize jointly the unexplained variance and pricing error min Λ,F N T (X ti F tλ i ) +γ N ( Xi NT N F ) Λ i i= t= i= }{{}}{{} unexplained variation pricing error with X i = T T t= X t,i and F = T T t= Ft and risk-premium weight γ A 45

70 The Model: Objective function Variation objective function: Minimize the unexplained variation: min Λ,F NT N i= t= T (X ti F tλ i ) ( = min Λ NT trace (XM Λ ) (XM Λ ) ) s.t. F = X (Λ Λ) Λ Projection matrix M Λ = I N Λ(Λ Λ) Λ Error (non-systematic risk): e = X F Λ = XM Λ Λ proportional to eigenvectors of the first K largest eigenvalues of NT X X minimizes time-series objective function Motivation for PCA A 46

71 The Model: Objective function Pricing objective function: Minimize cross-sectional expected pricing error: N = N i= N ) (Ê[Xi ] Ê[F ]Λ i i= N ( T X i T F Λ i ) = N trace ( ( T XM Λ ) ( T XM Λ ) ) s.t. F = X (Λ Λ) Λ is vector T of s and thus F T estimates factor mean Why not estimate factors with cross-sectional objective function? Factors not identified Spurious factor detection (Bryzgalova (6)) A 47

72 The Model: Objective function Combined objective function: Risk-Premium-PCA min Λ,F ( (( )) ( NT trace (XM Λ ) (XM Λ + γ ) ( ) ) N trace T XM Λ T XM Λ ( ( = min Λ NT trace M Λ X I + γ ) T ) XM Λ s.t. F = X (Λ Λ) Λ The objective function is minimized by the eigenvectors of the largest eigenvalues of X ( I NT T + γ ) X. T ˆΛ estimator for loadings: proportional to eigenvectors of the first K eigenvalues of ˆF estimator for factors: NT X ( I T + γ T ) X N X ˆΛ = X (ˆΛ ˆΛ) ˆΛ. Estimator for the common component C = F Λ is Ĉ = ˆF ˆΛ A 48

73 The Model: Objective function Weighted Combined objective function: Straightforward extension to weighted objective function: min Λ,F NT trace(q (X F Λ ) (X F Λ )Q) + γ ) ( N trace (X F Λ )QQ (X F Λ ) = min trace (M ( Λ Q X I + γ ) Λ T ) XQM Λ s.t. F = X (Λ Λ) Λ Cross-sectional weighting matrix Q Factors and loadings can be estimated by applying PCA to Q X ( I + γ ) XQ. T Today: Only Q equal to inverse of a diagonal matrix of standard deviations. For γ = corresponds to PCA of a correlation matrix. Optimal choice of Q: GLS type argument A 49

74 The Model Strong vs. weak factor models Strong factor model ( N Λ Λ bounded) Interpretation: strong factors affect most assets (proportional to N), e.g. market factor Strong factors lead to exploding eigenvalues RP-PCA always more efficient than PCA optimal γ relatively small Weak factor model (Λ Λ bounded) Interpretation: weak factors affect a smaller fraction of assets Weak factors lead to large but bounded eigenvalues RP-PCA detects weak factors which cannot be detected by PCA optimal γ relatively large A 5

75 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) A 5

76 Weak Factor Model Weak Factor Model Large eigenvalues converge either to A biased value characterized by the Stieltjes transform of the bulk spectrum To the bulk of the spectrum if the true eigenvalue is below some critical threshold Phase transition phenomena: estimated eigenvectors orthogonal to true eigenvectors if eigenvalues too small Onatski (): Weak factor model with phase transition phenomena Problem: All models in the literature assume that random processes have mean zero RP-PCA implicitly uses non-zero means of random variables New tools necessary! A 5

77 Weak Factor Model Assumption : Weak Factor Model Rate: Assume that N c with < c <. T Factors: F are uncorrelated among each other and are independent of e and Λ and have bounded first two moments. ˆµ F := T p F t µf ˆΣF := T T FtF t t= σ F p Σ F = σf K 3 Loadings: The column vectors of the loadings Λ are orthogonally invariant and independent of ɛ and F (e.g. Λ i,k N(, N ) and Λ Λ = I K 4 Residuals: e = ɛσ with ɛ t,i N(, ). The empirical eigenvalue distribution function of Σ converges to a non-random spectral distribution function with compact support and supremum of support b. Largest eigenvalues of Σ converge A 53

78 Weak Factor Model Definition: Weak Factor Model Average idiosyncratic noise σe := trace(σ)/n Denote by λ λ... λ N the ordered eigenvalues of T e e. The Cauchy transform (also called Stieltjes transform) of the eigenvalues is the almost sure limit: G(z) := a.s. lim T N N z λ i= i ( = a.s. lim T N trace (zi N ) T e e) B-function c B(z) :=a.s. lim T N c =a.s. lim T N trace N λ i (z λ i= i ) ( ( (zi N ) ( ) ) T e e) T e e A 54

79 Weak Factor Model Intuition: Weak Factor Model Signal matrix for PCA of covariance matrix (γ = ): Σ F + cσ e I K K largest eigenvalues θ PCA,..., θk PCA measure strength of signal Signal matrix for RP-PCA: ( ) Σ F + cσe Σ / F µ F ( + γ) µ F Σ / F ( + γ) ( + γ)(µ F µ F + cσe ) ( + γ) = + γ K largest eigenvalues θ RP-PCA,..., θk RP-PCA measure strength of signal RP-PCA signal matrix is close to Σ F + ( + γ)µ F µ F + cσ e I K A 55

80 Weak Factor Model Theorem : Risk-Premium PCA under weak factor model Assumption ( holds. ) The first K largest eigenvalues ˆθ i i =,..., K of T X I T + γ X satisfy T { ( ) p G if θ ˆθ i θ i > θ crit = lim z b i G(z) b otherwise The correlation of the estimated with the true factors converges to ρ Ĉorr(F, ˆF ) p ρ Ũ }{{} Ṽ }{{} rotation rotation ρ K with ρ i { p +θ i B( ˆθ i )) if θ i > θ crit otherwise A 56

81 Weak Factor Model Optimal choice or risk premium weight γ Critical value θ crit and function B(.) depend only on the noise distribution and are known in closed-form If µ F and γ > then RP-PCA signals are always larger than PCA signals: θ RP-PCA i > θ PCA i RP-PCA can detect factors that cannot be detected with PCA For θ i > θ crit correlation ρ i is strictly increasing in θ i. The rotation matrices satisfy Ũ Ũ I K and Ṽ Ṽ I K. Ĉorr(F, ˆF ) is not necessarily an increasing function in θ. Based on closed-form expression choose optimal RP-weight γ A 57

82 Strong Factor Model Strong Factor Model Strong factors affect most assets: e.g. market factor N Λ Λ bounded (after normalizing factor variances) Statistical model: Bai and Ng () and Bai (3) framework Factors and loadings can be estimated consistently and are asymptotically normal distributed RP-PCA provides a more efficient estimator of the loadings Assumptions essentially identical to Bai (3) A 58

83 Strong Factor Model Asymptotic Distribution (up to rotation) PCA under assumptions of Bai (3): (up to rotation) Asymptotically ˆΛ behaves like OLS regression of F on X. Asymptotically ˆF behaves like OLS regression of Λ on X. RP-PCA under slightly stronger assumptions as in Bai (3): Asymptotically ( ˆΛ behaves ) like OLS regression of FW on XW with W = I T + γ and is a T vector of s. T Asymptotically ˆF behaves like OLS regression of Λ on X. Asymptotic Efficiency Choose RP-weight γ to obtain smallest asymptotic variance of estimators RP-PCA (i.e. γ > ) always more efficient than PCA Optimal γ typically smaller than optimal value from weak factor model RP-PCA and PCA are both consistent A 59

84 Simplified Strong Factor Model Assumption : Simplified Strong Factor Model Rate: Same as in Assumption Factors: Same as in Assumption 3 Loadings: Λ Λ/N p I K and all loadings are bounded. 4 Residuals: e = ɛσ with ɛ t,i N(, ). All elements and all row sums of Σ are bounded. A 6

85 Simplified Strong Factor Model Proposition: Simplified Strong Factor Model Assumption holds. Then: The factors and loadings can be estimated consistently. The asymptotic distribution of the factors is not affected by γ. 3 The asymptotic distribution of the loadings is given by T ( H ˆΛ i Λ i ) D N(, Ω i ) ) ( ) Ω i =σe i (Σ F + ( + γ)µ F µ F Σ F + ( + γ) µ F µ F ( ) Σ F + ( + γ)µ F µ F E[e t,i ] =σ e i, H full rank matrix 4 γ = is optimal choice for smallest asymptotic variance. γ =, i.e. the covariance matrix, is not efficient. A 6

86 Strong Factor Model Theorem : Strong Factor Model Assumption holds and γ [, ). Then: For any choice of γ the factors, loadings and common components can be estimated consistently pointwise. If T N then ( T H ˆΛ ) D i Λ i N(, Φ) ( ) (Ω, Φ = Σ F + (γ + )µ F µ F + γµ F Ω, + γω, µ F + γ ) µ F Ω, µ F ( ) Σ F + (γ + )µ F µ F For γ = this simplifies to the conventional case Σ F Ω,Σ F. If N then the asymptotic distribution of the factors is not affected by the T 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 ) D ( ) T (Ĉt,i C t,i N, F t ΦF t A 6

87 Time-varying loadings Model with time-varying loadings Observe panel of excess returns and L covariates Z i,t,l : X t,i = F t K g K (Z i,t,,..., Z i,t,l ) + e t,i Loadings are function of L covariates Z i,t,l with l =,..., L e.g. characteristics like size, book-to-market ratio, past returns,... Factors and loading function are latent Idea: Similar to Projected PCA (Fan, Liao and Wang (6)) and Instrumented PCA (Kelly, Pruitt, Su (7)), but we include the pricing error penalty allow for general interactions between covariates A 63

88 Time-varying loadings Projected RP-PCA (work in progress) Approximate nonlinear function g k (.) by basis functions φ m(.): M g k (Z i,t ) = b m,k φ m(z i,t ) m= g(z t ) }{{} K N = }{{} B K M Φ(Z t ) } {{ } M N Apply RP-PCA to projected data X t = X tφ(z t ) X t = F tb Φ(Z t )Φ(Z t ) + e tφ(z t ) = F t B + ẽ t Special case: φ m = {Zt I m} X characteristics sorted portfolios Obtain arbitrary interactions and break curse of dimensionality by conditional tree sorting projection Intuition: Projection creates M portfolios sorted on any functional form and interaction of covariates Z t. A 64

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

90 Simulation 5. Factor. Factor 5 True factor RP-PCA = RP-PCA = RP-PCA = PCA Time Time 5 3. Factor 5 4. Factor Time Time Figure: Sample paths of the cumulative returns of the first four factors and the estimated factor processes.the fourth factor has a variance σ F =.3 and Sharpe-ratio sr =.5. A 66

91 Simulation: Multifactor Model. Factor Corr. (IS) for F =.3. Factor Corr. (OOS) for F =.3. Factor Corr. (IS) for F =.. Factor Corr. (OOS) for F = Corr.6 Corr.6 Corr.6 Corr Factor Corr. (IS) for F =.3. Factor Corr. (OOS) for F =.3. Factor Corr. (IS) for F =.. Factor Corr. (OOS) for F = Corr.6 Corr.6 Corr.6 Corr Factor Corr. (IS) for F =.3 3. Factor Corr. (OOS) for F =.3 3. Factor Corr. (IS) for F =. 3. Factor Corr. (OOS) for F = Corr.6 Corr.6 Corr.6 Corr Factor Corr. (IS) for F =.3 4. Factor Corr. (OOS) for F =.3 4. Factor Corr. (IS) for F =. 4. Factor Corr. (OOS) for F =. Corr.8.6. Corr.8.6. Corr.8.6. SR=.8 SR=.5 SR=.3 SR=. Corr Figure: Correlation of estimated with true factor. A 67

92 Simulation: Multifactor Model SR Factor SR (IS) for F =.3 SR Factor SR (OOS) for F =.3 SR Factor SR (IS) for F =. SR=.8 SR=.5 SR=.3 SR=. SR Factor SR (OOS) for F = Factor SR (IS) for F =.3. Factor SR (OOS) for F =.3. Factor SR (IS) for F =.. Factor SR (OOS) for F = SR SR SR SR Factor SR (IS) for F =.3 3. Factor SR (OOS) for F =.3 3. Factor SR (IS) for F =. 3. Factor SR (OOS) for F = SR SR SR SR Factor SR (IS) for F =.3 4. Factor SR (OOS) for F =.3 4. Factor SR (IS) for F =. 4. Factor SR (OOS) for F = SR SR SR SR Figure: Maximal Sharpe-ratio of factors. A 68

93 Corr Corr Simulation: Weak factor model prediction Statistical Model.5 PCA ( =-) RP-PCA ( =) RP-PCA ( =) RP-PCA ( =5).5..5 F Monte-Carlo Simulation F Correlations between estimated and true factor based on the weak factor model prediction and Monte-Carlo simulations. The Sharpe-ratio of the factor is.8. The normalized variance of the factors corresponds to σ F N. A 69

94 Weak Factor Model: Dependent residuals.8.6. dependent residuals i.i.d residuals signal Figure: Model-implied values of ρ i ( +θ i B( ˆθ if θ i )) i > σcrit and otherwise) for different signals θ i. The average noise level is normalized in both cases to σ e =. A 7

95 Simulation: Weak factor model prediction Statistical Model F =.3 Statistical Model F =.5 Statistical Model F =. Statistical Mode Corr.5 SR=.8 SR=.5 SR=.3 SR=. Corr.5 Corr.5 Corr Monte-Carlo Simulation F =.3 Monte-Carlo Simulation F =.5 Monte-Carlo Simulation F =. Monte-Carlo Simula Corr.5 Corr.5 Corr.5 Corr Monte-Carlo Simulation OOS F =.3 Monte-Carlo Simulation OOS F =.5 Monte-Carlo Simulation OOS F =. Monte-Carlo Simulatio Corr.5 Corr.5 Corr.5 Corr Correlation of estimated with true factors for different variances and Sharpe-ratios of SR Statistical Model F =.3.. the factor and for different RP-weights γ SR Statistical Model F =.5 SR. Statistical Model F =. 5 5 SR. Statistical Mode 5 A 7

96 Monte-Carlo Simulation OOS Simulation: Weak factor model prediction Corr F = Corr Monte-Carlo Simulation OOS F = Corr Monte-Carlo Simulation OOS F =. 5 5 Corr Monte-Carlo Simulatio.5 5 SR.8.6. Statistical Model F = SR.8.6. Statistical Model F = SR.8.6. Statistical Model F =. 5 5 SR.8.6. Statistical Mode 5 Monte-Carlo Simulation F =.3 Monte-Carlo Simulation F =.5 Monte-Carlo Simulation F =. Monte-Carlo Simula SR.6 SR.6 SR.6 SR Monte-Carlo Simulation OOS F =.3 Monte-Carlo Simulation OOS F =.5 Monte-Carlo Simulation OOS F =. Monte-Carlo Simulatio SR.6 SR.6 SR.6 SR Sharpe-ratio for different variances and Sharpe-ratios of the factor and for different RP-weights γ. The residuals have the empirical residual correlation matrix. A 7