Testing multifactor models

РЭШ EFM 2004/05 2 Plan Up to now: Up to now: –Testing CAPM –Single pre-specified factor Today: Today: –Testing multifactor models –The factors are unspecified!

РЭШ EFM 2004/05 3 Detailed plan Theoretical base for the multifactor models: APT and ICAPM Theoretical base for the multifactor models: APT and ICAPM Testing when factors are traded portfolios Testing when factors are traded portfolios Statistical factors Statistical factors Macroeconomic factors Macroeconomic factors –Chen, Roll, and Ross (1986) Fundamental factors Fundamental factors –Fama and French (1993)

РЭШ EFM 2004/05 4 APT K-factor return-generating model for N assets: K-factor return-generating model for N assets: R t = a + Bf t + ε t, –where errors have zero expectation and are orthogonal to factors –B is NxK matrix of factor loadings Cross-sectional equation for risk premiums: Cross-sectional equation for risk premiums: E[R] = λ 0 l + Bλ K –where λ is Kx1 vector of factor risk premiums ICAPM: another interpretation of factors ICAPM: another interpretation of factors –The market ptf + state variables describing shifts in the mean- variance frontier

РЭШ EFM 2004/05 5 Specifics of testing APT No need to estimate the market ptf No need to estimate the market ptf Can be estimated within a subset of the assets Can be estimated within a subset of the assets Assume the exact form of APT Assume the exact form of APT –In general, approximate APT, which is not testable The factors and their number are unspecified The factors and their number are unspecified –Factors can be traded portfolios or not –Factors may explain cross differences in volatility, but have low risk premiums

РЭШ EFM 2004/05 6 Testing when factors are traded portfolios With risk-free asset: With risk-free asset: –Regression of excess asset returns on excess factor returns r t = a + Br f,t + ε t, H 0 : a=0, F-test H 0 : a=0, F-test –Risk premia: mean excess factor returns Time-series estimator of variance Time-series estimator of variance

РЭШ EFM 2004/05 7 Testing when factors are traded portfolios (cont.) Without risk-free asset: need to estimate zero-beta return γ 0 Without risk-free asset: need to estimate zero-beta return γ 0 –Unconstrained regression of asset returns on factor returns: R t = a + BR f,t + ε t, –Constrained regression: R t = (l N -Bl K )γ 0 + BR f,t + ε t, –H 0 : a=(l N -Bl K )γ 0, LR test –Risk premia: mean factor returns in excess of γ 0 Variance is adjusted for the estimation error for γ 0 Variance is adjusted for the estimation error for γ 0

РЭШ EFM 2004/05 8 Testing when factors are traded portfolios (cont.) When factor portfolios span the mean- variance frontier: no need to specify zero- beta asset When factor portfolios span the mean- variance frontier: no need to specify zero- beta asset –Regression of asset returns on factor returns: R t = a + BR f,t + ε t, –H 0 : a=0 and Bl K =l N Jensens alpha =0 & portfolio weights sum up to 1 Jensens alpha =0 & portfolio weights sum up to 1 Otherwise, factors do not span the MV frontier of the assets with returns R t Otherwise, factors do not span the MV frontier of the assets with returns R t

РЭШ EFM 2004/05 9 Three approaches to estimate factors Statistical factors Statistical factors –Extracted from returns –Estimate B and λ at the same time Macroeconomic factors Macroeconomic factors –Estimate B, then λ Fundamental factors Fundamental factors –Estimate λ for given B (proxied by firm characteristics)

РЭШ EFM 2004/05 10 Statistical factors: factor analysis R t - μ = Bf t + ε t cov(R t ) = B Ω B + D –Assuming strict factor structure: Dcov(ε t ) is diagonal Dcov(ε t ) is diagonal –Specification restrictions on factors: E(f t )=0, Ωcov(f t )=I E(f t )=0, Ωcov(f t )=I Estimation: Estimation: –Estimate B and D by ML –Get f t from the cross-sectional GLS regression of asset returns on B

РЭШ EFM 2004/05 11 Statistical factors: principal components Classical approach: Classical approach: –Choose linear combinations of asset returns that maximize explained variance –Each subsequent component is orthogonal to the previous ones –Correspond to the largest eigenvectors of NxN matrix cov(R t ) Rescaled s.t. weights sum up to 1 Rescaled s.t. weights sum up to 1

РЭШ EFM 2004/05 12 Statistical factors: principal components Connor and Korajczyk (1988): Connor and Korajczyk (1988): –Take K largest eigenvectors of TxT matrix rr/N where r is NxT excess return matrix where r is NxT excess return matrix –As N, KxT matrix of eigenvectors = factor realizations The factor estimates allow for time-varying risk premiums! The factor estimates allow for time-varying risk premiums! –Refinement (like GLS): same for the scaled cross- product matrix rDr/N –Refinement (like GLS): same for the scaled cross- product matrix rD -1 r/N where D has variances of the residuals from the first-stage OLS on the diagonal, zeros off the diagonal where D has variances of the residuals from the first-stage OLS on the diagonal, zeros off the diagonal This increases the efficiency of the estimation This increases the efficiency of the estimation

РЭШ EFM 2004/05 13 Results 5-6 factors are enough 5-6 factors are enough –Based on explicit statistical tests and asset pricing tests Explain up to 40% of CS variation in stock returns Explain up to 40% of CS variation in stock returns –Better than CAPM Explain some (January), but not all (size, BE/ME) anomalies Explain some (January), but not all (size, BE/ME) anomalies

РЭШ EFM 2004/05 14 Discussion of statistical factors Missing economic interpretation Missing economic interpretation The explanatory power out of sample is much lower than in-sample The explanatory power out of sample is much lower than in-sample # factors rises with N # factors rises with N –CK fix this problem Static: slow reaction to the structural changes Static: slow reaction to the structural changes –Except for CK PCs

РЭШ EFM 2004/05 15 Macroeconomic factors Time series to estimate B: Time series to estimate B: R i,t = a i + b i f t + ε i,t Cross-sectional regressions to estimate ex post risk premia for each t: Cross-sectional regressions to estimate ex post risk premia for each t: R i,t = λ 0,t + b' i λ K,t + e i,t, –Risk premia: mean and std from the time series of ex post risk premia λ t

РЭШ EFM 2004/05 16 Chen, Roll, and Ross (1986) "Economic forces and the stock market" Examine the relation between (macro) economic state variables and stock returns Examine the relation between (macro) economic state variables and stock returns –Variables related to CFs / discount rates Data: Data: –Monthly returns on 20 EW size-sorted portfolios,

РЭШ EFM 2004/05 17 Data: macro variables Industrial production growth: MP t =ln(IP t /IP t-1 ), YP t =ln(IP t /IP t-12 ) Industrial production growth: MP t =ln(IP t /IP t-1 ), YP t =ln(IP t /IP t-12 ) Unanticipated inflation: UI t = I t – E t-1 [I t ] Unanticipated inflation: UI t = I t – E t-1 [I t ] Change in expected inflation: DEI t = E t [I t+1 ] – E t-1 [I t ] Change in expected inflation: DEI t = E t [I t+1 ] – E t-1 [I t ] Default spread: UPR t = Baa t – LGB t Default spread: UPR t = Baa t – LGB t Term spread: UTS t = LGB t – TB t-1 Term spread: UTS t = LGB t – TB t-1 Real interest rate: RHO t = TB t-1 – I t Real interest rate: RHO t = TB t-1 – I t Market return: EWNY t and VWNY t (NYSE) Market return: EWNY t and VWNY t (NYSE) Real consumption growth: CG Real consumption growth: CG Change in oil prices: OG Change in oil prices: OG

РЭШ EFM 2004/05 18 Methodology: Fama- MacBeth procedure Each year, using 20 EW size-sorted portfolios: Each year, using 20 EW size-sorted portfolios: Estimate factor loadings B from time-series regression, using previous 5 years Estimate factor loadings B from time-series regression, using previous 5 years R i,t = a i + b i f t + ε i,t Estimate ex post risk premia from a cross- sectional regression for each of the next 12 months Estimate ex post risk premia from a cross- sectional regression for each of the next 12 months R i,t = λ 0,t + b' i λ K,t + e i,t, –Risk premia: mean and std from the time series of ex post risk premia λ t

РЭШ EFM 2004/05 19 Results Table 4, risk premia Table 4, risk premia –MP: +, insurance against real systematic production risks –UPR: +, hedging against unexpected increases in aggregate risk premium –UTS: - in , assets whose prices rise in response to a fall in LR% are more valuable –UI and DEI: - in , when they were very volatile –YP, EWNY, VWNY are insignificant

РЭШ EFM 2004/05 20 Results (cont.) Table 5, risk premia when market betas are estimated in univariate TS regression Table 5, risk premia when market betas are estimated in univariate TS regression –VWNY is significant when alone in CS regression –VWNY is insignificant in the multivariate CS regression Tables 6 and 7, adding other variables Tables 6 and 7, adding other variables –CG is insignificant –OG: + in

РЭШ EFM 2004/05 21 Conclusions Stocks are exposed to systematic economic news and priced in accordance with their exposures Stocks are exposed to systematic economic news and priced in accordance with their exposures Market betas fail to explain CS of stock returns Market betas fail to explain CS of stock returns –Though market index is the most significant factor in TS regression No support for consumption-based pricing No support for consumption-based pricing

РЭШ EFM 2004/05 22 Discussion of macroeconomic factors Strong economic intuition Strong economic intuition Static Static –Slow reaction to the structural changes Bad predictive performance Bad predictive performance

РЭШ EFM 2004/05 23 Fundamental factors B is proxied by firm characteristics: B is proxied by firm characteristics: –Market cap, leverage, E/P, liquidity, etc. –Taken from CAPM violations Cross-sectional regressions for each t to estimate risk premia: Cross-sectional regressions for each t to estimate risk premia: R i,t = λ 0,t + b' i λ K,t + e i,t Alternative: factor-mimicking portfolios Alternative: factor-mimicking portfolios –Zero-investment portfolios: long/short position in stocks with high/low value of the attribute

РЭШ EFM 2004/05 24 Fama and French (1993) "Common risk factors in the returns on stocks and bonds" Identify risk factors in stock and bond markets Identify risk factors in stock and bond markets –Factors for stocks are size and book-to-market In contrast to Fama&French (1992): time series tests In contrast to Fama&French (1992): time series tests –Factors for bonds are term structure variables –Links between stock and bond factors

РЭШ EFM 2004/05 25 Data All non-financial firms in NYSE, AMEX, and (after 1972) NASDAQ in All non-financial firms in NYSE, AMEX, and (after 1972) NASDAQ in Monthly return data (CRSP) Monthly return data (CRSP) Annual financial statement data (COMPUSTAT) Annual financial statement data (COMPUSTAT) –Used with a 6m gap Market index: the CRSP value-wtd portfolio of stocks in the three exchanges Market index: the CRSP value-wtd portfolio of stocks in the three exchanges

РЭШ EFM 2004/05 26 Methodology Stock market factors: Stock market factors: –Market: RM-RF –Size: ME –Book-to-market equity: BE/ME Bond market factors: Bond market factors: –TERM = (Return on Long-term Gvt Bonds) – (T-bill rate) –DEF = (Return on Corp Bonds) – (Return on Long-term Gvt Bonds)

РЭШ EFM 2004/05 27 Constructing factor- mimicking portfolios In June of each year t, break stocks into: In June of each year t, break stocks into: –Two size groups: Small / Big (below/above median) –Three BE/ME groups: Low (bottom 30%) / Medium / High (top 30%) –Compute monthly VW returns of 6 size-BE/ME portfolios for the next 12 months Factor-mimicking portfolios: zero-investment Factor-mimicking portfolios: zero-investment –Size: SMB = 1/3(SL+SM+SH) – 1/3(BL+BM+BH) –BE/ME: HML = 1/2(BH+SH) – 1/2(BL+SL)

РЭШ EFM 2004/05 28 The returns to be explained 25 stock portfolios 25 stock portfolios –In June of each year t, stocks are sorted by size (current ME) and (independently) by BE/ME (as of December of t-1) –Using NYSE quintile breakpoints, all stocks are allocated to one of 5 size portfolios and one of 5 BE/ME portfolios –From July of t to June of t+1, monthly VW returns of 25 size- BE/ME portfolios are computed 7 bond portfolios 7 bond portfolios –2 gvt portfolios: 1-5y, 6-10y maturity –5 corporate bond portfolios: Aaa, Aa, A, Baa, below Baa

РЭШ EFM 2004/05 29 Time-series tests Regressions of excess asset returns on factor returns: Regressions of excess asset returns on factor returns: r i,t = a i + b i r f,t + ε i,t –Common variation: slopes and R 2 –Pricing: intercepts

РЭШ EFM 2004/05 30 Results Table 2: summary statistics Table 2: summary statistics –RM-RF, SMB, and HML: high mean and std, (marginally) significant –TERM, DEF: low mean, but high volatility –SMB & HML are almost uncorrelated (-0.08) –RM-RF is positively correlated with SMB (0.32) and negatively with HML (-0.38)

РЭШ EFM 2004/05 31 Results on common variation Table 3: explanatory power of bond-market factors Table 3: explanatory power of bond-market factors –The slopes are higher for stocks, similar to those for long-term bonds –TERM coefficients rise with bond maturity –Small stocks and low-grade bonds are more sensitive to DEF –R 2 is higher for bonds

РЭШ EFM 2004/05 32 Results on common variation (cont.) Table 4: explanatory power of the market factor Table 4: explanatory power of the market factor –R 2 for stocks is much higher, up to 0.9 for small low BE/ME stocks –The slopes for bonds are small, but highly significant, rising with maturity and riskiness Table 5: explanatory power of SMB and HML Table 5: explanatory power of SMB and HML –Significant slopes and quite high R 2 for stocks –Typically insignificant slopes and zero R 2 for bonds

РЭШ EFM 2004/05 33 Results on common variation (cont.) Table 6: explanatory power of RM-RF, SMB and HML Table 6: explanatory power of RM-RF, SMB and HML –Slopes for stocks are highly significant, R 2 is typically over 0.9 –Market betas move toward one –The SMB and HML slopes for bonds become significant Table 7: five-factor regressions Table 7: five-factor regressions –Stocks: stock factors remain significant, but kill significance of bond factors –Bonds: bond factors remain significant, stock factors become much less important RM-RF help to explain high-grade bonds RM-RF help to explain high-grade bonds SMB and HML help to explain low-grade bonds SMB and HML help to explain low-grade bonds

РЭШ EFM 2004/05 34 Results on common variation (cont.) Orthogonalization of the market factor: Orthogonalization of the market factor: RM-RF= SMB- 0.63HML+0.81TERM+0.79DEF+e –All coefficients are significant, R 2 =0.38 The market factor captures common variation in stock and bond markets! The market factor captures common variation in stock and bond markets! –Orthogonalized market factor: RMO = const + error Table 8: five-factor regressions with RMO Table 8: five-factor regressions with RMO –Stocks: bond factors become highly significant

РЭШ EFM 2004/05 35 Results on pricing Table 9a, stocks Table 9a, stocks –TERM, DEF: positive intercepts –RM-RF: size effect –SMB, HML: big positive intercepts –RM-RF, SMB, HML: most intercepts are 0 –Adding bond factors does not improve

РЭШ EFM 2004/05 36 Results on pricing (cont.) Table 9b, bonds Table 9b, bonds –TERM, DEF: positive intercepts for gvt bonds –RM-RF or SMB with HML make intercepts insignificant –Increased precision due to TERM and DEF explains positive intercepts in a five-factor model Table 9c, F-test Table 9c, F-test –Joint test for zero intercepts rejects the null for all models –The best model for stocks is a model with three stock factors

РЭШ EFM 2004/05 37 Diagnostics Time series regressions of residuals from the five- factor model on D/P, default spread, term spread, and short-term interest rates Time series regressions of residuals from the five- factor model on D/P, default spread, term spread, and short-term interest rates –No evidence of predictability! Table 10, time series regressions of residuals on January dummy Table 10, time series regressions of residuals on January dummy –January seasonals are weak, mostly for small and high BE/ME stocks –Except for TERM, there are January seasonals in risk factors, esp. in SMB and HML

РЭШ EFM 2004/05 38 Split-sample tests Each of the size-BE/ME portfolios is split into two halves Each of the size-BE/ME portfolios is split into two halves –One is used to form factors –Another is used as dependent variables in regressions Similar results Similar results

РЭШ EFM 2004/05 39 Other sets of portfolios Portfolios formed on E/P Portfolios formed on E/P –Zero intercepts Portfolios formed on D/P Portfolios formed on D/P –The only unexplained portfolio: D=0, a=-0.23

РЭШ EFM 2004/05 40 Conclusions There is an overlap between processes in stock and bond markets There is an overlap between processes in stock and bond markets –Bond market factors capture common variation in stock and bond returns, though explain almost no average excess stock returns Three-factor model with the market, size, and book-to- market factors explains well stock returns Three-factor model with the market, size, and book-to- market factors explains well stock returns –SMB and HML explain the cross differences –RM-RF explains why stock returns are on average above the T-bill rate Two bond factors explain well variation in bond returns Two bond factors explain well variation in bond returns –SMB and HML help to explain variation of low-grade bonds

РЭШ EFM 2004/05 41 Fama and French (1995) "Size and book-to-market factors in earnings and returns" There are size and book-to-market factors in earnings which proxy for relative distress There are size and book-to-market factors in earnings which proxy for relative distress –Strong firms with persistently high earnings have low BE/ME –Small stocks tend to be less profitable There is some relation between common factors in earnings and return variation There is some relation between common factors in earnings and return variation

РЭШ EFM 2004/05 42 Fama and French (1996) "Multifactor explanations of asset pricing anomalies" Run time-series regressions for decile portfolios based on sorting by E/P, C/P, sales, past returns Run time-series regressions for decile portfolios based on sorting by E/P, C/P, sales, past returns –The three-factor model explains all anomalies but one-year momentum effect Interpretation of the three-factor model in terms of the underlying portfolios M, S, B, H, and L: spanning tests Interpretation of the three-factor model in terms of the underlying portfolios M, S, B, H, and L: spanning tests –M and B are highly correlated (0.99) –Excess returns of any three of M, S, H, and L explain the fourth –Different triplets of the excess returns for M, S, H, and L provide similar results in explaining stock returns This is taken as evidence of multifactor ICAPM or APT This is taken as evidence of multifactor ICAPM or APT

РЭШ EFM 2004/05 43 Discussion of fundamental factors High predictive power High predictive power Dynamic Dynamic Though: data-intensive Though: data-intensive Widely applied: Widely applied: –Portfolio selection and risk management –Performance evaluation –Measuring abnormal returns in event studies –Estimating the cost of capital