THE NOT-SO-WELL-KNOWN THREE-AND-ONE-HALF-FACTOR MODEL Roger Clarke, - PowerPoint PPT Presentation

THE NOT-SO-WELL-KNOWN THREE-AND-ONE-HALF-FACTOR MODEL Roger Clarke, Harindra de Silva, and Steven Thorley Q-Group Spring Seminar April 8, 2014

How Many Factors Was That? “I know what you’re thinking. You're thinking, did he fire six shots or only five? Now to tell you the truth, I’ve forgotten myself in all this excitement. Do you feel lucky, punk?” Famous misquote of Harry Callahan (1971) 2 2

The Cross-section of Expected Stock Returns F ama and French (1992) R i = A + B1 beta i + B2 size i + B3 value i + ε i A = return on a “standard portfolio in which the weighted- average of the 1) explanatory variables are zero” B1 = return to individual stock betas (trailing 60-month time-series 2) regression estimate) B2 = return to size (beginning-of-month log market capitalization) 3) B3 = return to value (log book-to-market ratio) 4) 3

Modified Fama-MacBeth Regressions Monthly multivariate cross-sectional regression of stock returns on a list of stock characteristics … R i = A + B1 beta i + B2 small i + B3 value i + B4 mom i + ε i … with econometric enhancements now used in risk-modeling practice: 1) Capitalization-weighted regressions 2) Shift characteristics to a cap-weighted mean of zero 3) Scale characteristics (including beta) to a standard deviation of one With steps 1 and 2 above, the “standard portfolio” or the regression intercept term “A” is the cap-weighted market portfolio. 4 4

Scope All U.S. common stocks in the CRSP database except:  ETFs and REITs (require a CRSP share code of “10” or “11”)  Smallest size quintile (little impact because of cap-weighting)  Insufficient data (require at least 24 of 60 prior monthly returns) Half century ending December 2012: approximates Russell 3000  “Size” replaced by small (minus one times log market-cap)  Book value from Compustat informs start date of January 1963  Includes “Carhart” momentum as an additional factor  Factor names: z-Beta, z-Small, z-Value, and z-Mom 5 5

The Big Picture C u m u l a t i v e F a c t o r R e t u r n s f r o m 1 9 6 3 t o 2 0 1 2 Market z-Beta z-Small z-Value z-Mom 350% 300% 250% 200% 150% 100% 50% 0% -50% -100% 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 6

50 Years Of Factor Returns 1 9 6 3 t o 2 0 1 2 Market z-Beta z-Small z-Value z-Mom Average Return 5.64% -0.79% 0.86% 1.88% 4.99% Standard Deviation 15.54% 7.02% 3.61% 5.30% 6.45% Sharpe Ratio 0.363 -0.112 0.237 0.354 0.774 Correlation to: Market z-Beta z-Small z-Value z-Mom Market 1.000 0.684 0.226 -0.049 0.002 z-Beta 0.684 1.000 0.257 -0.131 -0.043 z-Small 0.226 0.257 1.000 -0.131 0.058 z-Value -0.049 -0.131 -0.131 1.000 -0.264 z-Mom 0.002 -0.043 0.058 -0.264 1.000 Market Beta 1.000 0.309 0.052 -0.017 0.001 Market Alpha 0.00% -2.53% 0.56% 1.97% 4.99% Active Risk 5.12% 3.52% 5.30% 6.45% Information Ratio -0.494 0.160 0.372 0.773 7

50 Years Of Factor Returns 1 9 6 3 t o 2 0 1 2 Market z-Beta z-Small z-Value z-Mom Average Return 5.64% -0.79% 0.86% 1.88% 4.99% Standard Dev. 15.54% 7.02% 3.61% 5.30% 6.45% Sharpe Ratio 0.363 -0.112 0.237 0.354 0.774 Market Beta 1.000 0.309 0.052 -0.017 0.001 Market Alpha 0.00% -2.53% 0.56% 1.97% 4.99% Average return - CAPM predicted return = Alpha of z-Beta factor -0.79% - (0.309) 5.64% = -2.53% 8

Capital Market Line (CML) Average Excess Return (%) Market 5.64 0.00 0.00 15.54 Return Standard Deviation (%) 9

Security Market Line (SML) Average Excess Return (%) 5.64 0.00 0.0 1.0 Market Beta 10

Security Market Line (SML) Average Excess Return (%) 8.20 5.64 0.00 0.0 1.0 Market Beta 11

Fama-MacBeth multivariate regression coefficients are “pure” factor-mimicking long-short portfolios 12

Formula For Weighted Fama-MacBeth Regression Portfolios W is an N-by-5 matrix of factor portfolio weights: where X is an N-by-5 matrix of stock characteristics (including a leading column of ones) and M is an N-by-5 matrix of the market weights (repeated in five columns). Portfolio returns for a given month are then calculated as where R is an N-by-1 vector of security returns. 13

Example of Factor Portfolio Weights J a n u a r y 2 0 1 2 Market z-Beta z-Small z-Value z-Mom 0.03% 0.00% 0.00% 0.00% 0.00% Average 0.13% 0.14% 0.15% 0.15% 0.13% Standard Deviation 2.90% 3.84% 0.11% 3.71% 2.11% Maximum Weight 0.00% -2.29% -4.65% -2.06% -2.23% Minimum Weight 100.00% 51.35% 41.78% 51.06% 58.86% Sum of Long Weights 0.00% -51.35% -41.78% -51.06% -58.86% Sum of Short Weights 3008 1150 2886 1443 1227 Long Security Count 0 1858 122 1565 1781 Short Security Count 14

Example of Factor Portfolio Weights J a n u a r y 2 0 1 2 z-Beta 5.0% XOM 4.0% 3.0% Factor Portfolio Weight 2.0% 1.0% 0.0% -1.0% AAPL -2.0% -3.0% -4.0% -5.0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% Market Portfolio Weight 15

Example of Factor Portfolio Weights J a n u a r y 2 0 1 2 z-Beta z-Small z-Value z-Mom 5.0% XOM 4.0% 3.0% Factor Portfolio Weight 2.0% 1.0% 0.0% -1.0% AAPL -2.0% -3.0% -4.0% -5.0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% Market Portfolio Weight 16

How do multivariate-weighted, regression-based portfolio returns compare to Fama-French sorted portfolio returns? 17

Cumulative Fama-French Factor Returns 1 9 6 3 t o 2 0 1 2 VMS SMB HML UMD 600% 500% 400% 300% 200% 100% 0% -100% -200% 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 18

Cumulative z-Factor Returns 1 9 6 3 t o 2 0 1 2 z-Beta z-Small z-Value z-Mom 300% 250% 200% 150% 100% 50% 0% -50% -100% 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 19

Fama-French Returns 1 9 6 3 t o 2 0 1 2 MRF VMS SMB HML UMD Average Return 5.63% 0.33% 3.00% 4.74% 8.42% Standard Deviation 15.57% 15.36% 10.80% 10.01% 14.81% Sharpe Ratio 0.361 0.021 0.277 0.473 0.568 Correlations: MRF VMS SMB HML UMD MRF 1.000 0.729 0.309 -0.301 -0.128 VMS 0.729 1.000 0.571 -0.442 -0.237 SMB 0.309 0.571 1.000 -0.227 -0.009 HML -0.301 -0.442 -0.227 1.000 -0.153 UMD -0.128 -0.237 -0.009 -0.153 1.000 Correlations: Market z-Beta z-Small z-Value z-Mom 1.000 0.913 0.811 0.728 0.850 20

Portfolio performance measurement with “pure” factor returns: The impact of Beta 21

Recent z-Factor Portfolio Returns 2 0 0 3 t o 2 0 1 4 Market approximates Russell 3000 Market z-Beta z-Small z-Value z-Mom Average Return 7.10% 1.11% 2.18% -1.73% 0.11% Standard Deviation 15.14% 7.12% 3.06% 5.01% 6.21% Sharpe Ratio 0.469 0.156 0.713 -0.344 0.018 Market Beta 1.000 0.336 0.074 0.123 -0.104 Market Alpha 0.00% -1.27% 1.65% -2.60% 0.85% Note: Alpha of the z-Beta factor measures the difference between the realized average return and the CAPM predicted return: 1.11 – (0.336) 7.10 = -1.27% 22

Portfolio Returns 2 0 0 3 t o 2 0 1 2 The 2003 to 2012 portfolio performance measurement period has the least “shortfall” (i.e., alpha) from the CAPM predicted return: Period z-Beta Market Alpha -------------------------------------------------------------------------------- 1963 to 2012 -0.79 – (0.309) 5.64 = -2.53% (50 years) 1963 to 1972 -0.11 – (0.226) 5.87 = -1.58% 1973 to 1982 -2.38 – (0.216) 0.78 = -2.54% 1983 to 1992 -0.93 – (0.277) 9.01 = -3.42% 1993 to 2002 -1.37 – (0.461) 5.43 = -4.23% 2003 to 2012 1.11 – (0.336) 7.10 = -1.27% 23

State Street Sector SPDR and MSCI Minimum Volatility ETFs Annualized excess (of risk-free rate) returns: 2003 to 2014 C. Discretionary Technology Healthcare C. Staples Financial Industrial Materials Utilities Energy XLY XLP XLE XLF XLV XLI XLB XLK XLU 8.39% 6.71% 14.09% 0.57% 4.80% 8.04% 9.23% 7.73% 8.72% MSCI Minimum Volatility ETF (USMV) 7.29% S&P 500 ETF (SPY) 6.25% Risk-free Rate (Ibbotson T-bill) 1.64% 24

Portfolio Performance Measurement by Time-Series Regression R P,t = α P + β P R M,t + … + ε P,t R P,t = α P + R M,t + ( β P - 1) R M,t + … + ε P,t R P,t = α P + R M,t + Z P R z β ,t + … + ε P,t The exposure to all stocks in the Fama-MacBeth regressions used to estimate R M,t (intercept term) is exactly one, so the portfolio exposure to R M,t in the subsequent time-series regression is known to be one. A: Traditional methodology Restrict the coefficient on R z β ,t to be zero (i.e., 0.000) or simply leave R z β ,t out of the regression. B: Alternative methodology Restrict the coefficient on R M,t to be one (i.e., 1.000) or simply subtract R M,t from R P,t . 25