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

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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)

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The Cross-section of Expected Stock Returns

F ama and French (1992)

1)

A = return on a “standard portfolio in which the weighted- average of the explanatory variables are zero”

2)

B1 = return to individual stock betas (trailing 60-month time-series regression estimate)

3)

B2 = return to size (beginning-of-month log market capitalization)

4)

B3 = return to value (log book-to-market ratio) Ri = A + B1 betai + B2 sizei + B3 valuei + εi

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Modified Fama-MacBeth Regressions

Monthly multivariate cross-sectional regression of stock returns on a list of stock characteristics … Ri = A + B1 betai + B2 smalli + B3 valuei + B4 momi + εi … 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.

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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

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• 100%
• 50%

0% 50% 100% 150% 200% 250% 300% 350% 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

Market z-Beta z-Small z-Value z-Mom

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

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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

50 Years Of Factor Returns

1 9 6 3 t o 2 0 1 2

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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%

50 Years Of Factor Returns

1 9 6 3 t o 2 0 1 2

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5.64 15.54 Average Excess Return (%) Return Standard Deviation (%) 0.00 0.00 Market

Capital Market Line (CML)

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5.64 Average Excess Return (%) 0.00 Market Beta 1.0 0.0

Security Market Line (SML)

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5.64 Average Excess Return (%) 0.00 Market Beta 1.0 0.0

Security Market Line (SML)

8.20

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

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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.

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Market z-Beta z-Small z-Value z-Mom

Average

0.03% 0.00% 0.00% 0.00% 0.00%

Standard Deviation

0.13% 0.14% 0.15% 0.15% 0.13%

Maximum Weight

2.90% 3.84% 0.11% 3.71% 2.11%

Minimum Weight

0.00%

• 2.29%
• 4.65%
• 2.06%
• 2.23%

Sum of Long Weights

100.00% 51.35% 41.78% 51.06% 58.86%

Sum of Short Weights

0.00% -51.35% -41.78% -51.06% -58.86%

Long Security Count

3008 1150 2886 1443 1227

Short Security Count

1858 122 1565 1781

Example of Factor Portfolio Weights

J a n u a r y 2 0 1 2

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• 5.0%
• 4.0%
• 3.0%
• 2.0%
• 1.0%

0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% Factor Portfolio Weight Market Portfolio Weight

z-Beta

AAPL XOM

Example of Factor Portfolio Weights

J a n u a r y 2 0 1 2

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• 5.0%
• 4.0%
• 3.0%
• 2.0%
• 1.0%

0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% Factor Portfolio Weight Market Portfolio Weight

z-Beta z-Small z-Value z-Mom

AAPL XOM

Example of Factor Portfolio Weights

J a n u a r y 2 0 1 2

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

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• 200%
• 100%

0% 100% 200% 300% 400% 500% 600% 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

VMS SMB HML UMD

Cumulative Fama-French Factor Returns

1 9 6 3 t o 2 0 1 2

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• 100%
• 50%

0% 50% 100% 150% 200% 250% 300% 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

z-Beta z-Small z-Value z-Mom

Cumulative z-Factor Returns

1 9 6 3 t o 2 0 1 2

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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

Fama-French Returns

1 9 6 3 t o 2 0 1 2

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

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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% Recent z-Factor Portfolio Returns

2 0 0 3 t o 2 0 1 4

Market approximates Russell 3000

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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%

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:

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• C. Discretionary
• C. Staples

Energy Financial Healthcare Industrial Materials Technology Utilities 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%

State Street Sector SPDR and MSCI Minimum Volatility ETFs

Annualized excess (of risk-free rate) returns: 2003 to 2014

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RP,t = αP + βP RM,t + … + εP,t RP,t = αP + RM,t + (βP - 1) RM,t + … + εP,t RP,t = αP + RM,t + ZP Rzβ,t + … + εP,t

The exposure to all stocks in the Fama-MacBeth regressions used to estimate RM,t (intercept term) is exactly one, so the portfolio exposure to RM,t in the subsequent time-series regression is known to be one. A: Traditional methodology Restrict the coefficient on Rzβ,t to be zero (i.e., 0.000)

• r simply leave Rzβ,t out of the regression.

B: Alternative methodology Restrict the coefficient on RM,t to be one (i.e., 1.000)

• r simply subtract RM,t from RP,t .

Portfolio Performance Measurement by Time-Series Regression

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XLY Return Market z-Beta z-Small z-Value z-Mom Alpha A Coefficients 1.015 0.000 0.718 0.030

• 0.377
• 0.29%

(t-statistics) (0.3) (3.2) (0.2) (3.5) B Coefficients 1.000 0.106 0.643 0.040

• 0.361
• 0.13%

(t-statistics) (1.1) (2.9) (0.3) (3.3) Note: t-statistic for the Market factor tests the coefficient against one, t-statistics for the z-Factors test the coefficient against zero, and italics indicate a restricted value. Consumer Discretionary Market coefficient not significantly different from one z-Beta coefficient not significantly different from zero Little change to estimated Alpha

Consumer Discretionary ETF

2 0 0 3 t o 2 0 1 2

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XLP Return Market z-Beta z-Small z-Value z-Mom Alpha A Coefficients 0.576 0.000

• 0.436
• 0.156
• 0.113

3.31% (t-statistics) (9.1) (2.1) (1.2) (1.1) B Coefficients 1.000

• 1.051
• 0.222
• 0.551
• 0.192

0.33% (t-statistics) (14.4) (1.3) (5.6) (2.4) Consumer Staples Market coefficient significantly lower than one z-Beta coefficient significantly negative Large reduction in estimated Alpha (exposure to “beta payoff shortfall” is acknowledged)

Consumer Staples ETF

2 0 0 3 t o 2 0 1 2

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XLK Return Market z-Beta z-Small z-Value z-Mom Alpha A Coefficients 1.200 0.000

• 0.337
• 0.863

0.106

• 1.56%

(t-statistics) (4.5) (1.7) (6.7) (1.1) B Coefficients 1.000 0.571

• 0.511
• 0.680

0.157

• 0.09%

(t-statistics) (7.1) (2.8) (6.2) (1.8) Technology Market coefficient significantly greater than one z-Beta coefficient significantly positive Large increase in estimated Alpha (exposure to “beta payoff shortfall” is acknowledged)

Technology ETF

2 0 0 3 t o 2 0 1 2

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USMV Return

Market z-Beta z-Small z-Value z-Mom Alpha A Coefficients 0.692 0.000

• 0.204

0.202

• 0.026

3.16% (t-statistics) (10.3) (1.5) (2.3) (0.4) B Coefficients 1.000

• 0.765
• 0.046
• 0.085
• 0.083

0.99%

(t-statistics) (18.0) (0.5) (1.5) (1.8)

Alpha Accounting: Portfolio Return – (Coefficients) Factor Returns = Alpha Specification A: Other factors 7.29 – (0.692) 7.10 + 0.78 = 3.16% 7.29 – (1.000) 7.10 – (– 0.308) 7.10 + 0.78 = 3.16% Specification B: 7.29 – (1.000) 7.10 – (– 0.765) 1.11 – 0.05 = 0.99%

MSCI Minimum Volatility Index

2 0 0 3 t o 2 0 1 2

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0.7 SPY Return Market z-Beta z-Small z-Value z-Mom Alpha A Coefficients 0.690 0.000

• 0.206

0.008

• 0.022
• 0.06%

(t-statistics) (59.1) (8.6) (0.5) (1.9) B Coefficients 1.000

• 0.435
• 0.370
• 0.297
• 0.017
• 1.95%

(t-statistics) (10.2) (3.8) (5.1) (0.4) Combined Portfolio: 70% SPY + 30% Cash Market coefficient is significantly lower than one (high t-stats because R-squared is 99.5%) z-Beta coefficient is significantly negative (erroneous calculation, low beta is due to cash) Large reduction in Alpha is misstated (best estimate of Alpha is -0.06%)

S&P 500 ETF and Cash

2 0 0 3 t o 2 0 1 2

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The Not-So-Well-Known Three-And-One-Half-Factor Model

The well-known three factor model has four factors (or three plus the Market “half-factor”) in Fama-MacBeth regressions used in the first tests of the CAPM. A separate beta factor is also used in enhanced Fama-MacBeth regressions now used in equity risk-factor modeling paractice. The long-term (half century) return to a pure (i.e., multivariate) z-Beta factor is strikingly lower than the large positive value predicted by the CAPM, and often

• negative. The realized SML is not just “too flat” but downward sloping.

The estimated alpha of fully invested (non-cash) portfolios may be distorted without the use of a separate z-Beta factor. The z-Beta factor explains most of the non-zero alphas of industry portfolios. The “low beta” anomaly explains most of minimum variance portfolio alpha.

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References

Assness, Cliford, and Andrea Frazzini, “The Devil is in HML’s Details” Journal of Portfolio Management, 39 (Summer 2013): 49-68. Back, Kerry, Nishad Kapadia, and Barbara Ostdiek, “Slopes as Factors: Characteristic Pure Plays” SSRN (2013) working paper. Black, Fischer, Michael C. Jensen and Myron Scholes, “The Capital Asset Pricing Model: Some Empirical Tests” in Studies in the Theory of Capital Markets (1972), pp. 79–121. Carhart, Mark M., “On Persistence in Mutual Fund Performance” Journal of Finance, 52 (1997), 57-82. Chan, Louis K. C., Jason Karceski, and Joseph Lakonishok, “The Risk and Return from Factors” Journal of Financial and Quantitative Analysis, 33 (1998), 159-188. Clarke, Roger, Harindra de Silva and Steven Thorley, “Minimum Variance Portfolios in the U.S. Equity Market” Journal

• f Portfolio Management, 33 (Fall 2006), 10-24.

Clarke, Roger, Harindra de Silva and Steven Thorley, “Know Your VMS Exposure” Journal of Portfolio Management, 36 (Winter 2010), 52-59. Fama, Eugene F., and Kenneth R. French, “The Cross-section of Expected Stock Returns” Journal of Finance, 47 (1992), 427–465. Fama, Eugene F., and Kenneth R. French, “Multifactor explanations of Asset Pricing Anomalies” Journal of Finance, 51 (1996), 55-84. Fama, Eugene F., and James D. MacBeth, "Risk, Return, and Equilibrium: Empirical Tests" Journal of Political Economy, 81 (1973): 607–636. Frazzini, Andrea, and Lasse Pedersen, “Betting Against Beta” forthcoming in the Journal of Financial Economics. Jegadeesh, Narasimham, and Sheridan Titman, “Returns to buying winners and selling losers: Implications for stock market efficiency” Journal of Finance, 48 (1993), 65-91. Sharpe, William F., “Capital Asset Prices: a Theory of Market Equilibrium”, Journal of Finance, 19 (1964), 259–263.