Download the not-so-well-known three-and-one-half-factor model

THE NOT-SO-W ELL -K NOW N
THREE-AND-ONE -HALF -FA CTOR MODEL
Roger Clarke, Harindra de Silva, and Steven Thorley
Q-Group Spring Seminar
April 8, 2014
H o w M a n y F a c t o r s Wa s T h a t ?
“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
The Cross-section of Expected Stock Returns
Fama and French (1992)
Ri = A + B1 betai + B2 sizei + B3 valuei + εi
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)
3
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.
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
The Big Picture
Cumulative Factor Returns from 1963 to 2012
Market
z-Beta
z-Small
z-Value
z-Mom
350%
300%
250%
200%
150%
100%
50%
0%
-50%
-100%
2012
2010
2008
2006
2004
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1976
1974
1972
1970
1968
1966
1964
1962
6
5 0 Ye a r s O f F a c t o r R e t u r n s
1963 to 2012
Market
z-Beta
z-Small
z-Value
z-Mom
5.64%
-0.79%
0.86%
1.88%
4.99%
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
Average Return
Standard Deviation
7
5 0 Ye a r s O f F a c t o r R e t u r n s
1963 to 2012
Market
z-Beta
z-Small
z-Value
z-Mom
5.64%
-0.79%
0.86%
1.88%
4.99%
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
Standard Dev.
Average return - CAPM predicted return = Alpha of z-Beta factor
-0.79% - (0.309) 5.64% = -2.53%
8
Average Excess Return (%)
Capital Market Line (CML)
Market
5.64
0.00
0.00
15.54
Return Standard Deviation (%)
9
Average Excess Return (%)
Security Market Line (SML)
5.64
0.00
0.0
1.0
Market Beta
10
Average Excess Return (%)
Security Market Line (SML)
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
January 2012
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
0
1858
122
1565
1781
14
Example of Factor Portfolio Weights
January 2012
z-Beta
5.0%
XOM
4.0%
Factor Portfolio Weight
3.0%
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
January 2012
z-Beta
z-Small
z-Value
z-Mom
5.0%
XOM
4.0%
Factor Portfolio Weight
3.0%
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
1963 to 2012
UMD
HML
SMB
VMS
600%
500%
400%
300%
200%
100%
0%
-100%
-200%
2012
2010
2008
2006
2004
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1976
1974
1972
1970
1968
1966
1964
1962
18
Cumulative z-Factor Returns
1963 to 2012
z-Mom
z-Value
z-Small
z-Beta
300%
250%
200%
150%
100%
50%
0%
-50%
-100%
2012
2010
2008
2006
2004
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1976
1974
1972
1970
1968
1966
1964
1962
19
Fama-French Returns
1963 to 2012
MRF
VMS
SMB
HML
UMD
5.63%
0.33%
3.00%
4.74%
8.42%
15.57%
15.36%
10.80%
10.01%
14.81%
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
Average Return
Standard Deviation
Sharpe Ratio
20
Portfolio performance measurement with
“pure” factor returns: The impact of Beta
21
Recent z-Factor Portfolio Returns
2003 to 2014
Market approximates Russell 3000
Market
z-Beta
z-Small
z-Value
z-Mom
7.10%
1.11%
2.18%
-1.73%
0.11%
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%
Average Return
Standard Deviation
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
2003 to 2012
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
Utilities
Technology
Materials
Industrial
Healthcare
Financial
Energy
C. Staples
C. Discretionary
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
RP,t = αP + βP RM,t
+ … + εP,t
RP,t = αP + RM,t + (βP - 1) RM,t
RP,t = αP + RM,t + ZP Rzβ,t
+ … + εP,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)
or simply leave Rzβ,t out of the regression.
B: Alternative methodology
Restrict the coefficient on RM,t to be one (i.e., 1.000)
or simply subtract RM,t from RP,t .
25
Consumer Discretionary ETF
2003 to 2012
XLY Return
A Coefficients
(t-statistics)
Market
1.015
(0.3)
z-Beta
0.000
z-Small
0.718
(3.2)
z-Value
0.030
(0.2)
z-Mom
-0.377
(3.5)
Alpha
-0.29%
B Coefficients
(t-statistics)
1.000
0.106
(1.1)
0.643
(2.9)
0.040
(0.3)
-0.361
(3.3)
-0.13%
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
26
Consumer Staples ETF
2003 to 2012
XLP Return
A Coefficients
(t-statistics)
Market
0.576
(9.1)
z-Beta
0.000
z-Small
-0.436
(2.1)
z-Value
-0.156
(1.2)
z-Mom
-0.113
(1.1)
Alpha
3.31%
B Coefficients
(t-statistics)
1.000
-1.051
(14.4)
-0.222
(1.3)
-0.551
(5.6)
-0.192
(2.4)
0.33%
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)
27
Te c h n o l o g y E T F
2003 to 2012
XLK Return
A Coefficients
(t-statistics)
Market
1.200
(4.5)
z-Beta
0.000
z-Small
-0.337
(1.7)
z-Value
-0.863
(6.7)
z-Mom
0.106
(1.1)
Alpha
-1.56%
B Coefficients
(t-statistics)
1.000
0.571
(7.1)
-0.511
(2.8)
-0.680
(6.2)
0.157
(1.8)
-0.09%
Technology
Market coefficient significantly greater than one
z-Beta coefficient significantly positive
Large increase in estimated Alpha
(exposure to “beta payoff shortfall” is acknowledged)
28
M S C I M i n i m u m Vo l a t i l i t y I n d e x
2003 to 2012
USMV Return
A Coefficients
(t-statistics)
Market
0.692
(10.3)
z-Beta
0.000
z-Small
-0.204
(1.5)
z-Value
0.202
(2.3)
z-Mom
-0.026
(0.4)
Alpha
3.16%
B Coefficients
1.000
-0.765
-0.046
-0.085
-0.083
0.99%
(18.0)
(0.5)
(1.5)
(1.8)
(t-statistics)
Alpha Accounting:
Portfolio Return – (Coefficients) Factor Returns = Alpha
Specification A:
7.29 – (0.692) 7.10
7.29 – (1.000) 7.10 – (– 0.308) 7.10
Specification B:
7.29 – (1.000) 7.10 – (– 0.765) 1.11
Other factors
+ 0.78 = 3.16%
+ 0.78 = 3.16%
– 0.05
= 0.99%
29
S&P 500 ETF and Cash
2003 to 2012
0.7 SPY Return
A Coefficients
(t-statistics)
Market
z-Beta
z-Small
z-Value
B Coefficients
(t-statistics)
z-Mom
Alpha
0.690
(59.1)
0.000
-0.206
(8.6)
0.008
(0.5)
-0.022
(1.9)
-0.06%
1.000
-0.435
(10.2)
-0.370
(3.8)
-0.297
(5.1)
-0.017
(0.4)
-1.95%
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%)
30
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.
31
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
of 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.
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