Signal Processing - May 2017 - 20
An Example CAPM Regression
Table S1 shows time-series regressions of excess stock portfolio
returns (in percent), R pt - rf, t , on the excess stock-market return,
R M, t - rf, t , from July 1963 to December 1991 (342 months)
for 25 portfolios using Center for Research in Security Prices
monthly return data of all U.S. market stocks. The 25 portfolios
are constructed by dividing firms using market caps (size) and
book-to-market (B/M) ratios into quintile buckets. The empirical
results are extracted from [23]. The t-values and s ( f ) are in
parentheses. The residual standard error s ( f ) is the root-meansquare error of the regression. As can be seen, only a few a
coefficients are statistically significant, i.e., t ( a ) 2 2. All of
the b coefficients are statistically significant.
2
3
Size
Low
2
Small
1.40
1.26
4
1.11
1.06
1.08
(26.33)
(28.12)
(27.01)
(25.03)
(23.01)
1.42
1.15
1.12
1.02
1.13
(26.33)
(28.12)
(27.01)
(25.03)
(23.01)
1.36
1.15
1.04
0.96
1.08
(26.33)
(28.12)
(27.01)
(25.03)
(23.01)
1.24
1.14
1.03
0.95
1.10
(26.33)
(28.12)
(27.01)
(25.03)
(23.01)
1.03
0.99
0.89
0.84
0.89
(28.12)
(27.01)
(25.03)
(23.01)
2
3
(26.33)
High
2
3
4
Big
4
High
Adjusted R2 and Residual Standard Error s (f)
a and t - value t (a)
Small
3
b and t-value t ( b)
Big
B/M Ratio
Low
B/M Ratio
4
Table S1. The time-series regressions from July 1963
through December 1991.
Size
Table S1. The time-series regressions from July 1963
through December 1991.
-0.22
0.15
0.30
0.42
0.54
(−0.90)
(0.73)
(1.54)
(2.19)
(2.53)
−0.18
0.17
0.36
0.39
0.53
(−1.00)
(1.05)
(2.35)
(2.79)
(3.01)
−0.16
0.15
0.23
0.39
0.50
(−1.12)
(1.25)
(1.82)
(3.20)
(3.19)
−0.05
−0.14
0.12
0.35
0.57
(−0.50)
(−1.50)
(1.20)
(2.91)
(3.71)
−0.04
−0.07
−0.07
0.20
0.21
(−0.49)
(−0.95)
(−0.70)
(1.89)
(1.41)
Small
2
3
4
Big
0.67
0.70
0.68
0.65
0.61
(−1.46)
(3.76)
(3.55)
(3.56)
(3.92)
0.79
0.79
0.76
0.76
0.71
(3.34)
(2.96)
(2.85)
(2.59)
(3.25)
0.84
0.84
0.80
0.79
0.74
(2.65)
(2.28)
(2.33)
(3.26)
(2.90)
0.89
0.90
0.87
0.80
0.76
(2.01)
(1.73)
(1.84)
(2.21)
(2.83)
0.89
0.91
0.54
0.79
0.69
(1.66)
(1.35)
(1.73)
(1.95)
(2.69)
FF Three-Factor Model and Cross-Sectional Returns
The FF three-factor model is an empirical asset-pricing
model featuring two cross-sectional return factors, the
small-minus-big (SMB) factor and the high-minus-low
(HML) factor:
E [R - rf ] = b M E [R M - rf ] + b s E [R SMB] + b h E [R HML] .
* The SMB factor is a firm size factor, represented by
the portfolio return of SMB (a portfolio of long
small-market-cap stocks and short big-cap stocks. Firm
size is measured by firm market capitalization (cap).
Small-size firms have excess returns in addition to the
market RP.
20
* The HML is a market-value factor, calculated by the portfolio return of HML (a portfolio of long high-value stocks
and short low-value stocks). Firm value is measured by
the B/M ratio or the price-to-earnings ratio. High B/M
ratios represent value stocks and low ratios growth
stocks. Value stocks have excess returns in addition to
the market RP.
Insight
SMB (or small-market cap) and HML (or value) portfolios
have significant nonzero alpha with respect to the CAPM,
i.e., excess returns in addition to the market RP. Note that
these two excess returns are predicted by the cross-sectional
factors determined by firm accounting features. Crosssectional means across various stocks/portfolios (sections).
IEEE Signal Processing Magazine
|
May 2017
|
Table of Contents for the Digital Edition of Signal Processing - May 2017
Signal Processing - May 2017 - Cover1
Signal Processing - May 2017 - Cover2
Signal Processing - May 2017 - 1
Signal Processing - May 2017 - 2
Signal Processing - May 2017 - 3
Signal Processing - May 2017 - 4
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Signal Processing - May 2017 - 112
Signal Processing - May 2017 - Cover3
Signal Processing - May 2017 - Cover4
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