1. The Value Premium and the CAPM
Fama and French (2006)

2. I. Introduction
FF(1993) find that the post-1963 value premium is left unexplained by the CAPM.
Ang and Chen (2005): the CAPM captures the value premium of the 1926 to They also argue that when the tests allow for time-varying market βs, even the post 1963 period produces no reliable evidence against CAPM for the value premium.
Loughran (1997): the value premium of 1963 to 1995 is particular to small stocks.

3. I. Introduction (con.) This paper has three goals.
1. How value premium vary with firm size.
Loughran (1997)的發現似乎僅適用於 (1) post-1963 period, (2) using the B/M as the value-growth indicator, (3) U.S. stocks.
2. If and when CAPM market βs explain observed value premium.
CAPM fails the tests for 1963~2004 (value stocks have lower β), but captures the value premiums for 1926~1963.

4. I. Introduction (con.)
3. Whether variation in β across stocks is related to average returns as predicted by CAPM.
Extends FF(1992): when forming portfolios on size, B/M and β, it is found that the variation in β unrelated to size and B/M goes unrewarded during 1928 to 1963 and throughout the sample period (1963~2004).
The above finding is as strong for big stocks as for small stocks.

5. II. Size and value premium
Examine the variants of VMG (or HML), the monthly value-growth returns of the three-factor models of FF (1993).
1. 建立三因子模型中的價值貼水(value premium)必須先建構價值型與成長型之投資組合,請參考p.2165的說明: 每年根據市值(S and B)與B/M (G, N and V)交叉形成六個value-weight size-B/M 投組. 此六個投組是計算三因子模型中規模貼水與價值貼水的基本元素(eq1,2).

6. II. Size and value premium
SMB: 三個小規模投組之monthly returns minus 三個大規模投組之average monthly returns
VMG: 兩個價值型投組之average monthly returns minus 兩個成長型投組之average monthly returns
2. Spilt VMG into its small stock and big stock components (eq.3) to test whether the value premium is special to small stocks.
3. Empirical results
表1為各因子與前述投組之月報酬率的敘述統計.

7. II. Size and value premium (con.)
樣本期間1926/07~2004/12(與sub-samples), 包含NYSE, AMEX, and Nasdaq 股票. 發現:
The value premiums (VMG) for the two sub-periods differ by just 0.38 standard errors. There is a value premium (0.4% per month) in expected returns.
Before 1963, the value premium is nearly identical for small and big stocks. Besides, the big stock value premium (VMGB) for 1963~2004 is just standard errors from the premium for 1926~1963. So there is little evidence of a change

8. II. Size and value premium (con.)
in the expected premium. The VMGB average return for the full 1926 to 2004 period is then solid evidence on the existence of a value premium in big stock expected returns (從full sample來看,小規模與大規模股票皆有價值貼水, 但小規模股票較高).
Finer size sorts (form 25 size-B/M portfolios as FF (1993))(This test is limited to the period 1963 to 2004)
表2為這25個投組的特性,表3為這25個投組的

9. II. Size and value premium (con.)
average returns, along with value premium within size quintiles, and size premium within B/M quintiles.
1. The value (size) premium for a size (B/M) quintile is the difference between the average return on the two highest B/M (smallest size) portfolios and the average return on the two lowest B/M (biggest size) portfolios of the size (B/M) quintile.

10. II. Size and value premium (con.)
2. When value and growth are defined by sorts on B/M, the value premiums decline monotonically from smaller to bigger size quintiles (但只有最大規模投組的價值貼水不顯著異於零).
3. When value and growth are defined by sorts on E/P, the small size quintile still produces the largest value premium, but any decline in value premium with size is far from monotonic.
The larger value premium we observe for small stocks (B/M as indicator) are due more to lower

11. II. Size and value premium (con.)
returns on small growth stocks than to higher returns on small value stocks.
Using E/P as indicator reduces average returns for the two extreme growth portfolios in the largest size quintile and increases average returns on the two extreme value portfolios.
4. There is a monotonic increase in size premiums from lower to higher B/M groups.
小結: p. 2172

12. III. The value premium and CAPM
FF (1993) conclude that CAPM can not explain the value premium in average returns for the post-1963 period.
First hint from figure 1: the plot of year-by-year betas for VMG (should be positive if CAPM can explain value premiums).
Time-series tests (迴歸)
Regress VMG, VMGS, VMGB returns on the excess market returns (Table 5).

13. III. The value premium and CAPM (con.)
1. VMG, VMGB and VMGS market betas are negative for 1963 to The intercept is significantly different from zero that easily rejects CAPM.
2. The power to reject CAPM for 1963 to seems to come more from the high returns on value portfolios than from the lower returns on growth portfolios.
3. Confirms the evidence of Ang and Chen (2005). CAPM can explain VMG, VMGS and VMGB

14. III. The value premium and CAPM (con.)
for the 1926 to 1963 period.
4. Table 5 also run SMB on the market excess returns: size premium in average returns is consistent with CAPM pricing (the SMB’s market beta does not change from the earlier to the later period).
Time-varying β
1. Estimate CAPM regressions for 1926 to 2004 that uses slope dummies to allow for periodic changes in
. Four alternatives are examined.

15. III. The value premium and CAPM (con.)
表6 panel A: for every portfolio, shortening the estimation interval for increase R2 or leaves it unchanged.
Allowing to change every year weakens the evidence against CAPM pricing of the value premium for the full sample period. However, the rejection is mostly due to small stocks.
Allowing annual change of continues to produce a strong rejection of the CAPM in the GRS test.

16. III. The value premium and CAPM (con.)
2. The rejection of CAPM pricing as an explanation may be entirely due to the 1963 to 2004.
表6 panel B: the regression contains a full period intercept, and a marginal intercept a63-a26, estimated by including a dummy variable for July 1963 to December 2004.
a63-a26 measures differences between CAPM intercepts for 1963 to 2004 and 1926 to The dummy variable thus tells us whether the CAPM’s ability to explain value premiums is different in the two periods. The answer is yes.

17. III. The value premium and CAPM (con.)
表6 panel C: estimating by including dummy variables for 1926 to 1963 and 1963 to 2004, but no full-period intercept.

18. III Sorts and the CAPM
Do the small CAPM intercepts for the value premiums of 1926 to 1963 imply that CAPM explains expected stock returns during this period?
1. Alternative explanation: expected returns vary with B/M (or a risk related to B/M), and seems to be rewarded in average returns only when it is positively correlated with B/M.
2. Since both B/M and are higher for value stocks, the positive value premiums are consistent with both the CAPM and B/M explanations.

19. III. Sorts and the CAPM (con.)
One way to distinguish between the above two explanations is to create variation in that is independent of the variation in B/M.
1. Split each of the six size-B/M portfolios into high and low each year. Then calculate the difference between the value weight returns on the six high
and the six low portfolios. An overall difference, the simple average of the six spreads, HBmLB is also calculated.

20. III. Sorts and the CAPM (con.)
2. 表7 顯示: splitting the size-B/M portfolios on estimates of beta produces small spreads in average returns. The tiny average spreads are ominous for the CAPM.
3. 表7 also runs the CAPM regression: regress the spread portfolio returns (high beta-low beta) on the market return in excess of the Treasury bill rate for each size-B/M portfolio. The results confirm that average returns on the beta-spread portfolios violate the CAPM.

21. IV. Conclusions
The CAPM has fatal problems throughout the 1926 to 2004 period. Specifically, size and B/M (or risk related to them) are important in expected returns, and CAPM has little or no independent role.