Investments, tenth edition

R

S

5

1

@

3

(R

S/L

1 R

S/M

1 R

S/H

); R

B

5

1

@

3

(R

B/L

1 R

B/M

1 R

B/H

)  

 Similarly, the returns on the high and low (Value and Growth  

20

  )  portfolios  are:   

R

H

5

1

@

2

(R

SH

1 R

BH

); R

L

5

1

@

2

(R

SL

1 R

BL

)  

 The returns of the zero-net-investment factors SMB (Small minus Big, i.e., Long Small 

and Short Big), and HML (High minus Low, i.e., Long High B/M and Short Low B/M) are 

created from these portfolios:   

R

SMB

5 R

S

2 R

B

; R

HML

5 R

H

2 R

L

  

 We measure the sensitivity of individual stocks to the factors by estimating the factor betas 

from first-pass regressions of stock excess returns on the excess return of the market index 

as well as on  R  

 SMB 

  and  R  

 HML 

 . These factor betas should, as a group, predict the total risk 

premium. Therefore, the Fama-French three-factor asset-pricing model is  

21

     

 

E(r

i

)

2 r

f

5 a

i

1 b

i

3E(r

M

)

2 r

f

4 1 s

i

E

3SMB4 1 h

i

E

3HML4 

 (13.8)   

19

Fama and French could have experimented with optimal break points for the three B/M groups, but such an 

approach might quickly give way to data mining.

20

High B/M stocks are called value assets because, for the large part, their market values derive from assets 

already in place. Low B/M are called growth stocks because their market values derive from expected growth in 

future cash flows. One needs to assume high growth to justify the prices at which the assets trade. At the same 

time, however, a firm that falls into hard times will see its market price fall and its B/M ratio rise. So some of 

the so-called value firms may actually be distressed firms. This subgroup of the value-firm portfolio may well 

account for the value premium of the B/M factor.

21

We subtract the risk-free rate from the return on the market portfolio, but not from the SMB and HML returns 

because the SMB and HML factors are zero-net-investment portfolios. Hence their entire return already is a pre-

mium. There is no opportunity cost from giving up the risk-free investment to switch into these portfolios.

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7/17/13   3:47 PM

7/17/13   3:47 PM

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428

P A R T   I I I

  Equilibrium in Capital Markets

 The coefficients  b  

 i 

 ,   s  

 i 

 , and  h  

 i 

  are the 

betas (also called  loadings  in this con-

text) of the stock on the three factors. 

If these are the only risk factors, excess 

returns on all assets should be fully 

explained by risk premiums due to 

these factor loadings. In other words, 

if these factors fully explain asset 

returns, the intercept of the equation 

should be zero. 

 Goyal  

22

 

 

 surveys asset pricing 

tests. He applies Equation 13.8 to the 

returns of 25 portfolios of all U.S. 

stocks sorted by size and B/M ratio. 

 Figure  13.1  shows the average actual 

return of each portfolio over the period 

1946–2010 against returns predicted 

by the CAPM (panel A) and by the FF 

three-factor model. In this test, the FF 

model provides a clear improvement 

over the CAPM.

   

 Notice in panel A that the predicted 

returns are almost the same for all 

portfolios. This is indeed a weakness 

of tests with portfolios that are sorted 

on size and B/M, but not on beta. As a 

result, all portfolios have betas near 1.0. 

Adding a sort on beta to a 5   3   5  sort 

on size and B/M will raise the number 

of portfolios from 25 to 125. This is 

unwieldy. But advances in economet-

rics and computing power will allow 

these types of tests to advance.