2012.05.10 - Fama-French Risk

Prévia do material em texto

1 
  
V
ol
at
ili
ty
 a
nd
 P
re
m
iu
m
s i
n 
U
S 
Eq
ui
ty
 R
et
ur
ns
 
Eu
ge
ne
 F
. F
am
a 
an
d 
K
en
ne
th
 R
. F
re
nc
h*
 
 
U
nd
er
st
an
di
ng
 v
ol
at
ili
ty
 i
s 
cr
uc
ia
l 
fo
r 
in
fo
rm
ed
 i
nv
es
tm
en
t 
de
ci
si
on
s. 
Th
is
 p
ap
er
 e
xp
lo
re
s 
th
e 
vo
la
til
ity
 o
f 
th
e 
m
ar
ke
t, 
si
ze
, a
nd
 v
al
ue
 p
re
m
iu
m
s 
of
 th
e 
Fa
m
a-
Fr
en
ch
 th
re
e-
fa
ct
or
 m
od
el
 f
or
 U
S 
eq
ui
ty
 
re
tu
rn
s. 
 Ta
bl
e 
1 
su
m
m
ar
iz
es
 th
e 
an
nu
al
 p
re
m
iu
m
s. 
Th
e 
da
ta
 a
re
 fr
om
 th
e 
C
en
te
r f
or
 R
es
ea
rc
h 
in
 S
ec
ur
ity
 
Pr
ic
es
 (C
R
SP
) o
f t
he
 B
oo
th
 S
ch
oo
l o
f B
us
in
es
s 
at
 th
e 
U
ni
ve
rs
ity
 o
f C
hi
ca
go
. T
he
 C
R
SP
 d
at
a 
on
 N
Y
SE
 
st
oc
ks
 s
ta
rt 
in
 1
92
6;
 A
m
ex
 a
nd
 N
A
SD
A
Q
 s
to
ck
s 
ar
e 
ad
de
d 
in
 1
96
3 
an
d 
19
73
, 
re
sp
ec
tiv
el
y.
 T
he
 
pr
em
iu
m
s 
in
 T
ab
le
 1
 a
re
 (i
) 
R
M
 –
 R
F,
 th
e 
di
ff
er
en
ce
 b
et
w
ee
n 
th
e 
m
ar
ke
t r
et
ur
n 
an
d 
th
e 
ris
kf
re
e 
re
tu
rn
, 
ca
lle
d 
th
e 
eq
ui
ty
 p
re
m
iu
m
; (
ii)
 S
M
B
 (s
m
al
l m
in
us
 b
ig
), 
th
e 
si
ze
 p
re
m
iu
m
; a
nd
 (i
ii)
 H
M
L 
(h
ig
h 
m
in
us
 lo
w
 
bo
ok
-to
-m
ar
ke
t e
qu
ity
), 
th
e 
va
lu
e 
pr
em
iu
m
. 
 
 
Ta
bl
e 
1 
al
so
 s
um
m
ar
iz
es
 t
he
 a
nn
ua
l 
re
tu
rn
s 
fo
r 
th
e 
co
m
po
ne
nt
s 
of
 t
he
 p
re
m
iu
m
s. 
Th
e 
m
ar
ke
t 
re
tu
rn
, l
ab
el
ed
 R
M
, i
s t
he
 c
ap
-w
ei
gh
te
d 
re
tu
rn
 fo
r a
ll 
U
S 
eq
ui
tie
s o
n 
C
R
SP
, a
nd
 th
e 
ris
kf
re
e 
re
tu
rn
, R
F,
 is
 
th
e 
an
nu
al
 r
et
ur
n 
on
 a
 o
ne
-m
on
th
 T
re
as
ur
y 
bi
ll 
ro
lle
d 
ov
er
 e
ve
ry
 m
on
th
 d
ur
in
g 
th
e 
ye
ar
. 
W
e 
us
e 
six
 
po
rtf
ol
io
s 
to
 c
on
st
ru
ct
 S
M
B
 a
nd
 H
M
L.
 T
he
 la
be
ls
 fo
r t
he
 s
ix
 p
or
tfo
lio
s 
ar
e 
de
sc
rip
tiv
e.
 T
he
 fi
rs
t l
et
te
r, 
S 
or
 B
, i
nd
ic
at
es
 th
at
 th
e 
st
oc
ks
 in
 a
 p
or
tfo
lio
 a
re
 s
m
al
l (
N
Y
SE
, A
m
ex
, a
nd
 N
A
SD
A
Q
 s
to
ck
s 
w
ith
 m
ar
ke
t 
ca
p 
be
lo
w
 th
e 
N
Y
SE
 m
ed
ia
n)
 o
r 
bi
g 
(m
ar
ke
t c
ap
 a
bo
ve
 th
e 
N
Y
SE
 m
ed
ia
n)
. T
he
 s
ec
on
d 
le
tte
r 
is
 G
 f
or
 
gr
ow
th
, N
 f
or
 n
eu
tra
l, 
or
 V
 f
or
 v
al
ue
. W
e 
de
fin
e 
gr
ow
th
 s
to
ck
s 
as
 N
Y
SE
, A
m
ex
, a
nd
 N
A
SD
A
Q
 s
to
ck
s 
w
ith
 b
oo
k-
to
-m
ar
ke
t (
B
/M
) e
qu
ity
 ra
tio
s 
in
 th
e 
bo
tto
m
 3
0%
 o
f B
/M
 fo
r N
Y
SE
 s
to
ck
s. 
N
eu
tra
l s
to
ck
s 
ar
e 
th
os
e 
w
ith
 B
/M
 in
 th
e 
m
id
dl
e 
40
%
 o
f B
/M
 fo
r N
Y
SE
 s
to
ck
s. 
A
nd
 v
al
ue
 s
to
ck
s 
ar
e 
th
os
e 
w
ith
 B
/M
 in
 th
e 
to
p 
30
%
 o
f 
B
/M
 f
or
 N
Y
SE
 s
to
ck
s. 
Fl
ip
pi
ng
 th
e 
ra
tio
, h
ig
h 
B/
M
 is
 lo
w
 M
/B
, s
o 
va
lu
e 
st
oc
ks
 h
av
e 
lo
w
 
pr
ic
es
 re
la
tiv
e 
to
 b
oo
k 
va
lu
e 
an
d 
gr
ow
th
 st
oc
ks
 h
av
e 
hi
gh
 p
ric
e-
to
-b
oo
k 
ra
tio
s. 
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
    
* B
oo
th
 S
ch
oo
l 
of
 B
us
in
es
s, 
U
ni
ve
rs
ity
 o
f 
C
hi
ca
go
 (
Fa
m
a)
 a
nd
 A
m
os
 T
uc
k 
Sc
ho
ol
 o
f 
B
us
in
es
s, 
D
ar
tm
ou
th
 C
ol
le
ge
 (
Fr
en
ch
). 
Fa
m
a 
an
d 
Fr
en
ch
 a
re
 a
ls
o 
st
oc
kh
ol
de
rs
 i
n,
 b
oa
rd
 m
em
be
rs
 o
f, 
an
d 
co
ns
ul
ta
nt
s t
o 
D
im
en
si
on
al
 F
un
d 
A
dv
is
or
s.  
  
2 
  
W
e 
w
ei
gh
t t
he
 s
to
ck
s 
in
 th
e 
si
x 
po
rtf
ol
io
s 
us
ed
 to
 c
on
st
ru
ct
 S
M
B
 a
nd
 H
M
L 
by
 th
ei
r m
ar
ke
t c
ap
s, 
bu
t S
M
B
 a
nd
 H
M
L 
ar
e 
eq
ua
l-w
ei
gh
t a
ve
ra
ge
s 
of
 th
e 
re
tu
rn
s 
on
 th
e 
si
x 
po
rtf
ol
io
s. 
Sp
ec
ifi
ca
lly
, t
he
 s
iz
e 
pr
em
iu
m
, S
M
B
, i
s 
th
e 
si
m
pl
e 
av
er
ag
e 
of
 th
e 
re
tu
rn
s 
on
 th
e 
th
re
e 
sm
al
l-s
to
ck
 p
or
tfo
lio
s 
m
in
us
 th
e 
si
m
pl
e 
av
er
ag
e 
of
 th
e 
re
tu
rn
s o
n 
th
e 
th
re
e 
bi
g-
st
oc
k 
po
rtf
ol
io
s, 
SM
B
 =
 (S
G
 +
 S
N
 +
 S
V
)/3
 –
 (B
G
 +
 B
N
 +
 B
V
)/3
. 
Li
ke
w
is
e,
 t
he
 v
al
ue
 p
re
m
iu
m
, 
H
M
L,
 i
s 
th
e 
si
m
pl
e 
av
er
ag
e 
of
 t
he
 r
et
ur
ns
 o
n 
th
e 
tw
o 
va
lu
e 
po
rtf
ol
io
s m
in
us
 th
e 
si
m
pl
e 
av
er
ag
e 
of
 th
e 
re
tu
rn
s o
n 
th
e 
tw
o 
gr
ow
th
 p
or
tfo
lio
s, 
H
M
L 
= 
(S
V
 +
 B
V
)/2
 –
 (S
G
 +
 B
G
)/2
. 
 
Th
e 
m
ot
iv
at
io
n 
fo
r e
qu
al
-w
ei
gh
tin
g 
th
e 
co
m
po
ne
nt
s 
of
 S
M
B
 a
nd
 H
M
L 
is
 to
 h
av
e 
a 
si
ze
 p
re
m
iu
m
 
th
at
 is
 n
eu
tra
l w
ith
 re
sp
ec
t t
o 
va
lu
e 
ve
rs
us
 g
ro
w
th
 (b
ec
au
se
 it
 e
qu
al
-w
ei
gh
ts
 it
s 
gr
ow
th
, n
eu
tra
l, 
an
d 
va
lu
e 
co
m
po
ne
nt
s)
 a
nd
 a
 v
al
ue
 p
re
m
iu
m
 th
at
 is
 n
eu
tra
l w
ith
 re
sp
ec
t t
o 
si
ze
 (b
ec
au
se
 it
 e
qu
al
-w
ei
gh
ts
 it
s 
sm
al
l 
an
d 
bi
g 
co
m
po
ne
nt
s)
. 
 
O
ur
 m
es
sa
ge
 c
en
te
rs
 o
n 
th
e 
im
po
rta
nc
e 
of
 t
he
 v
ol
at
ili
ty
 o
f 
R
M
 –
 R
F,
 S
M
B
, 
an
d 
H
M
L 
in
 
in
ve
st
m
en
t o
ut
co
m
es
. T
o 
se
t t
he
 s
ta
ge
, F
ig
ur
es
 1
 to
 3
 s
ho
w
 a
nn
ua
l v
al
ue
s 
of
 th
e 
th
re
e 
pr
em
iu
m
s 
fo
r t
he
 
19
27
–2
01
1 
sa
m
pl
e 
pe
rio
d.
 N
ot
e 
th
e 
sc
al
e 
of
 t
he
 v
er
tic
al
 (
R
et
ur
n)
 a
xi
s 
of
 t
he
 e
qu
ity
 p
re
m
iu
m
 p
lo
t 
in
 
Fi
gu
re
 1
, f
ro
m
 -6
0%
 to
 6
0%
. T
hi
s w
id
e 
sp
re
ad
 is
 n
ec
es
sa
ry
 to
 a
cc
om
m
od
at
e 
ye
ar
-b
y-
ye
ar
 o
bs
er
va
tio
ns
 o
f 
th
e 
an
nu
al
 e
qu
ity
 p
re
m
iu
m
. T
he
 m
os
t e
xt
re
m
e 
re
tu
rn
s 
ar
e 
in
 th
e 
19
30
s, 
bu
t e
ve
n 
if 
w
e 
st
ar
t i
n 
19
50
, t
he
 
an
nu
al
 r
et
ur
ns
 r
an
ge
 f
ro
m
 -
39
.9
9%
 (
20
08
) 
to
 4
9.
31
%
 (
19
54
). 
In
 s
ho
rt,
 r
et
ur
ns
 t
o 
st
oc
k 
in
ve
st
in
g 
ar
e 
vo
la
til
e.
 F
ig
ur
es
 2
 a
nd
 3
, w
hi
ch
 p
lo
t t
he
 e
qu
ity
 p
re
m
iu
m
 a
lo
ng
 w
ith
 th
e 
si
ze
 p
re
m
iu
m
 (F
ig
ur
e 
2)
 a
nd
 th
e 
va
lu
e 
pr
em
iu
m
 (
Fi
gu
re
 3
), 
sh
ow
 t
ha
t 
SM
B
 a
nd
 H
M
L 
ar
e 
sl
ig
ht
ly
 l
es
s 
vo
la
til
e 
th
an
 R
M
 –
 R
F,
 b
ut
 th
e 
vo
la
til
ity
 o
f t
he
 si
ze
 a
nd
 v
al
ue
 p
re
m
iu
m
s i
s n
ev
er
th
el
es
s h
ig
h.
 
 
N
ex
t w
e 
ne
ed
 a
 b
it 
of
 te
rm
in
ol
og
y.
 W
he
n 
st
at
is
tic
ia
ns
 ta
lk
 a
bo
ut
 th
e 
tru
e 
m
ea
n 
of
 a
 p
ro
ba
bi
lit
y 
di
st
rib
ut
io
n,
 th
ey
 c
al
l i
t t
he
 e
xp
ec
te
d 
va
lu
e.
 S
am
pl
e 
m
ea
ns
 o
r a
ve
ra
ge
s 
lik
e 
th
os
e 
in
 T
ab
le
 1
 a
re
 e
st
im
at
es
 
of
 e
xp
ec
te
d 
va
lu
es
. H
en
ce
fo
rth
, w
e 
us
e 
th
e 
te
rm
 “
ex
pe
ct
ed
 v
al
ue
” 
w
he
n 
re
fe
rr
in
g 
to
 tr
ue
 m
ea
ns
. 
 
3 
   
Fo
r e
ac
h 
po
rtf
ol
io
, T
ab
le
 1
 sh
ow
s t
w
o 
st
at
is
tic
s. 
Th
e 
fir
st
 is
 th
e 
av
er
ag
e 
an
nu
al
 re
tu
rn
 (s
im
pl
e,
 n
ot
 
co
m
po
un
de
d)
, l
ab
el
ed
 M
ea
n.
 T
he
 s
ec
on
d 
is
 th
e 
st
an
da
rd
 d
ev
ia
tio
n 
of
 a
 p
or
tfo
lio
’s
 a
nn
ua
l r
et
ur
ns
, l
ab
el
ed
 
St
d,
 w
hi
ch
 is
 a
 s
um
m
ar
y 
m
ea
su
re
 o
f 
th
e 
vo
la
til
ity
 o
f 
ye
ar
-b
y-
ye
ar
 r
et
ur
ns
. T
o 
fo
cu
s 
th
e 
st
or
y,
 T
ab
le1
 
sh
ow
s 
re
su
lts
 o
nl
y 
fo
r 
19
63
–2
01
1,
 w
hi
ch
 i
nc
lu
de
s 
N
Y
SE
, 
A
m
ex
, 
an
d 
(b
eg
in
ni
ng
 i
n 
19
73
) 
N
A
SD
A
Q
 
st
oc
ks
. T
hi
s 
is
 a
 l
on
g 
49
-y
ea
r 
in
ve
st
m
en
t 
lif
et
im
e.
 W
e 
st
re
ss
 t
he
 c
on
ce
pt
 o
f 
an
 i
nv
es
tm
en
t 
lif
et
im
e 
to
 
sh
ar
pe
n 
th
e 
m
ai
n 
m
es
sa
ge
. I
n 
a 
nu
ts
he
ll,
 th
e 
m
es
sa
ge
 (d
ev
el
op
ed
 in
 d
et
ai
l b
el
ow
) i
s t
ha
t l
on
g 
tim
e 
pe
rio
ds
 
ar
e 
re
qu
ire
d 
to
 b
e 
re
as
on
ab
ly
 s
ur
e 
th
at
 t
he
 a
ve
ra
ge
 e
qu
ity
 p
re
m
iu
m
, t
he
 a
ve
ra
ge
 s
iz
e 
pr
em
iu
m
, o
r 
th
e 
av
er
ag
e 
va
lu
e 
pr
em
iu
m
 w
ill
 b
e 
po
si
tiv
e.
 
 
Le
t’s
 m
ak
e 
th
e 
po
in
t c
on
cr
et
el
y.
 T
ab
le
 2
 p
re
se
nt
s 
es
tim
at
es
 o
f t
he
 li
ke
lih
oo
d 
of
 a
 n
eg
at
iv
e 
eq
ui
ty
 
pr
em
iu
m
 (
bi
lls
 b
ea
t s
to
ck
s)
 f
or
 p
er
io
ds
 o
f 
in
cr
ea
si
ng
 le
ng
th
. T
he
 c
al
cu
la
tio
ns
 u
se
 th
e 
ob
se
rv
ed
 a
ve
ra
ge
 
an
nu
al
 e
qu
ity
 p
re
m
iu
m
 f
or
 1
96
3–
20
11
 (
5.
95
%
 p
er
 y
ea
r)
 a
nd
 th
e 
st
an
da
rd
 d
ev
ia
tio
n 
of
 th
e 
ye
ar
-b
y-
ye
ar
 
re
tu
rn
s (
17
.8
5%
) a
s t
he
 tr
ue
 e
xp
ec
te
d 
va
lu
e 
an
d 
vo
la
til
ity
 (t
hu
s t
he
 la
be
l P
ro
b 
< 
0,
 E
(R
M
 –
 R
F)
 =
 5
.9
5%
). 
Th
e 
ca
lc
ul
at
io
ns
 a
ss
um
e 
th
at
 a
nn
ua
l o
bs
er
va
tio
ns
 o
n 
th
e 
eq
ui
ty
 p
re
m
iu
m
 a
re
 n
or
m
al
ly
 d
is
tri
bu
te
d;
 in
 fa
ct
, 
th
e 
di
st
rib
ut
io
n 
is
 f
at
-ta
ile
d 
(m
or
e 
ou
tli
er
s 
th
an
 p
re
di
ct
ed
 if
 r
et
ur
ns
 a
re
 n
or
m
al
) 
an
d 
rig
ht
-s
ke
w
ed
 (
m
or
e 
ex
tre
m
e 
po
si
tiv
e 
th
an
 n
eg
at
iv
e 
re
tu
rn
s)
. S
ta
tis
tic
al
 th
eo
ry
 (
th
e 
ce
nt
ra
l 
lim
it 
th
eo
re
m
) 
te
lls
 u
s, 
ho
w
ev
er
, 
th
at
 th
e 
di
st
rib
ut
io
n 
of
 th
e 
av
er
ag
e 
pr
em
iu
m
 c
on
ve
rg
es
 to
w
ar
d 
th
e 
no
rm
al
 a
s 
th
e 
pe
rio
d 
co
ve
re
d 
by
 th
e 
av
er
ag
e 
in
cr
ea
se
s. 
 
 
Th
e 
m
ea
n 
of
 th
e 
an
nu
al
 e
qu
ity
 p
re
m
iu
m
 f
or
 1
96
3–
20
11
 is
 la
rg
e,
 5
.9
5%
 p
er
 y
ea
r. 
Th
is
 is
 w
ha
t 
at
tra
ct
s 
in
ve
st
or
s 
to
 s
to
ck
s. 
Th
e 
st
an
da
rd
 d
ev
ia
tio
n 
of
 th
e 
pr
em
iu
m
 is
 a
ls
o 
la
rg
e,
 1
7.
85
%
, o
r 
th
re
e 
tim
es
 
th
e 
m
ea
n.
 T
he
 y
ea
r-b
y-
ye
ar
 v
al
ue
s 
of
 t
he
 p
re
m
iu
m
 a
re
 v
ol
at
ile
. 
A
ga
in
, 
in
ve
st
m
en
t 
in
 s
to
ck
s 
is
 r
is
ky
. 
Ta
bl
e 
2 
sh
ow
s 
th
at
 if
 th
e 
tru
e 
ex
pe
ct
ed
 v
al
ue
 o
f 
th
e 
an
nu
al
 e
qu
ity
 p
re
m
iu
m
 is
 5
.9
5%
 a
nd
 th
e 
st
an
da
rd
 
de
vi
at
io
n 
is
 1
7.
85
%
, a
 n
or
m
al
 d
is
tri
bu
tio
n 
pr
ed
ic
ts
 th
at
 th
e 
an
nu
al
 e
qu
ity
 p
re
m
iu
m
 is
 n
eg
at
iv
e—
st
oc
ks
 
un
de
rp
er
fo
rm
 T
-b
ill
s—
in
 3
7%
 o
f t
he
 y
ea
rs
 in
 1
96
3–
20
11
. I
n 
fa
ct
, 3
3%
 (1
6 
of
 4
9 
in
 A
pp
en
di
x 
Ta
bl
e 
A
1)
 
ar
e 
ne
ga
tiv
e.
 
Th
e 
pr
ob
ab
ili
ty
 th
at
 th
e 
av
er
ag
e 
st
oc
k 
re
tu
rn
 is
 b
el
ow
 th
e 
av
er
ag
e 
bi
ll 
re
tu
rn
 fa
lls
 if
 w
e 
le
ng
th
en
 
th
e 
in
ve
st
m
en
t p
er
io
d.
 W
e 
ar
e 
us
in
g 
si
m
pl
e 
av
er
ag
e 
(n
ot
 c
om
po
un
d)
 re
tu
rn
s, 
so
 th
e 
st
an
da
rd
 d
ev
ia
tio
n 
of
 
4 
  th
e 
av
er
ag
e 
an
nu
al
 p
re
m
iu
m
 f
or
 a
 p
er
io
d 
is
 th
e 
st
an
da
rd
 d
ev
ia
tio
n 
of
 y
ea
r-b
y-
ye
ar
 p
re
m
iu
m
s 
di
vi
de
d 
by
 
th
e 
sq
ua
re
 ro
ot
 o
f t
he
 n
um
be
r o
f y
ea
rs
 in
 th
e 
pe
rio
d.
 T
hu
s, 
if 
th
e 
pe
rio
d 
co
ve
re
d 
by
 th
e 
av
er
ag
e 
pr
em
iu
m
 
is
 g
re
at
er
 th
an
 a
 y
ea
r, 
th
e 
st
an
da
rd
 d
ev
ia
tio
n 
of
 th
e 
av
er
ag
e 
is
 le
ss
 th
an
 th
e 
st
an
da
rd
 d
ev
ia
tio
n 
of
 y
ea
r-b
y-
ye
ar
 a
nn
ua
l r
et
ur
ns
. T
ab
le
 2
 s
ho
w
s 
th
at
 th
e 
st
an
da
rd
 d
ev
ia
tio
n 
of
 th
e 
av
er
ag
e 
eq
ui
ty
 p
re
m
iu
m
 d
ro
ps
 fr
om
 
17
.8
5%
 fo
r a
 o
ne
-y
ea
r p
er
io
d 
to
 7
.9
8%
 fo
r f
iv
e 
ye
ar
s, 
5.
65
%
 fo
r t
en
 y
ea
rs
, 3
.5
7%
 fo
r 2
5 
ye
ar
s, 
an
d 
2.
53
%
 
fo
r 5
0 
ye
ar
s. 
N
ot
e 
th
e 
ef
fe
ct
 o
f t
he
 s
qu
ar
e 
ro
ot
 ru
le
. T
he
 s
ta
nd
ar
d 
de
vi
at
io
n 
of
 th
e 
av
er
ag
e 
re
tu
rn
 fa
lls
 a
t a
 
sl
ow
er
 a
nd
 sl
ow
er
 ra
te
 a
s t
he
 h
ol
di
ng
 p
er
io
d 
is
 in
cr
ea
se
d.
 
B
ec
au
se
 o
ur
 a
ss
um
ed
 e
xp
ec
te
d 
an
nu
al
 p
re
m
iu
m
, 5
.9
5%
, d
oe
s 
no
t c
ha
ng
e 
w
he
n 
th
e 
in
ve
st
m
en
t 
ho
riz
on
 c
ha
ng
es
, t
he
 d
ec
lin
e 
in
 th
e 
st
an
da
rd
 d
ev
ia
tio
n 
of
 th
e 
av
er
ag
e 
re
du
ce
s 
th
e 
lik
el
ih
oo
d 
of
 re
al
iz
in
g 
a 
ne
ga
tiv
e 
av
er
ag
e 
pr
em
iu
m
 a
s 
th
e 
ho
riz
on
 g
ro
w
s. 
Ta
bl
e 
2 
sh
ow
s 
th
at
 th
e 
pr
ob
ab
ili
ty
 o
f 
ne
ga
tiv
e 
av
er
ag
e 
pr
em
iu
m
 fa
lls
 fr
om
 3
7.
0%
 fo
r a
 o
ne
-y
ea
r p
er
io
d 
to
 2
2.
8%
 fo
r 5
 y
ea
rs
, 1
4.
6%
 fo
r t
en
 y
ea
rs
, 4
.8
%
 fo
r 2
5 
ye
ar
s, 
an
d 
0.
9%
 f
or
 5
0 
ye
ar
s. 
In
 o
th
er
 w
or
ds
, l
oo
ki
ng
 f
or
w
ar
d,
 th
er
e 
is
 a
lm
os
t a
 o
ne
-in
-fo
ur
 c
ha
nc
e 
th
at
 
th
e 
av
er
ag
e 
pr
em
iu
m
 fo
r a
 fi
ve
-y
ea
r p
er
io
d 
is
 n
eg
at
iv
e,
 th
at
 is
, b
ill
s 
be
at
 s
to
ck
s. 
A
t 2
5 
ye
ar
s—
a 
m
od
er
at
e 
in
ve
st
m
en
t l
ife
tim
e—
th
er
e 
is
 st
ill
 a
lm
os
t a
 o
ne
-in
-tw
en
ty
 c
ha
nc
e 
th
at
 th
e 
re
al
iz
ed
 a
ve
ra
ge
 e
qu
ity
 p
re
m
iu
m
 
is
 n
eg
at
iv
e.
 It
 ta
ke
s 
a 
lo
ng
 in
ve
st
m
en
t l
ife
tim
e,
 5
0 
ye
ar
s, 
to
 re
du
ce
 th
e 
pr
ob
ab
ili
ty
 o
f a
 n
eg
at
iv
e 
re
al
iz
ed
 
av
er
ag
e 
eq
ui
ty
 p
re
m
iu
m
 to
 0
.9
%
. 
 T
o 
em
ph
as
iz
e,
 e
ve
n 
if 
w
e 
kn
ow
 fo
r s
ur
e 
th
at
 th
e 
tru
e 
ex
pe
ct
ed
 v
al
ue
 o
f t
he
 e
qu
ity
 p
re
m
iu
m
 is
 a
 
he
al
th
y 
5.
95
%
 p
er
 y
ea
r, 
th
er
e 
is
 s
til
l a
 0
.9
%
 c
ha
nc
e 
th
at
 a
n 
in
ve
st
or
 w
ho
 h
ol
ds
 th
e 
m
ar
ke
t p
or
tfo
lio
 o
f 
st
oc
ks
 fo
r t
he
 n
ex
t 5
0 
ye
ar
s 
en
ds
 u
p 
w
ith
 a
 lo
w
er
 a
ve
ra
ge
 a
nn
ua
l r
et
ur
n 
th
an
 a
n 
in
ve
st
or
 w
ho
 ro
lls
 o
ve
r 
on
e-
m
on
th
 T
-b
ill
s. 
In
de
ed
, n
o 
m
at
te
r h
ow
 lo
ng
 th
e 
in
ve
st
m
en
t p
er
io
d,
 o
ne
 c
an
 n
ev
er
 b
e 
pe
rf
ec
tly
 c
er
ta
in
 
th
at
 th
e 
re
al
iz
ed
 a
ve
ra
ge
 p
re
m
iu
m
 w
ill
 b
e 
po
si
tiv
e.
 S
uc
h 
is
 th
e 
na
tu
re
 o
f r
is
k 
an
d 
re
tu
rn
! 
Th
er
e 
is
 a
no
th
er
 w
ay
 to
 in
te
rp
re
t T
ab
le
 2
. T
he
 d
is
cu
ss
io
n 
ab
ov
e 
tre
at
s 
th
e 
sa
m
pl
e 
av
er
ag
e 
eq
ui
ty
 
pr
em
iu
m
 fo
r 1
96
3–
20
11
 a
s 
th
e 
tru
e 
ex
pe
ct
ed
 p
re
m
iu
m
. B
ut
 w
e 
do
n’
t k
no
w
 th
e 
va
lu
e 
of
 th
e 
tru
e 
ex
pe
ct
ed
 
pr
em
iu
m
, a
nd
 T
ab
le
 2
 su
m
m
ar
iz
es
 o
ur
 u
nc
er
ta
in
ty
 a
bo
ut
 it
. T
he
 ta
bl
e 
sa
ys
 th
at
 if
 w
e 
ha
ve
 5
0 
ye
ar
s o
f d
at
a 
in
 w
hi
ch
 th
e 
av
er
ag
e 
pr
em
iu
m
 is
 5
.9
5%
 p
er
 y
ea
r a
nd
 th
e 
st
an
da
rd
 d
ev
ia
tio
n 
of
 th
e 
ye
ar
-b
y-
ye
ar
 p
re
m
iu
m
s 
is
 1
7.
85
%
, t
he
n 
ou
r b
es
t e
st
im
at
e 
of
 th
e 
tru
e 
ex
pe
ct
ed
 p
re
m
iu
m
 is
 5
.9
5%
, b
ut
 it
 is
 s
ub
ject
 to
 s
ub
st
an
tia
l 
5 
  un
ce
rta
in
ty
, 
su
m
m
ar
iz
ed
 b
y 
th
e 
es
tim
at
e 
of
 t
he
 s
ta
nd
ar
d 
de
vi
at
io
n 
of
 t
he
 5
0-
ye
ar
 a
ve
ra
ge
 p
re
m
iu
m
, 
2.
53
%
. R
ou
gh
ly
 s
pe
ak
in
g,
 w
ith
 t
hi
s 
50
 y
ea
rs
 o
f 
da
ta
, t
he
 p
ro
ba
bi
lit
y 
di
st
rib
ut
io
n 
of
 t
he
 t
ru
e 
ex
pe
ct
ed
 
pr
em
iu
m
 i
s 
no
rm
al
 w
ith
 m
ea
n 
5.
95
%
 a
nd
 s
ta
nd
ar
d 
de
vi
at
io
n 
2.
53
%
. T
hu
s, 
th
e 
tru
e 
ex
pe
ct
ed
 p
re
m
iu
m
 
m
ay
 b
e 
m
uc
h 
bi
gg
er
 o
r 
m
uc
h 
sm
al
le
r 
th
an
 5
.9
5%
, a
nd
 t
he
re
 i
s 
a 
0.
9%
 c
ha
nc
e 
th
at
 t
he
 t
ru
e 
ex
pe
ct
ed
 
pr
em
iu
m
 is
 n
eg
at
iv
e.
 I
n 
ge
ne
ra
l, 
ev
en
 if
 th
e 
tru
e 
ex
pe
ct
ed
 p
re
m
iu
m
 d
oe
s 
no
t c
ha
ng
e 
th
ro
ug
h 
tim
e,
 w
e 
ne
ve
r h
av
e 
en
ou
gh
 d
at
a 
to
 in
fe
r i
ts
 v
al
ue
 w
ith
 p
er
fe
ct
 c
er
ta
in
ty
. T
hi
s 
is
 a
no
th
er
 w
ay
 to
 th
in
k 
ab
ou
t r
is
k 
in
 
eq
ui
ty
 in
ve
st
in
g:
 W
e 
do
n’
t k
no
w
 th
e 
tru
e 
ex
pe
ct
ed
 v
al
ue
s o
f r
is
k 
pr
em
iu
m
s. 
 
U
nf
or
tu
na
te
ly
, u
nc
er
ta
in
ty
 a
bo
ut
 tr
ue
 e
xp
ec
te
d 
ris
k 
pr
em
iu
m
s 
m
ak
es
 u
s 
ev
en
 le
ss
 c
on
fid
en
t t
ha
t 
in
ve
st
or
s 
w
ill
 e
ar
n 
po
si
tiv
e 
pr
em
iu
m
s 
in
 a
ny
 g
iv
en
 fu
tu
re
 p
er
io
d.
 In
 e
ss
en
ce
, w
e 
ha
ve
 to
 a
dd
 u
nc
er
ta
in
ty
 
ab
ou
t e
xp
ec
te
d 
pr
em
iu
m
s 
to
 th
e 
es
tim
at
es
 o
f 
th
e 
un
ce
rta
in
ty
 a
bo
ut
 in
ve
st
m
en
t o
ut
co
m
es
 o
bt
ai
ne
d 
w
he
n 
w
e 
tre
at
 th
e 
sa
m
pl
e 
m
ea
n 
pr
em
iu
m
s a
s t
ru
e 
ex
pe
ct
ed
 v
al
ue
s. 
To
 b
e 
pr
ec
is
e 
(a
nd
 y
ou
 c
an
 sk
ip
 th
is
 p
ar
ag
ra
ph
 a
nd
 th
e 
ne
xt
 if
 y
ou
 a
re
 w
ill
in
g 
to
 ta
ke
 o
ur
 w
or
d 
fo
r 
it)
, t
he
 s
ta
nd
ar
d 
de
vi
at
io
n 
of
 a
 f
ut
ur
e 
in
ve
st
m
en
t o
ut
co
m
e 
ov
er
 s
om
e 
fix
ed
 h
or
iz
on
 is
 th
e 
sq
ua
re
 r
oo
t o
f 
th
e 
su
m
 o
f 
ou
r 
es
tim
at
e 
of
 th
e 
va
ria
nc
e 
of
 th
e 
ex
pe
ct
ed
 c
om
po
ne
nt
 a
nd
 th
e 
va
ria
nc
e 
of
 th
e 
un
ex
pe
ct
ed
 
co
m
po
ne
nt
, 
s(
In
ve
st
m
en
t O
ut
co
m
e)
 =
 [v
ar
(E
xp
ec
te
d 
Re
tu
rn
) +
 v
ar
(U
ne
xp
ec
te
d 
R
et
ur
n)
]1/
2 . 
W
e 
ta
ke
 th
e 
va
ria
nc
e 
of
 th
e 
ex
pe
ct
ed
 re
tu
rn
 in
 th
is
 e
qu
at
io
n 
to
 b
e 
th
e 
va
ria
nc
e 
of
 th
e 
es
tim
at
e 
of
 
th
e 
50
-y
ea
r a
ve
ra
ge
 p
re
m
iu
m
, t
ha
t i
s, 
th
e 
sq
ua
re
 o
f s
(M
ea
n)
 in
 T
ab
le
 2
 fo
r 5
0 
Y
ea
rs
 (t
he
 la
st
 c
ol
um
n)
. I
n 
ot
he
r w
or
ds
, w
e 
ta
ke
 th
e 
un
ce
rta
in
ty
 a
bo
ut
 th
e 
ex
pe
ct
ed
 p
re
m
iu
m
 to
 b
e 
lik
e 
th
e 
vo
la
til
ity
 o
f e
st
im
at
es
 o
f 
th
e 
av
er
ag
e 
pr
em
iu
m
 f
or
 a
 5
0-
ye
ar
 s
am
pl
e 
pe
rio
d 
(w
hi
ch
 i
s 
re
la
tiv
el
y 
lo
w
). 
Th
e 
va
ria
nc
e 
of
 t
he
 
un
ex
pe
ct
ed
 re
tu
rn
 fo
r a
 g
iv
en
 in
ve
st
m
en
t p
er
io
d 
(1
 to
 5
0 
Y
ea
rs
 in
 T
ab
le
 2
) i
s 
th
en
 th
e 
sq
ua
re
 o
f s
(M
ea
n)
 
fo
r 
th
e 
gi
ve
n 
in
ve
st
m
en
t p
er
io
d.
 T
he
 o
ve
ra
ll 
un
ce
rta
in
ty
, e
xp
re
ss
ed
 in
 th
e 
eq
ua
tio
n 
ab
ov
e,
 is
 th
e 
sq
ua
re
 
ro
ot
 o
f 
th
e 
su
m
 o
f 
th
e 
tw
o 
va
ria
nc
es
. N
ot
e 
th
at
 th
e 
va
ria
nc
e 
of
 th
e 
ex
pe
ct
ed
 r
et
ur
n 
is
 th
e 
sa
m
e 
fo
r 
al
l 
in
ve
st
m
en
t h
or
iz
on
s, 
bu
t t
he
 v
ar
ia
nc
e 
of
 th
e 
un
ex
pe
ct
ed
 re
tu
rn
 is
 lo
w
er
 fo
r l
on
ge
r i
nv
es
tm
en
t p
er
io
ds
. 
 
Ta
bl
e 
2 
re
po
rts
 t
he
 l
ik
el
ih
oo
d 
th
at
 e
ac
h 
of
 t
he
 t
hr
ee
 p
re
m
iu
m
s 
w
ill
 b
e 
ne
ga
tiv
e 
ov
er
 v
ar
io
us
 
in
ve
st
m
en
t h
or
iz
on
s 
w
he
n 
w
e 
re
co
gn
iz
e 
th
at
 u
nc
er
ta
in
ty
 a
bo
ut
 th
e 
tru
e 
ex
pe
ct
ed
 p
re
m
iu
m
s 
ad
ds
 to
 o
ur
 
6 
  un
ce
rta
in
ty
 a
bo
ut
 in
ve
st
m
en
t o
ut
co
m
es
 (l
ab
el
ed
 P
ro
b 
< 
0,
 E
(R
M
 –
 R
F)
 u
nc
er
ta
in
). 
U
nc
er
ta
in
ty
 a
bo
ut
 tr
ue
 
ex
pe
ct
ed
 p
re
m
iu
m
s 
ha
s 
lit
tle
 e
ff
ec
t o
n 
th
e 
un
ce
rta
in
ty
 a
bo
ut
 th
e 
pr
em
iu
m
s 
fo
r a
 o
ne
-y
ea
r h
ol
di
ng
 p
er
io
d.
 
Fo
r e
xa
m
pl
e,
 re
co
gn
iz
in
g 
ou
r u
nc
er
ta
in
ty
 a
bo
ut
 th
e 
ex
pe
ct
ed
 v
al
ue
 in
cr
ea
se
s 
th
e 
pr
ob
ab
ili
ty
 o
f a
 n
eg
at
iv
e 
re
al
iz
ed
 e
qu
ity
 p
re
m
iu
m
 i
n 
an
y 
pa
rti
cu
la
r 
ye
ar
 f
ro
m
 3
7.
0%
 t
o 
37
.1
%
. 
In
tu
iti
ve
ly
, 
fo
r 
a 
on
e-
ye
ar
 
in
ve
st
m
en
t 
ho
riz
on
, u
nc
er
ta
in
ty
 a
bo
ut
 t
he
 e
xp
ec
te
d 
pr
em
iu
m
 i
s 
sm
al
l 
re
la
tiv
e 
to
 t
he
 h
ig
h 
vo
la
til
ity
 o
f 
ye
ar
-b
y-
ye
ar
 r
et
ur
ns
. 
U
nc
er
ta
in
ty
 a
bo
ut
 t
he
 e
xp
ec
te
d 
pr
em
iu
m
 d
oe
s 
no
t 
ch
an
ge
 w
ith
 t
he
 i
nv
es
tm
en
t 
ho
riz
on
, b
ut
 u
nc
er
ta
in
ty
 a
bo
ut
 th
e 
un
ex
pe
ct
ed
 c
om
po
ne
nt
 o
f 
th
e 
in
ve
st
m
en
t 
ou
tc
om
e 
go
es
 d
ow
n.
 A
s 
a 
re
su
lt,
 u
nc
er
ta
in
ty
 a
bo
ut
 th
e 
ex
pe
ct
ed
 v
al
ue
 h
as
 a
 b
ig
ge
r r
el
at
iv
e 
im
pa
ct
 a
t l
on
ge
r h
or
iz
on
s, 
in
cr
ea
si
ng
 th
e 
pr
ob
ab
ili
ty
 o
f a
 n
eg
at
iv
e 
re
al
iz
ed
 a
ve
ra
ge
 e
qu
ity
 p
re
m
iu
m
 fr
om
 4
.8
%
 to
 8
.7
%
 fo
r a
 2
5 
ye
ar
 h
or
iz
on
 a
nd
 
fr
om
 0
.9
%
 to
 4
.8
%
 f
or
 5
0 
ye
ar
s. 
Th
us
, l
ar
ge
ly
 b
ec
au
se
 o
f 
th
e 
un
ce
rta
in
ty
 a
bo
ut
 th
e 
ex
pe
ct
ed
 p
re
m
iu
m
, 
ev
en
 o
ve
r 
a 
lo
ng
 5
0-
ye
ar
 i
nv
es
tm
en
t 
lif
et
im
e,
 a
n 
in
ve
st
or
 h
as
 a
 o
ne
-in
-tw
en
ty
 c
ha
nc
e 
of
 a
 n
eg
at
iv
e 
re
al
iz
ed
 a
ve
ra
ge
 e
qu
ity
 p
re
m
iu
m
. 
 
Th
e 
st
or
ie
s 
fo
r 
th
e 
si
ze
 a
nd
 v
al
ue
 p
re
m
iu
m
s, 
SM
B
 a
nd
 H
M
L,
 a
re
 s
im
ila
r 
to
 th
at
 f
or
 th
e 
eq
ui
ty
 
pr
em
iu
m
. D
ur
in
g 
19
63
–2
01
1 
th
e 
si
ze
 p
re
m
iu
m
 a
ve
ra
ge
s 
3.
69
%
 p
er
 y
ea
r, 
bu
t c
on
fir
m
in
g 
th
e 
im
pr
es
si
on
 
dr
aw
n 
fr
om
 F
ig
ur
e 
2,
 t
he
 v
ol
at
ili
ty
 o
f 
ye
ar
-b
y-
ye
ar
 v
al
ue
s 
of
 t
he
 p
re
m
iu
m
 i
s 
hi
gh
, 
w
ith
 a
 s
ta
nd
ar
d 
de
vi
at
io
n 
of
 1
4.
16
%
 p
er
 y
ea
r. 
A
s 
a 
re
su
lt,
 T
ab
le
 2
 s
ho
w
s 
th
at
 if
 th
e 
tru
e 
ex
pe
ct
ed
 p
re
m
iu
m
 is
 3
.6
9%
 p
er
 
ye
ar
, t
he
 p
ro
ba
bi
lit
y 
th
at
 th
e 
re
al
iz
ed
 p
re
m
iu
m
 w
ill
 tu
rn
 o
ut
 to
 b
e 
ne
ga
tiv
e 
fa
lls
 fr
om
 a
bo
ut
 3
9.
7%
 fo
r a
 
on
e-
ye
ar
 p
er
io
d 
to
 2
8%
 fo
r f
iv
e 
ye
ar
s, 
20
.5
%
 fo
r t
en
 y
ea
rs
, a
nd
 a
lm
os
t 1
0%
 fo
r 2
5 
ye
ar
s. 
In
 o
th
er
 w
or
ds
, 
ev
en
 if
 w
e 
kn
ow
 fo
r c
er
ta
in
 th
at
 s
m
al
l s
to
ck
s 
ar
e 
ex
pe
ct
ed
 to
 b
ea
t b
ig
 s
to
ck
s 
by
 3
.6
9%
 p
er
 y
ea
r, 
th
er
e 
is 
st
ill
 a
 o
ne
 in
 te
n 
ch
an
ce
 th
at
 b
ig
 s
to
ck
s 
pr
od
uc
e 
a 
la
rg
er
 a
ve
ra
ge
 re
tu
rn
 in
 a
 fu
tu
re
 2
5-
ye
ar
 p
er
io
d.
 E
ve
n 
fo
r 5
0-
ye
ar
 p
er
io
ds
, t
he
re
 is
 a
 3
.3
%
 p
ro
ba
bi
lit
y 
th
at
 th
e 
av
er
ag
e 
si
ze
 p
re
m
iu
m
 tu
rn
s o
ut
 to
 b
e 
ne
ga
tiv
e.
 
 A
lte
rn
at
iv
el
y,
 i
f 
w
e 
ac
kn
ow
le
dg
e 
th
at
 w
e 
do
n’
t 
kn
ow
 t
he
 t
ru
e 
ex
pe
ct
ed
 s
iz
e 
pr
em
iu
m
 a
nd
 5
0 
ye
ar
s 
ofd
at
a 
pr
od
uc
e 
an
 a
ve
ra
ge
 S
M
B
 re
tu
rn
 o
f 3
.6
9%
 p
er
 y
ea
r w
ith
 a
 s
ta
nd
ar
d 
de
vi
at
io
n 
of
 1
4.
16
%
 p
er
 
ye
ar
, t
he
n 
th
e 
di
st
rib
ut
io
n 
of
 th
e 
ex
pe
ct
ed
 p
re
m
iu
m
 is
 c
en
te
re
d 
at
 3
.6
9%
 w
ith
 a
 s
ta
nd
ar
d 
de
vi
at
io
n 
of
 2
%
, 
so
 a
ga
in
 th
er
e 
is
 s
ub
st
an
tia
l u
nc
er
ta
in
ty
 a
bo
ut
 th
e 
va
lu
e 
of
 th
e 
tru
e 
ex
pe
ct
ed
 a
nn
ua
l p
re
m
iu
m
. I
t c
ou
ld
 b
e 
m
uc
h 
bi
gg
er
 t
ha
n 
3.
69
%
 p
er
 y
ea
r, 
bu
t 
it 
co
ul
d 
be
 m
uc
h 
sm
al
le
r, 
an
d 
th
er
e 
is
 a
 3
.3
%
 c
ha
nc
e 
th
at
 it
 i
s 
7 
  ne
ga
tiv
e.
 A
ck
no
w
le
dg
in
g 
th
is
 u
nc
er
ta
in
ty
 in
cr
ea
se
s 
th
e 
lik
el
ih
oo
d 
of
 a
 n
eg
at
iv
e 
av
er
ag
e 
si
ze
 p
re
m
iu
m
 to
 
ab
ou
t 1
0%
 fo
r i
nv
es
to
rs
 w
ith
 a
 lo
ng
, 5
0-
ye
ar
 in
ve
st
m
en
t h
or
iz
on
. 
Pr
os
pe
ct
s 
ar
e 
a 
bi
t 
ro
si
er
 f
or
 t
he
 v
al
ue
 p
re
m
iu
m
. 
Th
e 
av
er
ag
e 
H
M
L 
re
tu
rn
 f
or
 1
96
3–
20
11
 i
s 
5.
29
%
 p
er
 y
ea
r, 
w
hi
ch
 is
 s
im
ila
r 
to
 t
he
 a
ve
ra
ge
 e
qu
ity
 p
re
m
iu
m
, 5
.9
5%
. T
ho
ug
h 
la
rg
e 
at
 1
3.
94
%
, t
he
 
st
an
da
rd
 d
ev
ia
tio
n 
of
 th
e 
H
M
L 
re
tu
rn
 is
 le
ss
 th
an
 th
at
 o
f t
he
 e
qu
ity
 p
re
m
iu
m
, 1
7.
85
%
. A
s 
a 
re
su
lt,
 if
 th
e 
tru
e 
ex
pe
ct
ed
 v
al
ue
 p
re
m
iu
m
 i
s 
5.
29
%
, t
he
 p
ro
ba
bi
lit
y 
th
at
 a
 n
eg
at
iv
e 
av
er
ag
e 
va
lu
e 
pr
em
iu
m
 w
ill
 b
e 
ob
se
rv
ed
 i
n 
a 
pe
rio
d 
of
 g
iv
en
 l
en
gt
h 
is
 l
ow
er
 t
ha
n 
fo
r 
th
e 
eq
ui
ty
 p
re
m
iu
m
. 
St
ill
, t
he
 p
ro
ba
bi
lit
y 
of
 a
 
ne
ga
tiv
e 
av
er
ag
e 
va
lu
e 
pr
em
iu
m
 fo
r a
 fi
ve
-y
ea
r p
er
io
d 
is
 a
lm
os
t 2
0%
, f
al
lin
g 
to
 1
1.
5%
 a
t t
en
 y
ea
rs
, a
nd
 
2.
9%
 a
t 2
5 
ye
ar
s (
Ta
bl
e 
2)
. 
A
lte
rn
at
iv
el
y,
 if
 w
e 
ac
kn
ow
le
dg
e 
th
at
 w
e 
do
n’
t k
no
w
 th
e 
tru
e 
ex
pe
ct
ed
 p
re
m
iu
m
 a
nd
 5
0 
ye
ar
s 
of
 
da
ta
 p
ro
du
ce
 a
n 
av
er
ag
e 
H
M
L 
re
tu
rn
 o
f 
5.
29
%
 p
er
 y
ea
r 
w
ith
 a
 s
ta
nd
ar
d 
de
vi
at
io
n 
of
 1
3.
94
%
 p
er
 y
ea
r, 
th
en
 th
e 
di
st
rib
ut
io
n 
of
 th
e 
ex
pe
ct
ed
 p
re
m
iu
m
 is
 c
en
te
re
d 
at
 5
.2
9%
 w
ith
 a
 s
ta
nd
ar
d 
de
vi
at
io
n 
of
 1
.9
7%
 
(c
lo
se
 to
 th
at
 o
f 
th
e 
si
ze
 p
re
m
iu
m
), 
so
 a
ga
in
 th
er
e 
is
 s
ub
st
an
tia
l u
nc
er
ta
in
ty
 a
bo
ut
 th
e 
va
lu
e 
of
 th
e 
tru
e 
ex
pe
ct
ed
 a
nn
ua
l p
re
m
iu
m
. I
t c
ou
ld
 b
e 
m
uc
h 
bi
gg
er
 th
an
 5
.2
9%
 p
er
 y
ea
r, 
bu
t i
t c
ou
ld
 b
e 
m
uc
h 
sm
al
le
r, 
an
d 
th
er
e 
is
 a
 0
.4
%
 c
ha
nc
e 
th
at
 it
 is
 n
eg
at
iv
e.
 C
om
bi
ni
ng
 b
ot
h 
so
ur
ce
s 
of
 u
nc
er
ta
in
ty
, t
he
 p
ro
ba
bi
lit
y 
of
 a
 
ne
ga
tiv
e 
va
lu
e 
pr
em
iu
m
 is
 6
.1
%
 o
ve
r t
he
 n
ex
t 2
5 
ye
ar
s a
nd
 2
.9
%
 o
ve
r t
he
 n
ex
t 5
0 
ye
ar
s. 
In
 s
um
, i
nv
es
tin
g 
in
 s
to
ck
s 
is
 r
is
ky
. E
ve
n 
ov
er
 a
 lo
ng
 in
ve
st
m
en
t l
ife
tim
e,
 th
er
e 
is
 n
o 
gu
ar
an
te
e 
th
at
 s
to
ck
s 
w
ill
 b
ea
t b
ill
s, 
or
 th
at
 s
m
al
l s
to
ck
s 
w
ill
 b
ea
t b
ig
 s
to
ck
s, 
or
 th
at
 v
al
ue
 s
to
ck
s 
w
ill
 b
ea
t g
ro
w
th
 
st
oc
ks
. 
M
or
eo
ve
r, 
ou
r 
an
al
ys
is
 is
 b
as
ed
 o
n 
hi
gh
ly
 d
iv
er
si
fie
d 
po
rtf
ol
io
s 
th
at
 a
re
 p
ro
ba
bl
y 
ra
th
er
 f
re
e 
of
 
di
ve
rs
ifi
ab
le
 v
ol
at
ili
ty
. F
or
 le
ss
 d
iv
er
si
fie
d 
po
rtf
ol
io
s, 
un
ce
rta
in
ty
 a
bo
ut
 o
ut
co
m
es
 is
 h
ig
he
r. 
Th
is
 is
 tr
ue
 
fo
r 
ac
tiv
el
y 
m
an
ag
ed
 s
to
ck
 p
or
tfo
lio
s, 
w
hi
ch
 a
re
 u
nd
iv
er
si
fie
d 
al
m
os
t 
by
 d
ef
in
iti
on
. 
A
nd
 i
t’s
 t
ru
e 
in
 
sp
ad
es
 fo
r r
is
ki
er
 in
ve
st
m
en
ts
 li
ke
 p
riv
at
e 
eq
ui
ty
 a
nd
 h
ed
ge
 fu
nd
s. 
Ca
ve
at
 e
m
pt
or
! 
 
8 
  
Figure 1: Annual Market Return in Excess of the T-Bill Rate 
 
9 
  
Figure 2: Annual RM – RF and SMB Returns 
 
10 
  
Figure 3: Annual RM – RF and HML Returns 
 
11
 
  
Ta
bl
e 
1:
 S
um
m
ar
y 
st
at
is
tic
s f
or
 a
nn
ua
l r
et
ur
ns
 fo
r 1
96
3–
20
11
, 4
9 
ye
ar
s 
 Th
e 
ta
bl
e 
sh
ow
s 
av
er
ag
es
 (M
ea
n)
 a
nd
 s
ta
nd
ar
d 
de
vi
at
io
ns
 (S
td
) o
f a
nn
ua
l v
al
ue
s 
of
 th
e 
eq
ui
ty
 p
re
m
iu
m
, 
R
M
 –
 R
F,
 th
e 
si
ze
 p
re
m
iu
m
, S
M
B
, a
nd
 th
e 
va
lu
e 
pr
em
iu
m
, H
M
L.
 T
he
 e
qu
ity
 p
re
m
iu
m
 is
 th
e 
di
ff
er
en
ce
 
be
tw
ee
n 
th
e 
m
ar
ke
t r
et
ur
n,
 R
M
, a
nd
 th
e 
T-
bi
ll 
re
tu
rn
, R
F.
 T
he
 s
iz
e 
pr
em
iu
m
 is
 th
e 
av
er
ag
e 
of
 th
e 
re
tu
rn
s 
on
 th
e 
th
re
e 
po
rtf
ol
io
s 
of
 s
m
al
l s
to
ck
s 
(S
G
, S
N
, a
nd
 S
V
) 
m
in
us
 th
e 
av
er
ag
e 
of
 th
e 
re
tu
rn
s 
on
 th
e 
th
re
e 
po
rtf
ol
io
s 
of
 b
ig
 s
to
ck
s 
(B
G
, B
N
, a
nd
 B
V
). 
Th
e 
va
lu
e 
pr
em
iu
m
 is
 th
e 
av
er
ag
e 
of
 th
e 
re
tu
rn
s 
on
 th
e 
tw
o 
va
lu
e 
po
rtf
ol
io
s (
SV
 a
nd
 B
V
) m
in
us
 th
e 
av
er
ag
e 
of
 th
e 
re
tu
rn
s o
n 
th
e 
tw
o 
gr
ow
th
 p
or
tfo
lio
s (
SG
 a
nd
 B
G
). 
 
 
 R
M
 –
 R
F 
 S
M
B
 
 H
M
L 
M
ea
n 
5.
95
 
3.
69
 
5.
29
 
St
d 
17
.8
5 
14
.1
6 
13
.9
4 
 
 R
M
 
 S
G
 
 S
N
 
 S
V
 
 B
G
 
 B
N
 
B
V
 
R
F 
M
ea
n 
11
.2
7 
11
.5
8 
16
.8
7 
19
.2
3 
10
.9
6 
11
.7
6 
13
.8
9 
5.
33
 
St
d 
17
.7
1 
28
.1
4 
22
.9
1 
24
.2
2 
18
.6
2 
16
.9
5 
18
.6
1 
2.
99
 
 
 
12
 
  
Ta
bl
e 
2:
 P
ro
ba
bi
lit
y 
th
at
 a
n 
av
er
ag
e 
pr
em
iu
m
 is
 n
eg
at
iv
e 
in
 p
er
io
ds
 o
f d
iff
er
en
t l
en
gt
h 
 Y
ea
rs
 i
s 
th
e 
nu
m
be
r 
of
 y
ea
rs
 i
n 
a 
pe
rio
d,
 a
nd
 s
qr
t(Y
ea
rs
) 
is
 t
he
 s
qu
ar
e 
ro
ot
 o
f 
Y
ea
rs
. s
(M
ea
n)
 i
s 
th
e 
st
an
da
rd
 d
ev
ia
tio
n 
of
 th
e 
av
er
ag
e 
pr
em
iu
m
, w
hi
ch
 is
 th
e 
st
an
da
rd
 d
ev
ia
tio
n 
of
 y
ea
r-b
y-
ye
ar
 v
al
ue
s 
of
 th
e 
pr
em
iu
m
 m
ul
tip
lie
d 
by
 1
/s
qr
t(Y
ea
rs
). 
Pr
ob
 <
 0
 is
 th
e 
pr
ob
ab
ili
ty
 th
at
 th
e 
re
al
iz
ed
 a
ve
ra
ge
 p
re
m
iu
m
 in
 a
 
pe
rio
d 
of
 le
ng
th
 Y
ea
rs
 is
 n
eg
at
iv
e.
 W
e 
co
m
pu
te
 tw
o 
pr
ob
ab
ili
tie
s f
or
 e
ac
h 
pr
em
iu
m
. T
he
 fi
rs
t a
ss
um
es
 th
e 
av
er
ag
e 
pr
em
iu
m
 fo
r 1
96
3–
20
11
 is
 th
e 
tru
e 
ex
pe
ct
ed
 p
re
m
iu
m
 a
nd
 th
e 
se
co
nd
 re
co
gn
iz
es
 o
ur
 u
nc
er
ta
in
ty
 
ab
ou
t t
he
 tr
ue
 e
xp
ec
te
d 
pr
em
iu
m
. T
he
 m
ea
ns
 o
f 
th
e 
ye
ar
-b
y-
ye
ar
 a
nn
ua
l 
pr
em
iu
m
s 
fo
r 
19
63
–2
01
1 
ar
e 
5.
95
%
 (
R
M
 –
 R
F)
, 3
.6
9%
 (
SM
B
), 
an
d 
5.
29
%
 (
H
M
L)
 p
er
 y
ea
r, 
an
d 
th
e 
st
an
da
rd
 d
ev
ia
tio
ns
 a
re
 1
7.
85
%
 
(R
M
 –
 R
F)
, 1
4.
16
%
 (S
M
B)
, a
nd
 1
3.
94
%
 (H
M
L)
. M
ea
n/
s(
M
ea
n)
 te
lls
 u
s 
ho
w
 fa
r t
he
 m
ea
n 
is
 fr
om
 z
er
o,
 
m
ea
su
re
d 
in
 u
ni
ts
 o
f 
th
e 
sta
nd
ar
d 
de
vi
at
io
n 
of
 th
e 
m
ea
n.
 F
or
 r
ea
de
rs
 in
te
re
st
ed
 in
 th
e 
de
ta
ils
, w
he
n 
w
e 
as
su
m
e 
w
e 
kn
ow
 th
e 
tru
e 
ex
pe
ct
ed
 p
re
m
iu
m
, t
he
 p
ro
ba
bi
lit
y 
th
at
 th
e 
re
al
iz
ed
 a
ve
ra
ge
 p
re
m
iu
m
 is
 n
eg
at
iv
e 
in
 a
 p
er
io
d 
ofl
en
gt
h 
(Y
ea
rs
) 
is
 e
st
im
at
ed
 a
s 
th
e 
pr
ob
ab
ili
ty
 t
o 
th
e 
le
ft 
of
 M
ea
n/
s(
M
ea
n)
 f
ro
m
 a
 u
ni
t 
no
rm
al
 d
is
tri
bu
tio
n.
 W
he
n 
w
e 
ac
kn
ow
le
dg
e 
ou
r 
un
ce
rta
in
ty
 a
bo
ut
 t
he
 t
ru
e 
ex
pe
ct
ed
 p
re
m
iu
m
s, 
ou
r 
un
ce
rta
in
ty
 a
bo
ut
 fu
tu
re
 re
al
iz
ed
 re
tu
rn
s i
s t
he
 sq
ua
re
 ro
ot
 o
f t
he
 su
m
 o
f t
he
 v
ar
ia
nc
e 
of
 th
e 
av
er
ag
e 
re
tu
rn
 
if 
w
e 
kn
ew
 th
e 
tru
e 
ex
pe
ct
ed
 p
re
m
iu
m
 a
nd
 th
e 
va
ria
nc
e 
of
 o
ur
 e
st
im
at
e 
of
 th
e 
pr
em
iu
m
. S
in
ce
 w
e 
as
su
m
e 
w
e 
us
e 
50
 y
ea
rs
 to
 e
st
im
at
e 
th
e 
ex
pe
ct
ed
 v
al
ue
, t
he
 v
ar
ia
nc
e 
of
 o
ur
 e
st
im
at
e 
is
 th
e 
sq
ua
re
 o
f s
(M
ea
n)
 a
t 5
0 
ye
ar
s. 
 Y
ea
rs
 
1 
5 
10
 
15
 
20
 
25
 
50
 
1/
sq
rt(
Y
ea
rs
) 
1.
00
 
0.
45
 
0.
32
 
0.
26
 
0.
22
 
0.
20
 
0.
14
 
Eq
ui
ty
 P
re
m
iu
m
, R
M
 –
 R
F 
s(
M
ea
n)
 
17
.8
5 
7.
98
 
5.
65
 
4.
61
 
3.
99
 
3.
57
 
2.
53
 
M
ea
n/
s(
M
ea
n)
 
0.
33
 
0.
74
 
1.
05
 
1.
29
 
1.
49
 
1.
67
 
2.
35
 
Pr
ob
 <
 0
, E
(R
M
-R
F)
 =
 5
.9
5%
 
37
.0
%
 
22
.8
%
 
14
.6
%
 
9.
9%
 
6.
8%
 
4.
8%
 
0.
9%
 
Pr
ob
 <
 0
, E
(R
M
-R
F)
 u
nc
er
ta
in
 3
7.
1%
 
23
.9
%
 
16
.8
%
 
12
.9
%
 
10
.4
%
 
8.
7%
 
4.
8%
 
Si
ze
 P
re
m
iu
m
, S
M
B
 
s(
M
ea
n)
 
14
.1
6 
6.
33
 
4.
48
 
3.
66
 
3.
17
 
2.
83
 
2.
00
 
M
ea
n/
s(
M
ea
n)
 
0.
26
 
0.
58
 
0.
82
 
1.
01
 
1.
16
 
1.
30
 
1.
84
 
Pr
ob
 <
 0
, E
(S
M
B
) =
 3
.6
9%
 
39
.7
%
 
28
.0
%
 
20
.5
%
 
15
.7
%
 
12
.2
%
 
9.
7%
 
3.
3%
 
Pr
ob
 <
 0
, E
(S
M
B
) u
nc
er
ta
in
 
39
.8
%
 
28
.9
%
 
22
.6
%
 
18
.8
%
 
16
.3
%
 
14
.4
%
 
9.
7%
 
V
al
ue
 P
re
m
iu
m
, H
M
L 
s(
M
ea
n)
 
13
.9
4 
6.
24
 
4.
41
 
3.
60
 
3.
12
 
2.
79
 
1.
97
 
M
ea
n/
s(
M
ea
n)
 
0.
38
 
0.
85
 
1.
20
 
1.
47
 
1.
70
 
1.
90
 
2.
68
 
Pr
ob
 <
 0
, E
(H
M
L)
 =
 5
.2
9%
 
35
.2
%
 
19
.8
%
 
11
.5
%
 
7.
1%
 
4.
5%
 
2.
9%
 
0.
4%
 
Pr
ob
 <
 0
, E
(H
M
L)
 u
nc
er
ta
in
 
35
.4
%
 
20
.9
%
 
13
.7
%
 
9.
9%
 
7.
6%
 
6.
1%
 
2.
9%
 
 
13
 
  
A
pp
en
di
x 
Ta
bl
e 
A
1:
 A
nn
ua
l R
et
ur
ns
, 1
96
3–
20
11
: 4
9 
Y
ea
rs
 
 
R
M
 
SG
 
SN
 
SV
 
B
G
 
B
N
 
B
V
 
R
M
-R
F 
SM
B
 
H
M
L 
R
F 
19
63
 
20
.8
9 
7.
54
 
16
.5
2 
29
.6
8 
21
.9
5 
17
.0
8 
32
.8
2 
17
.7
8 
-6
.0
4 
16
.5
1 
3.
11
 
19
64
 
16
.3
0 
8.
64
 
17
.7
3 
23
.8
7 
14
.4
1 
19
.8
9 
19
.1
5 
12
.7
8 
-1
.0
7 
9.
98
 
3.
53
 
19
65
 
14
.3
9 
34
.9
1 
33
.4
4 
42
.0
6 
14
.6
0 
8.
38
 
22
.5
1 
10
.4
7 
21
.6
3 
7.
53
 
3.
92
 
19
66
 
-8
.6
9 
-5
.9
4 
-6
.1
0 
-7
.0
7 
-1
0.
78
 
-5
.4
7 
-1
0.
93
 
-1
3.
44
 
2.
69
 
-0
.6
4 
4.
75
 
19
67
 
28
.5
6 
90
.5
9 
71
.8
3 
68
.1
3 
28
.9
1 
15
.6
9 
32
.1
0 
24
.3
5 
51
.2
8 
-9
.6
3 
4.
21
 
19
68
 
14
.1
6 
32
.4
2 
40
.6
5 
46
.8
2 
3.
84
 
15
.3
1 
27
.0
2 
8.
94
 
24
.5
7 
18
.7
9 
5.
22
 
19
69
 
-1
0.
85
 
-2
4.
65
 
-2
2.
76
 
-2
5.
39
 
2.
56
 
-1
6.
79
 
-1
6.
45
 
-1
7.
42
 
-1
4.
04
 
-9
.8
8 
6.
57
 
19
70
 
0.
06
 
-2
1.
49
 
-7
.6
5 
6.
44
 
-6
.0
9 
7.
87
 
10
.4
9 
-6
.4
5 
-1
1.
66
 
22
.2
5 
6.
52
 
19
71
 
16
.1
9 
26
.3
7 
21
.4
8 
14
.4
6 
24
.8
3 
5.
92
 
14
.4
2 
11
.8
0 
5.
72
 
-1
1.
16
 
4.
39
 
19
72
 
17
.3
3 
0.
77
 
7.
50
 
7.
33
 
21
.3
9 
11
.1
8 
18
.1
2 
13
.5
0 
-1
1.
70
 
1.
65
 
3.
84
 
19
73
 
-1
8.
75
 
-4
5.
24
 
-3
2.
72
 
-2
7.
17
 
-2
2.
31
 
-9
.0
7 
-4
.0
4 
-2
5.
68
 
-2
3.
24
 
18
.1
7 
6.
93
 
19
74
 
-2
7.
94
 
-3
2.
41
 
-2
5.
85
 
-1
8.
61
 
-2
9.
21
 
-2
2.
82
 
-2
3.
22
 
-3
5.
95
 
-0
.5
4 
9.
89
 
8.
01
 
19
75
 
37
.3
5 
60
.8
2 
58
.0
2 
58
.1
2 
34
.2
3 
41
.8
6 
55
.5
9 
31
.5
5 
15
.1
0 
9.
33
 
5.
80
 
19
76
 
26
.7
7 
38
.4
3 
47
.4
3 
60
.6
5 
17
.6
1 
40
.5
8 
44
.4
8 
21
.6
8 
14
.6
1 
24
.5
5 
5.
08
 
19
77
 
-2
.9
7 
18
.8
1 
18
.4
4 
23
.2
6 
-9
.1
4 
-0
.6
1 
1.
35
 
-8
.1
0 
22
.9
7 
7.
47
 
5.
13
 
19
78
 
8.
53
 
17
.7
7 
21
.0
6 
22
.2
7 
7.
07
 
6.
93
 
3.
76
 
1.
34
 
14
.4
5 
0.
59
 
7.
19
 
19
79
 
24
.3
9 
49
.6
1 
38
.0
0 
39
.5
3 
16
.4
1 
23
.4
3 
22
.5
0 
14
.0
1 
21
.6
0 
-2
.0
0 
10
.3
8 
19
80
 
33
.2
5 
53
.1
9 
30
.5
8 
22
.0
5 
35
.7
7 
36
.8
2 
16
.4
1 
21
.9
9 
5.
61
 
-2
5.
25
 
11
.2
6 
19
81
 
-3
.9
9 
-1
0.
65
 
14
.2
6 
17
.6
8 
-7
.7
0 
-7
.4
2 
14
.0
9 
-1
8.
71
 
7.
44
 
25
.0
6 
14
.7
2 
19
82
 
20
.4
3 
19
.4
7 
33
.4
1 
41
.1
7 
21
.6
8 
17
.7
9 
27
.4
1 
9.
90
 
9.
06
 
13
.7
2 
10
.5
3 
19
83
 
22
.6
6 
19
.7
8 
40
.6
1 
47
.9
4 
14
.5
5 
25
.4
0 
27
.4
3 
13
.8
7 
13
.6
5 
20
.5
2 
8.
80
 
19
84
 
3.
17
 
-1
3.
83
 
2.
55
 
8.
32
 
-0
.5
2 
5.
68
 
16
.1
0 
-6
.6
7 
-8
.0
7 
19
.3
8 
9.
84
 
19
85
 
31
.4
1 
29
.7
6 
35
.2
2 
32
.4
1 
32
.4
9 
32
.6
0 
31
.5
6 
23
.6
9 
0.
25
 
0.
86
 
7.
72
 
19
86
 
15
.5
5 
2.
47
 
10
.0
6 
14
.4
5 
14
.7
3 
20
.1
1 
21
.3
2 
9.
40
 
-9
.7
2 
9.
28
 
6.
16
 
19
87
 
1.
81
 
-1
3.
33
 
-4
.3
5 
-6
.6
2 
7.
57
 
3.
02
 
-2
.1
4 
-3
.6
6 
-1
0.
91
 
-1
.5
0 
5.
47
 
19
88
 
17
.5
6 
14
.6
1 
28
.6
1 
31
.0
6 
12
.6
7 
17
.8
7 
26
.0
5 
11
.2
0 
5.
89
 
14
.9
2 
6.
36
 
19
89
 
28
.4
2 
19
.4
4 
17
.9
1 
17
.0
5 
35
.9
3 
25
.7
3 
30
.1
2 
20
.0
4 
-1
2.
46
 
-4
.1
0 
8.
38
 
19
90
 
-6
.0
9 
-1
8.
35
 
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