OF
T HE
L O G - N O RMAL APPR O XIMATI ON
T O O PT IMAL P O RTF O LI O DE C I S I O N -MAK ING O VER
P ERI O D S
Rob er t C Mer t on
P au l A S amue ls on
.
.
62 3-7 2
No
S lo an S ch oo l
of
vemb e r
19 7 2
Mana gemen t and Dep a r tment o f E c onomi c s
M I T
.
.
.
FALLAC Y O F T H E L O G— N O RMAL APPRO X IMAT IO N
TO
O PT IMAL P O RTF O LI O DEC I S IO N- MAK ING O VER MANY PERI O D S *
Rob e r t C
P au l A
.
.
Me r t o n
S amue ls on
S l oan S cho o l o f Manag ement and D epa r tmen t o f E c onomi c s
Mas s a c h u s e t t s Ins t i t u t e o f Te c hno logy
I
Thanks t o t h e
I n t rod uc t i on
.
eviva l by von Ne umann and
r
Mo r g e n s t e r n ,
nmx 1 m i z a t i o n
o f th e exp e c t e d v a lue o f a c on c ave u t i l i t y fun c t i on o f ou t come s
l as t th i rd o f a c en tu ry g ene ra l ly b een a cc ep t ed
fo r
Op t ima l p or t f o l i o s e l e c t i on
O p e r at i ona l th eo rems
.
c as e we re d e layed in b e c oming re cogn iz ed
l9 5 0
s emina l b re akt hroughs o f the
'
s
/
l
c as e o f me an va r i anc e an a l y s i S u
—
mo d e l
,
th e we l l — known
o
f
No t
ab l e add i t i ona l as s ump t i ons
,
or
M
,
.
,
i t gave
In the mean— var i ance
u t ua l — Fund The o rem ho ld s ; and wi t h s u i t
.
,
and a numb e r
As a re s ul t
,
b een p oin t ed
out
o
f
emp i ri c a l ly t es t
an ove rwhe lming maj o r i ty
l i t e rat ure on p o r t f o li o t he ory have b een b as ed
h as
bu t in add i t i on
,
the mod e l c an b e us ed t o d e f ine a c omp le t e mi c r o
ab le hyp o thes e s c an b e de r ived
as
the gene ra l
on ly c ou ld the f rui t fu l Sharp e- L in tne r
e c onomi c f ramewo rk f o r the c ap i t a l ma rke t
Un f o r t una t e ly
fo r
c r i t e r i on
b e larg e ly p re o c cup i ed wi t h t he s p e c i al
p o r t fo l i o op t imi z ing
Sep ara t i on
c o rre c t
f o r the
and i t was app r op r i at e tha t the
,
Mos s in cap i t a l as s e t p r ic ing mod e l b e b as e d on i t
r i s e t o s imp le l ine ar ru le s
the
as
"
h as
on
th i s c r i t e r ion
rep e at ed ly
,
o
f
th e
2/
;
the mean — var i anc e
c r i t e ri on i s r ig o rous ly c ons i s t en t wi t h the g ene ra l expe c t e d — u t i li ty app roach
(33 6 65
6
on ly in the rathe r S pe c i a l ca s e s o f a quad r a t i c u t i l i ty f u n c t i on or o f ga u s
s i an d i s t r ib u t i ons on s e cur i t y p r i c e s
F u r the r
—
b o th invo lv ing d ub i ous imp l i c a t i ons
.
re c en t emp ir i ca l work has s hown that th e s imp le f orm o f th e mod e l
,
/ ad
2
d o e s no t s e em t o f i t the d a t a as we l l a s h ad b een p revi ous ly b e l i eved
n
,
re c en t dynami c
s im u l a t
io n
L/
é
have s h own tha t the b ehav i o r ove r t ime o f s ome
e f f i c i ent me an- var i anc e p o r t f o l i o s c an b e qu it e unr e as onab l e
As id e f rom
it s
a lgeb ra i c t r ac t ab i l i t y
be c aus e o f i t s s ep ar a t i on p rope r ty
t e re s t
e l u c id a t i on
in the C as s - S t ig l i t z
s uch a p rop e r ty mus t h o ld re gard le s s
t urns
f
th e me an- var i an c e mod e l has
The r e f o re
,
in
g r ea t int e res t inh e red
the b road e r c ond i t i ons und e r whi ch
of
o
.
,
.
th e p r o b ab i l i ty d i s t r ibu t i on o f
r
d
The S p e c i a l f ami l i e s o f ut i l i ty f un c t i on s wi t h c ons t an t - r e la t ive
.
r i s k ave rs i ons o r c ons tant ab s o lut e — r i sk ave rs i ons fur t her ga ined in in t er
-
es t
!
Bu t i t was rea l i z ed tha t re a l- li f e ut i l i t i es ne ed no t be o f s o s imp le
a f o rm
.
The d e s i re f o r s imp l i c i t y
app roxima t i on the o rems
m e anv
,
o
f
ana lys i s l ed nat ura l ly t o a s e ar ch fo r
p ar t i c u la rly
o
va r i an c e ana lys i s we re n o t exa c t
sma l l !
f
th e as ymp t o t i c t yp e
Thus
.
,
wou ld the e rr o r in us ing i t b e c ome
,
A d e f ens e o f i t wa s the d e mons t r a t i on tha t me an— var ianc e i s
c o r re c t i f the r i s ks ar e
t o t i cal ly
c l o s e ly re lat ed
w as
"
sma l l
the d emons t r a t ion
"
f o r c omp ac t p r obab i li t i es ) ;
con t inuous t rad ing )
i t has b e en shown th a t a s the numb e r o f as s e t s b e c ome s la rg e
t he me an var i anc e s o lu t i on i s as ymp t o t i c a l ly
—
,
a s ym p
the s ame a symp t o t i c e quiva l enc e when
of
the t rad ing int erva l be c omes sma l l
c ond i t ions
even i f
op
,
.
Mor e re c ent ly
u nd er c er ta i n
/
l
t im a l fi
A p ar t i cular ly t emp t ing hunt ing g r ound f o r a symp t ot i c t he ori e s was
t hough t t o b e p rovid ed by the c as e in wh i ch inve s t or s maximi z e the exp ec t ed
u t i li ty
o
f
t e rmina l we a l th when the t e rmina l da t e ( p lann ing h or i z on ) i s ve ry
f ar in the f utur e
.
Re c ours e t o the Law
o
f
La rge Numb e rs
,
as app l ied
to
t e
!
,
mu l t ip l i c
a t i v e var i a t e s
p e a t ed
s ums o f lo g ar i t hms
( cumu la t ive
chang es ) has indep enden t ly t emp t ed var i ous wr i te rs
,
of
p o r t f o l io
ho ld ing ou t t o t hem t he
hop e that one c an rep lac e an arb i t rary u t i l i ty fun c t i on wi th al l i t s
t ab i l i t y
by the f un c t i on
the exp e c t ed lo g
o
U (w
l o g (W
)
T
)
T
maximi z ing the ge ome t r i c mean or
:
f out c ome s was hop ed t o p r ovid e an a s ymp t ot i c al ly exac t
cr i t e ri on f o r r at i ona l ac t i on
imp ly ing as a b on u s the e f f i c i en c y o f a d i
,
ve rs i f i ca t i on- o f- p or t f o l i o s t ra t e gy c on s t ant th rough t ime
r u le
the s ame f or eve ry p e ri od
,
in t r a c
a
my op i c
even when p r ob ab i l i t i es o f d i f fe r ent p e r i od s
,
S o p ower f u l d i d the max- exp e c t ed — log cr i t e r i on app e ar
we re
to be
that i t s e emed ev e n t o s u p e r c ed e the gene ra l exp ec t ed ut i l i ty c r i t e r i on
,
in c as e s whe re t h e la t te r
lo g c r i t e r i on
w as
s hown t o b e inc ons i s t en t wi th the max- exp ec t ed
F or s ome w r i t e rs
.
,
i t was a c as e o f s imp le err ors in re as oning
d ri
t hey mis t akenly though t t ha t th e s ure - th ing p r inc ip le s anc t i f i ed the new
/
L
F o r o t he r s
t e r i o nf
an inde f in i t e ly lar ge pr ob ab i l i t y o f d o ing b e t t e r by
,
Me th od A than b y Me thod
B
w as
fo r
t aken as con c lus ive evid en c e
t he s up e r i or i ty
/
of
A
jy St i l l o t her c onv e r t s t o the new f a i th ne v e r re a l i z ed tha t i t c ou ld c on
wi th t he p la u s ib l e p os t u lat e s o f von Ne umann maximi z ing ; o r
f lic t
p h i s t i c at e d l y ,
have t r i ed t o s ave the app roxima t i on by app e a l ing
u t i li ty f u n c t i ons
,
.
.
o
f
the s ame
o
f
on
the as ymp t o t i c f a l
La rge Numb e rs
,
b ut r a the r the
I t i s we l l— known t hat un i f o rm p or t f o l i o s t ra t eg i e s
g ive r i s e t o a cumula t i v e
—
I t c an c onc en t rat e ins t ead
not p r imar i ly th e Law
C en t ra l L imi t The orem
b ut ion
b ound ed
th e p re s ent p ap e r ne ed n ot mo re than revi ew the s imp le
,
lacy tha t invo lves
,
so
:
max— exp e c t e d - log f a l la cy
when n orma l i z ed
to
mo r e
9/
Ex ce p t t o p rep are t he gr ound for a mo re s ub t le f a l l acy
as ymp t o t i c gen u s
,
s um
o f log ar i t hms
o
f
re t urns that d o app roa ch
unde r s p e c i f i ed cond i t i ons e as i ly me t
s ugge s t ing he u r i s t i ca l ly a
,
Log no rma l S urro gat e
!
a gaus s i an
for
,
d is t r i
the a c t ua l d i s
t r ib u t i o n
I f one c an rep la c e t h e t r u e d i s t r ib u t i on b y
.
on ly two p a r ame t e rs
Sur roga t e ,
var ian c e o f log o f re turn
-
'
e ach p e r i od s exp e c t ed l og o f re t urn
"
—
w i l l be come
f o r e f f i c i ent p o r t f o l i o manag ing
Log- no rmal
it s
an d
,
as ymp t o t i c al ly s u f f i c i en t p arame t ers
t ru ly an eno rmous s imp l i f i c a t i on in t h a t a l l
,
op t imal p o r t f o l i os w i l l l i e on a new e f f i c iency f ron t i er in wh i ch t he f i rs t
p arame t e r
max imi z ed f o r each d i f fe rent va lue o f t he s e cond
is
s o le ly by t h e f ami ly
b e g ene r a t e d
U (N
)
T
and f o r
away fr om the s imp l e max— exp e c t ed - l og p o r t f o l io
.
Th i s f ront i er c an
0,
Y
t hi s leads
S o t h is new me t hod d oe s
.
avo id the c rude f a l la c y th a t men wi th l i t t l e t o l eran c e fo r r i s k ar e t o have
the s ame long— r u n p or t f o l i o a s men wi t h much r is k t o l eranc e
F u r t he rmo re
.
,
s inc e t he mean and v a ri anc e o f ave rage— re t urn— p e r— p e r i od are as ymp t o t i c
'
s urrog a t e s f o r the log no rma l s f i rs t two moment s
p
e ct e d -
,
H ak an s s o n
'
s
[ 17 ]
re t u rn s e ems t o b e g iven a new leg i t imacy by the Cent ra l — L imi t Theorem
O ne p u rp o s e o f th i s pap e r
is
t o s how
,
,
t he f a l la c i es
vo lve d in t h e ab ove- de s c rib e d as ymp t o t i c log — no rma l app roxima t i on
is
t e rmina l we al ths
cons t ruc t i v e ly
Then
,
f ixed - leng th p e ri ods g oe s
in f ini ty
in
What ho ld s
.
shown t o b e no t ne ce s s ar i ly app l i c ab l e t o ac t u a l
f o r no rma li z e d va ri ab les
.
.
by c o u nt e rexamp l es and examina t i on
o f i l leg i t ima t e int e rchange o f l imi t s in d oub le l imi t s
to
a v e rag e - ex
to
,
we s how t ha t
,
no t
as t he n umb er o f
in f ini t y b e c a u s e the p lanning h or i z on
,
T
g o es
,
b u t rat her as any f ixed h o ri z on p lanning int e rval i s sub d ivid ed
,
int o a n umb e r o f s ub - in t e rva l p e ri od s tha t g o es t o in f ini ty ( c aus ing t he unde r
a
ly ing p rob ab i l i t i e s t o b e long t o /g a u s s i an in f in i t e ly- d ivi s ib le c on t inuous - t ime
p rob ab i l i ty d is t r i b ut i on )
-
then t he me an— l og and va ri anc e - l o g p ar ame t e rs are
inde ed as ymp t o t i c a l ly s u f f i c i ent p arame t e rs for the d e c i s i on ; s o tha t
c an
on e
p repa re a ( m o ) e f f ic ien c y f ron t i e r tha t i s q u i t e d i s t inc t f r om t he Markowi t z
,
me an— va r ian c e f ron t ie r ( tha t had invo lved ac tua l ra th er t han loga ri t hms
re t urns )
,
b ut wh i ch now p rovi de s
m
any
o
f
o
f
the s ame two - d imens iona l s imp l i f i ca
t i on ( s uch as s e p ara t i on p rop e r t ies )
.
The re i s s t i l l ano the r k ind o f asymp t o t i c a p p ro xima t i on th a t at t emp t s
t o deve lop in Le land s [ 2 4 ] happ y phr as e
'
- m
T
"
a
t u rnp ike
t
h eo rem
"
,
as
in wh i ch t h e
op t ima l p o rt f o li o p rop o r t i ons are w e l l ap p roxima t e d by a uni f orm s t ra t e gy that
i s app rop ri a t e t o one o f th e s p e c i a l f ami ly o f ut i l i ty f unc t i ons
whe re
l is t he l imi t ing va lue
Y
u t i l i ty wi t h re s pe c t t o we a l th as
1 to
-w
g ene ra t es a new kind
o
f
W
T
o
f
the e las t i c i ty
Le t t ing
m
.
Y
e f f i c iency f ron t i e r
tha t o f o rd inary me an— var i anc e or
o
[ E { lo g
f
W}
-
Y
The re i s s ome
exp e c t ed - l o g maximi
be
o
q ue s t i on in any dep th
f
"
f r om memb e rs
our p ape r i s
inve s t ed in the j
is
J
v ar iab le s
th
o
fo r
th e i s o— e las t i c f ami ly ;
f
unc ove r the b ooby t r ap s inv o lved in
to
and we
,
d
o
no t
examine th i s
Exa c t S o lu t i on
inve s t o rs f a c e n s e c ur i t i es
O ne d o l la r
,
s e curi ty res ul t s a t th e end
of
p e r i od in a value tha t
one
The j o int d i s t rib u t i on o f the s e
.
s p e c i f i ed as
P rob {2 ( l ) 5 z
1
F
.
a p os i t ive random va r i ab le
is
mean
.
II
,
ff
'
log — no rmal and o t h e r as ymp t o t i c app r oxima t i ons
In any pe r i od
o
L as t o the r ob us t ne s s o f the Le land theo rem
n
much d i f fe rent
b u t th e main p urp os e
whe re
b ut wh i ch obvi ous ly
d ep end ing on our own s ubj e c t ive r i s k t o l eranc e p ar ame t e r
,
q ue s t i o
ut i l i ty f un c t i ons
ma rg ina l
d i s t inc t f r om
gene ra l i z e s t h a t c r i t e r i on in tha t we now ra t i ona l ly t rad e
re tu rn aga ins t r i s k
.
-
f
then run th e gamut f rom
va r { l o g
,
gene ra l i z es the s ing le - p o in t c r i t e r i on o f the wou ld
zers
,
o
U (W )
T
,
F
n
has f in i t e momen t s
.
Z!
N
O
,
[z
n
F[z
]
( 2 -1 )
Any p o r t f o l i o de c i s i on in the f i rs t p e r i od i s
d e f ined by the ve c t o r
wi th in i t ia l we a l th o f
}
Z
his
w
j
(l )
we a lt h a t the end
1;
o
f
one
i f the inve s t or b e g ins
p e r iod i s g iven by the
random var i ab l e
w
w
1
th e p ro b ab i li ty
By the us ua l S t i e l t j e s in t e g r a t i on over
b ut i o n
of
W
c an b e de f ined
1
,
name ly
f x}
r e—
An inve s tmen t p r og ram
de f ined by i t e ra t ing
w
It
=
T
t
0 f
1
dep enden t ly
,
f Z (t i
o
P
x;
[
l
P
[x;
l
inve s t ed f o r T p er i od s has t e rmina l wea l th
,
W ,
T
t o ge t the rand om var i ab l e
w
J
k)
n
n z un d
x
1
J
of
as s umed t ha t th e ve c t o r
is
Henc e
w
d is tri
,
w
J
un x
n
Q1
r
J
un x
nl
m
J
J
r and om var i ab le s
d i s t rib ut ed
is
b u t s ub j e c t t o th e s ame d i s t r ib ut i on a s
the j o in t p r ob ab i li ty d i s t r ib u t i on
o
f
(L b
Z
(l )
in
in
a l l s e curi t i e s ove r t ime i s g iven
by th e p r od u c t
P rob { Z( l )
S ince
W
T
/w
0
P
P
P
3
[ x;
x;
[
T
2
f
Z
(T ) }
cons i s t s o f a p rod uc t o f ind ep endent var i a t e s
wi l l c ons i s t o f a
d i s t r ibu t i on i s
f
,
s um
fo r
[ x;
W
(1)
w
(l )
,
,
o f indep end en t va ri a t es
e ach T
W
(1 )
W
(2 )
,
.
The re f ore
d e f inab le re curs ive ly by
f x}
P
,
W
(2 ) ]
I:
P
,
w
(3)
]
[Z
P
w
(T )
]
L
:
P
t he
,
l o g (w
,
T
1
l
1
s
;
-s
x
[
;
x
[
s
;
w
W
0
i t s p rob ab i l i ty
f o l lowing c onvo lu t i ons
x;
[
T
x
[
/w )
(l) ]
(l)
,
W
(2 ) ]
,
He re
,
fo r
as a ma t t e r o f no t a t i on
St i e l t j e s
in t e g ra t i on
,
L
:
f:
The inve s t o r
va lue
o
f
e)
( c o n c av
Max
E {U
p os t u la t ed t o a c t in ord e r
is
t e rm in a l
to
maximi z e the exp e c t ed
ut i l i t y o f we a lt h
Max
T
{W (t )
N
He re
,
U
T
fo r
Fo r
.
a c onc ave func t i on th a t c an b e arb i t rar i ly s p e c i f i ed
is
(
]
O
e ach t
a gener a l
U
und e rs t ood t o b e c ons t ra ine d b y 2 w ( t )
2 j
is
,
t he op t ima l s o l u t i on
T
vo lve p o r t f o l i o de c i s i ons c ons t an t thr oug h t ime
and
,
1
.
b u t ra the r op t imal ly va ry ing
,
in a c c o rd anc e wi th t he re cur s ive re la t i ons o f B e l lman dynami c p rog ramming
d i s cus s e d by n ume rous aut ho r s
f or examp l e in S am u e ls on
as
,
in
wi l l no t
B ut
,
,
h e re
as
,
we
sha l l f o r the mos t par t con f ine our a t t en t i on t o un i f o rm s t r a t e g ie s
( 2 -8 )
I
!
r
Fo r
e ach s u ch uni f o rm s t ra t e gy
,
( WT / W )
log
0
p enden t and i d en t i c a l ly d i s t r ibu t ed var ia t e s
s t ra t egy as
w
P
(t )
w
[
T
Ac t ua l ly
U
T
T
x;w ]
[
T
W
U
w
E
,
of
ind e
We wr i t e the op t ima l uni f o rm
T
T
[x ;
w ]
[
T
,
W
]
O
U
w
[
T
W
]
0
.
f o r th e s p e c i al u t i li ty fun c t i on s
Y
(W )
w
i t is we l l known t ha t
.
,
1 . Y 34 0
/y , Y
10 s W . Y
fu l l op t ima l i ty
.
s um
and abb revi a t e
’
P
w i l l c ons i s t o f a
=
0,
E
w *
,
indep endent o f T
,
i s a ne c es s ary re s u lt
fo r
O the rs
e
,
.
g
Markowi t z [ 2 8
.
p
,
.
3
w
-
have c onj ec t ure d th a t the max exp e c t ed
op t imal
U
T
l arge T when
fo r
U
i t is a rgued
"
wi l l b e
,
T
fr o m
ab ove )
o
I!
f
o
( o r bound ed
f a l l a by
the s imp l e
f
p o l i cy wi l l b e
app roxima t ely
f r om ab ove )
I
.
.
e
.
"
if
,
th en the expe c t ed ut i l i ty maximi z e r
,
app roxima t e ly ind i f fe rent
p ro g ram s as T b e c omes la rg e
The exa c t meaning
lo g
i s b ounded
i s b ounded ( o r b ounded
(
—
are awa r e
ho
b e twe en the { w * } and { W
,
T
.
app roxima t e ind i f f e rence
"
i s op en
to
int e rp re t at i on
.
A t r ivia l me aning would b e
1 1 m EU
T
oo
+
T
(W
O
Z
w
[
T
1 1 m EU
M
T
whe re M i s t he upp e r b ound
o
f U
(
T
the uppe r b ound o f u t i l i ty
wi th p os i t i v
e re t urn p e r p e ri od R
arb i t rar i ly c l os e
n i t io n
o f ind i f f e renc e wer e
to
e o n t en t
no
wea l th eq u iva len t
Tha t i s
.
l
p ro g ram { w } re la t ive
EU
( ni
l )w
j
o
T
(n
i
i
Z
to
w
[
T
i s the amoun t
f
wi l l
,
.
f o r lar g e enough T
Henc e
le t
,
w
H
N
we c ou ld
the { w
m a x—
1
{w
us e
1a
H
T;w )
(
12
o
} p rogr am i s
ge t
'
one
defi
S
i t s imp l i c a t i on s
,
ii o
d e f ine d su ch th a t
W Z [
(
T
O
T
o
f
a
ini t i a l
b e the ini t i al weal th eq u iva lent
fo r
,
e a ch T
o
f
a
,
j
w
add it i ona l in i t i a l wea l t h the inve s t or would
req ui re t o b e ind i f f e ren t t o g iving up the { w } p rog ram
,
,
( 3 -4 )
even i f
,
ould b e ind i f fe renc e in t e rms
J
Thus
w
.
EU
o
ab s urd ) wi l l l ead as T
j us t h o ld ing the r i s kle s s as s e t
,
1,
e
p ro g ram
i
D
make the c onj ec tu r e t rue
A me an ing f u l in t e rp re t a t ion
"
r
too
t he b l i s s leve l o f u t i l i ty
to
h ave p rac t i c a l ly
w
[
T
Thi s de f ini t i on me re ly re f le c t s t he fa c t
For e xamp le
.
O
Z
( un le s s i t i s
tha t even a s ub - op t ima l s t ra t egy
to
oo
+
T
(W
!
11
as a me as ur e
to
the { w
2
/
o
f h ow
{w *}
fo r
c los e
p rog ram
.
n
I
th e { W } p rog ram
in Op t ima l i t y t erms
The c on j e c t ure tha t
exp e c t ed — lo g i s as ymp t o t i ca l ly op t ima l in th i s mod i f i ed s ens e wou ld b e
t rue on ly i f i t c ould b e s hown tha t
H
(T ; W )
12
O
.
i s a de cr eas ing func t i on
o
f
T
,
and l im H (T ; W )
o
2
1
T
oo
l
or e v en
,
if H
1
we r e s imp ly a b o unded func t i on
T ;W )
(
2
0
+
of T
.
U
C ons i de r t he cas e wh en
w z
(
T O T
an
w
Y
) /Y
(
T
s
O
1
Y
,
T hen
.
,
T
T
/Y
.
b y the ind epend ence and i d en t i c a l d is t r ib ut i on o f the p or t f o l i o re turn in e a ch
p e r i od
S imi lar ly
.
we h ave t ha t
,
(u
n
Now
fl
m
o
(z
1> 1P M
t
l
(3
.
{ w * } maximi z es the exp e c t ed u t i l i ty o f we a l th ove r one p e ri od
,
th e
w
Y
is o—
e las t i c f ami ly
E
m
w
fl
l
Fr om
w **
mE
E
and s in ce
[
(Z
and
H
whe re
A (Y )
T /Y
0,
T
(
12
.
w
0
[
l
w
W
Y
]
as
y
¥
w *
2
we have th at
T /Y
)
A(Y )
Fr o m
en t o f T and
A (Y )
O
Y
N
O
1
and
we h av e f rom
,
we ha v e tha t
0,
Y
o
S
for
++
v
v fo r
fo r
T /Y
t ha t
O,
,
f or
f or
C;
Y
0
Y
an d
,
s
in c e A
and eve ry
W
0
and
1
is
0
in d
e p en d
,
O
H
l im
i
Henc e
,
u
(T ; W
even f o r
U
0
(
T
wi th an up p e r b ound
( as
when
would req ui re an eve r- l arge r ini t i a l p aymen t t o g ive up
S imi lar re s u l t s ob t ain f or
/
z
b e low %
Th e r e f o r e
ma l f o r large T
F ur the r
large
,
,
,
th e {
f
f
d
U
(
T
an inve s t o r
Y
his
{w
p r o gram
.
f unc t i ons wh i ch are b ounde d f rom ab ove and
} p r og ram i s d e f ini t e ly not
"
app r oxima t e ly
"
op t i
.
the
—
s ub
op t ima l { d
b e in a c le ar s ens e
"
ff
b eh ind
} p o l i cy wi l l
,
f o r eve ry f ini t e T howeve r
the b e s t s t rat e gy
,
Inde ed
,
l e t us
- 11
app ly t he t es t u s ed in the E is e n h owe r Admini s t ra t i on t o comp are
g rowt h
How many ye ars a f t e r the
.
f ( G NP
)
t
Kr e m i n o l o g i s t s
and s ome
,
was no t de c lin ing in t ime
AT
T
fi
o f exp e c t ed ut i l i ty
,
ca l cu la t ed
exp e c t ed log s t r at e gy { w
—
,
rea ch ed each rea l GNP l ev e l
.
Th i s d e f ines a func t i on
le t us f o r e ach leve l
o
H
}
we can
,
p r ogr am :
b ad
,
b e fa l la c i ous
to
the ini t i a l wea l th equ iva len t
us e
lfi
in tha t t he { w
by the p rog r am o f h o ld ing no thing bu t
3
D e f ine { w } E [ O
,
l
0,
,
O
exp e c t ed lo g p r o g r am { w
—
lfi
0
Y
H
E [ (Z [w
T
13
} as b e f o re
m( m
n
whe re
—
T
1+
p ro g ram wi l l
and henc e
the
l ead
not
ap p roxima t e
wi l l b e domina t ed
,
r is kl es s as s e t
to
"
.
.
Then
,
R
,
1
l
.
Le t { w } b e the max
f o r the i s o— e las t i c fami l
y
D
Y
]
W
to
Y
T
Y
R
“
f
T o examine the p rop e r t i e s o f the ¢ (Y ) f unc t i on
5
[
,
w
rf
é
,
w
tf
g
]
d oe s maximi z e
E
,
the
the { w } p ro g ram
Y
0
,
3
z
r
f
wi
w
l ] t o b e the p r og ram wh i ch ho ld s no th ing
,
Y
Y
}
as T
i s d e f ine d by
,
W
{
w
d emons t ra t e tha t
to
in i t ia l we a l th e q u ivalent f o r the { w } p r ogram re la t ive
r
r
w
AT
the max
.
l
13 O
for
the max- exp e c t ed — l o g s t ra t egy i s a
bu t the r i s k le s s as s e t wi th re tu rn p e r p e r i od
W
v
t hen i t i s no t hard to s how that
,
op t imal i ty even in the t r ivi a l s ens e o f
H
AT
d e f ine
f
the Op t imal s t ra t egy { w * } and
fo r
fo r s u f f i c ien t ly r i sk- ave rs e inve s t or s
bad
and
o f tha t day t o ok s at i s f ac t i on tha t
Ag ain the ge ome t r i c mean s t rat e gy p rove s
"
.
as th e d i f fe renc e in t ime p er i od s ne ed ed t o s u rp as s tha t le e l
T*
F ina lly
S
.
I n a new c a l cula t i on s imi lar t o the in i t i a l
.
wealt h e q u iva len t ana lys is ab ove
n
.
t o re a c h th at leve l !
d id i t t ake the
AT
U
U S
we no t e tha t s inc e
,
A s e c ond mo re s ub t le
f a l la cy has g r own out o f th e mo re re c ent l i t e ra
,
t ur e on op t ima l p o r t f o l i o s e le c t i on f o r maxim iz a t i on o f ( d is tant t ime ) ex
p e ct ed
t e rmina l u t i l i ty o f wea l t h
Hakans s on
.
a f t e r g ivi n g argumen t s b as ed
on
maximi z ing exp ec t ed
ave rage rat e - o f - re tu rn th a t imp ly myop i c and un i fo rm s t ra t e g i e s
p r o ceed s t o
us e
t r ib u t i o n a s
t h e C en t ra l— Limi t the orem
T
Henc e
!
.
t emp t ed t o rep l a c e t he t rue rand om var i ab l e p o r t f o l i o re turn
i t s as s o c ia t ed log —no rma l r and om var i ab le when T i s lar ge
maximi z a t i on o f t h e exp e c t ed va lue
f un c t i on
Th i s d one
.
[l 7
H ak an s s o n
,
.
maximi z a t i on o f exp e c t ed
Y
W
/Y
o
f
,
t u rn t rade
o
ff
lo g
pu 8 7 fl
fo r
a l l ut i l i t i e s
,
lo g
f ixed value
the min i mum
[E
i t wi l l b e op t ima l
to
2 14 4
to t ic
In
2
0
u,
o
f
t o t he
no rma l d i s t r ib u t i on reduc es t o
o
f
lo g
o
re turn and
f
'
the inves t o r s ri s k— r e
0
th e s e cond
.
Var
t hen
have a maximum o f the f i rs t p ara
Wh i l e i t is
no t
t rue tha t
fo r
a
a l l c on ca v e ut i l i ty maximi z e rs wou ld ne c e s s ar i ly p re fe r
i t i s t rue tha t f or a f i xed va lue
o
f a
a l l c onc ave ut i l i ty maximi z e rs wou ld op t ima l ly ch oo s e the minimum
u
0
of
2
I
me t e r f o r any f ixed va lue
wi th
i f a p o r t f o l i o i s known t o have a logno rma l
,
,
is
i s ab le t o us e the p r op e r ty th a t
re t urn wi th Y b e ing a meas ure
d i s t r i b u t i on w i th p ar ame t e rs [ u
“
fig
th e p ar t i cu la r i s oe las t i c u t i l i t y
und e r the
Mo re gene ra l ly
.
5
one
,
,
2 2 12 5
a s imp l e l inear t r ad e o f f re l at i ons hip b e twe en the exp e c t ed
the var i anc e o f
(t )
ar gue tha t the as ymp t o t i c d i s
s uch p o r t f o l i os w i l l b e log no rma l
fo r
w
to
w
,
Hen ce
the Hakans s on d e riva t i on c an s ugg e s t tha t the re exi s t s an
re la t ed
s p ac e s
e f fi c i en t f ront i e r in e i the r o f the two/ p arame t e r
or
,
"
the s p e c i a l c as e
o
f
the
Y
W
/Y
f ami ly
,
t he Y de t e rmine s the p o in t
'
f ron t i e r whe re a g iven inve s t o r s o p t ima l p o r t f o l i o l i es
p enden t ly
a s ym p
,
fe l l i n t o th i s s ame t r ap
15
;
/
.
O ne
of
on
us
,
t ha t
inde
Un fo rt un a t e ly
s ub s t i t u t i on o f t h e as s o c ia t e d l og -no rma l f o r the t r u e
,
d is t r i b u t ion l e a d s t o in c orr e c t
e x amp le
.
T he e r ro r
r e s ul t s
,
as
wi l l b e demons t r at e d b y c oun t e r
t h e analy sis le ad ing t o t h i s f als e c on j e c t u re
in
re s u l t s f rom an imp rop e r in t e r ch ange
o
f
l imit s
F o r e a c h uni fo rm p or t fo l i o s t rat e gy { w }
p e r i od
d e f in e
as
in
.
the one
re t u rn in p e r i od t b y
n
p ort folio
Z[t ;w
,
a s w e now d emo ns t r a t e
,
]
2
E
w
1
z (t )
i i
G i v en t h e d i s t r ib u t i on al as s ump t i ons a b o u t as s e t re t u rns in s e c t i on I I
Z [ t ;w
the
] w i l l b e indep en d e nt ly and i den t i c a l ly d i s t ri b u t e d t h r o u gh t ime wi t h
x}
P
t
Th e
,
r e t u rn on t h e p o rt fo l i o
T - p e r i od
Z
1
l
T
]
5
H
Z
(x; w )
is
d e f in ed
,
f or any T
1,
by
[ t ;w ]
w i th
x}
as
in
wi th the
De f ine
i
;
uT
2
0
We c al l
is
Z
O
T
a
.
l ogno rma l ly d is t r ib u t ed rand om v ar i ab l e wi t h
c h os en s u ch t h at
s
T
T
E
Z
I
[w ]
the
s u r ro ga t e
the l o gno rma l ap p ro xima ti on t o
moment s in t he c l as s i c a l
d e fin i t ion
x; w )
(
T
in depen den t o f t ime
s
[w ] t o b e
an d
uT
p arame t e rs
w
'
a P
lo gno rmal t o the random va r i ab l e
Z
P e ars o n i an
w ]
[
T
Z
w ]
[
T
;
f i t t e d b y eq u at in g t he f i rs t tw o
c u rve - f i t t ing p ro c edure
.
No t e tha t b y
it
2
2
0
0
Sin c e
e a ch
I
(w )
(x
in Z
is
1
in dep end e n t v ar i a t e w i t h f in i t e v a ri an c e
t o g ive
t h e va l id
us
v ar i ab l e
,
t h e n o rma
as
Y
ymp t o t i c
4
io
T
a
l d i s t ri bu t i on
I
.
y
5 }
.
e
.
l im
T + oo
013
N(
is
t h e c um u lat i v e
d ar d no r ma l v a ri a te
S in c e
y mp t o t i c
o f t e rmi n a l w e al t h
In a
mo
m
b o th
,
P (x; w )
T
L
l im
T
L
l
0
=
l
.
2
no t th e c as e t h at
l im { [ L
T
+
cn
2
e
d i s t r ib u t ion
P
is
T
( x; w )
fo r
wh e re
is
( 2a )
-
t
2
d t
,
func t i on
fo r
a s t an
the v a l i d fo r mu l at i on
o f the
.
n ame ly
T
y
-1
pp ro x ima t i on fo r t h e p rop e r ly s t andard i z e d d is t ri b u t i on
N [ (x
lim
o (w ) JT y
[
T
a
r e t ri v i al s ens e
E
pp r o a ch es
,
P
5
th i s
as
pp l i es
a
.
lo g
lo gn o rma l
,
,
l
E
{
N ( y)
wh e r e
t h e C en t ral -Limi t t h e or em
5
l im P rob { Y
T
r e l at
an i dent i c ally d is t r ib u t e d
+
v
°
for
Z
w ]
and
[
T
it s
s u r r ogate
,
app roa ch t he s ame c o mmon l imi t
,
in the s ens e tha t
lim [
T o
00
ac
I
Bu t
f
1
-
w (w
x
o
T+
act u a l ly
,
is
theo r em wi l l show
A c orre c t
o
) 1 1m
e
m
q u i t e f a l s e a s c are f u l
n ly s i s immed i a t e ly shows tha t the heu r i s t i c arg ument lead ing
an in c o rre c t l imi t int e r change
t h e r and om v ar ia b l e
Y
T
and
U (W
T
f o r th e
,
s
0
x
e
P
(
T
,
w
,
)
f
F (
T
,
w
)
h av e d ens i t ie s
,
N
t o d e r iv e
'
,
O
e
/Ty
pT
,
we a l s o ha v e tha t
OP / 8x
T
P
1
T
(x
,
as
wi l l b e i l le g i t ima t e
,
)
and
OF / B y
T
F
v
T
,
/ Ty
[ l im
U (W
s een f rom e as y c o u n t e r - examp le s
is
w
i n t h e c as e whe re
t he f o l lo w ing l imi t in t e r chang e
0
s
I n g ene ra l
,
,
( y)
w o u l d h av e t o be v a lid f o r e a ch y
l im [ U (W e
0
T oo
(
y;w )
T
,
f r om
0
F
f or
r
/
m fy
U OJ e
O
f r om t h e C en t ra l -L imi t t h e or em
and
F rom
we ha v e tha t
urr og a t e - f u n c t ion c a l c u la t i on
l im
Ho we v e r
,
f U ( W0
I
Fu r t h er
.
wi ll h ave a p roba b i l i ty func t ion
By de f ini t ion
U
na ly s i s o f t h e C en t r a l -L imi t
a a
invo lv e s
,
a
.
to
e a ch T
.
and h en c e
whe r e t he l imi t int e r ch ange in
,
,
0
e
T+
v a l id
m
s uc h an in t e r c hange o f l imi t s
the F a l s e C oro l lary
is
JT y
is
inva l id
U is
.
I n tho s e c a s e s
a b o u nd ed f u nc t ion )
the F a l s e Co ro l lary ho ld s only in the t r iv ia l s ens e o f
in s e c t ion I I I
a s a l re ady no t e d
the l imi t a s T
t h e re w i l l exi s t an inf ini t e numbe r o f p o r t f o l io
.
,
*
’
w
,
,
p rog r ams ( in c l u d ing ho ld ing one hund red p e r c en t
p o s i t i v e -yi e ld ing
e qua l t o the
,
of
the p or t f o l io in the
r i s kle s s a s s e t ) wh i c h wi l l g i v e exp e c t e d u t i l i ty lev e l s
up p e r b o und
of U
we now s how by c o u nt er - examp le
As
.
t rue tha t p o r t f o l io p rop o r t ions
,
w*
,
it
i s no t
cho s en t o max imi z e exp e c t ed u t i l i ty ov er
,
1
.
the
P
(
T
;w
d i s t r ibu t i on w i l l b e e q u a l t o the p ropo r t ions
)
maximi z e expe c t ed u t i l i ty o v e r th e su r ro ga t e
w
,
ev en in t he
n(
To d emo ns t ra t e our c ou n t e r - examp le t o the F a l s e C o ro l lary
tha t f o r the i s oe la s t i c f ami ly
,
t he
c ho s en t o
,
f ir s t
l im i t l
,
no t e
xpe c t ed u t i l i ty l ev e l f o r the s u rr oga t e
e
l o gno rma l c an b e wr i t t en a s
As
Hakans s on [ 1 7 ]
ha s
shown
2 2
l
Y
W
e xp [ Yu(w ) T
O
Y
F
o
(W m / Y
ma x imi z a t i on o f
,
is
eq u iva len t t o t h e
max imi z a t ion o f
[ u(w )
L
Hen c e
f rom
,
the maximi z ing
w
1
for
d ep end only on the me an
and v ar ian c e o f the l o gar i t h m o f one -p e r i o d r e t u rn s
Se c ond
,
.
no t e tha t b e c aus e the p or t f o l io re t u rns f o r ea ch p e r i od are
indep end en t ly and i dent i c al ly d i s t r ibu t ed
leve l f or the t rue d i s t r i b u t ion
we c an w r i t e t he exp e c t ed u t i l i ty
,
as
T
} /Y
Henc e
max imi z a t i on o f
,
is
equ iva lent t o the maximi z a t i on o f
whi c h a ls o dep end s only on the one pe r iod
—
r e t ur n s
C ons ide r the s imp le two - a s s e t c a s e whe re
whe re the
y
P rob { y
A}
t
t
are ind epend ent
Prob { y
t
6}
,
Z
gf
l
[ t ;w ]
w
(yt
R)
R
and
Be rno u l l i - d i s t r ibu t ed random var i ab le s wi th
and
A
R
6
0
.
Sub s t i tu t ing in t o
we ha v e tha t th e Op t ima l p o rt f o l i o ru le
mg
x
uw
m
+
k
fl
,
w*
a
r
w
n
+
wi l l s o lv e
,
R
+
fi
wm
(t u )
W
wh i c h b y the u s u a l c a l c u l u s f i rs t - o rd e r c ond i t i on imp l i e s tha t
Y 1
-1
_
(l
0
R)
R]
ur
wo
w
w *
w *
Y
R]
)
we h ave th a t
Re a rrang ing t e rms in
-
(d
-R
w i l l s a t i sfy
w*
M J
(y )
=
[
M
O
-
(A
m
mm w
+
R)
m
m
-
+
i
mfi
a) 1
A (R
u
un
I,
25)
l
whe re
H
A
F rom
Q K
-
R
A
mfl
-
d
.
th e s u r r oga t e lo gno rma l f o r
u
u( )
R)
w
2
0
(w )
Z
W ] wi l l ha v e p arame t e r s
[
T
R]
R)
lo g [w
R]
1
—R
(6
log
2
)
[w ( 6
R]
R)
}
R 1}
.
The op t ima l p o r t fo l i o r u le r e la t ive t o
0
'
l l2 i
w
lL
—
+
(
-
4
l —R ) + R
—
'
w
+
w
£tj &
R)
o
[1
+
(6
as
g
R)
-R + R
6
(
)
lo g [w
wh i c h c an b e r ewr i t t en
wi l l b e the s o l u t i on t o
l o g (B
)]
B
[l
lo g (B
)]
R)
2
11
_
non -no rma l i z ed
lim
V ar ia te s
P
(x ; w ) d
T
Comb in ing
21
_
th e p ro b ab i l i t ie s s p re ad o u t and
,
m
x
x
we
and
s ee
the i l leg i t ima c y
Of
int er ch anging l imi t s
in th e f o l l owing f ash i on
9
1
lim
P
-> o o
T
S imi la r ly
as
,
w as
T
(x ; w ) d
P
é
(x ; w ) d x
dx
O
0
.
a lre a dy ind i c a t ed f o r a non- dens i ty d is c re t e - p r ob a
examp le in the p r ev i o u s
b i l i ty
l im
x
s e c t io n
,
t h e f o ll owing t ru e r e la t i on
f r om
,
1
l im lP
T+
it
1 1m
is
T
(x ; W )
0.
w
f a ls e t o c on c l u de t h a t
e
YX
,
for
0
Y
1,
I
[ P T (x ; W )
e
n
1
l im [ P
T-+ cc
0
Henc e
large T
,
,
t ry ing t o c a lcul at e the c o rre c t
U
w ]
[
T
,
T
(x ; W )
0
dx
in
even f or very
b y re lyin g on i t s s u r ro ga t e
2
T) d x
le ad s t o the wr ong p o r t f o l i o ru l e s
p [ Yu(w )
ex
l / 2Y O
i
,
namely t o w
w
(w ) ] T
z
w
(t)
.
A sp ec i f i c examp le f o r th e de n s i ty c as e can demons t ra t e th e
t r ead h e ry
invo lve d in rep la c ing the a c tua l p rob ab i l i t i e s b y the ir as ymp t o t i c su r r o ga t e
.
C ons i de r
2 xe
dx
(x
c onvo lu t i on
P
2
'
P
wh i ch
'
T
1/ 2 )
2
2
v a ria t e s s at i s fy
we e as i ly d educe tha t the
,
( x)
2
( x)
2
2x
a
2x
2x
6:
( 2 x)
T- 1
a g amma p rob ab i l i ty law
is
-2x
e
d x
/ (T
0
x
By the C ent r a l - L imi t th eo rem
.
o f th e p rob ab i l i ty dens i ty f o r th e no rma l i z ed v a r ia t e
f
(
y)
T
E
P
( O / Ty
'
T
l im
—) O O
T
f
T
Y
T
,
th e l imi t
E [X
T
wi l l b e
( y)
l im
l / 2T ; w
y
l im { ff
-
( 2 11)
e xp
l/Z
e
-l
[
/ 2y
y
r
)
1 / 2 Jf
'
uff y
2
N
'
(y )
'
a s c an b e ve r i f i ed b y S t i r l ing s app roximat i on t o the f ac to ri a l f o r la rge T
.
Howeve r
rep la c ing
,
l /4T )
e xp e c te d v a l u e
2
by i t s no rmal s u r ro g a t e
e
lead s t o d i s as t er wh en c omp u t ing
,
of
its e l f
t e rmina l money we a l th
c omp u t ed and in c o rr e c t ly c omp u t ed b y u s e
d i s c rep an c y go e s t o i nf ini ty
Th u s
no t
,
of
z er o
to
[
a]
Cons i de r the
.
as
,
co rr e c t ly
l og — no rma l s u rro ga te : the i r
its
T g oes t o in f ini ty !
as
,
B
yx
,
a l th oug h
,
e
x
l im
—l
[2e
)
dx
!
0,
T
it
is
s t i l l th e c as e th a t
l im
dx
T—) O O
l im [ ( 2 )
lim [ e
T
e
Tlo gZ
T (1/ 2
e
5T/8
T—) oo
2
s inc e lo g
Thu s
,
=
f
.
6931
.
6 25
.
we h op e t h at th e f a l la c y o f th e s u r rog a t e lo gno rma l ap p roxima t i on
wi th res p ec t t o o p t ima l p or t f o l i o s e le c t i on
b ee n lai d t o re s t
h as
.
I n c on c l u d ing th i s d e bu nking o f imp rop e r lo gno rma l ap p roxima t i ons
,
we s hou ld ment i on th at t h is s ame fa l la cy p op s up wi th mono t ono u s re g u l a r i t y
in a l l b ranch e s o f s t o ch as t i c inve s tment ana ly s i s
S amu e ls on
had t o wa rn o f
it s
in c or re c t
us e
.
Th u s
,
one o f u s
,
in r a t i ona l war ran t p r i c i ng
Supp os e a c ommon s t o ck s fu t u re p r i c e c omp a red t o i t s p res en t p r i c e
'
V (t
T)
/
V(t)
E
Z [ T]
uni f o rm p r ob ab i li t i e s
is
,
P
[Z]
d i s t r ibu t ed l ike th e
.
For T la rge
,
the r e
p
r o d uc t o f T independen t
is
,
,
a l ogno rma l s u r ro ga t e
for
.
25
t ive
var iab le s o f u t i l i ty ( o r o f
rand om
m
oney )
,
wr i t t en a s
res p e c t ive ly and s a t i s fy i n g p ro b ab i li ty d i s t ri b u t i ons
P
U
I
U ; T)
(
I
and
an
U
II
d
T oo o f t en th e fa l s e in f e renc e i s m
ad e tha t
P r ob { U
l im
T
—) O O
U
I
}
II
l
imp li e s ( o r i s imp l i ed by ) t h e c ondi t ion
l im { E [ U
T—) O O
]
I
E [U
}
]
II
0
shown in o u r r e fu t ing the fa l la cy o f max - e xp ec t ed — l o g
tha t c ome s
y i e ld i ng U I ra the r than t h e c o r re c t op t ima l U / f r o m u s e o f w
II
Ind e e d
is
,
as
w as
E [U
U
I
1
11
0
fo r a l l T howeve r l arge
1+
O r c ons ide r ano th er p rop er ty o f the w
x}
P rob
whe r e
M (T ; w
)
P r ob { Z
s tr a t e gy : Name ly
w ]
[
f
T
x}
fi
T+
I
.
.
M (T ; w
M (T ;
w)
)
w
.
Yet
m
no r d oe s i t imp ly a symp t o t i c F i r s t O rde r S t o ch as t i c
doe s no t imp ly
Dominan c e
,
f or a l l x
an in c rea s ing func t i on o f T wi th lim
is
it
s a t i s f i ed and ye t i t i s a ls o tr u e th at
is
the c as e t ha t
,
e
.
,
i t d oes no t f o l l ow f rom
f_
Th e mora l
x}
tha t
x}
,
as
T
m
,
f or a l l x indep enden t ly
of T
Never c onfus e exac t l imi t s inv o l v ing no rma l i z e d
Al t hough
va ri ab le s wi th th e i r na ive f orma l e xt r ap o l a t i o n s / a
b and b
c imp l i e s a
is t h i s
:
,
0
;
still
a
x
b and b
2
c c anno t r e l iab ly imp ly a
s t r i c t i on s pu t on th e int erp r e t a t i on o f
z
c wi thou t c are fu l
re
'
26
knows
I f one
U
Y
(WT ) / Y
W )
(
T
T
s is
th a t t h e re l ev an t u t i li ty
fami ly
b e l ongs t o t h e
f unc t i on
s ome s w eep ing s imp l i f i c at i ons o f p or t f o l i o
,
are p os s i b le w i th ou t re ga rd t o th e lo g - no rma l s imp l i f i c a t i ons
mu ch in t e re s t res i de s in re c e nt wo rk b y Le land
la rge T
the
,
F i g u re
1a
—w
z er o p r ob ab i l i ty o f c omp le t e r u i n
lo cu s
0A
Cas s - S t i g l i t z [ 7
s ep ara t i on the o rem
p o s i t ive f r a c t i ona l Y
a t B whe re Y
a ne ga t i v e Y
By c on t ras t
in th e (p
cho i c e
,
0
)
1 / 2 0 y)
.
Any po in t b e twee n A and
as
,
s p ac e
L
tha t cho s en by the
c or res p ond s t o a
Any p o int b e twe en
the l o g -no rma l e f f i c i ency l ocu s
WY /Y
I f we c omb ine w i t h l og no rma l i ty the
-
.
0
and L
f o r a mo re c au t i o u s inve s t o r wi th Y
in F i g u re l b
,
is
is
-1
.
drawn
fami ly
,
th i s f ron t i e r c ome s whe r e i t i s t ang en t i a l t o a con t o u r o f e q u a l
on
2
(p
lb
2
as
,
The p oin t L
.
.
to
Al l op t ima l p o r t f o l ios l i e on the
.
ma x -exp e c t ed - l og c r i t er i on
co r r e sp on d s
me r e ly tha t e a c h r i sky as s e t
,
a s t rai gh t l ine th r oug h the o r i g in b y vi r tue o f th e
is
wh i ch
,
.
r i sky as s e t s and one s a f e as s e t a re as s umed ;
Tw o
.
no lo g—no rma l i ty p rop er t i es a re as s ume d
h as
a s ymp
i l l u s t ra t e s the e f f i c i en c y f ron t ie r gene ra t ed b y le t t ing
.
th ro u gh
1
run f r om
,
s u gg es t ing tha t f o r
i f they ha d i s o e l as t i c u t i l i ty f u n c t i ons
as
Hen c e
.
ximi z e r s o f a b ro ad c la s s o f u t i l i ty f u nc t ions wi l l
b eh ave
t o t i c a l ly
Y
ma
an a l y
,
wi t h s lope equ a l t o
c o rre s p ond t o
(l / 2
y
,
-l
,
o)
Y/ 2
—
Thu s
.
p o in t s
.
po in t s o f Fi gu re
,
Rememb e r th o u gh tha t the
p o in t s in la we re g ene ra t ed f o r non— lo g - no rma l p rob ab i l i t i es
.
c o rr e s p ond ing lo c us f o r
lb f s
log — no rma l p rob ab i l i t i e s
,
Hen c e
.
th e
,
wh en p lo t t ed in
1a
.
w i l l b e the new lo cus
Ju s t
as
t h i s l as t lo gno rma l lo cu s
c an p lo t th e ( n
,
0
2
)
is
lo c us t ra c ed o u t f o r
ir r e le v an t in la
al l
‘
Yal ong
.
th e t r u e
s o in lb
OA
.
we
op t ima l i ty
28
We deno te by T the t ime ho r i z on
o
f
the inve s t or
N ow
.
deno t e by N th e numb e r o f p o r t fo l i o revi s i ons ove r th a t h ori z on
h
I
I
I
T /N
of
wi l l b e the leng th
p revi ous s e c t i ons
t im e
we e x amine d the
,
N t ende d t o i nf i ni ty
for
,
b e twe en p or t f o l i o revi s i ons
a s ump t o t i c
a f ixe d
we
s o tha t
,
In the
.
p o r t f o l i o b eh avi or as T and
I n th i s s e c t i on
the as ymp t o t i c p o r t f o l i o b eh avi o r f or a f ixe d T
and h t end s t o z er o
,
,
we c ons i d er
,
N t end s t o inf ini ty
as
The l imi t i ng b ehavi o r i s in t e rp r e t e d a s th a t o f an
.
inves t o r who can c ont inuous ly revi s e h i s p or t f ol i o
We leave f o r la t e r
as s ump t ion
the d i s cus s i on o f h ow rea l i s t i c i s t h i s / an d o th er as s ump t i ons as a d e s c r ip
.
t i on o f ac t u a l as s e t ma rke t c ond i t i ons
t o s ee wha t r e su l t s ob t a i n
fo r
and
,
the moment
,
p r oc e ed f o rma l ly
.
Wh i le th e as s u mp t i ons o f p rev i ou s s e c t i ons ab ou t th e d i s t rib ut i ons
o f as s e t re turns a re kep t
int e rva l leng th
t ime s
t -T
wh e r e
T
h
,
Z
j
.
e
.
,
we make exp l i c i t th e i r dep end en c e
the p er — d o l la r r e t u rn on th e
and t when the t r a d ing i nt e rva l
int e gr a l i n h
is
S up p os e
dua l
I
.
,
'
,
as
is
is
o f l eng th h
na tu ra l once t ime i s c ont inuo u s
u
j
N (x ; u T
j
i
2
and
O
and
as s e t b e tween
,
is
,
th a t the
ar e d i s t r ibu t e d lo g - no rma l ly wi th c ons t an t p arame t er s
wh ere
u
the t r ad ing
.
x}
w i th
j
th
on
0
inde pende nt o f h
.
j
F u r th er
T
,
,
d e f ine
2
,
o
j
T
)
.
v
in d i i
I
.
e
.
,
29
E
0L T
J
o
(
“
2
J
J
T
T
As i n
th e n
th
as s e t
We d eno t e b y
is
r i sk le s s wi th
of
the ve c t o r
a t t ime t when t h e t rad ing i nt erva l i s
o
O
0
.
n
Op t ima l po r t f o l i o p r op o r t i ons
h+ 0
and i t s l imi t as
,
"
"
mu t u a l fund ) th e orem
Th en th e f o l l owi ng s ep ara t i on ( or
by
n
leng th h
f
r and
a
2
o b t a ins
Th e orem V I I 1 G i v en n as s e t s whos e r e t urns are l og - no rma l ly
d i s t ri b u t e d and g iven c on t inuous - t r ad ing opp o rt un i t i e s ( i e h= 0 )
th en th e
are s u ch th a t : ( 1 ) The re exi s t s a unique
"
( u p t o a non s ingu la r t rans fo rma t i on ) p a i r o f mu tua l fund s
c ons tr u c t ed f r om l ine ar c omb ina t i ons o f th e s e as s e t s s u c h th a t
indep endent o f p re f erenc e s we a l th d i s t r ib u t i on o r t ime
h o r i z on inve s t o rs wi l l b e i nd i f f er ent b e tween ch o o s ing f rom a
l ine a r c omb ina t i on o f t he s e two f u nd s o r a l ine ar c omb i na t i on o f
t h e or i g ina l n as s e t s
( 2 ) I f Z f i s th e r e tu rn o n e i th er fund
th en Z f i s lo g — no rma l ly d i s t r ib u t ed
( 3 ) Th e f ra c t i ona l p r o
p or t i ons o f re sp e c t ive as s e t s c on t a ine d i n e i th er f u nd a re s o le ly
"
=
a f unc t i on o f th e i and O i ( i j l 2
n ) an
e f f i c i ency
j
c ondi t i o n .
.
.
.
,
,
,
,
.
,
.
“
,
,
,
.
,
P r o o f o f th e th eo r em c an b e f ound in Me r to n [
whe th er one o f th e as s e t s i s r i sk le s s o r
no t
30
.
,
3 84
‘
p
,
.
I t ob t a in s
From th e th eo r em we c an a lway s
,
wo rk h er e wi th j u s t two as s e t s : the r i s k le s s as s e t and a ( c omp o s i t e ) r is ky
as s e t wh i ch
is
lo g - no rma l ly d i s t r ib u t e d w i th pa rame t e rs
w e c an r edu ce th e ve c t o r
inve s t e d in th e r i sky as s e t
(
Ew *
2
2
O
t o a s c a la r
.
w *
(t)
He nc e
,
equa l t o th e f ra c t i on
—
I f we d eno t e by
r
)
r ) and
9
“
O
*
th e ( ins t an t ane ous ) me a n ga in and va r ianc e o f a g iven i nve s t o r s
op t ima l p o r t f o li o
'
,
an e f f i c i ency f ron t i er in t erms
o
f
*
a
(
O
or
c an b e t r ac ed
f r ont i er
is
wh e re a
p er i o d
ou t
s h own i n e i th er f igur es
as
p e r i od
"
is
an ins t ant
(t )
w *
,
a c ons t an t
u t i l i ty func t i ons
,
Hen c e
we have
,
is
the s ame in eve ry
,
h i s cu rren t wea l th an d T
on
of
is
o-
/
g
e las t ic ut i l i t y g
3
in f i gu re
as
the d i s t r ib u t i on
w i l l b e l ogno rma l f o r a l l
ing
Al t h o u gh th e f ron t i er
.
t
Fu r th e r
.
.
Y
T
W
vl
/Y ,
<
of
wea l th und er the op t ima l p o l i cy
ggj
f r om th e as s u mp t i ons
lo g —no rma l i ty and c ont inuou s t rad
at a ll
even f o r T f ini t e and n o t/ l a r g e a c omp le t e as ymp t ot i c th e ory
,
of
,
as s u mp t i ons
.
Fur the r
,
b u t wi thou t
,
it s
th e s e r e s u l t s s t i l l ob t ain even i f
Hav ing der iv ed the th e ory
r e as onab lene s s o f th e a s s u mp t i ons
th e answe r w i l l dep end on
,
f or every
for
th er
f ac t
.
d oe s th er e exi s t an
s ome no rm
,
we now tu rn t o th e que s t i on
"
,
j
f
ec
a l lows
V (c )
,
zé
/
th e
S ince t ra d ing c on t inuous ly i s no t a
h ow c l os e
h> 0
,
"
is to
I
.
e
s u ch th a t
and wh a t i s th e na tu re o f th e
6
6
6 (h )
f unc t i on !
s inc e l o gno rma li t y as t h e d i s t r ibu t i on f or r e tu rns i s no t a
,
o
ob
one
i nt e rme d i a t e c onsump t i on e v a l u a t e d a t s ome c onc ave u t i l i ty f u nc t i on
re a l i ty
,
f o r th i s S p e c i a l c la s s
,
w i th a l l th e S imp li c i ty o f c las s i c a l mean -vari ance
t i o n ab l e
U
t
and the en t i re p o r t f o l i o s e le c t ion p rob lem c an
,
b e p re s ent e d g raph i ca l ly
f
Th e
.
a given inve s t o r w i l l in g ene ral ch o o s e a d i f f er ent p o in t on th e
,
I n th e sp e c i a l c as e
o
2b
or
exac t ly akin t o th e c las s i ca l Ma rkowi t z s ing le — p er i od f r on t i er
f ron t i e r e a c h p e r io d d ep end in g
w *
2a
"
Fur
known
wh a t a re t h e c ond i t i ons su ch tha t one c an va li d ly us e the l ogno rma l
a s a s u rro g a t e !
S in c e th e answer t o b o th que s t i ons wi l l tu rn on th e d is t rib ut i ona l
as s ump t i ons f o r the re tur ns
f o r th e
Z
j
(
;h)
,
,
we
n ow
dr op th e as s ump t i on o f lo gno rma l i ty
b u t r e t ain the as s ump t i ons ( main t aine d th rough ou t th e
pap e r ) tha t
for
,
a
/
fi
hg
ve n
gi
th e
'
o
ne - p er i
j o int d i s t r ibu t i ons id en t i c a l th r ough t ime
r e turns
Z
,
j
(
,
h ;h)
have
and the vec t or
,
i s d i s t r ib u t ed inde p end en t ly o f
for
s>
h
.
By d e f ini t i on
j
th e r e tu rn on the
,
s e cu r i ty ove r a
t
ime p e r i od
o f leng th T wi l l b e
T
H
J
Le t
X (k , h )
j
J
k= 1
Th en
!
l
l
,
a g i ven h
fo r
the { X
,
indep end en t ly and i den t i c a l ly di s t r ib u t e d w i th non— c ent r a l momen t s
o,
1
J
J
k
1,
2,
-
and momen t g e nera t ing f u n c t i on
w (1 ; h)
j
I
x
i
E{
De f ine th e
n o n—
l
j
j
(l ; h )
d
j
(h ) h
z
v (h ) h
(2 ; h )
f
re t u rn p e r p er i od
,
1}
i
2
1]
j
i
h)
J
R
(h )
o
by
,
M
a
j
c en t ra l momen t s o f th e ra t e
M
wher e
[Z
11
}
1
,
3
,
4
2
,
2
i s th e expe c t e d r a t e o f re turn p e r uni t t ime and v ( h ) i s
th e va ri ance o f the r a t e
j
o
f
re turn p er uni t t ime
.
The s e co nd a u tho r [ 4 3 ] has demon
a t ed tha t i f the moment s
h ) s a t i s fy
2
k
w*
th en
(t
0)
o (h )
g (h )
0 (h )
g (h )
o (h )
,
"
wh e re
O
"
and
"
0
"
are de f ined by
i f ( g / h ) i s b ound ed f o r a ll h
l im( g / h )
if
O
0
h+ 0
Th u s
,
i f th e d is t r ib u t i ons o f re tu rns s a t i s fy
o us -
1L
,
O,
fo r
6
eve ry
0,
and the c on t i nu
t ime s o l u t i on wi l l b e a va l id a s ymp t o t i c s o lu t i on t o the d i s cr e t e
in t e rva l c as e
2
s u ch th a t I
0
the re ex i s t s an h
then
No t e th a t i f
.
(h )
l at i o n s h ip
and
a
G iv en t h a t
M
for
m
(
m
(l ; h )
3
j
; h)
.
,
th en
d
j
(h )
lima ( h ) wi l l b e f ini t e a s wi l l
,
,
(
S
i s s a t i s f ie d
; h)
,
E { 10 g
E{
M
j
by Tay l o r
(l
[Z
s e rie s
ay
f
j
z
J
( l ; h)
0 (h )
,
2
V;
l im V ;
we can de r ive a S imi la r
s a tis f ies
Name ly
and
o
(h )
,
f r om
re
2
(h )
I f we
d
e f ine
Y
j
(T ; h )
lo g [ Z
H
[
j
(
then f rom
we
have t h a t
N
Z
J
k= 1
wh er e N
J
T /h
E
,
and f r om the indep enden c e and i den t i ca l d i s t r ib u t io n o f th e
X ( ; h)
,
J
,
the
moment - g ene ra t ing fun c t i on o f Y j w i l l s a t i s fy
5
T h
Taking lo gs o f b o th s i de s o f
and us ing Tay l o r s e r ie s
we have
,
tha t
z
3
k= 0
Z
J
k— O
m
j
(1 ; h ) 1
o
(h ) ]
2
S ub s t i tu t i ng
u (h ) h
th e l imi t as
hx o
in
for
m
j
(1 ; h )
and
O
2
(h ) h
j
u
2
.
J
(h ) h
2
for
m
j
(2 ; h)
and t ak ing
we h ave tha t
l im
h+ 0
lu T
j
and th ere f o re
,
1 / 2o
2
j
Ti
2
,
is th e momen t — g ene ra t i ng func t i on
d i s t r ibu t ed rand om var i ab le wi th mean
u T
,
J
and var i anc e
2
0
,
J
fo r
T
.
a no rma l ly
Thu s
a
in wh a t i s e s s ent i a l ly a v
,
app l i c a t ion o f th e Cent r a l
L imi t the o rem we have s h own t h a t th e limi t d i s t r ib u t i on f o r Y
j
unde r th e p o s i t e d as s ump t i ons
h t end s t o z e r o i S / g aus s i an and hen ce the l imi t d i s t r ib u t i on f o r
as
,
,
,
,
w i l l b e log — no rma l fo r a l l f ini t e T
s u r ro ga t e l o g - no rma l
wi l l
for
,
f o r sma l le r and sma l le r h
,
Z
f i t t ed in the
Fu r the r f rom
.
f as h ion
P e a r s o n i an
b e in th e l imi t
,
t he
,
o
f
ea r l i e r s ec t ions
th e t rue l imi t d i s t r ib u t i on
1
I t i s s t ra igh t f o rwa rd t o S how th a t i f the d i s t r ibu t i on
the
Z
th e
"
j
(
j
j
(t ) E
t ime
l
s ing le - p e ri o d
However
w
W
j
)
,
,
2
,
,
,
s a t i s fy
r e t u rns
p or t f o l i o
fo r
e a ch o f
th en f o r b ounded
wi l l f o r ea ch t s at i s fy
,
u nles s the p o r t f o li o we i gh t s a re cons t ant th rough t ime
the r es u l t ing l imi t d i s t r ibu t i on f o r the p o r t f o l i o over f ini t e
w i l l no t b e log - no rma l
E s s en t i a l ly
o ne )
n
,
How r eas on ab le
o us —
,
is a
is
s et
.
wi l l b e s a t i s f i ed !
i t t o as s ume tha t
o f s u f f i c i en t c ondi t i ons
fo r
th e l imi t i ng c ont i nu
t ime s t o c has t i c p ro c e s s t o have a c ont inuous s amp le p a th (wi th p rob ab i l i ty
It
.
is
c los e ly re la t ed
to
the
"
lo ca l markov p r op er ty
"
o f d i s cr e t e
t ime s t o ch as t i c p r o c es s e s whi ch a llows only movemen t s t o ne i ghb o r ing s t a t es
in
one
p e r i od
the S imp le r andom wa lk )
.
A s omewh a t we ak e r su f f i c i en t c ond i t i on ( imp l i ed by
p
.
3 2 1]
i s th a t f o r ev ery
P rob { — 6
wh i ch c le ar ly ru l es ou t
0
O
X (
j
,
;h)
l
f or a l l k and S imi lar ly
o (h )
.
6}
l
o
(h )
j ump — typ e p r o ce s s e s s u ch a s th e P o i s s on
eas y t o S h ow f o r th e P o i s s on tha t
o (h )
,
11 ,
.
i s no t s a t i s f i ed b e caus e M
i s no t s a t i s f i e d s inc e P rob { — 6
I t is
(k ; h )
X
j
(
;h
)
In th e gene ra l cas e wh en
s
a t i s f i ed bu t th e d i s t r ib u t i o n
o f r e t u rns ar e no t c omp le t e ly indep enden t no r iden t i ca l ly d i s tr ibu t ed
th e
,
l imi t d i s t r i b u t i on
wi l l n o t b e lo g - no rma l
d i f f us i on p r o c e s s
Al th o u gh c e r t a in q u ad ra t i c s imp li f i ca t i ons s t i l l oc cur
.
th e s t r ong th e o rems o f t h e e a r li e r p ar t o f
o b t a in
b u t wi l l b e gener at e d by a
,
t
,
hi s s e c t i on wi l l no longe r
.
Th e a c cur acy o f the c on t in u ous s o lu t ion wi l l dep end
f o r reas onab le t rad ing int erva l s
,
whe th er
on
,
c omp ac t d i s t r i bu t i ons a r e an a c cur a t e
rep re s en t a t i on f o r as s e t r e t urns and whe the r
d i s t r ib u t i ons c an b e t aken t o b e inde p endent
,
.
fo r thes e int ervals
the
,
Examina t ion o f t ime s er i e s
fo r c ommon s t o c k re t urns S how s tha t skewne s s and h i gher —o rd er moment s t end
t o b e ne g l i g ib le re la t ive t o the f i rs t two momen t s f o r d a i ly
s er
va t i o n s
d a t a t e nd
,
wh i ch
is
c ons i s t en t wi th
as s ump t i ons
a l s o t o s h ow s ome ne ga t ive s er i a l c o r re la t i on
a b o u t two weeks
.
,
or
Howeve r
,
d a i ly
s i gni f i c an t
h i le th is f ind ing i s inc ons i s t en t wi th inde pend ence
W
.
ob
we ekly
fo r
,
th e s i z e o f the c o rre la t i on c o e f f i c i en t i s no t l ar g e and th e sh o r t — dur a t ion
o f th e c o r re la t i on s u g ge s ts a
re l at i on func t i on
.
He nc e
,
"
h i gh - s p eed o f ad j us tmen t
in the au t o - co r
wh i le we c oul d modi fy the c ont inuous ana lys i s
t o inc l u d e an O r n s t e in —Uh lenb e ck typ e p ro c es s t o cap ture thes e e f fe c t s
the res u l t s may no t d i f fe r much f rom the s t anda rd mod e l wh en emp ir i c a l
t im a t e s
o f th e c o rr e la t i on a re p lug g ed in
.
27
,
es
F OO TN O TE S
We th ank M B Go ldman f o r s c i en t i f i c as s i s t anc e
Ai d f rom the
Na t i ona l S c i enc e F o u nda t i on i s g ra t e f u l ly acknow led g ed
tha t de a l wi th
No t ab l e exc ep t i ons /
th e g ene ra l c as e c an b e f o u nd in th e works
o f A r row [ 1
Ro ths ch i ld and S t i g l i t z
and Samu e l s o n l4 0 1
Al ong wi th me an- var i anc e ana ly s i s the the o ry o f p o r t f o l i o
s e l e c t ion when th e d i s t r ib u t i ons are P ar e t o -Levy h a s b e en d e
F ama [ 8
and S amue ls on
ve l o p e d and t es t ed by Mand e lb r o t
.
.
.
.
,
,
[ 4H
.
Th e l i t e ra t u r e i s s o ext ens ive tha t we r e f er th e r ead e r t o the
b ib l io g raphy
Add i t i ona l re f er enc e s c an b e f ound in th e s urvey
a r t i c les by Fama [ 9
and Jens en [ 2 1 ] and th e b o ok by Sha rp e
.
S e e B la c k
,
J ens en
,
and S ch o le s [ 3
and Fr iend and B lume
S ee Hakans s on
C as s and S t i g l i t z [ 7
S h owed th a t th es e u t i l i ty func t i ons wer e
among th e f ew wh i ch s a t i s f i e d th e s ep ar a t i on p r op e r ty f or arb i
O th er a u th o rs [ 1 5 2 3 2 9 3 0 4 2 ]
t r a r y d i s t r ibu t i ons
have p revi ous ly made ex t ens iv e us e o f the s e func t i ons
.
,
,
,
,
.
Fo r
t
[ 29 ,
Se e
he f i r s t s e e S amue ls on
f o r th e las t s e e Ro s s
f o r th e s e cond
,
s ee
,
Mer t on
,
f o r examp le
,
,
Au c amp
[
2
S e e Hakans s on
S e e Markowi t z [ 2 7 ] and [ 2 8]
Ros s [ 3 6] p r ovid es a ra the r S imp le examp l e wh e r e th e Le land r e s ul t
d oe s no t ob t a in
.
H
co u ld b e us ed t o rank p o r t f o l i os b e c aus e i t p rovid es a
then
1
c om
p le t e and t rans i t ive o rd e r ing
No t e : i f H li JM<
l /H
Howeve r H i (
n 1<
1
(
)
i
j
.
.
.
Se e
Go ldman [
,
1 3]
,
.
j
j
f o r s ome examp le s
.
Go ldman [ 1 3] d e r ives a S imi lar r es ul t f or a b ound ed u t i l i ty func t i on
Thus th e s e c ond au th o r [ 4 4 p 2 4 9 5 ] c o n c e d e d t o o much in h i s
c r i t i c i sm o f the g eome t r i c mean po l i cy wh en he s t a t ed tha t su ch a
p o l i cy wou ld as ymp t o t i c a l ly ou tp er f o rm any o th e r uni f o rm p o l i cy
f o r u t i l i ty fun c t i ons b ounde d f rom ab ove and b e low
,
,
.
.
S e e Me r t on [
33
ap p end ix B ]
fo r
a p ro o f
.
.
2
Foo tno t e s
The nea tne s s o f th e heu r i s t i c a rgum
en t s and th e a t t r ac t ivenes s
o f th e re s u l t s h a s led a numb e r o f au th or s t o a s s er t and / o r
c on j ec t u r e t h e i r t ru t h
Se e S amue ls on [ 4 4
p 2 4 9 6 ] w h e r e u(w )
"
2
and 0 (w ) ar e mi s le ad ing ly s a id t o b e as ymp t o t i c a l ly s u f f i c ien t
"
m
p a r ame t er s
an a s s er t i o n n o t c o rr e c t f o r T
bu t ra th er f o r
in cr eas ing s ub d i v i s i ons o f t ime p e r i o d s l ead ing t o d i f fus i on
typ e S t o c h as t i c p ro ce s s e s
.
.
,
,
i
.
To ap p ly th e Cen t r a l —Limi t th eo rem c o rr ec t ly the cho i c e f or { w }
mu s t b e su c h th at Z [ t ; w ] i s a p o s i t ive r andom var i a t e wi th i t s
l o gar i t hm we l l d e f ined
O ur la t e r d i s c u s s ion o f op t ima l p o l i c i es
i s unaf f e c t ed by th i s res t r i c t i on s inc e th e c la s s o f u t i l i ty
f u nc t i ons cons i de r ed wi l l ru l e o u t Z [ t ; w ] wi th a po s i t ive p roba
b i l i t y o f ru in
C f Hakans s on [ 1 7
p 86 8 ]
,
.
.
.
.
.
Wi t h re f e ren c e t o f o o tno t e 1 6 the c o ro l l ary i s f a ls e even i f we r é
s t r i c t the s e t o f u n i f o rm s t r a t eg i e s c ons id er ed t o th os e s uch tha t
P r ob { Z [ w ]
0}
O f o r al l f in i t e T
T
,
.
Th e a rgumen t i s tha t a c ons t an t t ime s a no rma l va r i a t e p lus a c on
s t ant i s a no rma l var i a t e
,
.
Th i s i s an imp o r t ant p o in t b e c au s e i t imp l i es tha t i f the c onj e c tu r e
*
=
l
tha t { w } { w } co u ld ho l d f o r the i s o - e la s t i c fami ly f o r la rg e T
th en i t wou ld ho l d f o r T sma l l o r T
l! I e
w (w ) and o (w )
wou l d b e su f f i c i ent p arame t e r s f o r th e p o r t f o l i o d ec i s i on f o r any
t ime ho ri z on
,
.
.
,
.
[ 45
4
B la ck —S cho le s
wa r rant p r i c ing i s b as ed s q uar e ly
l og —no rma l d e f ini t i ons i t i s inexa c t f or n o n - l og — n o rma l
o n exac t
f o r fur th er d i s c u s s i on
sur ro g at e re as oning
S ee Me r t on [ 3 3
Sinc e
,
.
.
S e e Me r t on [ 3 0
p
a s a fun c t i on o f th e
.
3 88 ]
fo r
m
i
and
S ee Mer t on [ 2 9
2
1
5
]
p
and
S ee Me r t on [ 3 0
p
.
O
an exp l i c i t exp re s s i on f o r
j
l 2
n
i
ij
’
30
,
,
,
p
.
a
and
2
0
.
,
3 88
.
Fur the r the l o g - no rmal i ty as s ump t i on c an b e weakene d and s t i l l
s ome s ep a ra t i on and e f f i c i ency c ond i t i ons wi l l ob t a in
See
Me r t on [ 3 2
,
.
I t i s no t r equ ir ed th a t th e d i s t r ib ut i on o f o n e — p e r io d r e tu rns wi th
t rad ing in t e rva l o f l eng th h l wi l l b e in the s ame f ami ly o f d i s
t r i b u t i o n s as th e one —p er i o d r e t urns w i th t rad ing int e rva l o f l eng t h
the d i s t r ibu t i ons ne ed n o t b e inf ini t e ly d iv i s ib l e
I e
in t ime
Howev er we d o r e q u i re s u f f i c i en t r egu lari ty that the
d is t r ibu t i on o f Xj ( T ; h ) i s in the doma in o f a t t ra c t i on o f th e no rma l
0 wi l l b e
d i s t r ibu t i on and hen c e th e l imi t di s t r ibu t i on a s h
in f in i t e ly d ivi s ib l e in t ime
.
.
.
,
,
,
,
.
Foo tno t e s
Z
s e r i es exp an s i on i s on ly va l i d
as sh own la t er in
f o r sma l l enough h
th is rang e wi th p rob ab i l ity o n e
.
27
.
Cf
.
Me r t on
30
p
.
4 01]
.
2
,
Z
However
wi l l b e in
.
,
3
Re f e rence s
14
.
15
.
16
.
Had a r J and W Ru s s e l l
Ame ri can E conom i c Revi ew
.
,
H akans s on
N
,
"
.
2
Ru le s f o r O rde r ing Un c e r t ain P ro s p e c t s
59
( Ma rch
,
,
,
"
,
,
O p t imal I nves tment and C ons ump t i on S t r at e g i es u nde r Ri s k
,
( J anuary ,
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