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Performance analysis of log-optimal portfolio

strategies with transaction costsMihly Ormos

a& Andrs Urbn

a

aDepartment of Finance , Budapest University of Technology and Economics ,

Muegyetem rkp. 9, Bld. R. 202, Budapest , HungaryPublished online: 04 Aug 2011.

To cite this article:Mihly Ormos & Andrs Urbn (2013) Performance analysis of log-optimal portfolio strategies withtransaction costs, Quantitative Finance, 13:10, 1587-1597, DOI: 10.1080/14697688.2011.570368

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Performance analysis of log-optimal portfoliostrategies with transaction costs

MIHA L Y O RM OS * a nd A ND RA S U RB A N

D e p a rt m e n t o f F i n an c e , B u d a p e s t U n i v e r s i ty o f T e c h no l o gy a n d E c o no m i cs ,

M u eg y et e m r k p. 9 , B ld . R . 2 0 2, B u da p es t , H un g ar y

(Received 4 July 2009; in final form 7 March 2011)

I n t h i s p a p e r w e i n t r od u c e a n e m p ir i c a l a p p r o x im a t io n o f t h e l o g - op t i ma l i n v es t m e nt s t r a te g yt h at g u ar a nt e es a n a l mo s t o p ti ma l g r ow t h r a te o f i n ve s tm e nt s . T h e p r op o se d s t ra t eg y a l soc o n s i d e r s t h e e f f e c t s o f p o r t f o l i o r e a r r a n g e m e n t c o s t s o n g r o w t h o p t i m a l i t y a n d r e c o m m e n d s as u bo p ti ma l p o rt f ol io f o r d i sc r et e i n ve s tm en t p e ri o ds . W e d o n o t a s su me a n y p a ra m et r ics t r u c t u r e f o r t h e m a r k e t p r o c e s s , o n l y a f i r s t - o r d e r M a r k o v p r o p e r t y . T h e m o d e l i n t r o d u c e d i sb a s e d o n k e r n el - b as e d a g e n ts ( e x pe r t s ) a p p r ox i m at i o n o f t h e m a x im u m t h e or e t i ca l g r o w thr a t e w i t h t r a n s ac t i on c o s t s. A l t ho u g h t h e o p t im a l s o l u t i on i s t h e or e t i ca l l y a c o m pl e x B e l lm a np r og r am mi n g p r ob l em , o u r s u bo p ti ma l e m pi r ic a l r e su lt a p pe a rs t o b e a t tr a ct i ve f o r D owJ o n e s 3 0 s h a r e s. T h e p a p er p r e s en t s a p e r f or m a nc e a n a l y s i s w h e r e t h e r e t u rn o f t h e e m p ir i c a ll o g - op t i ma l p o r t fo l i o i s c o m pa r e d w i t h p a s s i v e p o r tf o l io c o u n te r p a r ts c o m pi l e d f r o m s i m il a rc om po ne nt s u si ng t he C AP M, t he t hr ee -f ac to r m od el a nd t he f ou r- fa ct or m od el . T hep ro po se d m et ho ds , i n t he p re se nc e o f t ra ns ac ti on c os ts , p ro vi de a s ig ni fi ca nt p os it iv ea b n o rm a l r e t u r n c o m pa r e d w i t h t h e p r e c e d i ng e q u il i b r iu m m o d el s , a n d i s e v e n a s u r v i vo r s h ipb i a s -f r e e s e t u p.

Keywords: A g e nt b a s e d m o d el l i ng ; P o r tf o l io o p t im i z at i o n; T r a n sa c t i on c o s ts ; M u l ti - f a ct o rm o d el s ; L e a r ni n g i n f i n a nc i a l m o de l s ; P e r f or m a n ce e v a l ua t i o n

JEL Classification: C 1, C 4, C 5, C 14 , C 4 4 , C 5 1, G 1, G 11 , G 1 2

1. Introduction

S ev er al r ec en t s tu di es ( Gy o rfi et al. 2 00 6, 2 00 7) h av e

s h o w n t h a t a n e m p ir i c a l, l o g - op t i ma l i n v e st m e n t s t r a t e g y

r es ul ts i n a h ig he r r et ur n t ha n t he r et ur n o f t he s t oc k i n

t he p or t fo li o w it h t he h ig he st r et ur n. B ot h t he m at he -m at ic al p ro of a nd t he e mp ir ic al i nv es ti ga ti on , a mo ng

o th er b ou nd ar y c on di ti on s, a ss um e t ha t t he re i s n o

t ra ns ac ti on c os t. T hi s s up po si ti on i s a cc ep ta bl e f or

m od el f or mu la ti on , a s c an b e s ee n i n t he c as e o f, f or

e x am pl e , S h ar p e ( 1 96 4 ) a n d M e rt o n ( 1 96 9 , 1 9 73 ) , w h il e

t he c os t o f t r an sa ct io ns o n t he c ap it al m ar ke t i s i nc om -

p a ra b ly s m al le r t h an o n a n y o t he r m a rk e t i n ve s ti g at e d i n

e co no mi cs . H ow ev er , t ra di ng s tr at eg ie s t ha t u se l ar ge

n u mb e rs o f b i ds a n d a s k s c a n r e s ul t i n v e ry l a rg e v a l u es .

T h e l o g - op t i ma l p o r t f ol i o s t r a t eg y r e b a la n c e s t h e p o r t f o-

l i o i n e a ch p e ri o d, t h us n e gl e ct i ng t h e c o st a c co m pa n yi n g

t h e t r a d e s, a n d c a n r e su l t i n l a rg e e r ro r s i n t h e c a l c ul a te d

r e tu r ns . S i mi l ar l y t o M a gi l l a n d C o ns t an t in i de s ( 1 97 6 )

a n d C o ns t an t in i de s ( 1 99 7 ), w e i n tr o du c e p r op o rt i on a l

t r a n s a c t i o n c o s t s w h e n r e v i s i t i n g t h e l o g - o p t i m a l p o r t f o l i o

s t ra t eg y . I n t h e t h eo r y o f l o g- o pt i ma l s t ra t eg i es , t h e g o al

o f t he i nv es to r i s t o m ax im iz e h is w ea lt h i n t he l on g r un

w i t ho u t k n o wi n g t h e u n d e rl y i n g d i s t r ib u t i on g e n e r at i n g

t h e s t oc k p r ic e s. U n de r t h is a s su m pt i on , t h e a s ym p to t icr at e o f g ro wt h h as a w el l- de fi ne d m ax im um t ha t c an b e

a c h ie v e d i n f u l l k n o w l e dg e o f t h e u n d e r l y i ng d i s t r ib u t i on

g e n e ra t e d b y t h e s t o c k r e t u r ns . W i t h ou t t r a n sa c t i on c o s t s

t h e r e e x i s t e m p i r i c al l o g - op t i ma l p o r t f ol i o s t r a t eg i e s t h a t

a r e a b le t o a c hi e ve t h e t h eo r et i ca l m a xi mu m g r ow t h r a te

(Gyo rfi et al. 2 00 6, 2 00 8) . W e p re se nt a m od el u si ng

k er ne l- ba se d a ge nt s ( ex pe rt s) t ha t a pp ro xi ma te s t he

m a x im u m t h e o r e t i c al g r o w th r a t e w i t h t r a n s a c t io n c o s t s .

A lt h ou g h t h e o p ti ma l s o lu t io n i s t h eo r et i ca l ly a c o mp l ex

B e ll m an p r og r am mi n g p r ob l em , o u r e m pi r ic a l r e su l t i s

attracti ve.

T h e p r o p os e d i n v e st m e n t s t r a t e g y c o n s i d e r s t h e e f f e c ts

o f p o rt f ol i o r e ar r an g em e nt c o st s o n g r ow t h o p ti m al it y

a nd r e co mm en ds a s ub op ti ma l p or tf ol io f or d is cr et e

i nv es t me nt p er io ds . W e d o n ot a ss um e a ny p ar am et r ic*Correspondi ng author. Em ai l : orm os@fi nance. bm e. hu

2 01 T a yl o r & F r an c is

Quantitative Finance, 2013

Vol. 13, No. 10, 15871597, http://dx .doi.or g /1 0.1 0 8 0 /1 4 69 7 6 8 8 .2 0 11 .5 7 0 3 6 8

3

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s tr uc tu re f or t he m ar ke t p ro ce ss , o nl y a f ir st -o rd er

M a rk o v p r op e rt y . A l th o ug h t h e p r op o se d m e th o d g i ve s

o n ly a s u bo p ti ma l s o lu t io n , o u r n o ve l a p pr o ac h i s a b le t o

c a pt u re r i sk f a ct o rs t h at t h e c l as s ic a l e q ui li b ri u m m o de l s

do no t c ov er , e ve n in t he p re se nc e o f p ro po rt ion al

t r a n sa c t i on c o s t s .

T h er e a r e f e w p a pe r s d e al i ng w i th t h e f o rm u la t io n o f

g ro wt h o pt im al i nv es tm en t w it h t ra ns ac ti on c os t s i n a

d i sc r et e t i me s e tt i ng . I y en g ar a n d C ov e r ( 2 00 0 ) f o rm u -

l a te d t h e p r ob l em o f h o rs e r a ce m a rk e ts , w h er e , i n e v er y

m ar k et p e ri o d, o n e o f t h e h o r s es ( a ss e ts ) p a ys o f f a n d a l l

t h e o t he r s p a y n o th i ng , w it h p r op o rt i on a l t r an s ac t io n

c o st s , u s in g l o ng - ru n e x pe c te d a v er a ge r e wa r d c r it e r ia .

T h er e a r e r e su l ts f o r m or e g e ne r al m a rk e ts . B o br y k a n d

S t et t ne r ( 1 99 9 ) c o ns i de r ed t h e c a se o f p o rt f ol i o s e le c ti o n

w it h c on su mp ti on , w he n t he re a re t wo a ss e ts : a b an k

a cc ou nt a nd a s to ck . F ur th er mo re , l on g- ru n e xp ec te d

d i sc o un t ed r e wa r d a n d i .i . d. a s se t r e tu r ns w e re a s su me d .

I n t h e c a s e o f d i sc r et e t i me , t h e m o s t f a r- r ea c hi n g s t u d y

w a s t h a t o f S c h a f e r ( 2 0 0 2) , w h o c o n s id e r e d m a x im i z at i o no f t h e l o ng - ru n e x pe c te d g r ow t h r a te w it h s e ve r al a s se t s

a n d p r o p o r t i on a l t r a n s a c t i on c o s t s w h e n t h e a s s e t r e t u r ns

f o l lo w a s t a t i on a r y M a r k o v p r o c es s .

T he p er fo rm an ce m ea su re o f a ny a ct iv e o r p as si ve

p o rt f ol i o s t r a t eg y , o r s i mp l y m u t u al f u nd s o r a n y o t he r

a s se t s, h a s a w e ll - de f in e d w a y i n t h e w o rl d o f e q ui l ib r iu m

a ss et p ri ci ng . I t i s n ot s uf f ic ie nt t o s ta te t ha t a s tr at eg y

r es ul te d i n t he h ig he st h ig h r et ur n, b ut o ne m us t a ls o

c on si de r t he r is k o f t he r et ur n. I n t he c as e o f e mp ir ic al

s t u d ie s o n t h e l o g -o p t i ma l p o r t f ol i o s t r a t e g y , t h e s e r e s u lt s

a n d c o m pa r i s on s a r e s t i l l m i s s i n g.

T h is p a pe r u n am bi g uo u sl y a n sw e rs t h e q u es t io n o f t h es u cc e ss u s in g t h e l o g- o pt i ma l p o rt f ol i o s t r at e gy . I n o u r

e m p ir i c a l i n v e s t ig a t i on u s i n g t h e c o m po n e n ts o f t h e D o w

J on es I nd us tr ia l A ve ra ge I nd ex , f ro m w hi ch a d ai ly

r e ba l an c ed l o g- o pt i ma l p o rt f ol i o i s c o mp i le d , a 1 5 -y e ar

p e ri o d i s i n ve s ti g at e d i n a n e f fi c ie n t c a pi t al m a rk e t. F o r

p e rf o rm a nc e m e as u re m en t p u rp o se s w e u s e t h e s i mp l e

C a p i t a l A s s e t P r i c i n g M o d e l ( S h a r p e 1 9 6 4 ) , t h e F a m a a n d

F r e n ch ( 1 9 9 3) t h r e e -f a c t o r m o d e l , a n d t h e C a r ha r t ( 1 9 9 7)

f o u r -f a c t o r m o d e l , w h e r e w e a r e s e a r ch i n g f o r s i g n if i c a nt

J e ns e n a l ph a s. O b vi o us l y, i n t h e e m pi r ic a l s t ud y w e u s e

t h e l o g -o p t i ma l p o r t fo l i o s t r a t eg y i n t r od u c e d p r e v io u s ly

w it h t r an s ac t io n c o st s . W e p e rf o rm t e s ts o n t h re e d i st i nc t

p o rt f ol i os : ( i ) a p o rt f ol i o c o nt a in i ng t h e c o mp o ne n ts o f the DJIA observed in December 2005, whi ch is a

s u r v iv o r s hi p - b ia s e d s e t u p ; ( i i ) a p o r t fo l i o c o n t ai n i ng t h e

c om po ne nt s o f t he D JI A i n J an ua ry 1 99 1, w hi ch i s a

s ur vi vo rs hi p- bi as -f re e s et up ; a nd ( ii i) a p or tf ol io t ha t

t ra ck s t he a ct ua l c om po ne nt s o f t he D JI A. E ac h t es t

p o r t f o l i o c o n t a i n s 3 0 s h a r e s . I n a d d i t i o n , w e c o m p a r e o u r

r e s u l t s f o r t h e l o g - o p t i m a l p o r t f o l i o s t r a t e g y w i t h t h e b u y -

a n d- h ol d ( p as s iv e ) s t ra t eg y . F r om t h e r e su l ts , o n e c a n s e e

t ha t t hi s n ov el t ra di ng s tr at eg y r es ul ts i n s ig ni fi ca nt

p o s it i v e a l p h a s , s i g n i f ic a n t ly l a r g er t h a n t h o s e o f t h e b u y -

a n d - ho l d s t r a t eg y .

T h e s e r e s u l t s s t r e n g t he n o u r b e l i e f t h a t t h e l o g - o p t i ma lp o rt f ol i o s t ra t eg y y i el d s h i gh e r r e tu r ns t h an t h e e q ui l ib -

r iu m s tr at eg ie s, w hi ch c an b e e xp la in ed f ro m t hr ee

d ir ec ti on s. O ne c ou ld s ay t ha t t he e qu il ib ri um m od el s

u se d t o e xp la in t he r et ur n o f o ur d yn am ic p or tf ol io

c o n t a i n o n e o r m o r e m i s s i n g e x p l a n a t o r y v a r i a b l e s , w h i c h

c au se s ou r r es ul ts t o s ho w h ig he r r et ur ns t han t he

e q ui li b ri u m. T h e o t he r a t te m pt t o e x pl a in t h e a b no r ma l

r e tu r n i s t h at t h e s a m p le ( s t oc k s a n d / or t h e p e ri o d) u s ed

i n o u r e x p e r i m e n t i s n o t a d e q u a t e , a n d w e a r e t a l k i n g o n l y

a bo ut a n a no ma ly t ha t d is ap pe ar s i f a no th er d at as et i s

u se d. T hi s e xp la na ti on s ee ms t o b e w ea k b ec au se o th er

s tu di es ( e. g. , G yo rfi et al. ( 20 06 , 2 00 7, 2 00 8) ) u si ng

d if f er en t s to ck s i n d if fe re nt t im e i nt er va ls g iv e r es ul ts

s i mi la r t o o u rs , h o we v er t h es e s t ud i es n e gl e ct t h e t r an s -

a c ti o n c o s t . T h e t h i r d a r gu m en t s t em s f r om t h e t h eo r et -

i c al d i ff e re n ce o f t h e o n e- p er i od e q ui l ib r iu m m o de l s a n d

m u l ti - p e r io d i n v e st m e n t s t r a t eg i e s .

2. Log-optimal strategies with transaction costs

2.1. Trading model

T h e m o d e l s t o c k m a r k e t i n v e s t i g a t e d i n t h i s s e c t i o n i s t h a t

c o n s id e r e d, a m o ng o t h e rs , b y B r e im a n ( 1 9 6 1 ) a n d A l g o e t

a nd C ov er ( 19 88 ). C on si de r a m ar ke t o f d a ss et s. A

market vector x x1 , . . . , xdT 2 Rd i s a v ec to r of d

n o n- n eg a ti v e n u mb e rs r e pr e se n ti n g p r ic e r e la t iv e s f o r a

g i v e n t r a d i n g p e r i o d . T h a t i s , t h ejt h c o m p o n e n tx(j) 0 of

x e x pr e ss e s t h e r a ti o o f t h e c o ns e cu t iv e c l os i ng p r ic e s o f

asset j. I n o th er w or ds ,x(j) i s t h e f a c t or b y w h ic h c a pi t al

i n ve s te d i n t h ejt h a s se t g r ow s d u ri n g t h e t r ad i ng p e ri o d.

T he i nv es to r i s a ll ow ed t o d iv er s if y h is c ap it al a t t he

b e gi n ni n g o f e a ch t r ad i ng p e ri o d a c co r di n g t o a p o rt f ol i o

vector b (b(1), . . . , b(d))T. The jt h c o mp o ne n t b(j) of b

d e n o t e s t h e p r o p o r t i o n o f t h e i n v e s t o r s c a p i t a l i n v e s t e d i n

asset j. Th rou gho ut t he p ap er we a ss ume t ha t t he

p o rt f ol io v e ct o r b h a s n o n- n eg a ti v e c o mp o ne n ts w i thPdj1b

j 1 . T he f ac t t ha tPd

j1bj 1 m ea ns t ha t t he

i n v e st m e n t s t r a t e g y i s s e l f - fi n a n ci n g a n d t h e c o n s um p t io n

o f c a pi t al i s e x cl u de d . T h e n o n- n eg a ti v it y o f t h e c o mp o -

n e n t s o f b m e a n s t h a t s h o r t s e l l i n g s h a r e s o n m a r g i n i s n o t

p e rm it t e d. L e t S0 d en o te t h e i n ve s t or s i n it i al c a pi t al .

T h e n a t t h e e n d o f t h e t r a d i n g p e r i o d t h e i n v e s t o r s w e a l t h

becomes

S1 S0 Xd

j1

bj

1 x

j

1 S0hb1 , x1i,

where h, i d e no t e s t h e i n n er p r o d uc t .

T h e e v o l u t i o n o f t h e m a r k e t i n t i m e i s r e p r e s e n t e d b y a

s e q u e nc e o f m a r ke t v e c t o rs x 1 , x2 , . . .2 Rd , w h e r e t h e jth

component xji of xi d e no t es t h e a m ou n t o b ta i ne d a f te r

i n ve s ti n g a u n it c a pi t al i n t h e jt h a ss et i n t he it h t r ad i ng

p e r i od . F o r j i w e a b br e vi a te b yxij t h e a r ra y o f m a rk e t

v e ct o rs (xj, . . . , xi) a nd d en ot e b y Dd t he s im pl ex o f a ll

vectors b2 Rd w i t h n o n - ne g a t iv e c o m po n e n t s s u m mi n g

up to one. An investment strategy i s a s eq ue nc e B of

functi ons

bi :Rdi1 ! Dd , i 1 , 2 , . . . ,

s o t h a tbixi11 d e n ot e s t h e p o rt f ol i o v e c t o r c h os e n b y t h e

i n v e s t o r i n t h e it h t r a d i n g p e r i o d , u p o n o b s e r v i n g t h e p a s t

2 M. Ormos and A. Urban1588

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b e h a v i o r o f t h e m a r k e t . W e w r i t e bxi11 bixi11 t o e a s e

t h e n o ta t io n . T h er e fo r e, w e o b ta i n b y i n du c ti o n t h at

Sn S0Yni1

hbxi11 , xii S0ePn

i1 lnhbxi1

1 , xii S0 e

nWnB ,

where Wn(B) d en ot es t he average growth rate o f the

i n v e st m e n t s t r a t eg y B fbng1

n1

:

WnB 1

nl n Sn

1

n

Xni1

l nhbxi11 , xii:

Remark 1: W i t h ou t t r a n sa c t i on c o s t s , t h e f u n d a m e nt a l

l i m i t s , d e t e r m i n e d b y A l g o e t a n d C o v e r ( 1 9 8 8 ) , r e v e a l t h a t

o n s ta ti on ar y a nd e rg od ic m ar ke ts t he s o- ca ll ed log-

optimum portfolio B {b() } i s t h e b e s t p o ss i bl e c h oi c e.

M o re p r ec i se l y, i n t r ad i ng p e ri o dn l et b() b e s u c h t h a t

bnXn11 a r g m a x

b

Efl nhbXn11 , Xni jXn11 g: 2:1

If Sn SnB d en ot es t he c ap it al a ch ie ve d b y a l og -o p t i mu m p o r t fo l i o s t r a t eg y B, a ft e r n t r a d in g p e r i od s ,

t he n f or a ny o th er i nv es t me nt s tr at eg y B w i t h c a pi t al

Sn Sn(B) a nd f or a ny s ta ti on ar y a nd e rg od ic r et ur n

process fXng11 ,

l i m i n f n!1

1

nl n

SnSn

0 a lm os t su re ly

and

l i mn!1

1

nl n Sn W

a l mo s t s u re l y ,

where

W E maxb

Efl nhbX11, X0 i jX11g

i s t h e m a xi mu m p o ss i bl e g r ow t h r a te o f a n y i n ve s t me n t

s t r a t e g y . M o r e o v e r , G y o r f i a n d S c h a f e r ( 2 0 0 3 ) a n d G y o rfi

et al. ( 2 0 0 6 , 2 0 0 7 ) c o n s t ru c t e d e m p ir i c a l ( d a t a- d e p en d e n t)

l o g - op t i mu m s t r a t eg i e s f o r u n k no w n d i s t r ib u t i on s . N o t e

t h at , f o r a f i rs t -o r de r M a rk o vi a n r e tu r n p r oc e ss ,

bnXn11 b

nXn1 a rg m a x

b

Efl nhbXn1, Xni jXn1g:

T o m ak e t he a na ly si s f ea si bl e, s om e s im pl if yi ng

a s su m pt i on s a r e u s ed t h at n e ed t o b e t a ke n i n to a c co u nt

i n t he u su al m od el o f l og -o pt im al p or tf ol io t he or y.

A s s u me t h a t

. t h e a s s e t s a r e a r b i t r a r il y d i v is i b l e,

. t h e a s se t s a r e a v ai l ab l e i n u n li mi t ed q u an t it i es

a t t h e c u r r e n t p r i c e i n a n y g i v e n t r a d i n g p e r i o d ,

and

. t h e b e h av i or o f t h e m a rk e t i s n o t a f f ec t e d b y t h e

a c ti o ns o f t h e i n ve s t or u s in g t h e s t r at e gy u n de r

i nvesti gati on.

2.2. Investment with transaction costs

T h e t r an s ac t io n c o st p r ob l em p r of i te d f r om t h e f o rm ul a -

t i on o f I y en g ar a n d C o v e r ( 2 00 0 ), S c ha f e r ( 2 00 2 ), G yo rfi

a n d V a j d a ( 2 00 8 ) a n d S t e t t ne r ( 2 00 9 ). L e tSn d e n ot e t h e

w ea lt h a t t he e nd o f m ar ke t d ay n, n 0 , 1 , 2 , . . . , w h er e ,

w it h ou t l o ss o f g e ne r al it y , l e t t h e i n ve s to r s i n it i al c a pi t al

S0 b e o ne d ol la r. A t t he b eg in ni ng o f a n ew m ar ke t d ay

n 1 , t he i nv es to r s et s u p h is n ew p or tf ol io , i .e . p ur -

c h a s e s / s e l l s s h a r e s a c c o r d i n g t o t h e a c t u a l p o r t f o l i o v e c t o r

bn1. D u ri n g t h i s r e ar r an g em e nt , h e h a s t o p a y t r an s ac -

t i o n c o s t s , t h e r e f o r e a t t h e b e g i n n i n g o f a n e w m a r k e t d a y

n 1 t h e n e t w e a l t h Nn i n t h e p o rt f ol i obn1 i s n o t l a r g er

t ha n t he g ro ss Sn. U si ng t he a bo ve n ot at io n t he g ro ss

weal th Sn a t t he e nd o f m ar ke t d ay n i s

Sn Nn1hbn , xni: 2:2

T h e r a te o f p r op o rt i on a l t r an s ac t io n c o st s ( c om mi s si o n

f ac to rs ) l ev ie d o n o ne a ss et i s d en ot ed b y 05cs51 a nd

05cp51 , i . e . t h e s a le o f o n e d o ll a r s w o r t h o f a s se ti nets

o n l y 1 cs d ol l a r s, a n d s i m i l a rl y w e t a k e i n t o a c c o u n t t h e

p ur ch as e o f a n a ss et i n s uc h a w ay t ha t t he p ur ch as e o f

o n e d o l l ar s w o rt h o f a s se ti c o s t s a n e x t r a cp d o l l ar s . W e

c o n s i d e r t h e s p e c i a l c a s e w h e n t h e r a t e o f c o s t s i s c o n s t a n to ve r t he a ss et s. I n t he c ur re nt a pp ro ac h t he i nv es to r s

w e a l t h i s a l w a y s i n v e s t e d i n s e c u r i t i e s , t h e r e f o r e t h e s a l e o f

a s se t s i s a l wa y s f o ll o we d b y p u rc h as i ng o t he r s.

L et u s c al cu la te t he t ra ns ac t io n c os t t o b e p ai d w he n

s el ec ti ng t he p or tf ol io bn1. B ef or e r ea rr an gi ng t he

c ap it al , f or t he jt h a ss et t he re is bjn xjn Nn1 dol l ars,

w hi le a ft er r ea rr an gi ng w e n ee d bjn1 Nn d o ll ar s. I f

bjn xjn Nn1 4b

jn1Nn , then we have to sell and the

t r an s ac t io n c o st f o r t h ejt h a ss et i s

cs bjn x

jn Nn1 b

jn1Nn,

o t he r wi s e n o t r an s ac t io n i s m a de .Let x d e no t e t h e p o si t iv e p a rt o f x . T hu s, t he s um o f

c o st s o n t h e s a l e o f a p pr o pr i at e a s se t s i sXdj1

csbjn x

jn Nn1 b

jn1Nn

: 2:3

T h e m o n e y s t e m m i n g f r o m t h e s a l e i s u s e d t o a u g m e n t t h e

i nv es to r s p or tf ol io w it h n ew a ss et s. I n t hi s c as e t he

i nc om e f ro m t he s al e r ed uc ed b y t he t ra ns ac ti on c os t

c ov er s t he va lue o f t he s ha re s t o p ur cha se a nd t he

c o l la t e r a l c o s t , i . e .

Xdj1

bjn xjn Nn1 bjn1Nn csbjn xjn Nn1 bjn1NnXdj1

b

jn1Nn b

jn x

jn Nn1

cp bjn1Nn b

jn x

jn Nn1

:

I n e q ui v al e nt f o rm w e h a ve

1csXdj1

bjn xjn Nn1 b

jn1 Nn

1cp

Xd

j1

bjn1Nn b

jn x

jn Nn1

: 2:4

U s i ng t h e i d e nt i t y

ab a b ba ,

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e q ua t io n ( 2 .4 ) h a s t h e e q ui v al e nt f o rm

1csXdj1

bjn xjn Nn1 b

jn1Nn

1cp(Xd

j1

bjn1Nn b

jn x

jn Nn1

Xdj1

bjn xjn Nn1 b

jn1Nn

):

Si ncePd

j1bjn1Nn Nn and

Pdj1b

jn x

jn Nn1 Sn ,

cp cs

1cp

Xdj1

bjn xjn Nn1 b

jn1Nn

Sn Nn: 2:5

D iv i de b o th s i de s b ySn a n d i n tr o du c e t h e r a ti o

wn

Nn

Sn,

w h er e 05wn 1 . U s in g ( 2 .2 ) w e o b ta i n

cp cs

1cp

Xdj1

bjn xjn

hbn , xnib

jn1wn

1wn: 2:6

Remark 2: E quat ion (2. 6) i s used i n the sequel .

E xa mi ni ng t hi s c os t e qu at io n, i t t ur ns o ut t ha t, f or

a r b it r a r y p o r t fo l i o v e c t o rs bn and bn1 a n d r e tu r n v e ct o r

xn, t h er e e x is t s a u n iq u e c o s t f a ct o rwn 2 [ 0 , 1 ) . T h e v a l ue

o f c os t f ac to r wn o n d ay n i s d e te r mi n ed b y p o rt f ol io

vectors bn and bn1 a s w el l a s b y r et ur n v ec to rxn, i .e .

wn wbn , bn1 , xn,

f o r s o m e f u n c t i o nw. T h a t i s , t u n i n g o f wn i s a l l o w e d a t t h e

e nd o f t ra di ng p er io d n, a ft er xn i s o bs er ve d a nd bn i s

c al cu la te d a t t he e nd o f t ra di ng p er io d n 1. bn1 i s a

t a r g e t v a r i a b l e t h a t i s t u n e d i n p a r a l l e l w i t hwn. I f w e w a n t

t o r ea rr an ge o ur p or t fo li o s ub st an ti al ly , t he n o ur n et

w e al t h d e cr e as e s c o ns i de r ab l y, h o we v er i t r e ma i ns p o si -

t iv e. N ot e a ls o t ha t t he c os t d oe s n ot r es tr ic t t he s et o f

n ew p or tf ol io v ec to rs , i .e . t he o pt im iz at io n a lg or it hm

s e a r ch e s f o r o p t i ma l v e c t o rbn1 w i t hi n t h e w h o le s i m pl e x

Dd. T h e m a x i m u m v a l u e o f wn i s 1 w h e n n o t r a n s a c t i o n i sc ar ri ed o ut . I t r ea ch es i ts m in im um i f t he d if fe re nc e

b e t w e e n t h e g r o s sSn a n d t h e n e tNn r e a c h e s i t s m a x i m u m ,

i n t h e c a s e w h e r e t h e s u m i n ( 2 . 5 ) i s e q u a l t o 1 . T h u s , t h e

v al ue o f t he c os t f ac t or f al ls i n t he r an ge

1cs

1cp wn 1:

A s su m e a g e ne r al c o nd i ti o n t h at c cp cs, t h e r ef o r e

1c

1c wn 1,

i n m os t c a se s . S i nc e wn c a n no t b e f o rm ul a te d i n e x pl i ci t

f or m, i t c an on ly b e c al cul at ed n ume ric all y us in g

e q u a ti o n ( 2 . 6 ).

S t a r t in g w i t h i n i t i a l w e a lt hS0 1 a n dw0 1 , w e a l t hSna t c l o se o f t h e nt h m a rk e t d a y b e co m es

Sn Nn1hbn , xni wn1Sn1hbn , xni

Yni1

wbi1 , bi, xi1hbi, xii:

I n t r o du c e t h e n o t a t io n

gbi1 , bi, xi1 , xi l nwbi1 , bi, xi1hbi, xii,

t h en t h e a v er a ge g r ow t h r a te b e co me s

1

nl n Sn

1

n

Xni1

l nwbi1 , bi, xi1hbi, xii

1

n

Xni1

gbi1 , bi, xi1 , xi: 2:7

O ur a im i s t o m ax im iz e t hi s a ve ra ge g ro wt h r at e. I n t he

sequel ,xi w i l l b e a r a n d o m v a r i a b l e a n d i s d e n o t e d b y Xi.

L e t u s u s e t h e d e co mp o si t io n

1

nl n Sn In Jn , 2:8

where

In1

n

Xni1

gbi1 , bi, Xi1 , Xi Efgbi1 , bi, Xi1 , Xi j Xi11 g

and

Jn 1

nXn

i1

Efgbi1 , bi, Xi1 , Xi jXi11 g:

In i s t h e a v e r a g e o f t h e m a r t i n g a l e d i f f e r e n c e s . U n d e r m i l d

c o nd i ti o ns o n t h e s u pp o rt o f t h e d i st r ib u ti o n o f X,g(bi1,

bi, Xi1, Xi) i s b ou nd ed , t he re fo re In is a n a ve rag e o f

b o u n d e d m a r t i n g a l e d i f f e r e n c e s t h a t c o n v e r g e t o 0 a l m o s t

surely, since, according to the Chow Theorem

( S t o u t 1 9 7 4 ),X1i1

Efgbi1 , bi, Xi1 , Xi Efgbi1 , bi, Xi1 , Xi j Xi11 g

2g

i2

51

i m p li e s t h a t

In ! 0

a l mo s t s u re l y. T h us , t h e a s ym pt o ti c m a xi m iz a ti o n o f t h e

a ve ra ge g ro wt h r at e ( 1/n) l nSn i s e qui val ent t o t he

m a x im i z at i o n o f Jn.

I f t he m ar ke t p ro ce ss {Xi} i s a homogeneous and first-

order Markov process, t he n, f or a pp ro pr ia te p or t fo li o

s e l e c ti o n {bi} , w e h a v e

Efgbi1 , bi, Xi1 , Xi jXi11 g

Efl nwbi1 , bi, Xi1hbi, Xii jXi11 g

l n wbi1 , bi, Xi1 Efl nhbi, Xii jXi1

1 g l n wbi1 , bi, Xi1 Efl nhbi, Xii jbi, Xi1g

d ef

vbi1 , bi, Xi1,

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t h e r e f o r e m a x i m i z a t i o n o f t h e a v e r a g e g r o w t h r a t e ( 1 / n) l n

Sn i s a s y m pt o t i c al l y e q u i va l e nt t o t h e m a x im i z at i o n o f

Jn 1

n

Xni1

vbi1 , bi, Xi1: 2:9

T he f ir st -o rd er M ar ko v p ro pe rt y i mp li es t ha t, i n t he

a b o v e e q u a t i o n s a n d i n t h e e m p i r i c a l m e t h o d p r e s e n t e d i ns e c t i o n 2 . 4 , t h e c o n d i t i o n a l e x p e c t e d v a l u e o f t h e p o r t f o l i o

r e tu r n i s b a se d o n ly o n t h e i mm e di a te l y p r ec e di n g d a y s

r e tu r n. T im e -h o mo g en e it y i mp l ie s t h at E{l nhbi, Xii|bi,

Xi1} i s i n d e p e n d e n t o f i. F o r a m o re d e ta i le d d e sc r ip t io n

o f t he a pp li ca ti on o f t he M ar ko v p ro pe rt y i n e mp ir ic al

m e th o ds , s e e s e ct i on 2 . 4.

2.3. Suboptimal portfolio selection algorithm

A ss um in g a n h om og en eo us a nd f ir st -o rd er M ar ko v

p r oc e ss w e i n tr o du c e t h e s u bo p ti ma l s o lu t io n o f G yo rfi

a n d V a jd a ( 2 00 8 ) b y a o n e- s t ep o p ti mi z at i on a s f o ll o ws :put b1 {1/d, . . . , 1/d} a n d , f o r n41,

bnXn1 a r g m axb0 2Dd

vbn1Xn2, b0 , Xn1

a r g m axb0 2Dd

fl n wbn1Xn2, b0 , Xn1

Efl nhb0 , Xni jb0 , Xn1gg: 2:10

T hi s s ol ut io n i s s ub op ti ma l o nl y b ec au se t he m od el

p r o p o s e d o n l y g i v e s a n o p t i m a l p o r t f o l i o v e c t o r r e g a r d i n g

t h e n e xt t r ad i ng d a y a s o p po s ed t o a l o ng - ru n o p ti mi z a-

t io n, w hi ch , o n e ac h d ay , c on si de rs t he o pt im al it y

r e ga r di n g t h e i n fi n it e s e qu e nc e o f t h e f o ll o wi n g t r ad i ngp e r i o d s . I f t h e d i s t r i b u t i o n o f t h e r e t u r n p r o c e s s i s k n o w n ,

Gyo r f i a n d V a jd a ( 2 00 8 ) p r ov e d t h at , u n de r a n h o mo g e-

n eo us a nd f ir st -o rd er M ar ko v p ro ce ss , t he re e xi st s a

t h eo r et i ca l ly o p ti ma l s o lu t io n i n t h e p r es e nc e o f p r op o r-

t i o n a l t r a n s a c t i o n c o s t s , a l t h o u g h a n e f f i c i e n t a l g o r i t h m i s

a v a i la b l e o n l y f o r t h e s u b o pt i m al a p p r ox i m at i o n .

I n t he n ex t s ec ti on , t he p er fo rm an ce o f t he e mp ir ic al

a p pr o xi ma t io n f o r t h e s u bo p ti ma l m o de l i s i n ve s ti g at e d

o n U .S . s t oc k e x ch a ng e d a ta s e ri e s.

2.4. Empirical portfolio selectionI n o r de r t o c o ns t ru c tb, o n e h a s t o k n o w t h e c o n d i t i o n a l

d i s t ri b u t io n o f Xn gi ven Xn11 . T h e c l a s si c al M a rk o wi t z

m e a n v a r i an c e a p p r oa c h ( M a r ko w i t z 1 9 5 2 ) t o p o r t fo l i o

o p t i mi z a t io n f o r s i n g le - p e r io d i n v e st m e n t s e l e ct s p o r t fo l i o

b b y p e r f or m a n ce E{hb, Xni} a nd r is k V ar {hb, Xni} i n a

s u c h w a y t h a t o n l y t h e f i r s t a n d s e c o n d m o m e n t s o f Xn are

u s ed i n t h e c a lc u la t io n s ( F ra n ci s 1 9 80 ) . S i mi l ar l y, i f t h e

p r oc e ss {Xn} i s l o g- n or m al ly d i st r ib u te d , t h en a g ai n o n ly

the fi rst and second moments are needed i n the

d e r i va t i o ns ( S c h a f e r 2 0 02 ) . O t tu c sa k a nd V aj da ( 20 07 )

i nv es ti ga te d t he c on ne ct io n b et we en t he M ar ko wi tz

m ea n va ri an ce a pp ro ac h a nd t he l og -o pt im al c ho ic e.T he y f ou nd t ha t t he l og -o pt im al p or tf ol io s m ay b e

d es cr ib ed a s s pe ci al m ea n va ri an ce -b as ed p or tf ol io s

w i t h t i m e- v a r y in g r i s k -a v e r s io n f a c t o r.

T h e e m p ir i c a l k e r n e l - b as e d s t r a t eg i e s i n t r od u c e d b e l o w

esti mateb b a s e d o n p r e v i o u s m a r k e t d a t a . A l t h o u g h t h e y

w o rk w i th n o k n ow l ed g e o f t h e t h eo r et i ca l c o nd i ti o na l

d i st r ib u ti o n o f t h e m a rk e t p r oc e ss , t h ey p e rf o rm w e ll o n

f in an ci al t im e s er ie s. W e i nv es ti ga te t he ir e mp ir ic al

b e ha v io r i n t h e n e xt s e ct i on .

T he p re se nt ed e mp ir ic al m et ho ds a re b as ed o n t he

h om og en eo us a nd f ir st -o rd er M ar ko v p ro pe rt y m en -

t i on e d a b ov e . T h e f i rs t - or d er p r op e rt y i mp l ie s t h at t h e

i n tr o du c ed k e rn e l e s ti ma t or s o n ly u s e r e tu r ns f r om t h e

p r e c e d i n g t r a d i n g d a y s t o g i v e f o r e c a s t s o n f u t u r e r e t u r n s .

T h e t i me - ho m og e ne i ty a l lo w s u s t o b a se o u r e s ti ma t io n s

o n o bs er va ti on s f ro m t he p as t, s in ce , d ue t o t he h om o-

ge ne it y, t he s tat e t ran sit io ns pr ob abi lit y i s t ime

i ndependent.

F o r e m pi r ic a l e s ti m at i on l e t u s d e fi n e a n i n fi n it e a r ra y

o f e x p e r t s B() {b()() } , w h e r e i s a p o s i t i v e i n t e g e r . F o r

f i x e d p o s i ti v e i n t e ge r s, c h o o s e t h e r a d i u sr40 i n s u c h a

w a y t h at

l i m!1

r 0:

T h e n, f o rn41 , d e fi n e t h e e x pe r tb() a s f o ll o ws . L e tP nb e t h e l o ca t io n s o f m a tc h es

P n fi5n :kxi1 xn1 k rg,

where k k d en ot es t he E uc li de an n or m. D en ot e t he

k e r n el - b a se d s u b o pt i m a l ( s e e s e c t io n 2 . 3 ) p o r t f o l i o d e t e r -

m in e d b y e x pe r tb() f o r t r ad i ng d a yn as

b n a r g m axb0 2Dd

Xfi2P

n g

l nfwbn1 , b0 , xn1hb0 , xiig 2:11

i f t he p ro du ct i s n on -v oi d, a nd b0 (1/d, . . . , 1/d) o th er -

w i s e. T h u s ,b n s e e k s m a rk e t v e c t o rs s i m i l ar t o xn1, t h e n

d e te r mi n es t h e f i xe d p o rt f ol io t h at m a xi m iz e s t h e r e tu r n

o n m ar ke t v ec t or s f ol lo wi ng t he p re vi ou sl y s el ec te d

s imi la r ve ct or s. Th e s imil ar it y is me as ur ed b y t he

E u c li d e a n n o r m.

T he se e xp er ts a re mix ed a s f ollo ws : le t {q} be a

p r o b ab i l it y d i s t ri b u t io n o v e r t h e s e t o f p o s i ti v e i n t e g e r s

s uc h t hat , f or a ll , q40. If Sn(B()) i s the capital

a cc um ul at ed b y t he e le me nt ar y s tr at eg y B()

after np e ri o ds w h en s t ar t in g w i th a n i n it i al c a pi t al S0 1 , t h en ,

a f t e r p e r i od n, t h e i n ve s to r s c a pi t al b e co m es

Sn X

qSnB : 2:12

N ot e t h at ( 2 .1 2 ) i s a n o n- p ar a me t r ic e s ti ma t io n o f ( 2 .1 0 ).

A ll t he p ro po se d a lg or it hm s u se a n i nf in it e a rr ay o f

e x p e r t s . I n p r a c t i c e , w e t a k e a f i n i t e a r r a y o f s i z e L. I n a l l

c a se s , s e le c t L 1 0. C ho os e t he u ni fo rm d is t ri bu ti on

{q} 1/(L) o ve r t he e xp er ts i n u se , a nd t he r ad iu s

r2l 0:0001d, 2:13

where 1, . . . , L and d d e no t es t h e n u m b er o f a s se t s i n

t h e p o r t fo l i o.

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3. Empirical results

T h e g o a l o f o u r e m pi r ic a l s t ud y i s t o i n ve s ti g at e w he t he r

o r n o t c o n v en t io n al r i sk f a ct o rs , s u ch a s t h emarket risk

premium, the momentum factor ( J e ga d ee s h a n d T it m an

1 99 3, C ar ha rt 1 99 7) a nd t he t wo FamaFrench factors

( F a m a a n d F r e n c h 1 9 9 3 ) , a r e a b l e t o e x p l a i n t h e s c a t t e r o f

l o g - op t i ma l r e t u r ns u s i ng e q u i li b r i um m o d e ls . W e d o n o t

i nt en d t o j ud ge t he e qu il ib ri um m od el s t he ms el ve s,

h ow ev er w e a re i nt er es t ed i n m ea su ri ng t he a bn or ma l

retur n compared wi th t he equil ibrium returns.

A dd i ti o na l ly , t o e v al u at e t h e r e su l ts o f t h e l o g- o pt i ma l

p o rt f ol i os , w e a l so e x am i ne t h e p e rf o rm a nc e o f p a ss i ve

s t ra t eg i es w i th t h e s a me c o mp o ne n ts . I n t h e c o mp a ri s on

w e c an no t n eg le ct t he e ff ec t o f t ra ns ac ti on c os t f or t he

l o g - op t i ma l s t r a t eg y , s i n c e t h i s f a c t o r s i g n i f i c an t l y a f f e c ts

t he p er f or ma nc e o f a ny a ct iv e p or tf ol io s tr at eg y c om -

p a r e d w i t h t h e p a s s iv e b u y - an d - h ol d c o u n te r p a r t.

3.1. Data and methodology

T o t e st t h e l o g- o pt i ma l m e th o d e m pi r ic a ll y , t h e r e tu r n

da ta of t he Dow J on es I nd us tr ia l Av er age ( DJ IA)

c om po ne nt s i s a pp li ed f or a 1 5- ye ar -l on g p er io d f ro m

J a nu a ry 1 9 91 t h ro u gh D e ce m be r 2 0 08 . W e p e rf o rm t e st s

o n t h re e d i st i nc t p o rt f ol i os : ( i ) a p o rt f ol i o c o nt a in i ng t h e

c o mp o ne n ts o f t h e D J IA o b se r ve d i n D e ce m be r 2 0 05 ; ( i i)

a p o rt f ol i o c o nt a in i ng t h e c o mp o ne n ts i n J a nu a ry 1 9 91 ;

a nd ( ii i) a p or tf ol io t ha t t r ac ks t he a ct ua l c om po ne nt s.

E a ch t e st p o rt f ol io c o nt a in s 3 0 s h ar e s.

T he s ou rc e o f t he a ss et r et ur ns i s t he d at ab as e o f The

Center for Research in Security Prices ( C RS P ) . T h e r i s k -f re e r at e i s t he y ie ld o f t he 1 m on th U .S . T re as ur y b il l

c o l l e c t e d a l s o f r o m t h e C R S P d a t a b a s e . T h e C R S P v a l u e -

w e i gh t e d r e t u r n i n d e x, i n c l ud i n g d i s t r i b ut i o n s, i s m a d e u p

o f a ll N ew Y or k S to ck E xc ha ng e ( NY SE ), A me ri ca n

S t o c k E x c h a n g e ( A M E X ) a n d N A S D A Q s h a r e s s e r v e d a s

m ar ke t p or tf ol io s. T he l is t o f D JI A c om po ne nt s i s

o b t a in e d f r o m D o w J o n e s o f f i ci a l w e b p a g e . T h e p r o p or -

t i o n a l t r a n s a c t i o n c o s t i s s e t t o 0 . 1 % o f t h e t r a d e d v o l u m e

(c 0 .0 01 ), b ot h f or s al e a nd p ur c ha se . W e s ho rt ed t he

f i x e d c o s t f a c t o r . T h e a p p l i e d e m p i r i c a l s t r a t e g y i s d e f i n e d

a s a m i xt u re o f 1 0 l o g- o pt i ma l e x pe r ts , d i st r ib u ti n g t h e

i n i ti a l w e a l t h a m o n g t h e e x p e rt s a c c o r di n g t o t h e u n i f o r m

di stri buti on q 1 /1 0, f or a ll . T he r ad iu s r i s d e fi n edi n ( 2 . 1 3) .

F o r t e s t i n g p u r p o s e s w e c o n s tr u c t e d f o u r l i n e a r m o d e l s

a nd i nv es ti ga te t he c oe ff ic ie nt s o f c om mo n f ac to rs f or

m on t hl y l o g- o pt i ma l r e tu r ns . A lt h ou g h t h e i n tr o du c ed

e m pi r ic a l l o g- o pt i ma l s t ra t eg y p r ov i de s t h e f a ci l it y f o r

d a il y p o rt f ol i o r e ar r an g em e nt , i n o r de r t o e l im in a te t h e

h a rm f ul e f f ec t o f a u to c or r el a ti o n a n d h e te r os c ed a st i ci t y

o n l i n e a r r e g r e s s i o n s , m o n t h l y r e t u r n s w e r e f o r m e d o f t h e

d a il y l o g- o pt i ma l r e tu r ns , w h ic h m e an s 1 8 0 d a ta p o in t s

f or t he 1 5- ye ar -l on g p er io d. T he m on th ly m ar ke t p re -

m iu m s, m om e nt u m f a ct o rs a n d F a ma Fr e nc h f a ct o rs

w e re c o ll e ct e d f r om t h e C RS P d a ta b as e .W e c o n s tr u c t e d t h e m o s t c o m m o n e q u i l i b ri u m m o d e l s .

I n o rd er t o e st im at e t he r eg re ss io n c oe ff ic ie nt s, f ou r

m od e ls a r e a p pl i ed s e qu e nt i al l y t he c l as s ic a l Capital

Asset Pricing Model ( C A P M ) ( S h a r p e 1 9 6 4 , L i n t n e r 1 9 6 5 ,

M os si n 1 96 6) , t he C AP M a me nd ed w it h m om en tu m

f ac to r ( Ca rh ar t 1 99 7) , t he F am a Fr en ch t hr ee -f ac to r

m o de l ( F am a a n d F r en c h 1 9 93 ) a n d a f o ur - fa c to r m o de l

( C ar h ar t 1 9 97 ) . M o re p r ec i se l y, o n e c a n e s ti m at e t h e l o g-

o p t i ma l p r e mi u m s i n t h e f o l l ow i n g w a y s e q u en t i a ll y :

rtl r

tf l lr

tm r

tf "

tl, 3:1

rtl rtf l lr

tm r

tf mlMOM

t "tl, 3:2

rtl rtf l lr

tm r

tf slSMB

t hlHMLt "tl,

3:3

rtl rtf l lr

tm r

tf slSMB

t hlHMLt

mlMOMt "tl, 3:4

wherel, t, rf, rtm r

tf and" a r e s h a r e l, t h e t i m e, t h e r i s k-

f r ee r a te , t h e m ar k et p r em iu m a n d t h e e s ti m at i on r e si d -

u a l s , r e s p e c t iv e l y . A c c o r d in g t o F a m a a n d F r e n ch ( 1 9 9 3) ,SMB ( s ma ll -m in us -b ig ) m ea su re s t he a ve ra ge r et ur n

d i ff e r en c e b e tw e en s m al l a n d l a rg e c a pi t al i za t io n a s se t s,

and HML ( h i gh - mi n us - lo w ) i s t h e a v er a ge r e tu r n d i ff e r -

e n ce b e tw e en h i gh a n d l o w b o o k -t o -m a rk e t e q ui t y (B/M)

compani es. MOM i s t he m om en tu m f ac to r ( Ca rh ar t

1 99 7) , w hi ch g iv es t he a ve ra ge e xc es s r et ur n o f t he p as t

w in ne rs a bo ve t he r et ur n o f p as t l os er s ec ur it ie s. T he

r e g r e ss i o n c o e f f ic i e n ts , , s, h and m w e re e s t im a te d

b a se d o n e q ua t io n s ( 3 .1 ) , ( 3 .2 ) , ( 3 .3 ) a n d ( 3 .4 ) . F o r b e tt e r

u nd er st an di ng o f t he p er fo rm an ce m ea su re , t he f ou r

e s ti m at i on s a r e a l so p e rf o rm e d o n t h re e d i st i nc t p a ss i ve

p o r t f ol i o s t h a t c o n t a in t h e D e c e m b e r 2 0 0 5 , J a n u a r y 1 9 9 1a nd t he a ct ua l D JI A c om po ne nt s, r es pe ct iv el y. T he

r e su l ts a r e c o mp a re d w i th t h e r e su l ts o f t h e l o g- o pt i ma l

portfol i o.

T he a na ly si s o f r eg re ss io n c oe ff ic ie nt s i nv es t ig at e s

s t a t i s t i c a l l y t h e q u e s t i o n a s t o w h e t h e r o r n o t a n e m p i r i c a l

l og -o pt im al s tr at eg y h as a n a ve ra ge r et ur n t ha t c an b e

e xp ec te d f ro m a ny i nv es tm en t a t t he r is k l ev el o f t he

p r op o se d e m pi r ic a l s t ra t eg y . I n o t he r w or d s, w e i n ve s ti -

g a te i f t h e l o g- o pt i ma l s t r at e gy a l so h a s g o od e m pi r ic a l

p ro pe rt ie s b ey on d t he t ra di ti on al m od el s, b es id es t he

a t tr a ct i ve t h eo r et i ca l a c co m pl i sh me n ts . O t he r wi s e, t h is

t yp e o f l og -o pt im al i nv es tm en ts c an b e e mp ir ic al ly

rej ected.

3.2. DJIA components of the portfolios of

December 2005

W e i n v e st i g a te i n v es t m e nt s c o n t ai n i ng s t o c k s t h a t f o r m ed

t he D JI A i n D ec em be r 2 00 5. S in ce t he 1 5- ye ar -l on g

p e r i od s t a r t s i n J a n u ar y 1 9 9 1 , t h e s e i n v e s t ig a t i on s a r e n o t

f r ee o f s u rv i vo r sh i p b i as . W e p r es e nt t h e b i as - fr e e r e s ul t s

i n t h e f o l l ow i n g s u b s e c t i o ns .

T o o bt ai n a b en ch ma rk w he n e va lu at in g t he p er fo r-

m a n c e o f t h e l o g - o p t i m a l i n v e s t m e n t , w e a l s o e v a l u a t e t h e

p er fo rm an ce o f a b uy -a nd -h ol d s tr at eg y, w hi ch i s ap a ss i ve l y m a na g ed a l te r na t iv e o f o u r a c ti v e m e th o d, a n d

h e n c e i t d o e s n o t i n f e r t r a n s a c t i o n c o s t s . S i n c e t h e o r i g i n a l

DJ IA is n ot a t ot al- re tur n i nd ex , i t c an not be u se d

6 M. Ormos and A. Urban1592

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a s a b e nc h ma r k. T h e p a ss i ve l y m an a ge d p o rt f ol i o a l lo -

c at es c ap it al a cc or di ng t o t he c om po ne nt s p ri ce i n

J a n u a r y 1 9 9 1 , w h i c h c o r r e s p o n d s t o t h e D J I A s w e i g h t i n g

m e th o d. T h e s t r a t eg y s w e al t h i s t h e w e ig h te d s u m o f t h e

a g gr e ga t ed c a pi t al o f e a ch i n di v id u al a s se t . W e a s su m e

c o st l es s d i vi d en d r e in v es t me n t t o t h e i s su e r f i rm s s t oc k .

T h e l o g - o p t i m a l s t r a t e g y a p p l i e s a d y n a m i c , s e l f - f i n a n c i n g

a nd s ho rt - sa le -f re e a ll oc at io n o f t he c ap it al r eg ar di ng

s e c t i on 2 . 4 .

T ab le 1 s ho ws t he c oe ff ic ie nt s o f t he m os t c om mo n

e q ui l ib r iu m m o de l s a n d t h ei r t e st s t at i st i cs . E s ti m at e d

c oe ff ic ie nt s a re d if fe re nt ia te d b y a h at f ro m t he ir t he o-r e t i c al v a l ue s . W e m e a s ur e dR2 n e a r t o 0 . 5 5 f o r e a c h c a s e

f o r t h e l o g- o pt i ma l s t ra t eg y , w h i c h r e f e r s t o t h e f a ct t h at

c la ss ic al m od el s a re n ot a bl e t o e xp la in l og -o pt im al

p r e m iu m s s u f f ic i e n tl y . H o w ev e r , t h e e x p l an a t o ry p o w e r

o f t he i nt ro du ce d m od el s c an no t b e r ej ec te d d ue t o t he

si gni fi cant F- s ta t is t ic s . T he l a rg e r R2 f or t he p as si ve

s t ra t eg y (40 .7 8 f or e ac h c as e) s ug ge st s t ha t t he e qu il ib -

r i um m od e ls a r e m o re p o we r fu l i n e x pl a in i ng t h e e x ce s s

r e tu r ns o f t h e p a ss i ve i n ve s tm e nt . S i nc e t h e m a rk e t p r ox y

i s a ls o a p as s iv el y m an ag ed p or tf ol io , c on ta in in g t he

majority of U.S. stocks, i t i s abl e to capture the

m o ve m en t s o f a p a ss i ve i n ve s tm e nt b e tt e r t h an a n a c ti v es tr at eg y, w hi ch m ay i nv es t i n o nl y o ne a ss et ( an d o ft en

c al ls f or a s ma ll er d eg re e o f d iv er si fi ca ti on t ha n t he

p a s s iv e c o u n te r p a r t) . W h i leb i s t h e c o ef f ic i en t o f m a rk e t

premi um rtm rtf , i ts v al ue n ea r 1 i mp li es t ha t, ceteris

paribus, t he i nv es ti ga te d p re mi um s m ov e t og et he r w it h

t h e m a r ke t .

T h i s f a c t i s n o t s u r p r i s i n g i f o n e t a k e s i n t o a c c o u n t t h a t

t he p or tf ol io s ar e f or me d of r ec en t Do w J one s 30

c om po ne nt s, w hi ch a re a ll l ar ge -c ap c om pa ni es i n t he

m a j or i t y o f t h e i n v e st i g a te d p e r i od . W h e n t h e m o m en t u m

f a ct o r i s a p pl i ed , w e m e as u re s l ig h tl y , b u t m os t ly s i gn i f-

i c an t , n e ga t iv ebm v al ue s, w hi ch m ea ns t ha t t he l og -o pt im al p re mi um s, a nd l es s s o t he p as si ve i nv es tm en t

p re mi um s, m ov e c ou nt er si de t he r et ur ns o f t he p ri or 1

y e ar w in n er s , o n a v er a ge . T h e i m p ac t o f f i rm s i ze a n d t h eB/M e q u i t y r a t i o i s s t r o n g e r f o r t h e p a s s i v e i n v e s t m e n t , a s

w e m e a s u r e s i g n i f i c a n t n e g a t i v e l o a d i n g s o n t h e SMB and

HML f a ct or s. A lt ho ug h f or t he l og -o pt im al s tr at eg y

coeffi ci entsbs andbh a r e n o t s i gn i fi c an t , a c co r di n g t o t h em o d el s e l e c ti o n m e a s ur e m e nt , t h eadjusted R2, u s e o f t h e

SMB and HML f ac to rs i nc re as es t he p ow er o f o ur

m od e ls . E c on o mi c c o ns i de r at i on s a l so s u pp o rt t h is c o n-

c e p t b e c a u s e t h e s e f a c t o r s a r e a b l e t o c a p t u r e t h e n e g a t i v e

r e t u r n e f f e c t s o f l a r g e c a p i t a li z a t io n , l o w B/M compani es.

T h e p o rt f ol i os c o ns i st o n ly o f s h ar e s o f l a rg e c a pi t al iz a -

t i on c o mp a ni e s, w hi l e 2 3 c o mp a ni e s o f t h e 3 0 h a ve a l o w

b oo k- to -m ar ke t e qu it y r at io . T hi s i s i mp li ed b y t henegati vebs andbh. E xc ep t f or t he C AP M, e ac h m od els h ow s s i gn i fi c an t e x ce s s r e tu r ns a b ov e t h e e q ui l ib r iu m

v a l u e f o r t h e l o g - o p t i m a l s t r a t e g y . T h a t i s , t h e l o g - o p t i m a l

T a b le 1 . R e g r es s i o n c o e f f ic i e nt s o f p o r t f o l io s f o r m e d o f D o w J o n e s I n d u s t r ia l A v e r a g e c o m p o n en t s i n D e c em b e r 2 0 0 5 . T h e t a b l ec o n t a i n s c o e f f i c i e n t s a n d t e s t s t a t i s t i c s r e g a r d i n g t h e v a l i d i t y o f t h e i n t r o d u c e d c o e f f i c i e n t s a n d e q u i l i b r i u m m o d e l s . C o e f f i c i e n t s a n ds t at i st i cs a r e p r es e nt e d f o r b o th t h e l o g- o pt i ma l s t ra t eg y a n d a p a ss i ve b u y- a nd - ho l d s t ra t eg y c o ns t ru c te d f r om t h e s a me s t oc k s.E st im at ed c oe ff ic ie nt s a re d if fe re nt ia te d b y a h at f ro m t he ir t he or et ic al v al ue s. I n e ac h m od el t he e xp ec te d p re mi um s a rer e p r es e n t ed , w h i ch a r e t h e e q u il i b ri u m v a l u e s t h a t t h e i n v es t i g at e d s t r a t eg i e s s h o u l d a c h ie v e o n a v e r a ge i f m a r ke t e q u il i b r iu m h o l dsi n t h e l o n g t e r m . T h e l a s t s e c t i o n c o n t a i n s m e a s u r e m e n t s o n t h e p r o p o s e d e m p i r i c a l l o g - o p t i m a l s t r a t e g y a n d t h e p a s s i v e i n v e s t m e n t ,

i n c lu d i ng a v e r ag e m o nt h l y a n d a n n u al r i s k p r e m i u ms a n d s t a n d a r d d e v i a t i on s .

L og -o pt P as si ve bL bP bL bP bsL bsP bhL bhP bmL bmPCAPMR2 0.55 0. 78 Coeff. 0.55 0.25 1.09 1.05F-Stat. 214.4 631. 5 t-Stat. 1.77 1.42 14.64 25.13

p-Val ue 0.00 0. 00 p-Value 0.08 0.16 0.00 0.00A d j. R2 0.54 0. 78

CAPM MOMR2 0.56 0. 79 Coeff. 0.69 0.33 1.07 1.04 0. 13 0. 08F-Stat. 111.5 325. 0 t-Stat. 2.18 1.85 14.18 24.61 2. 12 2. 20

p-Val ue 0.00 0. 00 p-Value 0.03 0.07 0.00 0.00 0.04 0. 03A d j. R2 0.55 0. 78

Three-factor modelR2 0.56 0. 84 Coeff. 0.67 0.47 1.07 1.03 0. 17 0. 36 0. 11 0. 21

F-Stat. 73.2 306. 1 t-Stat. 2.05 2.98 12.36 24.33 1. 84 8. 03 1. 00 3. 84p-Val ue 0.00 0. 00 p-Value 0.04 0.00 0 .00 0 .00 0.07 0.00 0.32 0. 00A d j. R2 0.55 0. 84

Four-factor modelR2 0.56 0. 84 Coeff. 0.81 0.53 1.03 1.01 0. 15 0. 35 0. 14 0. 22 0. 13 0. 05F-Stat. 56.8 232. 5 t-Stat. 2.45 3.29 11.64 23.37 1. 60 7. 80 1. 21 4. 01 1. 98 1. 65

p-Val ue 0.00 0. 00 p-Value 0.02 0.00 0.00 0.00 0.11 0.00 0.23 0. 00 0.05 0. 10Adj. R

2 0.55 0. 84

Log-opt Passi ve

Avg monthly pr em. 1. 56% Std. dev. 6.53% Avg monthly prem. 1. 21% Std. dev. 5. 01%Avg annual prem. 18. 72% Std. dev. 21.29% Avg annual prem. 14. 52% Std. dev. 22. 91%

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p re mi um s a re h ig he r t ha n t he y s ho ul d b e a t s uc h r is k

l e ve l s. A s u rp r is i ng f a ct i s t h at w h il e t h e f o re t ok e ns a n d

m a g ni t u d es o f t h e c o e f fi c i e nt s r e f l ec t t h e p r o p e r t i es o f t h e

m aj o ri t y o f s h ar e s m ak i ng u p t h e l o g- o pt i ma l p o rt f ol i o,

t he m od el s a re u na bl e t o e xp la in t he a bn or ma l e xc es s

r et ur ns . I n t he c as e o f t he p as si ve s t ra te gy , t heb val uesa r e s i gn i fi c an t o n ly f o r t h e t h re e -f a ct o r a n d f o ur - fa c to r

m od e ls . I n a d di t io n , e a ch o f t h eb v a l ue s i s p o si t iv e , a n dt h e l o g- o pt i ma l s t ra t eg y o u tp e rf o rm s i t s p a ss i ve l y m an -

a g e d c o u n te r p a rt r e g a r di n g a l l e q u i li b r i um m o d el s .

Theb v a lu es o f t he e qu il ib ri um m od el s m ea su re t hea bn or ma l r et ur n a bo ve t he r is k- ad ju st ed e qu il ib ri umv a lu e . B e si d es t h eb v al u e, w e a l so p r es e nt t h e a b so l ut e( n o n -r i s k -a d j u st e d ) a v e r ag e m o n th l y a n d a n n ua l r e t u r ns

a n d s t a n d a r d d e v i a t io n s . T h e r e s u lt s a r e s i m i l a r , s i n c e t h e

l og -o pt im al m et ho d a ch ie ve s a m uc h h ig he r a ve ra ge

r e tu r n t h an i t s p a ss i ve c o un t er p ar t , a l be i t w i th a g r ea t er

m on th ly d ev ia ti on , w hi ch i s c lo se t o b ei ng e qu al a t t he

a n n ua l l e v e l.

3.3. DJIA components of the portfolios of January 1991

I n t hi s s ub se ct io n w e p er fo rm s im il ar t es t s o n t he l og -

o p ti m al a n d a p a ss i ve l y m an a ge d p o rt f ol i o c o ns t ru c te da cc or di ng t o t he s am e m et ho do lo gy a s i n t he p re vi ou s

s ub se ct io n, e xc ep t f or t he f ac t t ha t t he p or tf ol io s a re

f or me d o f s t oc ks m ad e u p o f t he I nd us tr ia l A ve ra ge i n

J a nu a ry 1 9 91 . T h e i n ve s ti g at e d p e ri o d a l so h o ld s f r om

J an ua ry 1 99 1 t hr ou gh D ec em be r 2 00 8. T ha t i s, t he

a r ra n ge m en t o f t h is a n al y si s i s f r ee o f s u rv i vo r sh i p b i as .

T h e p a ss i ve p o rt f ol i o i s a g ai n p r ic e w e ig h te d a t p o rt f ol i o

l aunch.

T a bl e 2 p r es e nt s t h e c o ef f ic i en t s o f t h e s a me c o mm o n

e qu il ib ri um m od el s a nd t he ir t es t s ta ti st ic s a s i n t he

p re vi ou s s ub se ct io n. T he m os t i mp or ta nt d if fe re nc e

b et we en t he p re vi ou s a nd c ur r en t r es ul ts i s t ha t t hebv alu es a re l owe r b y 0. 24 %, on a ve rag e, f or t he lo g-

o p ti ma l s t ra t eg y , h o we v er t h e d i ff e re n ce i s e v en h i gh e r,

0 .5 0% , r eg ar di ng t he s t ro ng es t ( by a dj us te d R2) f ou r-f a ct o r m o de l . A l th o ug h w e a l so m e as u re d t h e d i ff e re n ce s

f o r t h e p a s s i v e s t r a t e g y , o n l y o n e m o d e l ( C A P M MOM)

h a s a s i gn i fi c an tb p a ra me t e r, s o t h e f i gu r es d o n o t r e f e rt o t h e e x is t en c e o f a b no r ma l r e tu r ns . T h e s u pe r io r it y o f

t h e p o r t f o l i o f o r m e d o f I n d u s t r i a l A v e r a g e c o m p o n e n t s i n

2 0 05 i s d u e t o s u rv i vo r sh i p b i a s .

H o w e v e r , t h e l o g - o p t i m a l p o r t f o l i o a l s o o u t p e r f o r m s i t s

p a s s iv e c o u nt e r p a rt r e g a r di n g b o t h t h eb v a lu e s a n d t h ea bs ol ut e v al ue s o f t he a ve ra ge m on th ly a nd a nn ua l

r et ur ns . T he n ot ab le d if fe re nc e i n t he a ve ra ge r et ur n

h e lp s t h e l o g- o pt i ma l s t ra t eg y t o a c hi e ve m o re t h an t w ic e

t he p as si ve s tr at eg y s f in al w ea lt h d ur in g t he 1 5 y ea rs ,a l th o ug h i t r e su l ts i n a l a rg e r s t an d ar d d e vi a ti o n.

E x ce p t f o r t h eb v a lu e s , a l l c o e f fi c i e nt s a r e s i g n if i c a nt ,h o w e ve r t h eR2 v al ue s a re s ma ll er t ha n i n t he c as e o f t he

T a bl e 2 . R e gr e ss i on c o ef f ic i en t s o f p o rt f ol i os f o rm e d o f D ow J o ne s I n du s tr i al A v er a ge c o mp o ne n ts i n J a nu a ry 1 9 91 . T h e t a bl ec o n t a i n s c o e f f i c i e n t s a n d t e s t s t a t i s t i c s r e g a r d i n g t h e v a l i d i t y o f t h e i n t r o d u c e d c o e f f i c i e n t s a n d e q u i l i b r i u m m o d e l s . C o e f f i c i e n t s a n ds t at i st i cs a r e p r es e nt e d f o r b o th t h e l o g- o pt i ma l s t ra t eg y a n d a p a ss i ve b u y- a nd - ho l d s t ra t eg y c o ns t ru c te d f r om t h e s a me s t oc k s.E st im at ed c oe ff ic ie nt s a re d if fe re nt ia te d b y a h at f ro m t he ir t he or et ic al v al ue s. I n e ac h m od el t he e xp ec te d p re mi um s a rer e p r es e n t ed , w h i c h a r e t h e e q u il i b ri u m v a l u e s t h a t t h e i n v e st i g a te d s t r a te g i e s s h o u l d a c h i e v e o n a v e r a ge i f m a r k et e q u il i b ri u m h o l d si n t h e l o n g t e r m . T h e l a s t s e c t i o n c o n t a i n s m e a s u r e m e n t s o n t h e p r o p o s e d e m p i r i c a l l o g - o p t i m a l s t r a t e g y a n d t h e p a s s i v e i n v e s t m e n t ,

i n c lu d i ng a v e r a ge m o n th l y a n d a n n ua l r i s k p r e m i u ms a n d s t a n da r d d e v i at i o ns .

L og -o pt P as s iv e bL bP bL bP bsL bsP bhL bhP bmL bmPCAPMR2 0.40 0.65 Coeff. 0. 56 0.09 0 .86 0 .82F-Stat. 118.8 330.9 t-Stat. 1. 68 0.49 10.90 18.19

p-Value 0.00 0.00 p-Value 0. 09 0.62 0.00 0.00Adj. R2 0.40 0.65

CAPM MOMR2 0.45 0.69 Coeff. 0. 82 0.27 0 .80 0 .78 0. 26 0. 18F-Stat. 72.3 199.3 t-Stat. 2. 52 1.51 10.46 18.16 3. 99 4. 93

p-Value 0.00 0.00 p-Value 0. 01 0.13 0.00 0.00 0. 00 0.00Adj. R2 0.44 0.69

Three-factor modelR2 0.51 0.81 Coeff. 0. 09 0.12 1.13 1.01 0. 01 0.25 0.62 0.34

F-Stat. 61.6 247.5 t-Stat. 0. 28 0 .8 2 1 3. 55 2 5. 85 0. 14 6.20 5.66 6.57p-Value 0.00 0.00 p-Value 0. 78 0.41 0 .00 0 .00 0. 89 0.00 0.00 0.00Adj. R2 0.50 0.81

Four-factor modelR2 0.54 0.82 Coeff. 0. 31 0.00 1.07 0.97 0.02 0.24 0.58 0.31 0. 20 0. 11F-Stat. 51.4 204.4 t-Stat. 0. 99 0.02 12.75 25.19 0. 26 5.91 5.43 6.38 3. 25 3. 89

p-Value 0.00 0.00 p-Value 0. 33 0.99 0.00 0.00 0. 80 0.00 0.00 0.00 0. 00 0.00Adj. R

2 0.53 0.82

Log-opt Passive

Avg monthly prem. 1.13% Std. dev. 5. 62% Avg monthly prem. 0.79% Std. dev. 4.20%Avg annual pr em. 13.56% Std. dev. 26. 81% Avg annual prem. 9.48% Std. dev. 15.96%

8 M. Ormos and A. Urban1594

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p r e v io u s p o r t f ol i o , e s p e c i a l ly f o r t h e l o g - op t i ma l s t r a t e gy .

S in ce a n ot ab le p ar t o f t he s to ck s h as l os t s om e o f i ts

e ar li er c ap it al iz at io n o r h as n ot b ee n t ra de d y et , t he

e q u i l i b r i u m m o d e l s a r e e v e n w e a k e r i n e x p l a i n i n g t h e l o g -

o p t i ma l p r e m iu m s . R e g a r d in g t h e C A P M a n d t h e C A P M

w i t h m o m e n t u m f a c t o r , t h e p o r t f o l i ob v a l u e s a r e c l o s e t o0 . 8 , w h i l e t h e t h r e e -f a c t or a n d f o u r - f a c t or m o d el s y i e ldbv al ue s a ro un d 1 . C on tr ar y t o t he f or me r a na ly si s, t he

p o r t f o l i o s n o w h a v e a s i g n i f i c a n t l y p o s i t i v e l o a d i n g o n t h e

HML f a ct or . T ha t i s, t he se p or tf ol io s i nv es t r at he r i n

s t oc k s w i th a h i gh b o ok - to - ma r ke t e q ui t y r a ti o .

3.4. Portfolios of actual DJIA components

T h e p o r t f o l i o s i n v e s t i g a t e d h e r e a r e b l e n d s o f t h e p r e v i o u s

i n v e s t m e n t s a s t h e y a l w a y s h o l d s t o c k s f r o m t h e s e t o f t h e

a ct ua l D JI A c om po ne nt s . T hi s m ea ns t ha t t he p as si ve

p or t fo li o i s n ot a p ur e b uy -a nd -h ol d s tr at eg y s in ce t he

p or tf ol io i s r eb al an ce d w he n a c ha ng e o cc ur s i n t he

c o m po n e n t s l i s t . A t t h e b r e a k p o i nt s t h e p a s s i v e p o r t f o l i o

r ea ll oc at es t he c ap it al a cc or di ng t o t he c om po ne nt s

p ri ce s. S in ce i n t hi s c as e p or tf ol io r eb al an ci ng i s v er y

r a re , t h e t r an s ac t io n c o st s a r e n e gl ig i bl e .y B e tw e e n t w o

b r ea k p o in t s t h e s t ra t eg y w o rk s a s w e ll a s t h e p r e v io u sl y

i n tr o du c ed p a ss i ve m e th o ds . A f te r t h e b r ea k p o in t s t h e

l o g- o pt i ma l s t ra t eg y i s s i mp l y r e st r ic t e d t o a n ot h er s e t o f

3 0 s to ck s. A lt ho ug h t he se t yp es o f i nv es tm en ts r ep et i-

t iv el y e xc lu de t he s to ck s t ha t l os t s om e o f t he ir e ar li er

a p pe a l, t h ey a r e n o t a f fe c te d b y t h e s u rv i vo r sh i p b i a s t h at

w or se ns t he j ud ge me nt o f t he f ir st i nv es tm en ts s in ce

t r a c k i n g t h e D J I A i s a c a u s a l s t r a t e g y . W e s u m m a r i z e t h e

r e su l ts o f t h e f o u r r e gr e ss i on s i n t a b l e 3 . I t i s n o te w or t hy

t h at , e x ce p t f o r t h e t h re e -f a ct o r m o de l , t h e

b v a l ue s a r e

s i gn i fi c an t ly p o si t iv e f o r t h e l o g- o pt i ma l s t r at e gy , w hi l e

t h ey d o n o t d i f f e r s i g n if i ca n tl y f r om z e ro f o r t h e p a s s iv ei nvestment.z I t i s a l s o w o r t h n o t i n g t h a t t h e R2 v a l u e s a r e

t he l ar ge st i n t hi s c as e a s t he s t ra te gi es t ra ck a s et o f t he

m o s t i m p o r t a n t s t o c k s i n e a c h s u b p e r i o d . T h e b v a l u e s a r en o ta b ly h i gh e r f o r t h e l o g- o pt i ma l s t ra t eg y , w h ic h i s a l so

m ir r or e d i n t h e l a rg e r v a lu e s o f t h e s t an d ar d d e vi a ti o n.

I n c o nt r as t t o t h e o t he r p o rt f ol i os , t h ebh v a lu e s a r e n o n-s ig ni fi ca nt f or t he l og -o pt im al s tr at eg y, w hi le t he y a re

s i g n if i c a nt l y p o s i ti v e f o r t h e p a s s iv e i n v e st m e n t. B o t h t h e

l o g- o pt i ma l a n d t h e p a ss i ve s t ra t eg y h a ve n e ga t iv e l o ad -

i ng s o n t he m om en tu m f ac to r, a lt ho ug h

bm i s not

s ig ni fi ca nt f or t he l og -o pt im al s tr at eg y r eg ar di ng t he

s t r o ng e s t f o u r -f a c t o r m o d e l.

yF o ur c h an g es o c cu r re d d u ri n g t h e 1 5 -y e ar p e ri od i n t h e D J IA .zT h e t h r e e- f a c to r m o d el a l s o y i e l d s p o s i ti v eb v a lu e s a t t h e 0 . 1 s i g n if i c a nc e l e v el .

T a b le 3 . R e g r es s i o n c o e f f ic i e nt s o f p o r tf o l io s f o r me d o f a c t u al D J I A c o m p on e n ts . T h e t a b l e c o n t a i ns c o e ff i c ie n t s a n d t e s t s t a t i s t ic sr e g a r di n g t h e v a l i d i t y o f t h e i n t r o d u ce d c o e ff i c ie n t s a n d e q u il i b r iu m m o d el s . C o e f f i c ie n t s a n d s t a t is t i c s a r e p r e s en t e d f o r b o t h t h el o g - o p t i m a l s t r a t e g y a n d a p a s s i v e b u y - a n d - h o l d s t r a t e g y c o n s t r u c t e d f r o m t h e s a m e s t o c k s . E s t i m a t e d c o e f f i c i e n t s a r e d i f f e r e n t i a t e db y a h a t f r o m t h e ir t h e o re t i ca l v a l u e s . I n e a c h m o d e l t h e e x p e ct e d p r e m i u ms a r e r e p r e s e n te d , w h i c h a r e t h e e q u i l i br i u m v a l u e s t h a tt h e i nv e st i ga t ed s t ra t eg i es s h ou l d a c hi e ve o n a v er a ge i f m a rk e t e q ui l ib r iu m h o ld s i n t h e l o ng t e rm . T h e l a st s e ct i on c o nt a in sm e a s ur e m e nt s o n t h e p r o po s e d e m p i r i ca l l o g - op t i ma l s t r a te g y a n d t h e p a s s i v e i n v e st m e nt , i n c lu d i ng a v e r a ge m o n th l y a n d a n n ua l

r i s k p r e mi u m s a n d s t a n da r d d e v ia t i o ns .

L og -o pt P as si ve bL bP bL bP bsL bsP bhL bhP bmL bmPCAPMR2 0.62 0. 78 Coeff. 0. 53 0.03 1 .12 0.93F-Stat. 287.6 622. 3 t-Stat. 1. 92 0.16 16.96 24.95

p-Val ue 0.00 0. 00 p-Value 0. 06 0.87 0.00 0.00Adj. R2 0.62 0. 78

CAPM MOMR2 0.63 0. 81 Coeff. 0. 68 0.19 1 .09 0.89 0. 14 0. 16F-Stat. 151.6 376. 7 t-Stat. 2. 43 1.28 16.50 25.50 2. 57 5. 48

p-Val ue 0.00 0. 00 p-Value 0. 02 0.20 0.00 0.00 0. 01 0. 00Adj. R2 0.63 0. 81

Three-factor modelR2 0.65 0. 85 Coeff. 0. 47 0.14 1.22 1.07 0. 21 0.17 0. 13 0.26

F-Stat. 108.3 341. 0 t-Stat. 1. 67 1 .0 6 1 6. 35 3 0. 14 2. 72 4.57 1. 37 5.50p-Val ue 0.00 0. 00 p-Value 0. 10 0.29 0.00 0.00 0.01 0.00 0. 17 0.00Adj. R2 0.64 0. 85

Four-factor modelR2 0.66 0. 87 Coeff. 0. 58 0.01 1.18 1.03 0. 19 0.15 0. 11 0.23 0. 10 0. 11F-Stat. 83.19 289. 06 t-Stat. 2. 05 0 .1 0 1 5. 56 2 9. 80 2. 48 4.21 1. 17 5.29 1. 84 4. 52

p-Val ue 0.00 0. 00 p-Value 0. 04 0.92 0.00 0.00 0.01 0.00 0. 24 0.00 0. 07 0. 00Adj. R

2 0.65 0. 87

Log-opt Passi ve

Avg monthly pr em. 1. 14% Std. dev. 5 . 93% Avg monthl y prem. 0.87% Std. dev. 4. 35%Avg annual prem. 13. 68% Std. dev. 22. 69% Avg annual prem. 10.44% Std. dev. 16. 24%

Performance analysis of log-optimal portfolio strategies 91595

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I n s u m ma r y , w e h a v e d e m o n s tr a t e d t h a t n o i n v e st m e n t

s tr at eg y h as a h ig he r a sy mp to ti c g ro wt h r at e t ha n l og -

o pt im al s tr at eg ie s o n s ta ti on er y a nd e rg od ic m ar ke ts

w it ho ut t ra ns ac ti on c os ts . H ow ev er , w e n ow a ls o h av e

e m pi r ic a l e v id e nc e t h at , r e ga r di n g t h e c o mm o n e q ui l ib -

r i um m o de l s, t h ey a r e a b le t o a c hi e ve s u rp a ss i ng r e t ur n s

m a n if e s t e d i n p o s it i v eb v al ue s, e ve n i n t he p re se nc e o f t r a n sa c t i on c o s t s .4. Extensions and concluding remarks

T h e r e a r e s e v e r al d i r e ct i o n s i n w h i ch t h e m o d e l p r e s e nt e d

i n t h i s p ap er c an b e e xt e nd ed . O n t he o ne h an d, o ne c an

t a k e i n t o a c c o u nt a r i s k le s s i n v es t m e nt i n s t ru m e n t, w h i ch

g i ve s t h e p o ss i bi l it y o f r e sc u in g o u r i n ve s tm e nt w h en a l l

t he e xp er t s f or ec as t a n eg at iv e p ri ce c ha ng e f or t he n ex t

p er io d. F or t hi s r ea so n, t he m od el c an a ls o b e e xt en de d

w it h t h e p o ss i bi li t y o f s h or t s e ll i ng . I f e i th e r t h e r i sk - fr e ei n v es t m e nt p o s s ib i l it y o r t h e s h o r t -s e l l in g o p t i on i s s o l v ed

b y f i n d in g t h e b u ll o r t h e b e ar p e ri o ds o f t h e m a rk e t, t h e

e st im at io n o f t he e xp er ts m ay b e m or e a cc ur at e. O n t he

ot he r ha nd, t he mo de l a ss ume s t ha t t he d eman de d

s e cu r it y i s a l wa y s a v ai l ab l e i n t h e m a rk e t o n t h e c l os i ng

p r i c e , a n d t h i s c o n d i t i o n c a n b e s o l v e d u s i n g a t r a d e s a n d

q u o t es d a t a ba s e .

I n t h i s p a p e r w e a b s t r a c t f r o m m a n y r e a l i s t i c f e a t u r e s o f

t he m ar ke t i n o rd er t o h ig hl ig ht t he i mp li ca ti on s o f a n

e m p ir i c a l l o g - op t i m al p o r t f ol i o s t r a t eg y . O u r c o n c lu s i on

i s t h a t t h e l o g - o p t i m a l p o r t f o l i o s t r a t e g y s h o w s s o m e k i n d

o f m a r k e t i n e f f i c i e n c y , i n t h e s e n s e t h a t t h e a p p l i e d k e r n e l -

b as ed e xp er ts a re a bl e t o g iv e b et te r a dv ic e t ha n p ur e

r a n d o m s e l e c t i o n b y w h i c h a p o s i t i v e a b n o r m a l r e t u r n c a n

b e g a in e d. T h e m a i n e x pl a na t io n s f o r t h is f a ct s t em f r om

t h r e e d i r e c t i o n s : t h e d a t a s e t u s e d i n o u r a n a l y s e s , t h e j o i n t

h yp ot he si s, w hi ch i s t he e qu il ib ri um m od el u se d t o

e xp la in t he r et ur n, a nd t he d if fe re nc e be tw ee n t he

s i n gl e - p e ri o d a n d m u l ti - p e r io d a p p r oa c h e s. T h e f i r s t t w o

e x p l a n a t i o n s h a v e a n e x t e n s i v e l i t e r a t u r e , w h e r e a s t h e l a s t

a r gu m en t d o es n o t. T h e g o al o f t h e o n e- p er i od p o rt f ol io

t he or y ( Ma rk ow it z 1 95 2) a nd t he r iv al e qu il ib ri um

m od el s ( su ch a s C AP M, A PT , F am a Fr en ch , a nd t he

C a rh a rt m o de l ) i s t h e o p ti mi z at i on o f a s se t a l lo c at i on i n

o rd er t o a ch ie ve t he o pt im al t ra de -o ff b et we en t hee xp ec te d o ne -p er io d r et ur n a nd r is k. T hi s a ss um es a

w o rl d w h er e i n ve s to r s o p ti mi z e t h ei r c o ns u mp t io n a n d

i n v es t m e nt s t r a t eg i e s f o r t h a t g i v e n o n e p e r i o d . H o w e v e r ,

m os t m ea n va ri an ce a na ly si s d ea ls o nl y w it h s ta ti c

m od e ls , c o nt r ar y t o t h e e x pe c te d u t il i ty m o de l s, w ho s e

l it e ra tu re i s r ic h i n m ul ti -p er io d m od el s, a ss um in g a n

i n di v id u al w it h a l o ng e r i n te r va l t h an s i mp l y o n e- p er i od

t hi nk in g. I n t he m ul ti -p er io d m od el s, i nv es to rs a re

a ll ow ed t o r eb al an ce t he ir p or tf ol io s i n e ac h t ra di ng

p e ri o d, t h er e fo r e t h ei r i n ve s tm e nt s m a y b e c h ar a ct e ri z ed

i n d if f er en t w ay s i n o ne a nd m ul ti pl e p er io ds d ue t o t he

m u l ti p l ic a t i ve e f f e c t o f c o n s ec u t i ve r e i n ve s t m en t s . I n o u re x p e ri m e nt t h e a p p r ox i m at i o n u s e s a n i n f i ni t e a p p r oa c h ,

w h e r ea s 1 5 y e a r s w o u l d d i c t at e a m u l ti - p e ri o d i n v e s t m e nt

m od e l. H ow e ve r , w e u s e a d a il y r e ba l an c in g s t ra t eg y a n d

t h e n u m b er o f p e ri o ds i s m o re t h an 3 7 00 , w hi c h b e h a ve s

a l mo s t a s i t d o es i n t h e i n f i ni t e m od e l.

Acknowledgements

T h e a u t h or s t h an k L a szl o Gyo r f i f o r h i s c a re f ul r e ad i ngo f t he m an us cr ip t a nd u se fu l a dv ic e. T hi s p ap er h as

g r ea t ly b e ne f it e d a n d b e en i m pr o ve d b y t h e m a ny h e lp f ul

s ug ge st io ns a nd c om me nt s f ro m t he t wo a no ny mo us

r e vi e we r s. T h is w o rk f o rm s p a rt o f t h e s c ie n ti f ic p r oj e ct

De ve lo pm en t o f q ua li ty -o ri en te d a nd h ar mo ni ze d

RDI s t ra te gy a nd f un ct io na l m od el a t B ME . T hi s

p r oj e ct i s s u pp o rt e d b y t h e N e w H u ng a ry D e ve l op m en t

Plan (project ID: TA MOP-4. 2. 1/B-09/1/KMR-2010-

0002).

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