Decision Suppor Sysems 37 (24) 485 5 www.elsevier.com/locae/dsw rbirage pricing heory-based Gaussian emporal facor analysis for adapive porfolio managemen Kai-hun hiu*, Lei Xu Deparmen of ompuer Science and Engineering, he hinese Universiy of Hong Kong, Shain, New erriories, Hong Kong, PR hina vailable online 7 July 23 bsrac Ever since he incepion of Markowiz s modern porfolio heory, saic porfolio opimizaion echniques were gradually phased ou by dynamic porfolio managemen due o he growh of populariy in auomaed rading. In view of he inensive compuaional needs, i is common o use machine learning approaches on Sharpe raio maximizaion for implemening dynamic porfolio opimizaion. In he lieraure, reurn-based approaches which direcly used securiy prices or reurns o conrol porfolio weighs were ofen used. Inspired by he arbirage pricing heory (P), some oher effors concenrae on indirec modelling using hidden facors. On he oher hand, wih regard o he proper risk measure in he Sharpe raio, downside risk was considered a beer subsiue for variance. In his paper, we invesigae how he Gaussian emporal facor analysis (F) echnique can be used for porfolio opimizaion. Since F is based on he classical P model and has he benefi of removing roaion indeerminacy via emporal modelling, using F for porfolio managemen allows porfolio weighs o be indirecly conrolled by several hidden facors. Moreover, we exend he approach o some oher varians ailored for invesors according o heir invesmen objecives and degree of risk olerance. D 23 Elsevier B.V. ll righs reserved. Keywords: emporal facor analysis; rbirage pricing heory; Porfolio opimizaion; Sharpe raio; Downside risk; Upside volailiy. Inroducion * orresponding auhor. E-mail addresses: kcchiucse.cuhk.edu.hk (K.-. hiu), lxucse.cuhk.edu.hk (L. Xu). Porfolio managemen has evolved as a core decision-making aciviy for invesors and praciioners in he financial marke nowadays. Prior o he incepion of Markowiz s modern porfolio heory [], heoreical research on invesmens has concenraed on modelling expeced reurns [2]. During he early sage of is developmen, porfolio opimizaion was ofen consrained by is saic implemenaion. Unlike dynamic porfolio opimizaion by which he opimal porfolio weighs were racked over ime based on updaed marke informaion, he weighs deermined using saic opimizaion echniques could no adap o marke changes wihin he invesmen horizon. Despie dynamic porfolio opimizaion being powerful, i urned ou o be a problem ha required inensive compuaion. Recall ha he mos naural echnique for solving dynamic porfolio opimizaion problems was sochasic dynamic programming. However, his approach was ofen compromised by several facors such as he curse of dimensionaliy when oo many sae variables were involved [7]. In 67-9236/3/$ - see fron maer D 23 Elsevier B.V. ll righs reserved. doi:.6/s67-9236(3)82-4
486 K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 general, pracical consideraions such as axes and ransacions coss also increased he number of sae variables in he objecive funcion. In fac, his problem could be beer solved via some popular machine learning approaches [3,2,3, 2] which required he opimal parameers o be adapively learned over ime, and consequenly, we have he erm adapive porfolio managemen. mong he various mehodologies suggesed, he mos popular one is based on maximizing he well-known Sharpe raio [7]. In implemenaion, rading could be based on raining a rading sysem on labelled daa [2] or direcly maximizing he expeced profi via he socalled adapive supervised learning decision neworks [8,2]. In his paper, hese approaches were generally referred o as reurn-based porfolio managemen because hey eiher explicily reaed he weighs as consans or depend direcly on he securiy price or reurns. Inspired by he arbirage pricing heory (P) in finance, which assumes ha he cross-secional expeced reurns of securiies is linearly relaed o k hidden economic facors, ypical saisical echniques such as principal componen analysis (P), independen componen analysis (I) [,22] and maximum likelihood facor analysis [] have been used. However, should we adop eiher P or I for esimaing he hidden facors, we have o compromise on he erms of zero noise. Likewise, we have o make a compromise on roaion indeerminacy if we use convenional facor analyic echniques. In fac, many researchers also realized ha variance was no appropriae for quanifying risk in he Sharpe raio because i couned posiive reurns as risk. For insance, Fishburn used he lower parial momen (LPM) [5] of reurns called downside risk o replace he radiional variance measure. Moreover, similar ideas were adoped for implemening porfolios opimizaion [8,9]. In his paper, we aim o invesigae using he echnique emporal facor analysis (F) [8] for porfolio opimizaion. Since F is based on he classical P model and has he benefi of removing roaion indeerminacy via emporal modelling, using F for porfolio managemen allows porfolio weighs o be indirecly conrolled by several hidden facors. Moreover, we can exend he approach o some oher varians ailored for invesors according o heir risk and reurn objecives. he res of he paper is organized in he following way. Secions 2 and 3 briefly review he P and he Gaussian F models, respecively. Secion 4 illusraes how he P-based adapive porfolio managemen can be effeced wih algorihms proposed in his paper. hree varians of he P-based Sharpe raio maximizaion echnique are sudied in Secion 5. Secion 6 concludes he paper. 2. Review on arbirage pricing heory he P begins wih he assumpion ha he n vecor of asse reurns, R, is generaed by a linear sochasic process wih k facors [4 6]: R ¼ R þ f þ e ðþ where f is he k vecor of realizaions of k common facors, is he n k marix of facor weighs or loadings and e is an n vecor of asse-specific risks. I is assumed ha f and e have zero expeced values so ha R is he n vecor of mean reurns. he model addresses how expeced reurns behave in a marke wih no arbirage opporuniies and predics ha an asse s expeced reurn is linearly relaed o he facor loadings or R ¼ R f þ p ð2þ where R f is an n vecor of consans represening he risk-free reurn, and p is k vecor of risk premiums. Similar o he derivaion of PM, Eq. (2) is based on he raionale ha unsysemaic risk is diversifiable and herefore should have a zero price in he marke wih no arbirage opporuniies. 3. Overview of emporal facor analysis Suppose he relaionship beween a sae y ar k and an observaion x ar d is described by he firs-order sae-space equaions as follows [8,9]: y ¼ By þ e ; ð3þ x ¼ y þ e ; ¼ ; 2;...; N ð4þ where e and e are muually independen zeromean whie noises wih E(e i e j )=R e d ij, E(e i e j )=R e d ij,
K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 487 E(e i e j )=, R e and R e are diagonal marices and d ij is he Kronecker dela funcion: 8 < ; if i ¼ j; d ij ¼ : : ; oherwise ð5þ We call e he driving noise upon he fac ha i drives he source process over ime. Similarly, e is called measuremen noise because i happens o be here during measuremen. he above model is generally referred o as he F model. In he conex of P analysis, Eq. () can be obained from Eq. (4) by subsiuing (R R ) for x and f for y. he only difference beween he P model and he F model is he added Eq. (3) for modelling emporal relaion of each facor. he added equaion represens he facor series y={ y } = in a mulichannel auoregressive process, driven by an i.i.d. noise series {e } = ha are independen of boh y and e. Specifically, i is assumed ha e is Gaussian disribued. Moreover, F is defined such ha he k sources y (), y (2),..., y (k) in his sae-space model are saisically independen. he objecive of F is o esimae he sequence of y s wih unknown model parameers H={,B,R e,r e } hrough available observaions. 3.. learning algorihm In implemenaion, an adapive algorihm has been suggesed. each ime uni, facor loadings are esimaed by cross-secional regression, and facor scores are esimaed by maximum likelihood learning. Xu proposed an algorihm in Ref. [9] as shown below. Sep : Fix, B, R e and R e, esimae he hidden facors y by ŷ =(R e + R e ) ( R e x + R e Bŷ ), e = y Bŷ, e = x ŷ. Sep 2: Fix y, updae, B, R e and R e by he gradien ascen approach as follows: B new = B old + g diag[e y ], new = old + g e y, R new e =( g)r old e + g diag[e e ], R new e =( g)r old e + g diag[e e ]. where g denoes he learning rae. 3.2. F driven by RH( p) process In he finance lieraure, effecs of auoregressive condiional heeroscedasiciy (RH) were considered in modelling unobserved componens [6] as well as hidden facors [4]. In fac, he F model can be direcly exended so as o explicily consider he presence of RH effec. For example, we may jus assume ha each facor series has RH( p) effec. Mahemaically, we have e ð jþ w ð jþ2 ¼ m ð jþ w ð jþ ; m ð jþ ¼ a ð jþ2 þ Xp s¼ fnð; Þ a ð s jþ2 e ð jþ2 s o accommodae for he learning of RH effec, updaing of R e a ime can be alernaively done via ( updaing a j) ( and {a j) p s } s = as shown below: a ð jþnew ¼ a ð jþold þ a ð jþnew s B ¼ a ð jþold s B ga ð jþ a ð jþ2 þ Xp e ð jþ2 a ð jþ2 þ Xp þ s¼ s¼ a ð s jþ2 e ð jþ2 s a ð s jþ2 e ð jþ2 s ga ð s jþ e ð jþ2 s a ð jþ2 þ Xp e ð jþ2 a ð jþ2 þ Xp s¼ s¼ a ð s jþ2 e ð jþ2 s a ð s jþ2 e ð jþ2 s w ðþ2 ::: w ð2þ2 ::: R e ¼ B ] O ::: ::: w ðkþ2 w ð jþ2 ¼ a ð jþ2 þ Xp s¼ a ð s jþ2 e ð jþ2 s ; j ¼ ; 2; ::: ; k ð6þ ð7þ
488 4. Gaussian F for adapive porfolio managemen When he P-based Gaussian F model is adoped for porfolio managemen, porfolio weighs adjusmen can be made under he conrol of independen hidden facors ha affec he porfolio. In he sequel, we illusrae how his can be achieved under he following four scenarios: K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi raio (S p ) [8] wih S p ¼ MðR Þ= VðR Þ given by Ref. [9]. In oher words, he objecive funcion o maximize is: max S p ¼ MðR Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi w;/ VðR Þ ransacion cos Scenario I no no Scenario II yes no Scenario III no yes Scenario IV yes yes Shor sale permission 4.. Scenario I: No ransacion cos and shor sale no permied 8 a ¼ expðf Þ; f ¼ gðy ; wþ; >< subjec o ¼ exp >: b ð jþ n ð jþ n ¼ f ðy ; /Þ:, exp n ðrþ ; ð9þ he assumpions underlying his scenario are no ransacion cos and shor sale no permied. onsequenly, we consider he reurn of a ypical porfolio which is given by Ref. [9] R ¼ð a Þr f þ a 8 a > ; >< Vb V; subjec o >: b ð jþ ¼ b ð jþ x ð jþ ; : ð8þ where r f denoes he risk-free rae of reurn, x denoes reurns of risky securiies, a he proporion of oal capial o be invesed in risky securiies and ( j) b he proporion of a o be invesed in he jh risky asse. Insead of focussing on he mean variance efficien fronier, we seek o opimize he porfolio Sharpe where MðR Þ¼ P ¼ R is he condiional expeced reurn and VðR Þ¼ P ¼ ½R MðR ÞŠ 2 is a measure of risk or volailiy, { y } = is he ime series of N independen hidden facors ha drives he observed reurn series {x } N =, g( y,w) and f (y,/) are some nonlinear funcions ha map y o, respecively, f and n which, in urn, adjuss he porfolio weighs a and ( b j), respecively. Maximizing he porfolio Sharpe raio in effec balances he rade-off beween maximizing he expeced reurn and a he same ime minimizing he risk. In implemenaion, we can simply use he gradien ascen approach. he ime series { y } = can be N esimaed via he Gaussian F algorihm in Ref. [9]. lhough he funcions g( y,w) and f( y,/) are no known a priori, i may be approximaed via he adapive exended normalized radial basis funcion (ENRBF) algorihm in Ref. [2]. Like radial basis funcion (RBF) nework, ENRBF is one of he popular models adoped for funcion approximaion. he general form of RBF is f k ðxþ ¼ Xk w j uð½x l j Š R j ½x l j ŠÞ ðþ
K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 489 ENRBF is an improved modificaion of RBF by replacing w j wih a linear vecor funcion W j x + c j and dividing he erm u([x l j ] R j [x l j ]) over he aggregae of all erms o arrive a f k ðxþ ¼ X k ðwj x þ c j Þuð½x l j Š R j ½x l j ŠÞ X k uð½x l j Š R j ½x l j ŠÞ ðþ where W j is a parameer marix. Basically, each W j x + c j represens a local linear segmen. he ENRBF nework approximaes a globally nonlinear funcion by joining all piecewise linear segmens weighed by probabiliy. he se of parameers o be esimaed is H={l j, R j, W j, c j } j =. k Specifically, g( y,w) and f( y,/) can be modelled by he ENRBF shown below. gðy ; wþ ¼ Xk p¼ f ðy ; /Þ ¼ Xˆk where p¼ uðl p ; R p ; kþ¼ ðw p y þ c p Þuðl p ; R p ; kþ ðŵ p y þ ĉ p Þuð ˆl p ; ˆR p ; ˆkÞ X k expð :5ðy l p Þ R p ðy l p ÞÞ ð2þ ð3þ expð :5ðy l r Þ R p ðy l r ÞÞ he se of parameers in Eqs. (2) and (3) o be k esimaed is H where H = w[/, w={l p,r p,w p,c p } p = kˆ and /={lˆp,rˆp,ŵ p,ĉ p } p =. In general, for each hah, updaing akes place adapively in he following form: h new ¼ h old þ g r h S p : ð4þ where g is he learning sep size, j h S p denoes he gradien wih respec o h in he ascen direcion of S p. ypically, he adapive algorihm shown in able can be adoped for implemenaion. 4.2. Simulaion 4.2.. Daa consideraions ll simulaions in his paper are based on he pas average fixed deposi ineres rae, sock and index daa of Hong Kong. Daily closing prices of he -week bank average ineres rae, 3 major sock indices as well as 86 acively rading socks covering he period from January, 998 o December 3, 999 are used. he number of rading days hroughou his period is 522. he hree major sock indices are, respecively, Hang Seng Index (HSI), Hang Seng hina-ffiliaed orporaions Index (HSI) and Hang Seng hina Enerprises Index (HSEI). Of he 86 equiies, 3 of hem are HSI consiuens, 32 are HSI consiuens and he remaining 24 are HSEI consiuens. he index daa are direcly used for adapive porfolio managemen while he sock prices are used by Gaussian F for recovering independen hidden facors y. 4.2.2. Mehodology We consider he ask of managing a porfolio which consiss of four securiies, he average fixed deposi ineres rae and he hree major sock indices in Hong Kong. he fixed deposi ineres rae is used as he proxy for he risk-free rae of reurn r f. he firs 4 samples are used for raining and he las 2 samples for esing. In he es phase, we firs make predicion on ŷ and xˆ wih ŷ c By and xˆ c ŷ. Moreover, learning is carried ou in an adapive fashion such ha he acual value of x a ime is used o exrac y and modify he parameers once i is known (i.e., once he curren ime is passed ino + ). he Pbased algorihm in able is adoped ha uses hidden independen facors exraced by F for conrolling porfolio weighs. We refer o his approach P-based porfolio managemen. Boh F algorihms wih or wihou RH effec consideraion could be used for his purpose. For simpliciy, in he following experimens, we only adop he one
49 K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 able n adapive algorihm for implemenaion of he P-based porfolio managemen Updaing rules for he parameer se w l new p = l old p + g(j f S p )u(l p,r p,k)s(l p,r p,w p,c p,k)( y l p ) R new p = R old p + g(j f S p )u(l p,r p,k)s(l p,r p,w p,c p,k)j(l p,r p ) W new p = W old p + g(j f S p )y u(l p,r p,k) c new p = c old p + g(j f S p )u(l p,r p,k) Updaing rules for he parameer se / lˆ pnew = lˆ pold ( j) + ĝ(j n Sp )( y lˆ p)u(lˆp,rˆp,kˆ )v(lˆp,rˆp,ŵ p,q,ĉ p,q,kˆ) Rˆ pnew = Rˆpold ( j) + ĝ(j S n p)j(lˆp,rˆp)u(lˆp,rˆp,kˆ)v(lˆp,rˆp,ŵ p,q,ĉ p,q,kˆ) Ŵ new p,q = Ŵ old ( j) p,q + ĝ(j n Sp )y u(lˆp,rˆp,kˆ) ĉ new p,r =ĉ old ( j) p,r + ĝ(j n Sp )u(lˆp,rˆp,kˆ) where g and ĝare learning raes, M(R )=/R = R, V(R )=/R = [R M(R )] 2 " # VðR Þ MðR Þ R MðR Þ X ðr MðR ÞÞ ¼ r f S p ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½VðR ÞŠ 3 B r ð jþ n " # VðR Þ MðR Þ R MðR Þ X ðr MðR ÞÞ ¼ S p ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ½VðR ÞŠ 3 m uðl p ; R p ; kþ ¼ X k expð :5ðy l p Þ R p ðy l p ÞÞ expð :5ðy l r Þ R r ðy l r ÞÞ expðf Þx ð jþ j(l p,r p )=R p ( y l p )( y l p ) R p.5diag [R p ( y l p )( y l p ) R p ], sðl p ; R p ; W p ; c p ; kþ ¼ vðl p ; R p ; W p;q ; c p;q ; kþ ¼ ðwp y þ c p Þ Xk ðwr y þ c r Þuðl r ; R r ; kþ X k expð :5ðy l r Þ R r ðy l r ÞÞ ðwp;q y þ c p;q Þ Xk ðwp;r y þ c r Þuðl r ; R r ; kþ X k expð :5ðy l r Þ R r ðy l r ÞÞ ; ; ÞxðrÞ Þ r f expðf Þ; jþ Þ expðnð Þ expðn ð jþ Þ Þ 2 ; W p,q denoes he ph column of he qh marix, diag[m] denoes a diagonal marix ha akes he diagonal par of a marix M, f = g( y,w) as defined in Eq. (6) and n ( j) is he jh oupu of f ( y,/) as defined in Eq. (7). ; wihou RH consideraion. For each y under es, we can adapively ge f = g( y,w) and n = f( y, / ), and hen he porfolio weighs a = exp (f ) and ¼ expðn ðjþ Þ= P m exp(n (r) ). Finally, reurns can be compued via Eq. (8). For he sake of comparison, we also implemen a radiional approach ha direcly uses sock reurns x insead of hidden facors y [8]. We refer o his approach reurn-based porfolio managemen. 4.2.3. Resuls Fig. shows he reurns of individual securiies ha make up he porfolio during he es phase, wih relevan risk-reurn saisics given in able 2. Graph-
K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 49 Fig.. Reurns of individual securiies in he porfolio. ical comparison of profi gain beween he wo approaches using es daa is shown in Fig. 2. Daily risk-reurn saisics of he porfolios are given in able 3. 4.3. Scenario II: has ransacion cos bu shor sale no permied Scenario II differs from Scenario I in aking ino accoun he effec of ransacion cos. Since any able 2 Daily risk-reurn saisics of consiuens of porfolios omponen name Mean reurn (%) verage ineres rae Risk (%).48. HSI.8.48 HSI.3 2.5 HSEI.2 2.55 Fig. 2. omparaive profi gain of P-based and reurn-based porfolios for Scenario I.
492 K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 able 3 Daily risk-reurn saisics of he porfolio for Scenario I change on b ( j) leads o a ransacion ha incurs a cos on reurn c given by c ¼ a ¼ a r c r c pðjþ =p ðjþ ð þ xðjþ Þ ð5þ where r c is a consan denoing he rae of ransacion cos. onsequenly, we consider he porfolio reurn adjused for ransacion cos given by Ref. [9] R ¼ð a Þr f þ a ½ x ðjþ r c ð þ xðjþ ÞŠ; 8 a > ; >< Vb V; subjec o >: ¼ : ð6þ he P-based algorihm in able could sill be adoped in his case, excep ha he wo erms j f S p and j n ( j) Sp become, respecively, " # VðR Þ MðR Þ R MðR Þ ðr MðR ÞÞ ¼ r f S p ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½VðR ÞŠ 3 2 Xm expðn ðjþ ÞxðjÞ expðn ðjþ B 6 r Þ c 4 Þ Þ 3 expðnðjþ Þ ð þ x ðjþ Þ Þ 7 5 r f expðf Þ; Reurn-based porfolio P-based porfolio X hange in Sharpe raio DS p Mean reurn.6%.4% Risk.48%.8% Sharpe raio.25.728 z38.24% r ðjþ n S p ¼ " VðR Þ MðR Þ X R MðR Þ # ðr MðR ÞÞ expðf Þ ¼ h i x ðjþ r csign expðn ðjþ Þ expðnðj Þ Þ " # Xm Þ expðnðjþ Þ, 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ½VðR ÞŠ 3 expðn ðjþ Þ 3 2 Þ 5 4.3.. Simulaion For he purpose of simulaion, we fix he rae of ransacion cos a r c =.%. Graphical comparison of profi gain beween he wo approaches using es daa is shown in Fig. 3, while daily risk-reurn saisics of he porfolios are given in able 4. 4.4. Scenario III: no ransacion cos bu shor sale is permied Scenario III differs from Scenario I in ha shor sale is now permied. By removing he Fig. 3. omparaive profi gain of P-based and reurn-based porfolios for Scenario II.
K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 493 able 4 Risk-reurn saisics of he porfolio for Scenario II nonnegaive consrains on a and b in Eq. (8), we ge R ¼ð a Þr f þ a r c subjec o Xm Reurn-based porfolio j x ðjþ ð þ xðjþ Þ ¼ ð7þ and he new objecive funcion max S p ¼ MðR Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi w;/ VðR Þ 8 a ¼ f ¼ gðy ; wþ; ><, X subjec o ¼ n ðjþ m n ðrþ >: n ¼ f ðy ; /Þ ; : ð8þ In implemenaion, he algorihm in able could be adoped, excep he wo erms j f S p and j n( j) S p become, respecively, " # VðR Þ MðR Þ R MðR Þ r f S p ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½V ðr ÞŠ 3 2 Xm n ð jþ jþ xð n ð jþ B 6 r c X n ðrþ m 4 X n ðrþ m P-based porfolio X ¼ jþ nð n ðrþ " # V ðr Þ MðR Þ R MðR Þ X ðr MðR ÞÞ f x ð jþ ¼ r ð jþ n S p ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½VðR ÞŠ 3 n ðrþ hange in Sharpe raio DS p Mean reurn.4%.2% Risk.42%.73% Sharpe raio.952.644 z72.69% ðr MðR ÞÞ 3 ðþx ð jþ Þ 7 5 r f n ðrþ jþ nð 4.4.. Simulaion For he purpose of simulaion, shor selling is no applicable o he reurn-based approach. Graphical comparison of profi gain beween he wo approaches using es daa is shown in Fig. 4, while daily risk-reurn saisics of he porfolios are given in able 5. 4.5. Scenario IV: has ransacion cos and shor sale is permied Scenario IV differs from Scenario I in ha he effecs of boh ransacion cos and shor sale on porfolio selecion have o be reaed appropriaely. s a resul, we have R ¼ð a Þr f þ a subjec o Xm ð þ xðjþþš; ½ x ðjþ r c ¼ ð9þ Here, we have he objecive funcion he same as Eq. (8). he P-based algorihm in able could Fig. 4. omparaive profi gain of P-based and reurn-based porfolios for Scenario III.
494 K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 able 5 Risk-reurn saisics of he porfolio for Scenario III sill be adoped in his case, excep ha he wo erms j f S p and j n ( j) Sp become, respecively, " # VðR Þ MðR Þ R MðR Þ r f S p ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½V ðr ÞŠ 3 n ð jþ jþ xð B r f ; r ð jþ n S p ¼ n ð jþ " VðR Þ MðR Þ X ðr MðR ÞÞ ¼ h x ðjþ r csign n ðjþ " # Xm, 2 q 4 Reurn-based porfolio n ðrþ nðjþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½VðR ÞŠ 3 P-based porfolio X ¼ R MðR Þ # f nðj Þ n ðrþ 2 ðr MðR ÞÞ i 3 5 hange in Sharpe raio DS p Mean reurn.6%.9% Risk.48%.92% Sharpe raio.25.265 z65.2% 4.6. Performance evaluaion o summarize he experimenal resuls of he above four scenarios, we have noed he following wo phenomena. Firs, he P-based porfolio in general performs beer han he reurn-based porfolio if he scope of comparison is limied o wihin each scenario, as evidenced by higher S p aained in ables 3 6. I should be noed ha higher S p may arise as a consequence of one of he following siuaions: (i) higher expeced reurn, lower overall volailiy; (ii) higher expeced reurn, same overall volailiy; (iii) same expeced reurn, lower overall volailiy; (iv) boh expeced reurn increase or decrease, wih expeced reurn increases (decreases) a a faser (lower) rae han overall volailiy. Second, if we compare he performance of P-based porfolios across all he four scenarios, especially he porfolio Sharpe raio of scenario III agains I (z9.5%) and scenario IV agains II (z.58%), we may conclude ha performance may be furher improved whenever shor sale is permied. he firs phenomenon reveals he fac ha independen hidden facors may be more effecive in conrolling porfolio weighs. Possible raionales include dimensionaliy reducion, as here are usually only a few hidden facors for a large number of securiies. Wha seems o be a more imporan revelaion is ha he classical P [6] model is sill helpful here. 4.5.. Simulaion In simulaion, we fix he rae of ransacion cos a r c =.%, and shor selling is no applicable o he reurn-based approach. Graphical comparison of profi gain beween he wo approaches using es daa is shown in Fig. 5, while daily risk-reurn saisics of he porfolios are given in able 6. Fig. 5. omparaive profi gain of P-based and reurn-based porfolios for Scenario IV.
K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 495 able 6 Risk-reurn saisics of he porfolio for Scenario IV Reurn-based porfolio P-based porfolio hange in Sharpe raio DS p Mean reurn.4%.6% Risk.42%.88% Sharpe raio.952.88 z9.97% In addiion o considering he downside risk, he so-called porfolio expeced upside volailiy V + can be defined similarly as V þ ¼ X ¼ Z l Z l x ðiþ x ðjþ p lhough shor selling is expensive for individual invesors and no generally permissible for mos insiuional invesors [2] in many markes, relevan experimenal resuls reveal he hypoheical poenial benefi such faciliy migh add o he porfolio reurns. he benefi mainly arises from he exploiaion of downside rend in marke price in addiion o upward movemen. his, in urn, reduces he chance ha he fund is lef idle due o declining sock prices for mos socks, which is more or less a phenomenon when he general marke amosphere is gloomy. 5. P-based porfolio managemen by modified porfolio Sharpe raio In his secion, we consider hree varians of he porfolio Sharpe raio. Specifically, we consider porfolio expeced downside risk V which is represened by ¼ ðx ðiþ X ; x ðjþ ¼ Þdx ðiþ dx ðjþ U ði;jþ ð2þ where b i and x i denoe he porfolio weigh and reurn of he ih risky securiy, respecively, and U (i, j) is a consan. 5.. Modified Sharpe raio wih minimum downside risk and maximum upside volailiy Given ha porfolio variance can be broken down ino porfolio downside risk and upside volailiy, i is desirable o consider he maximizaion of he upside volailiy and minimizaion of he downside risk a he same ime in calculaing he opimal porfolio. In oher words, we can consider maximizaion of he following improved Sharpe raio S p V V ¼ X ðx ðiþ ¼ ; x ðjþ Þdx ðiþ dx ðjþ Z Z l l x ðiþ x ðjþ p max w;/ S pv¼ MðR ÞþV þ V ¼ X ¼ D ði;jþ ð2þ where b i and x i denoe he porfolio weigh and reurn of he ih risky securiy, respecively, and D (i,j) is a consan. 8 a ¼ expðf Þ; >< f ¼ gðy ; wþ; subjec o, X ¼ expðn ðjþ m Þ >: n ¼ f ðy ; /Þ : Þ; ð22þ
496 K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 In implemenaion, he algorihm in able could be adoped, excep he wo erms j f S p and j n ( j) Sp become, respecively, r f S p V¼ expðn ðjþ Þ expðn ðjþ X ¼ ÞxðjÞ r f Xm ÞexpðnðjÞ ÞDði;jÞ expðn ðjþ expðf Þ Þ ; r ð jþ n S p V¼ 2 6 4 X ¼ Bexpðf Þx ðjþ þ X ¼ U ði;jþ Þ, R þ X Xm ÞexpðnðjÞ ÞDði;jÞ ¼ ÞDði;jÞ Þ ði;jþ ÞU Þ ÞexpðnðjÞ Þ 3 7 5 " X # m Þ expðnðjþ Þ expðn ðjþ Þ X ¼ Xm Þ Š ÞexpðnðjÞ ÞDði;jÞ 5... Simulaion We implemen he modified Sharpe raio simulaion using he same se of daa described before and he P-based approach in Scenario I as benchmark for comparison. Graphical comparison of profi gain beween he wo approaches using es daa is shown in Fig. 6, while daily risk-reurn saisics of he porfolios are given in able 7. # 2 Fig. 6. omparaive profi gain under original and modified Sharpe raio. 5.2. Risk minimizaion wih conrol of expeced reurn Some conservaive invesors are more concerned abou risk han reurn. herefore, a more appropriae invesmen sraegy may be o minimize risk while conrolling he expeced reurn. Paricularly, his can be achieved by seing he expeced reurn in Eq. (3) o be a consan specified by he invesor, and he opimizaion essenially becomes a minimizaion of downside risk and a maximizaion of upside volailiy. max w;/ S pv¼ r þ V þ V >< subjec o 8 a ¼ expðf Þ; f ¼ gðy ; wþ; X ¼ expðn ðjþ m Þ Þ; n ¼ f ðy ; /Þ; >: MðR Þ¼r : ð23þ o solve he above opimizaion problem wih equaliy consrains, we adop he augmened La-
K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 497 able 7 Daily Risk-reurn saisics of porfolio under original and modified Sharpe raio grangian mehod. Specifically, for he equaliy consrained problem, maximize subjec o hðxþ ¼; f ðxþ wih respec o x ð24þ he augmened Lagrangian funcion can be wrien as Lðx; kþ ¼f ðxþ khðxþ 2 c½hðxþš2 ð25þ where k is he Lagrange muliplier, c is he penaly parameer. hen, a sequence of minimizaions of he form maximize L ck ðx; k k Þ wih respec o x subjec o xar n ð26þ is performed, where {c k } is a sequence of posiive penaly parameers sequence saisfying < c k < c kþ bk : ð27þ c k l as k l he muliplier sequence {k k } is generaed by he ieraion k kþ ¼ k k þ c k hðˆxþ Original Sharpe raio Mean reurn.4%.24% Risk.8%.3% Upside volailiy.43% Downside risk.35% Sharpe raio S p.728.943 where xˆ is he soluion of Eq. (26). Modified Sharpe raio ð28þ L ¼ Here, he augmened Lagrangian is given by r þ X ¼ X D ði;jþ ¼ k X ¼ R r c 2 U ði;jþ X 2 R r ð29þ ¼ In implemenaion, he algorihm in able could be adoped, excep he wo erms j f S p and j n ( j) Sp are replaced by j f L and j n ( j) L, respecively, where r f L ¼ expðf Þ r ð jþ n L ¼ Br f ðk þ cðmðr Þ rþþ 2 6B 4 B X ¼ ¼ X, X ÞxðrÞ Þ ÞexpðnðjÞ ÞDði;jÞ ði;jþ ÞU Þ ¼ ÞexpðnðjÞ ÞDði;jÞ Þ ði;jþ ÞU ÞexpðnðjÞ ÞDði;jÞ 2
498 K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 expðf Þx ðjþ 3 ðk þ cðmðr Þ rþþ 7 5 Þ expðnðjþ Þ expðn ðjþ Þ " X # m 2 Þ 5.2.. Simulaion We simulae he modified Sharpe raio wih conrol of expeced reurn approach and use he modified Sharpe raio approach in he previous subsecion as benchmark. he predeermined expeced reurn used for he simulaion is r =.5%. Graphical comparison of profi gain beween he wo approaches using es daa is shown in Fig. 7, while daily risk-reurn saisics of he porfolios are given in able 8. 5.3. Reurn maximizaion wih conrol of expeced downside risk Some aggressive invesors are more concerned abou reurn han risk. herefore, a sraegy ha able 8 Risk-reurn saisics of porfolio wih conrol of expeced reurn may beer serve hem is o maximize he expeced reurn while conrolling he expeced downside risk. In paricular, his can be achieved by seing he expeced downside risk in Eq. (3) o be a consan specified by he invesor, and he opimizaion essenially becomes a maximizaion of expeced reurn and upside volailiy. max S pv¼ MðR ÞþV þ w;/ v 8 a ¼ expðf Þ; >< subjec o >: f ¼ gðy ; wþ; X ¼ expðn ðjþ m Þ Þ; n ¼ f ðy ; /Þ; V ¼ v Modified Sharpe raio Mean reurn.24%.7% Risk.3%.79% Upside volailiy.43%.3% Downside risk.35%.23% Sharpe raio S p.943 2.435 Modified Sharpe raio wih conrol of expeced reurn : ð3þ Here, he augmened Lagrangian is given by L ¼ v k c 2 X ¼ X R þ X ¼ ¼ X ¼ U ði;jþ D ði;jþ v D ði;jþ v 2 ð3þ Fig. 7. omparaive profi gain of porfolio wih conrol of expeced reurn. In implemenaion, he algorihm in able could be adoped, excep he wo erms j f S p and
K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 499 j n ( j) Sp are replaced by j f L and j n ( j) L, respecively, where r f L ¼ expðf Þ v r ð jþ n B 2 expðf Þv Xm L ¼ 6 4 Xm ÞDði;jÞ ÞxðrÞ v Xm Þ jþ Þxð þ Xm Þ Þ ðk þ cðv vþþ 7 5 r f jþ Þ expðnð Þ expðn ð jþ Þ " X # m 2 Þ ði;jþ ÞU 5.3.. Simulaion We simulae he modified Sharpe raio wih conrol of expeced downside risk approach and use he modified Sharpe raio approach in he previous subsecion as benchmark. he predeermined expeced downside risk used for he simulaion is v =.2%. 3 able 9 Risk-reurn saisics of porfolio wih conrol of expeced downside risk Modified Sharpe raio Mean reurn.24%.5% Risk.3%.7% Upside volailiy.43%.23% Downside risk.35%.9% Sharpe raio S p.943.2 Graphical comparison of profi gain beween he wo approaches using es daa is shown in Fig. 8, while daily risk-reurn saisics of he porfolios are given in able 9. 5.4. Performance evaluaion Modified Sharpe raio wih conrol of downside risk he invesmen sraegy wih conrol of expeced reurn is well suied for risk-averse invesors. By comparing he saisics shown in able 8 wih ha of able 7, we can see ha no only is he expeced reurn under conrol, bu also is risk lowered. s a resul, S p remains more or less consan. his observaion agrees wih he ene in finance ha risk and reurn go hand in hand wih each oher. Similar reasoning could also be exended o include he case of aggressive profi-seeking invesors by comparing he saisics shown in able 9 wih ha of able 7. 6. onclusion Fig. 8. omparaive profi gain of porfolio wih conrol of expeced downside risk. In his paper, we inroduce how o uilize he Pbased Gaussian F model for adapive porfolio managemen. Since F is based on he classical P model and has he benefi of removing roaion indeerminacy via emporal modelling, using F for porfolio managemen would allow porfolio weighs o be indirecly conrolled by several hidden facors. Moreover, he approach is exended o ailor for invesors according o heir risk and reurn objecives. Simulaion resuls reveal ha P-based porfolio managemen in general excels reurn-based porfolio managemen and porfolio reurns may be somehow enhanced by shor selling, especially when he general marke climae is no ha favorable.
5 K.-. hiu, L. Xu / Decision Suppor Sysems 37 (24) 485 5 cknowledgemens We would like o express our graiude o he anonymous reviewers for heir commens and suggesions ha improved he original manuscrip. he work described in his paper was fully suppored by a gran from he Research Gran ouncil of he Hong Kong SR (Projec No. UHK 4297/98E). References [].D. Back,.S. Weigend, firs applicaion of independen componen analysis o exracing srucure from sock reurns, Inernaional Journal of Neural Sysems 8 (4) (997) 473 484. [2] L. han, J. Karceski, J. Lakonishok, On porfolio opimizaion: forecasing covariances and choosing he risk model, he Review of Financial Sudies 2 (5) (999) 937 974. [3] M. hoey,.s. Weigend, Nonlinear rading models hrough Sharpe raio opimizaion, Inernaional Journal of Neural Sysems 8 (3) (997) 47 43. [4] M. Dungey, V. Marin,. Pagan, mulivariae laen facor decomposiion of inernaional bond yield spreads, Journal of pplied Economerics 5 (2) 697 75. [5] P.. 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Xu, RBF nes, mixure expers, and Bayesian Ying Yang learning, Neurocompuing 9 (998) 223 257. [2] L. Xu, Y.M. heung, dapive supervised learning decision neworks for raders and porfolios, Journal of ompuaional Inelligence in Finance 5 (6) (997) 5. [22] F. Yip, L. Xu, n applicaion of independen componen analysis in he arbirage pricing heory, Proceedings of he Inernaional Join onference on Neural Neworks (IJNN 2) 5 (2) 279 284. Kai-hun hiu received his BB (Hon.) in accounancy from he hinese Universiy of Hong Kong in 2. He is a member of Bea Gamma Sigma. He is currenly a PhD suden of he Deparmen of ompuer Science and Engineering, he hinese Universiy of Hong Kong. Lei Xu is a chair professor in he Deparmen of ompuer Science and Engineering of he hinese Universiy of Hong Kong. He has been concurrenly a full professor a Peking Universiy since 992, and gues professor a hree oher universiies in hina and he Unied Kingdom. fer compleing his PhD hesis in singhua Universiy by he end of 986, he joined Peking Universiy in 987 firs as a posdoc and hen became one of a few excepionally promoed young associae professors of he universiy in 988. During 989 993, he worked as posdoc or senior research associae in several universiies in Europe and Norh merica, including Harvard and MI. Prof. Xu is a governor of he Inernaional Neural Nework Sociey, he chair of he ompuaional Finance echnical ommiee of he IEEE Neural Neworks Sociey, a pas presiden of sia Pacific Neural Neworks ssembly, and an associae edior for six inernaional journals on neural neworks, including Neural Neworks and IEEE ransacions on Neural Neworks. He has published more han more han 24 papers in refereed journals, edied books and inernaional conferences, wih a number of hem being well-cied conribuions. He has received several hinese naional presigious academic awards, including 994 hinese Naional Naure Science ward, and 988 hinese Sae Educaion ouncil Fok Ying ung ward, and inernaional awards (including an 995 INNS Leadership ward). In addiion, he has given over 4 keynoe/plenary/invied/uorial alks in inernaional major neural neworks conferences, such as IONIP, WNN, IEEE- INN, IJNN, ec. He was an IONIP 96 Program ommiee hair, a Join-INN-IONIP3 Program ommiee co-chair and a general chair of IDEL 98, IDEL, IEEE IFER 3 and IEEE FE4. Prof. Xu is an IEEE fellow, a fellow of he Inernaional ssociaion for Paern Recogniion and a member of he European cademy of Sciences.