PASSAUER DISKUSSIONSPAPIERE

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1 ofolio elecion wih ime onsains and a Raional xplanaion of Insufficien Divesificaion and xcessive ading Amin Dolze/ Benhad iee AAUR DIKUIOAIR

2 Heausgebe: Die Guppe de beiebswischaflichen ofessoen de Wischafswissenschaflichen Fakulä de Univesiä assau assau ofolio elecion wih ime onsains and a Raional xplanaion of Insufficien Divesificaion and xcessive Amin Dolze / Benhad iee Diskussionsbeiag. B--06 Beiebswischafliche Reihe I Adesse des Auos / de Auoen: Amin Dolze c/o Lehsuhl fü Beiebswischafslehe mi chwepunk Finanzwischaf und Bankbeiebslehe Wischafswissenschafliche Fakulä de Univesiä assau assau elefon: 085/ elefax: 085/ amin.dolze@uni-passau.de Benhad iee c/o Lehsuhl fü Beiebswischafslehe mi chwepunk Finanzieung und Banken Wischafswissenschafliche Fakulä de Univesiä Mabug Univesiaessasse 4, Mabug, elefon: 064/8-377 elefax: 064/ niee@wiwi.uni-mabug.de. Fü den Inhal de assaue Diskussionspapiee is de jeweilige Auo veanwolich. s wid gebeen, sich mi Anegungen und Kiik diek an den Auo zu wenden.

3 3 ofolio elecion wih ime onsains and a Raional xplanaion of Insufficien Divesificaion and xcessive ading Amin Dolze and Benhad iee Absac ivae invesos have limied ime available fo leaning abou socks as hey need o divide hei ime beween sock analysis and wok. his pape analyzes he influence of leaning consains in he fom of ime consains on pofolio selecion and deives boh opimal pofolio holdings and ime allocaion. Unde ime consains, aional pivae invesos make pofolio choices simila o hose of invesos wih bounded aionaliy, i.e., insufficien divesificaion and excessive ading. hus, ime consains offe an alenaive, fully aional explanaion fo hese eal-wold invesmen phenomena, which have o dae been inepeed pimaily in he ligh of behavioal finance. JL lassificaion: G; G Keywods: xcessive ading; Insufficien Divesificaion; Leaning; ofolio elecion; ime onsain Amin Dolze is wih assau Univesiy, Depamen of Business Adminisaion and conomics, hai of Finance and Banking, Innsasse 7, 9403 assau, Gemany; amin.dolze@uni-passau.de. Benhad iee is wih Mabug Univesiy, Depamen of Business Adminisaion and conomics, hai of Finance and Banking, Univesiaessasse 4, Mabug, Gemany; niee@wiwi.uni-mabug.de.

4 4 ofolio elecion wih ime onsains and a Raional xplanaion of Insufficien Divesificaion and xcessive ading. eliminaies.. Inoducion o he poblem he Inene and financial news on elevision equip pivae invesos, a no chage, wih an abundance of daa concening socks, including hisoical sock quoes, companies fundamenal daa, and analyss epos. Moeove, ansacion coss fo ading ae low. heefoe, i comes as no supise ha Goezmann/Kuma s 004 empiical sudy finds ha ansacion coss, as well as daa acquisiion coss, do no significanly limi pofolio selecion. Howeve, daa canno be used fo decision making; infomaion is equied, i.e., messages ha ae elevan o decision making. hus, daa mus be ansfomed ino infomaion so ha pivae invesos can use a poseioi insead of a pioi disibuions of sock pices. his ansiion fom a pioi o a poseioi disibuions consiues a leaning pocess and, obviously, leaning akes ime. ime is a scace esouce and is scaciy is seen as one of he majo poblems in decision making see, e.g., Juse/affod, 99; Mankins, 004. Hence, limied ime means leaning consains fo decision makes and he quesion aises as o how he ime ha is available should be allocaed beween leaning abou socks via sock analysis and ohe aciviies such as wok. aing fom his descipion of he poblem, he objecives of ou pape ae wofold. Fis, we aim o deemine he opimal soluion o he pofolio selecion and ime allocaion poblems. econd, we wan o demonsae ha nomaive pofolio selecion wih ime consains his undesanding of infomaion combines Mag s 977, p. 4 definiion of infomaion wih he disincion beween infomaion and knowledge in Kuhlen 995, p. 38.

5 5 can be applied o explain wo eal-wold invesmen phenomena, namely, insufficien divesificaion and excessive ading. o dae, hese invesmen phenomena have been inepeed pimaily in he ligh of behavioal finance. o achieve hese wo objecives, we conside a leaning pocess whee a pivae inveso can influence he a pioi disibuion of sock pices by invesing ime in sock analysis. he alenaive use of ime i.e., insead of leaning abou sock pices involves woking longe hous han conacually equied so as o ean bonus paymens. Based on his famewok, he following esuls ae obained. ime consains inoduce inveso-specific componens ino he sucue of opimal pofolio holdings. Moeove, ime consains make i opimal fo decision makes o neihe analyze one sock compleely no o inves an equal amoun of ime in he analysis of each sock. heefoe, decision makes have diffeen infomaion on diffeen socks a diffeen poins of calenda ime even hough he amoun of publicly available daa has no changed. onsequenly, as i is easonable o adap he pofolio saegy o his unequal level of infomaion, insufficien divesificaion and fequen pofolio esucuing can be seen as aional behavio. o bee illusae he conibuions of ou pape, we conas i wih he lieaue. Ou pape s nomaive pofolio model wih ime consains disinguishes iself fom he lieaue on leaning consains van ieuwebugh/veldkamp, 005; eng, 005 in hee majo aspecs. Fis, his lieaue consains leaning by use of an enopy consain following ims 003; we use ime consains insead. ince enopy consains ae ofen jusified based on limied compue capaciy and compue capaciy migh be inceased via invesmen in I echnology, enopy consains can be appoximaed via wealh consains. Howeve, decision makes can do lile o incease limied ime see, e.g., Minzbeg, 973, p. 73 and hus ime consains emain a poblem even if decision makes ae no confoned wih sicly binding budge consains. econd, van ieuwebugh/veldkamp 005 and eng 005

6 6 wok wih model-exogenous leaning consains wheeas ou model employs endogenous leaning consains: ime mus be opimally divided beween eihe sock analysis o exa wok ha will ean bonus paymens, hus making he ime budge fo sock analysis endogenous. hid, van ieuwebugh/veldkamp 005 and eng 005 deal only wih insufficien divesificaion, bu neglec excessive ading; we look a boh. Ou pape s nomaive pofolio model is also diffeen fom Ahn/Kim/Yoon s 006 pofolio selecion wih ime consains. hey use a model-exogenous ime consain o penalize holdings of he isky asse, simila o ansacion coss, and heeby explain invesos limied paicipaion in he sock make. Howeve, hei ime consain is no designed o cope wih he influence of ime consains on invesos leaning. In paicula, Ahn/Kim/Yoon 006 neihe deive he opimal ime allocaion beween seveal socks no do hey analyze he ineacions beween pofolio selecion and ime allocaion. Finally, ou pape s applicaion aspec, he use of nomaive pofolio selecion wih ime consains o explain he eal-wold phenomena of insufficien divesificaion and excessive ading, disinguishes i fom ha pa of behavioal finance lieaue ha deals wih pofolio selecion see, e.g., Babeis/hale, 003. As opposed o behavioal finance, which uses elaively ficionless makes and bounded aional invesos o explain insufficien divesificaion and excessive ading, his pape employs leaning consains in he fom of ime consains and fully aional invesos. he pape is oganized as follows. he emainde of ecion oulines he model seup. In ecion, he opimal soluion o pofolio selecion and ime allocaion is deived. ecion 3 applies he nomaive model of ecion o insufficien divesificaion and excessive ading. he pape ends wih a conclusion ecion 4 and a fomal appendix.

7 7.. Model seup wo foces dive he selecion of ou model seup. Fis, ime consains mus be adequaely poayed; second, explici soluions fo invesmen decisions should be obainable o enable economic inepeaions. An adequae epesenaion of ime consains calls fo a discee-ime model. In a coninuous-ime model such as eng 005, a ime consain canno be inegaed since he leaning pocess mus be insananeous by definiion. A lowe speed of leaning can be capued only wih discee-ime models. Unfounaely, discee-ime models ofen canno be solved in explici fom e.g., Beeden, 004, so in his espec coninuous-ime models ae pefeable because hey yield an easy-o-handle µ-σ-calculus. o deal wih hese conflicing equiemens, we chose a compomise famewok ha is oulined by he following assumpions. Assumpion : Objecive funcion of he decision make Ou decision make is a pivae inveso who does no wok as a pofessional pofolio manage. Ohewise, decisions abou how much ime o spend on wok as opposed o invesmen analysis would no be elevan. he pivae inveso has exponenial uiliy and maximizes expeced uiliy ove eminal wealh. His objecive funcion eads: W max e whee denoes he pivae inveso s absolue isk avesion, {} he uncondiional expecaion opeao, and W is eminal wealh a planning hoizon. Assumpion : Invesmen oppouniy se he pivae inveso can choose beween n isky socks and one iskless asse. ocks ae no subjec o sho selling consains, and hei pices ae joinly nomally disibued. he iskless ae is consan hough ime, idenical fo boowing and lending, and he em sucue is assumed o be fla.

8 8 In addiion o sochasic income fom capial invesmens, he pivae inveso eceives deeminisic income fom employmen. his income consiss of wo pas: conacual income fom employmen and income fom bonus paymens. he conacual income fom employmen is independen of addiional woking hous and by definiion fixed. Bonus paymens equal zeo if no addiional hous ae spen woking, bu ae some posiive amoun when exa hous ae woked. Bonus paymens can each a maximum since companies do no usually offe indefiniely high bonus paymens. Bonus paymens ae a concave funcion of addiional woking hous due o assumed deceasing maginal labo poduciviy. Finally, o simplify noaion, we assume ha he lengh of he ime ineval duing which he pivae inveso canno ebalance his pofolio holdings is he same as he ime ineval ha is he basis fo he deeminaion of bonus paymens. Fo example, if pivae invesos eceive monhly bonus paymens, hey ebalance pofolios on a monhly basis as well. Assumpion 3: Leaning pocess I is assumed ha all invesos have he same fee access o daa such as hisoical sock quoes, companies fundamenal daa, and analyss epos and ha no inveso is pivy o inside infomaion. Hisoical sock quoes allow deiving a pioi disibuions of sock pices. ock quoes, companies fundamenal daa, and analyss epos can be egaded as signals fom which invesos deive a poseioi disibuions. If a poseioi disibuions ae moe infomaive han a pioi disibuions, infomaion is diffeen fom daa; ohewise, daa and infomaion ae equal. he ansiion fom a pioi o a poseioi disibuions consiues he leaning pocess by which pivae invesos can influence how much he a poseioi disibuion is moe infomaive han he a pioi disibuion by invesing ime. oe ha invesmen in he iskless asse does no equie leaning.

9 9 Moe fomally, he leaning pocess develops as follows. I is assumed ha signals and sock pices ae joinly nomally disibued. his means ha he a poseioi disibuion is nomally disibued and can be chaaceized compleely via he veco of condiional means and he condiional vaiance/covaiance maix. Fuhemoe, boh he condiional mean { } he condiional vaiance/covaiance maix and ae funcions of coelaion coefficiens be- ween he andom vaiables signals and sock pices see, e.g., Madia/Ken/Bibby, 99, p. 63 { } { } OV { } OV OV 3 whee denoes he m veco of signals, { } he m veco of uncondiional expeced values of signals, { } he n veco of uncondiional expeced values of sock pices a calenda ime, and OV is he n m uncondiional covaiance maix beween sock pices a calenda ime and signals. is he m m uncondiional vaiance/covaiance maix of signals, he n n uncondiional vaiance/covaiance maix of sock pices a calenda ime. heefoe, in ou model, leaning means ha he pivae inveso impoves he coelaion coefficien beween signals and sock pices via ime invesmen and i holds OV insead of OV as in quaions and 3. If no ime is invesed, coelaion coefficiens beween signals and sock pices equal zeo and a pioi and a poseioi disibuions coincide. he moe ime he pivae inveso invess in sock analysis, he close he absolue values of coelaion coefficiens convege owad and he moe infomaive a poseioi disibuions of sock pices become. his incease in absolue values of he coelaion coefficiens is concave in he ime invesed because i seems easonable o assume ha leaning exhibis de-

10 0 ceasing maginal poduciviy. Keep in mind ha once he a poseioi disibuion has been deived fom sock analysis, his knowledge can be applied o any numbes of socks bough and sold. Fo example, o obain infomaion on 0 pieces of sock i, he same amoun of ime mus be invesed as fo obaining infomaion on one piece of sock i. Finally, i is assumed ha he pivae inveso is small in he sense ha his ansacions do no influence sock pices. heefoe, sock pices do no eflec he infomaion via leaning gleaned by he pivae inveso. Assumpion 4: ime consain he pivae inveso mus mee his physiological needs and wok he conacually equied numbe of hous. He wans o spend any addiional ime available eihe woking moe hous so as o ean a bonus and/o leaning abou socks. In summay, alhough he pivae inveso is aional, he is subjec o leaning consains in he fom of he following ime consain: m n i, j 4 θ,, θ j i h θ whee denoes he ime available fo sock analysis and acquiing bonus paymens a calen- θ da ime θ {cuen calenda ime τ, τ,,, planning hoizon, i, j, θ he ime invesed in he analysis of signal j ou of m signals fo sock i ou of n socks a calenda ime θ, and h,θ is he ime invesed in acquiing bonus paymens via woking longe han conacually equied a calenda ime θ. wo hings mus be kep in mind when consideing quaion 4. Fis, is he ime avail- θ able afe ime fo physiological needs e.g., eaing, sleeping, ec. and conacual woking hous ae deduced fom oal ime available. oal ime available equals he lengh of he ime ineval in ou model as oulined in Assumpion. econd, is inveso-specific; e.g., a pi- θ vae inveso conacually equied o wok eigh hous pe day will have less ime fo sock

11 analysis and fewe hous available in which o ean bonus paymens han a pivae inveso who is conacually equied o wok only six hous pe day.. ofolio selecion and ime allocaion o analyze opimal pofolio selecion and ime allocaion, we poceed in wo seps. Fis, we develop he geneal pofolio and ime allocaion model. econd, we discuss special cases o illusae opimal pofolio and ime allocaion since he geneal pofolio and ime allocaion model does no have closed-fom soluions... Geneal pofolio selecion and ime allocaion Inuiively, he pocess of pofolio and ime allocaion evolves as follows. A cuen calenda ime τ he pivae inveso chooses, fis, he ime o be invesed in he analysis of each sock. hen he obseves a ealizaion of each signal. econd, based on he ealizaion of each signal, he selecs his pofolio of socks. A calenda ime τ, he obains wealh as a consequence of his pofolio decision a calenda ime τ. Using wealh a calenda ime τ as he saing poin, he pocess of pofolio selecion wih ime consains sas anew he pivae inveso deemines his ime allocaion based on he wealh level a calenda ime τ, obseves new signals a calenda ime τ, and may even be able o use he signals obseved a calenda ime τ in he fom of ineempoal leaning. In accodance wih he signals obseved a calenda imes τ and τ and he wealh level achieved a calenda ime τ, he selecs his pofolio holdings. his pocess is epeaed evey calenda ime θ unil calenda ime. o implemen his pocess, he pivae inveso uses backwad inducion. He fis deives pofolio holdings a calenda ime fo evey possible ealizaion of wealh a calenda ime and evey possible ealizaion of signals beween calenda imes τ and. Based on his opimal condiional pofolio selecion, he pivae inveso nex deemines he opimal ime allocaion a calenda ime. Using opimal condiional pofolio selecion and ime

12 allocaion a calenda ime, he calculaes opimal pofolio selecion a calenda ime fo evey possible ealizaion of signals beween calenda imes τ and. hese pofolio holdings ae he saing poin fo deemining he opimal ime allocaion a calenda ime. his pocess is epeaed unil calenda ime τ. Fomally, he decision poblem eads as follows: W Max Max L Max Max e, τ,,w τ,,w L τ τ τ inne poblem a ime oue poblem a ime inne poblem a ime τ oue poblem a ime τ 5 s..: m n i, j fo all θ {τ, τ,., } θ,, θ j i h θ i, j, θ 0 fo all {,...,n} i and j {,..., m} fo all θ {τ, τ,., } 0 fo all θ {τ, τ,., } wih: h, θ h, θ W h W θ θ θ θ h, θ θ whee θ denoes he n veco of numbes of socks bough o sold a calenda ime θ, W θ wealh a calenda ime θ, ` ansposiion of vecos o maices, θ he n veco of sock pices a calenda ime θ, he iskless ae, h h,θ deeminisic bonus paymens a calenda ime θ as a funcion of addiional woking hous a calenda ime θ, and is he h, θ ime invesmen a calenda ime θ ha leads o maximum bonus paymens. - is he m veco of signals fo all socks a calenda ime, τ,- sands fo signals fo all socks and a all calenda imes beween τ and encompasses ineempoal leaning, and θ is he veco of ime invesed in he analysis of all signals fo all socks a calenda ime θ.

13 3 In he geneal model, ime consains ae subjec o an assumpion concening sock pice movemens duing he pocess of sock analysis. ince leaning does no happen insananeously bu, by definiion, akes ime, sock pices will change beween he beginning and end of he sock analysis peiod, i.e., duing he leaning pocess. Addiionally, a new sock pice migh conain new infomaion, meaning ha he leaning pocess mus begin anew, meaning ha moe ime will pass, duing which, possibly, he sock will change pice again, conaining ye moe new infomaion ec. o avoid his cicula pah, we assume, in ou model, ha sock analysis happens ouside of ading hous. his assumpion solves he cicula-pah poblem because sock pices will no longe change duing he pocess of sock analysis. We believe his o be a easonable assumpion since pivae invesos will usually be a hei egula employmen duing sock ading hous. he soluion o decision poblem 5 is as follows. he opimal pofolio holdings a calenda ime sem fom he soluion o he inne poblem a calenda ime :, W τ,,. heefoe, pofolio holdings a calenda ime ae condiional on W, τ,-, and -, as well as on he ime invesed in sock analysis a all calenda imes beween τ and. heefoe, he dependence of, W τ,, on τ,- indicaes ineempoal leaning. he opimal ime invesmen in sock analysis a calenda ime can be deived fom he soluion o he oue poblem a calenda ime : W τ,,. his makes he opimal ime invesmen a calenda ime condiional on W - and τ,- and, as such, condiional on he ime invesed a all calenda imes beween τ and. he soluion o he inne poblem a calenda ime yields opimal pofolio holdings a calenda ime :, W τ,3, ha ae condiional on boh τ,- and he ime invesed a each calenda ime beween τ and. Finally, he opimal ime invesmen a calenda ime - sems fom he soluion o he oue poblem a calenda ime.

14 4 his pocess of deemining opimal pofolio holdings and ime allocaions is epeaed unil calenda ime τ. Obviously, decision poblem 5 is impossible o solve in is geneal fom. On he one hand, he opimal ime invesmen canno be deived in explici fom since leaning is nonlinea due o deceasing maginal poduciviy see Assumpion 3. On he ohe hand, condiional expecaions conain opimal pofolio holdings and ime invesmens and, as such, ae highly nonlinea funcions of he andom vaiables sock pices and signals. heefoe, he epeaed calculaion of condiional expecaions fo calenda imes,,,, τ is beyond an explici soluion. onsequenly, we analyze ineempoal leaning in moe deail insead of deiving he fomal chaaceisics of opimal pofolio holdings and ime allocaions in he geneal case. he foms of ineempoal leaning ae:. he coelaion beween θ and τ,θ allows he pivae inveso o exe diec influence on ρ via ime invesmen, i.e., he a poseioi disibuion of θ τ, θ θ θ can be impoved hough signals ha have occued a leas wo peiods ealie. his fom of leaning, howeve, sesses he ime consain a calenda ime θ.. he pivae inveso may lean abou coelaion coefficiens beween signals and sock pices in he fom of ρ θ θ. his means ha he pivae inveso does no compleely θ θ foge wha he leaned in pevious peiods abou he connecion beween sock pices and signals and hus leaning becomes easie a lae calenda imes, hee emed ineempoal infomaional synegies. 3. A coelaion beween signals τ,θ and θ makes he a poseioi disibuion of θ moe infomaive han is a pioi disibuion. Appendix conains some calculaions o illusae he soluions o he special case of a wo-peiod poblem.

15 5 4. A coelaion beween θ und θ also impoves he a poseio disibuion of θ. he second, hid, and fouh foms of leaning do no sess ime consains a calenda ime θ. he fac ha infomaion abou individual socks can change due o ineempoal leaning, even hough he amoun of daa has no necessaily changed, has an ineesing consequence fo pofolio selecion. he pivae inveso mus updae his pofolio via ading o ake advanage of he new infomaion... pecial cases of pofolio selecion and ime allocaion ince decision poblem 5 canno be solved in explici fom, i is difficul o gain an adequae undesanding of opimal pofolio holdings and ime allocaions wih ime consains. hus we nex conside special cases ha bing us close o o even achieve explici soluions of he opimal pofolio and ime allocaion poblem.... Fis special case: ofolio selecion and ime allocaion in he las peiod In he fis special case i is assumed ha he pivae inveso has eached calenda ime τ so ha he is jus one peiod pio o his planning hoizon. his means ha geneal decision poblem 5 simplifies o: 3 W Max Max e inne poblem a oue poblem a 6 m n j i s..: i,, j h, i, j, 0 fo all {,...,n} i and j {,..., m} 3 oe ha his special case is no idenical o focusing on he peiod beween and of decision poblem 5. In he lae case, opimal pofolio holdings and ime allocaions a calenda ime ae condiional on

16 6 0 h, h, wih: W W h Opimal pofolio holdings follow fom he soluion o he inne poblem, which soluion can be found by solving he following equivalen poblem: 4 h, Max W { } va W wih: { W } { } W h h, 7 va W Relying on he definiions of condiional expecaions quaion and vaiance/covaiances maices quaion 3, he following opimal pofolio holdings ae obained as he soluion o decision poblem 7: [ OV ] { } OV 8 [ OV OV ] OV { } he pofolio holdings 8 consis of hee componens. Fis, a volume componen ha deemines he allocaion of funds beween isky and iskless asses. econd, a sucual componen ha allocaes he isky invesed funds o single socks. his sucual componen iself consiss of wo pas. he fis pa fis line of quaion 8, is he adeoff beween expeced value and isk of each sock ha can be influenced hough leaning OV. he second pa second line of quaion 8 is composed of a coecion pofolio ha adaps pofolio holdings o signal obsevaions. oe ha isk OV dependen on leaning he opimal pofolio and ime allocaion decisions a all calenda imes pio o. In special case 6, he one-peiod decision is, by definiion, uncondiional on all calenda imes pio o. 4 ince signals occu only a calenda ime, we hencefoh dop he signals ime index o simplify noaion.

17 7 and he coecion pofolio ae exacly hose componens ha disinguish he pofolio holdings 8 fom neoclassical opimal pofolio holdings in he hybid model, i.e., { } 9 he decomposiion of pofolio holdings ino an inveso-dependen volume componen and an inveso-independen sucual componen is known as obin sepaaion. quaion 8 shows ha he obin sepaaion beaks down in he even of ime consains. he sucual componen conains leaning-dependen isk OV as a funcion of ime invesed in sock analysis. he ime invesed in sock analysis, howeve, is inveso-specific because i depends on boh he inveso-specific speed of leaning and he ime consain. Inseing he opimal pofolio holdings 8 ino he inne decision poblem 6 povides he foundaion fo calculaing opimal ime allocaions. In ohe wods, he oue poblem of decision poblem 6 eads see Appendix A.: Min e W h h, e { } { } de B 0 m n j i s..: i,, j h, i, j, 0 fo all {,...,n} i and j {,..., m} 0 h, h, wih: B Id cholesky ρ [ ] ρ ρ ρ ρ ρ ρ cholesky ρ whee de. denoes he deeminan of a maix, Id denoes he m m ideniy maix, ρ he m m maix of coelaion coefficiens beween signals, coefficiens beween sock pices a calenda ime, and ρ he n n maix of coelaion ρ is he n m maix of co-

18 8 elaion coefficiens beween sock pices a calenda ime and signals. oe he dependence of ρ on he ime invesed in sock analysis and, hus, he poenial o impove sock analysis hough invesing ime. he necessay condiion of he ime invesed in sock i s analysis hough leaning i,, j abou is connecion o signal j eads ineio soluion: B d de d h d h, i, j, 0 d i,, de j B "income effec" "disibuional effec" Accoding o quaion, he opimum ime allocaion is deemined in a wo-sep pocedue. oe ha in acualiy, boh seps occu simulaneously and ae sepaaed hee fo illusaive puposes only. In he fis sep, i is deemined how he ime budge is divided beween leaning abou socks on he one hand, and woking exa hous o ean bonus paymens on he ohe hand. In he opimum, he negaive impac of invesing ime in sock analysis on iskless bonus paymens income effec mus be exacly offse by is posiive effec on socks a poseioi disibuions disibuional effec. he income effec sems fom he fac ha a highe ime invesmen in sock analysis leads o a decease in iskless bonus paymens because ime invesed in sock analysis canno be used o ean bonus paymens by woking exa hous. oe, howeve, ha boh effecs have diffeen saing poins. he income effec descibes diec, he disibuional effec indiec consequences of leaning on he pivae inveso s objecives. he indiec consequences sem fom he fac ha he disibuional effec needs a ansfomaion vehicle, namely, opimal pofolio holdings -, o ene he pivae inveso s objecives. Fuhemoe, sock pices a calenda ime ae andom vaiables and a bee a poseioi disibuion is no guaanee ha he pivae inveso achieves highe uiliy ex pos. onsequenly, a pivae inveso wih a highe absolue isk avesion invess moe ime in

19 9 sock analysis o impove coelaion coefficiens beween signals and sock pices. he disibuional effec is especially ponounced in he even of infomaional synegies. Infomaional synegies occu when infomaion abou seveal socks can be obained by analyzing jus one sock. Moe fomally, he ime invesed in analysis of signal j exes influence on he coelaion coefficiens of socks i and i, e.g., ρ and ρ i, j j, i, j absence of infomaional synegies whee ρ and i, j i, j, i, j i, j, j, ρ holds., compaed o he he second sep involves dividing he ime budge fo sock analysis as a whole, deemined in he fis sep, beween individual socks.... econd special case: ofolio selecion and ime allocaion in he las peiod wih specified leaning and bonus paymen funcions o chaaceize he opimal ime allocaion fuhe and, in paicula, o examine he disibuional effec beyond he geneal saemens made in ecion.., i is necessay o solve quaion. his ask can be achieved only by paiculaizing he bonus paymen and leaning funcions. In a fis sep, assume h h, W max h, h, W max h, m n j i i, j,, i.e., a linea bonus paymen funcion. hen, quaion simplifies o Wmax h, 3 B d de d i, j, 0 de B "income effec" "disibuional effec" quaion shows ha boh he opimal ime invesmen in he acquisiion of bonus paymens and in he analysis of signal j of sock i is independen of. Obviously, a pivae inveso wih a high ime budge due o, e.g., a low numbe of conacual woking hous, will choose an allocaion of ime beween he acquisiion of bonus paymens and sock analysis ha is idenical o ha chosen by a pivae inveso wih a low ime budge. his is because a

20 0 linea bonus paymen funcion means ha he income effec is independen of, and, h, hus, via he ime consain in poblem 0, independen of. his consan income ef- fec is complemened by a disibuional effec ha depends by definiion only on, i,, no on. In a second sep, assume uncoelaed signals and uncoelaed socks in addiion o linea bonus paymen funcions. Fuhemoe, specify a leaning envionmen whee one sock i has only one signal i, hee ae no infomaional synegies, and leaning in he fom of sock analysis develops accoding o j i,i, ρ i, i i,i, 3 X i,i, whee X is he ime ha mus be invesed in analysis of signal i,, i of sock i so ha he i coelaion coefficien beween signal i and sock i s pice equals. he highe X i,, i is, he moe daa ae available on sock i and he highe he ime invesmen mus be o each a ceain coelaion coefficien beween sock pices and signals compaed o i a lowe X. heefoe, i is easonable o se i,, X lage han i,, because hen coela- ion coefficiens beween sock pices and signals canno each, i.e., socks canno be analyzed compleely. Despie he dependence of X on sock i, signal i,, i, and calenda ime, i i X is independen of he individual pivae inveso, fo X i,, i,i, i is elaed o daa and he amoun of daa is idenical fo all invesos accoding o Assumpion 3. Individual aspecs do affec leaning howeve, accoding o 3, hough he speed of leaning. A pivae inveso wih a leaning funcion accoding o quaion 3 akes i,i, ρ X o each a ceain coelaion level i, i i,i, ρ i,i ; a pivae inveso wih a leaning funcion i,i, ρ i, i i,i, akes longe, namely, i,i, ρ X. i, i i,i, X i,i,

21 Based on he leaning funcion of quaion 3, he opimal ime invesmen i,, i is see Appendix A..4: h, i,, X i i,i, 4 Wmax quaion 4 povides seveal insighs ino opimal ime allocaions. Fis, he pivae inveso does no analyze one sock compleely. his is because he holds a pofolio of socks and wans o lean somehing abou each sock in he pofolio. his is especially ue as hee ae no infomaional synegies in he sense ha infomaion abou all socks canno be obained by analyzing any one sock. 5 econd, he pivae inveso does no spend an equal amoun of ime analyzing each sock. Insead, he invess moe ime analyzing hose socks fo which moe daa ae available socks wih highe X. ocks fo which less daa ae available socks wih lowe i,, X do i,, i no need as high a ime invesmen o achieve an adequae ρ as do socks fo which i,i i,i, moe daa ae available. o ge a feeling which ypes of socks have a high and which have a low X, conside eal-wold sock analysis. mallcap and midcap companies, which have i,, i gea difficuly in aacing analys coveage see, e.g., heae, 003, p., ceae less daa han lage companies o exciing high-gowh companies. Less daa esul in a smalle X i,, small cap fo smallcap and midcap companies: X small cap,, < X. Moeove, complex sig- l age cap,, nals like balance shees ae moe difficul o analyze han simple signals like ode flow of a company; heefoe, X i,balance shee, > X. Finally, he amoun of daa available abou i,ode flow, i i socks can change ove ime. Fo example, in he fouh quae of 005, sola enegy socks eceived a gea deal of coveage by analyss, which ceaed a huge amoun of daa ha had o l ag e cap i i 5 his ype of leaning behavio is in conas o he one in van ieuwebugh/veldkamp 005 whee invesos choose o lean abou one sock. he diffeence aises because in van ieuwebugh/veldkamp 005, sock pices have common facos ha can be leaned by analyzing any sock wha we call infomaional synegies and, also, hei invesos canno lean abou socks isk, bu only abou socks means.

22 be ansfomed ino infomaion. heefoe, sola enegy socks changed fom being quick o analyze socks befoe he fouh quae 005 o being moe slowly o analyze socks fom he fouh quae 005 on, i.e., X < X. sola,i,3d sola,i,4h hid, quaion 4 demonsaes ha ime invesmen in he analysis of sock i inceases wih lowe h, and highe W max. ince he slope of he bonus paymen funcion W max h, inceases wih lowe h, and highe W max, i becomes easie o achieve bonus paymen. heefoe, pivae invesos feel less pessue o inves ime in bonus paymens and he saved ime can be invesed in sock analysis. 3. ime consains and a aional explanaion of insufficien divesificaion and excessive ading his secion deals wih he second goal of he pape he applicaion aspec. We will demonsae ha leaning consains in he fom of ime consains offe a fully aional explanaion fo wo of he mos discussed eal-wold invesmen phenomena: insufficien divesificaion and excessive ading. hose phenomena ae o dae no adequaely explained by neoclassical pofolio selecion see, e.g., Babeis/hale, 003, ecion Insufficien divesificaion Insufficien divesificaion is chaaceized by pofolio holdings ha ae much less divesified han ecommended by nomaive pofolio selecion models see, e.g., Babeis/hale, 003, pp. 0. Howeve, i is no exacly clea how one would define much less divesified han ecommended by nomaive pofolio selecion models. In he secions ha follow, we paiculaize insufficien divesificaion and illusae how adding ime consains o he neoclassical model of pofolio selecion conibues o explaining insufficien divesificaion.

23 es cieion We define he es cieion o deec poenial connecions beween insufficien divesificaion and leaning consains in he fom of ime consains as follows: he developmen of he quoien of neoclassical pofolio holdings fo wo socks i and j compaed wih ha of he pue leaning componens of pofolio holdings wih ime consains. o apply his es cieion, we have o paiculaize is componens. In his connecion, we employ he special case of ecion... heefoe, we specify he quoien of neoclassical pofolio holdings as i,,neocl using pofolio holdings 9. he quoien of pue leaning j,,neocl componens of pofolio holdings wih ime consains consiss of he adeoff beween expeced value and isk OV dependen on leaning, i.e., he fis pa of pofolio hold- ings 8: i,,lean. j,,lean If a deceasing ime budge yields i,,lean fo all sock i j fahe away fom han j,,lean i,,neocl j,,neocl fo all sock i j, hen igh ime consains can conibue o a aional explanaion of insufficien divesificaion. he es cieion is jusified as follows. Babeis/hale 003, p. 0 associae nomaive pofolio models wih neoclassical pofolio heoy. eoclassical uncondiional pofolio holdings, as in obin 965 and Meon 969, do no conain a efeence o leaning and, hus, do no disinguish beween a pioi and a poseioi disibuions. heefoe, hey can be descibed wih he help of he pofolio holdings 9. ince a poseioi condiional pofolio holdings 8 conain leaning consains in he fom of ime consains, hey migh be a good saing poin in he compaison wih uncondiional pofolio holdings. Howeve, cauion is needed egading wo aspecs. Fis, condiional po-

24 4 folio holdings 8 ae chaaceized by isk OV dependen on leaning and he coecion pofolio. he coecion pofolio conains a combinaion of limied leaning due o ime consains and signal-induced coecion ems and heefoe is a mixue of wo compleely diffeen componens. o analyze he elaion beween leaning consains in he fom of ime consains and insufficien divesificaion, i is necessay o concenae on pue leaning effecs and, hus, on he adeoff beween expeced value and isk OV dependen on leaning. econd, a diec compaison of he pofolio holdings 9 wih he pue leaning effecs of quaion 8 is inadequae. quaion 9 conains an infomaion level of zeo, wheeas quaion 8 is chaaceized by vaious infomaion levels depending on he ime budge. o ge aound his poblem, i is easonable o focus on he developmen of i,,lean elaive o j,,lean i,,neocl fo seveal ime budges. eihe j,,neocl i,,lean compaed o j,,lean i,,neocl fo a fixed ime budge no he size of he pofolio holdings 8 compaed o ha j,,neocl based on quaion 9 ae adequae measues Resuls and inepeaion he connecions beween leaning consains in he fom of ime consains and insufficien divesificaion can be bes illusaed by means of a numeical example. o do his, we will employ he famewok of ecion.. pofolio selecion and ime allocaion in he las peiod and he leaning envionmen of ecion.. one sock i has only one signal i, hee ae no infomaional synegies, and leaning in he fom of sock analysis develops accoding o quaion 3. o fuhe simplify he analysis, we assume ha signals ae uncoelaed and ha hee ae no paymens fom conacual wok and no bonus paymens.

25 5 he following paamees ae he basis fo ou numeical analysis. 6 he pivae inveso can choose beween wo socks and one iskless asse, wih sock pices a calenda ime,, 00, expeced values {, } 05 and {, } 07. 5, and vaiance/covaiance maix σ, va σ,, ρ,,, σ, σ va ρ,,,,, he iskless ae equals % 058 pe annum, and he pivae inveso has an exogenous income of W - 5,000 UR. 7 he pivae inveso s absolue isk avesion is Wih espec o sock analysis, wo scenaios ae consideed. In he fis scenaio, hee ae moe daa available fo ock han fo ock, i.e., X,, > X,, In he second scenaio, X,, < X,, Using hese peliminaies, we plo he es cieion quoien i,,lean j,,lean, based on quaion 8 condiional holding, vesus i,,neocl j,,neocl, based on quaion 9 uncondiional holdings, as a funcion of he ime budge and obain: 6 We do no sive o explain pofolio holdings found in he empiical lieaue. In paicula, we do no claim ha he paamees dealing wih he ime consain ae empiically valid alhough we believe hey ae ealisic. 7 A iskless ae of % is in accodance wih he cuen em sucue of inees aes in Gemany. 5,000 UR is appoximaely he goss naional income pe capia fo Gemany in 004 accoding o Wold Bank saisics. 8 he absolue isk avesion is chosen so ha he pofolio weighs w i, i, i, do no conain a sho W sale of one isky o he iskless asse: w,- 5.3%, w,- 9.3%, and w 0, %.

26 6 Fig. a. ocks es cieion quoiens when X,, > X,, 0. 8 cenaio Fig. b. ocks es cieion quoiens when X,, 0.64 < X,, 0. 8 cenaio Figues a and b illusae ha he ineacion beween availabiliy of daa diffeen levels of X in cenaios and and ime budges povides ich divesificaion paens, including insufficien divesificaion: In cenaio, he es cieion quoien fo condiional pofolio holdings is close o han ha fo uncondiional holdings, iespecive of he ighness of he ime consain. Fo ime budges aound 0.4, condiional pofolio holdings even show naïve divesificaion, i.e., he es cieion coefficien is aound. By conas, in cenaio, condiional pofolio holdings ae significanly moe unequal han uncondiional pofolio holdings fo all ime budges consideed. his means ha invesos wih diffeen ime budges follow compleely diffeen levels of divesificaion even hough hey have idenical daa, isk avesions, and wealh. Moeove, socks wih a diffeen amoun of daa diffeen X induce diffeen divesificaion paens, as cenaios and illusae, alhough hei uncondiional pofolio holdings ae independen of he amoun of daa X. ince a pivae inveso opimally invess a diffeen amoun of ime analyzing each sock, he possesses diffeen infomaion on each sock in he opimum. onsequenly, insufficien divesificaion of pofolio holdings can be explained hough a nomaive pofolio selecion

27 7 model, namely, pofolio selecion wih leaning consains in he fom of ime consains. hee is no need o aibue i solely o bounded aionaliy. 3.. xcessive ading xcessive ading occus when pofolios ae esucued moe ofen han can be jusified by he availabiliy of new infomaion see, e.g., Babeis/hale, 003, p. 03. Howeve, once again, i is no exacly clea how one would define esucued moe ofen han can be jusified by he availabiliy of new infomaion. In he secions ha follow, we paiculaize excessive ading and illusae how adding ime consains o he neoclassical model of pofolio selecion conibues o explaining excessive ading es cieion We define he es cieion o deec poenial connecions beween excessive ading and leaning consains in he fom of ime consains as follows: he quoien of he pue leaning componens of pofolio holdings wih ime consains fo one sock i a diffeen calenda imes and afe he incenive o ebalance neoclassical pofolio holdings has been eliminaed. o apply his es cieion, we have o paiculaize is componens. Based on he special case of ecion.., we specify he quoien of pue leaning componens of pofolio holdings wih ime consains a diffeen calenda imes as i,,lean j,,lean, he muli-peiod analogue 9 of he pue leaning componen of he pofolio holdings 8. If i,,lean fo all socks i diffes fo diffeen ime budges even hough he incenive fo j,,lean ebalancing neoclassical pofolio holdings has been eliminaed, hen ime consains can be successfully conneced wih excessive ading.

28 8 he es cieion is jusified as follows. he easonableness of concenaing on he pue leaning componen of he pofolio holdings 8 in ode o sudy he effecs of ime consains was peviously jusified see ecion 3... o elaboae he fequency aspec of excessive ading i is, in addiion, necessay o measue he fequency of pofolio esucuings wih ime consains agains he fequency of pofolio ebalancing in a neoclassical wold, i.e., o sepaae infomaion-induced ading fom noninfomaion-induced ading. All neoclassical dynamic pofolio selecion models advocae pofolio esucuings. Fo example, he discee-ime models of Fama 970 and Hakansson 970 esucue hei opimal pofolio holdings a evey poin in calenda ime. he coninuous-ime models of, e.g., Meon 969, 97, 973, even ebalance pofolio holdings coninuously and hus make excessive ading impossible. ofolio ebalancing in neoclassical dynamic pofolio selecion is based on he fac ha calculaed and acual pofolio holdings usually deviae when he andom vaiable sock pice becomes known. he eason fo his noninfomaion-induced ebalancing is ha he calculaed pofolio holdings ae based on momens of he sock pice disibuion, wheeas acual pofolio holdings ae based on acual sock pices. his means ha neoclassical pofolio holdings ae no esucued only if a ceain ealizaion of he andom vaiable sock pice occus. his ealizaion of he andom vaiable sock pice is wha we call compensaed sock pice. Using compensaed sock pices and calculaing pofolio holdings 8, we can be sue ha evey esucuing of i,-,lean mus be infomaion induced, i.e., elaed o leaning consains in he fom of ime consains alone Resuls and inepeaion he numeical analysis in his secion is based on he paamees of ecion 3... In addiion, we use he following paamees o exend ou example o he dynamic wold. 9 A deivaion of pofolio holdings fo his special case is conained in Appendix A.

29 9 he pivae inveso is pu ino a wo-peiod famewok. ock pices expeced values a calenda ime ae {, } 0. 5 and {, } he vaiance/covaiance maix a calenda ime eads 5 0 0, ha of calenda ime , and all ineempoal 6 coelaion coefficiens beween sock pices ae se o zeo. he compensaed sock pices can be calculaed wihin his envionmen as follows: 0, and, o simplify noaion, we fuhe assume and X i, X. i,, i i Using hese paamees, we plo he es cieion quoien i,,lean, based on quaion 8, j,,lean i,,neocl vesus, based on quaion 9, as a funcion of and obain: i,,neocl Fig. a. ocks es cieion quoiens when X,, X,, > X,, X,, 0. 8 cenaio Fig. b. ocks es cieion quoiens when X,, X,, 0.64 < X,, X,, 0. 8 cenaio Figues a and b demonsae ha diffeen ime budges and diffeen availabiliy of daa diffeen levels of X in cenaios and lead o diffeen pofolio esucuings since he es cieion quoiens ae usually unequal o. In fac, in his numeical example, he moe ime ha is available, he moe he pivae inveso can lean and he moe ponounced 0 he calculaions ae available fom he auhos as Maple file.

30 30 he pofolio esucuing will be. Only fo one paicula is no pofolio esucuing opimal es cieion quoien equals. his means ha invesos wih diffeen ime budges will esucue hei pofolios diffeenly even hough hey have idenical daa, isk avesions, and iniial wealh. Moeove, socks wih diffeen amoun of daa diffeen levels of X induce diffeen ebalancing paens see cenaios and even hough hei neoclassical pofolio holdings ae no esucued in he opimum. ince, a each poin of calenda ime, a pivae inveso opimally spends a diffeen amoun of ime analyzing each sock, he possesses diffeen infomaion on each sock a diffeen calenda imes. onsequenly, fequen pofolio ebalancing can be explained hough a nomaive pofolio selecion model, namely, pofolio selecion wih leaning consains in he fom of ime consains. hee is no need o aibue i solely o bounded aionaliy o o label as excessively fequen pofolio ebalancing. 4. onclusion We began his pape wih he obsevaion ha in his age of he Inene and he eady availabiliy of financial news on elevision, pivae invesos can obain, wihou cos, an abundance of daa concening socks, including hisoical sock quoes, companies fundamenal daa, and analyss epos. Howeve, pivae invesos do no have enough ime o ansfom daa ino infomaion because hey mus mee physiological needs and wok, leaving lile ime o obain infomaion abou socks via sock analysis. aing fom his famewok, he following esuls ae obained. ime consains inoduce inveso-specific componens ino he sucue of pofolio holdings. Moeove, due o ime consains, i is no opimal fo decision makes o eihe analyze one sock compleely o inves an equal amoun of ime o he analysis of each sock. heefoe, decision makes have diffeen infomaion on diffeen socks a diffeen calenda imes even hough he amoun of

31 3 publicly available daa has no changed. onsequenly, i is easonable o adap he pofolio saegy o his unequal level of infomaion, which migh esul in insufficien divesificaion and fequen pofolio esucuing. By basing ou model on a fully aional insead of a bounded aional pivae inveso, we offe a new explanaion of eal-wold invesmen phenomena, phenomena ha have, o dae, pimaily been inepeed in ligh of behavioal finance. We do no ejec he findings based on behavioal finance; ahe, we poin ou ha hee ae ohe explanaions fo eal-wold invesmen phenomena. We believe we have aken a fis sep owad he unificaion of mainly descipive behavioal finance and nomaive pofolio heoy. Also, we believe we may have found an answe o he quesions posed by hleife 000, p. 95: Why do diffeen invesos have diffeen models of wha ae good invesmens and why do hey ade so much wih each ohe? ehaps i is because hey ae subjec o diffeen ime consains and, hus, have diffeen amouns of infomaion available o guide hem. Appendix A.. Opimal ime allocaion in he saic model o ansfom decision poblem 6 ino he basis fo deemining opimal ime allocaion in poblem 0 using he opimal he pofolio holdings 8, seveal inemediae seps ae necessay. A... Fis sep: alculaion of he expeced value of he inne poblem 6 using opimal he pofolio holdings 8 ince W and signals ae joinly nomally disibued, he expeced value of he inne poblem 6 eads

32 3 { } W va W W e e A. wih: { } { } h, h W W W va Using he opimal pofolio holdings 8 in he ineess of simplificaion, he expession fo has no been subsiued ino pofolio holdings { } A. { } OV { } W and W va in quaion A. can be calculaed. { } W eads, afe using is definiion in quaion and pefoming some simplificaions { } W h, h W A.3 { } { } { } { } OV { } { } OV OV W va eads, afe using is definiion in quaion 3 and pefoming some simplificaions W va { } { } A.4 { } { } OV { } { } OV OV

33 33 Inseing he expeced value fom quaion A.3 and he vaiance fom quaion A.4 ino quaion A., we obain fo he expeced uiliy A. W e h, h W e A.5 { } { } exp { } { } OV { } { } OV OV A... econd sep: alculaion of he expeced value of he oue poblem 6 using he expession fo he inne poblem A.5 o calculae he expeced value of he oue poblem and, hus, o have a foundaion fo deemining he opimal ime allocaion, he following expecaion wih espec o signals mus be compued: W e A.6 alculaions will be simplified by swiching fom nomally disibued vaiables o sandad nomally disibued vaiables Y by using he ansfomaion { } cholesky Y. Fo ha eason, i holds W e h, h W e A.7 { } { } exp his ansfomaion sems fom he consideaion ha { } { } cholesky cholesky should be ansfomed ino Id Y Y and has in he even of jus one andom vaiable he well-known manifesaion { } Y sd.

34 34 { } { cholesky Y cholesky cholesky Y OV OV Y Using he signals m-dimensional sandad nomal densiy funcion o calculae he expeced value in quaion A.7 las wo lines of quaion A.7, quaion A.7 can be wien as W e h, h W e A.8 { } { } exp π m Id exp y y L { } cholesky exp y OV cholesky cholesky exp y OV OV y m y d y d L whee y denoes he ealizaions of he andom vaiables Y, and Id is he ideniy maix. o compue he inegals in quaion A.8, we sive o ansfom he inegans in quaion A.8 o densiy funcions so ha he inegals equal. his equies, fis, ha he ems Id exp y y and cholesky exp OV OV y cholesky y ae combined and, second, ha hey ae ansfomed ino a new sandad nomally disibued andom vaiable wih ealizaions z ; fomally Id exp z z Id exp y y A.9

35 35 exp y cholesky OV OV cholesky y exp y Id cholesky OV OV cholesky y B Applying he agumen se ou in foonoe leads o y cholesky B z. Along wih changing he exponen fom y o z, we also need o adap he inegaion vaiables. oe ha d y conains he fis ow of he veco y and hus equals he fis ow of y cholesky B z. Fo ha eason, d y L d y can be obained by muliplying he m elemens of he main diagonal of cholesky B and swiching o d z L d z. m heefoe, he nex poblem is wiing muliplying he elemens of he main diagonal of cholesky B in a moe concise fom. Fom he ansfomaion of nomally disib- ued andom vaiables ino sandad nomally disibued andom vaiables see he agumen in foonoe and he ex accompanying same, we know ha he poduc of he elemens of he main diagonal of cholesky equals de, whee de. denoes he deeminan of a maix. By analogy, we obain fo he poduc of he elemens of he main diagonal of B he shoe expession de cholesky B. Based on he above findings, quaion A.8 simplifies o e W W h h, e A.0 exp { } { }

36 36 π m Id exp z z L { } cholesky exp OV cholesky z B m z d z d L de B Finally, o finish he ansfomaion o densiy funcions, we need o complee he squae of he exponen in quaion A.0. Define { } OV K cholesky cholesky B, hen he squae can be compleed by adding ± K K o he exponen of he em in he second and hid o las lines of quaion A.0. Afe hese ansfomaions, quaion A.0 eads W e h, h W e A. { } { } exp e K K π Id exp m K z K z L m z d z d L de B which yields W e h, h W e A.

37 37 { } { } exp e K K de B A..3. hid sep: implificaion of he componens of quaion A. o deive quaion 0 oblem 0 disinguishes iself fom quaion A. in wo especs. On he one hand, he second line of quaion A. mus be meged wih K K ; on he ohe hand, B - mus be simplified. edious calculaions lead o { } { } exp A.3 e K K { } { } e o expess cholesky cholesky Id OV OV B wih he help of coelaion coefficiens, poceed as follows. Wie all vaiance/covaiance o covaiance maices as he poduc of he maix of sandad deviaions and he maix of coelaion coefficiens, i.e., diag ρ, diag diag ρ OV, and diag ρ, whee diag denoes he diagonal maix of sandad deviaions of sock pices, diag he diagonal maix of sandad deviaions of signals, and ρ denoes he maix of coelaion coefficiens beween andom vaiables. Wih hese subsiuions, we obain B Id A.4 cholesky diag diag diag diag cholesky ρ ρ