The mean variance portfolio

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1 Valeriy Zakamuli is a professor of fiace a he Uiversiy of Agder i Krisiasad Norway. valeri.zakamoulie@uia.o Opimal Dyamic Porfolio Risk Maageme Valeriy Zakamuli The mea variace porfolio model developed by Markowiz [95] cosiues a corersoe of moder porfolio heory. However mea variace porfolio opimizaio is difficul o impleme i pracice because of he challeges associaed wih forecasig he mea reurs. As a resul Markowiz s porfolio opimizaio echique has ever bee widely used. Isead mos porfolio maagers have focused o ucoverig udervalued securiies wih poeially high expeced reurs. The volaile ad urbule markes wiessed durig he 000s raised ieres i ivesme sraegies ha focus o opimal porfolio risk corol across asses. These porfolio opimizaio mehods do o require forecasig he mea reurs. The mai examples of risk corol sraegies of his ype are he equally weighed (or aive /N) porfolio (DeMiguel e al. [009] ad Duchi ad Levy [009]) he miimum-variace porfolio (Clarke e al. [006 0]) he maximum-diversificaio porfolio (Choueifay ad Coigard [008]) he risk-pariy porfolio (Maillard e al. [00] Chaves e al. [0] ad Asess e al. [0]) ad he volailiyimig porfolio (Flemig e al. [00 003] ad Kirby ad Osdiek [0]). To impleme hese risk corol sraegies he porfolio maager eeds o forecas a mos oly he covariace marix of reurs. Uil recely he cosesus was ha forecasig he covariace marix could be compleed easily usig a rollig-sample covariace-marix esimaor. A sadard approach is o use he mohly rollig -year covariace marix where varies from 5 o 0 years as a forward-lookig esimae of he fuure covariace marix (DeMiguel e al. [009] Duchi ad Levy [009] Krizma e al. [00]). This is a valid approach if oe assumes ha he covariace marix is eiher ime ivaria or varyig very slowly over ime. However his assumpio does o seem o hold for fiacial asse reurs. O he corary a large srad of lieraure demosraes ha fiacial asse reurs exhibi heeroskedasiciy wih volailiy cluserig. The correlaios amog he reurs of fiacial asses also vary wih ime. May uivariae ad mulivariae geeralized auoregressive codiioal heeroskedasiciy (GARCH) models perform very well i capurig he sylized facs of he variaces ad covariaces of fiacial ime series (for a review see for example Bauwes e al. [006]). Compared wih he sample covariace marix forecas he mulivariae GARCH models deliver superior covariace marix forecas accuracy (Zakamuli [05a]). Moivaed by he fac ha porfolio volailiy is ime varyig may praciioers bega o corol porfolio risk over ime usig he volailiy-argeig sraegy (Collie e al. [0] Fall 06 The Joural of Porfolio Maageme 85

2 Buler ad Philbrick [0] Albeverio e al. [03] ad Perche e al. [06]). The idea behid his sraegy is o cosruc a porfolio wih cosa risk over ime. This porfolio risk corol echique was exesively back esed usig hisorical daa ad exhibied superior performace relaive o ha of is passive couerpar. Iiially he volailiy-argeig sraegy seemed o be a ad hoc sraegy wih is superior performace aribued o he fac ha he coemporaeous correlaio bewee he reur ad volailiy is usually egaive (Frech e al. [987]). Tha is i a period wih aboveaverage volailiy he reur is usually below average; ad i a period wih below-average volailiy he reur is usually above average. Subsequely Hallerbach [0] demosraed ha he volailiy-argeig mechaism improves performace as measured by he Sharpe raio eve if he porfolio mea reur is cosa over ime. Therefore he success of he volailiy-argeig sraegy ca be explaied by wo facors: he egaive relaioship bewee he reur ad volailiy ad he performaceehacig abiliies of risk corol over ime. I a sraegy wih porfolio risk corol over ime he composiio of he risky porfolio is o chaged; oly he capial allocaio bewee he risky porfolio ad he risk-free asse is chaged. I such a sraegy he risky porfolio is ypically a passive sock marke idex. I coras i a sraegy wih porfolio risk corol across asses he composiio of he risky porfolio is revised o he basis of ew covariace marix forecass; he capial allocaio bewee he risky porfolio ad he risk-free asse is o chaged. If oe wishes o reap all he beefis promised by boh porfolio risk corol echiques he mos sraighforward approach is o combie porfolio risk corol across asses wih porfolio risk corol over ime. However a prese here is o heoreical suppor for his combied sraegy. I addiio alhough Hallerbach [0] demosraed ha maiaiig porfolio volailiy a a arge level allows a porfolio maager o improve he performace of a passive porfolio his does o mea ha volailiy argeig represes he opimal form of risk corol over ime. Give he icreasig populariy of porfolio risk corol echiques we begi his aricle by formulaig ad solvig he opimal muliperiod porfolio choice problem. Similar ypes of muliperiod (or ieremporal) porfolio choice problems have received due aeio i fiacial lieraure (a few examples are Mossi [968] Mero [97] ad Li ad Ng [000]). However he mai focus i hese aricles was o fidig he composiio of he opimal risky porfolio. I oher words he goal was o deermie he opimal porfolio diversificaio across asses. The ovely of our approach is o disiguish bewee deermiig he composiio of he opimal risky porfolio ad ideifyig he opimal capial allocaio bewee he risky porfolio ad he riskfree asse. Uder a simplified assumpio we obai he explici soluios o he opimal risky porfolio problem ad opimal capial allocaio problem. These soluios represe he firs coribuio of his aricle. Our soluios reveal ha he ivesor behaves myopically which meas ha he muliperiod porfolio choice problem reduces o a sequece of sigle-period problems. Therefore he soluio o he muliperiod opimal risky porfolio choice problem is give by a sequece of he familiar soluios o he Markowiz sigle-period problems. The soluio o he opimal capial allocaio problem is a porfolio wih risk exposure ha is proporioal o he iverse of he (risky) porfolio variace. Overall he mai message ha our soluios should covey o porfolio maagers is ha i a world i which he mea reurs ad covariace marix vary wih ime i is isufficie o revise he composiio of he opimal risky porfolio. I addiio as ime passes oe mus also revise he capial allocaio bewee he risky porfolio ad he risk-free asse. The secod coribuio of his aricle is o discuss he pracical applicaio of our aalyical soluios. I a real world i which he forecasig of mea reurs is subjec o subsaial esimaio risk porfolio maagers ca shif heir focus from he opimal dyamic mea variace porfolio choice o opimal dyamic porfolio risk maageme. Implemeig such dyamic porfolio risk maageme requires simulaeously corollig porfolio risk boh across asses ad over ime. Our aalyical soluios moivae he risk corol echiques. I he spiri of Kirby ad Osdiek [0] we propose a geeralized ad flexible sraegy wih a mechaism for risk corol over ime. The volailiy-argeig sraegy represes a specific versio of his geeralized sraegy. The hird coribuio of his aricle is o demosrae he beefis promised by opimal porfolio risk corol across asses ad over ime. We coduc wo empirical sudies i which we examie he ouof-sample performace of porfolios wih dyamic risk corol. I he firs sudy we demosrae he superioriy 86 Opimal Dyamic Porfolio Risk Maageme fall 06

3 of porfolio risk corol over ime. We compare () he performace of he passive sraegy wih () he ou-ofsample performace of he dyamic sraegy ha provides heoreically opimal risk corol over ime ad (3) he sraegy wih a volailiy-argeig mechaism. We fid ha he sraegies wih risk corol over ime sigificaly ouperform heir passive couerpar bu here are margial differeces i he performace of he wo compeig sraegies ha iclude risk corol over ime. I he secod sudy we demosrae he superioriy of simulaeously corollig porfolio risk boh across asses ad over ime. I his sudy we compare () he ou-of-sample performace of he sraegy wih risk corol across asses oly wih () he ou-of-sample performace of he sraegy wih risk corol boh across asses ad over ime. We show ha he porfolios wih risk corol boh across asses ad over ime ouperform hose wih risk corol across asses oly. THE SETUP We cosider a capial marke wih k risky asses ad oe risk-free asse. The ivesor eers he marke a ime zero ad ca allocae his wealh amog (k + ) asses. Subsequely he ivesor ca re-allocae his wealh amog he same (k + ) asses a he begiig of each of he followig ( - ) periods. The ivesor exis he marke a ime T which correspods o he T ed of period. All periods are of equal legh: =. For simpliciy of exposiio we assume ha he risk-free rae of reur r f is cosa over ime ad ha he vecor of period risky asse reurs follows he mulivariae ormal disribuio r N ~ ( µ Σ ) () where μ is he k vecor of asse mea reurs ad Σ is he k k variace covariace marix of asse reurs. Noe ha we assume ha boh he vecor of mea reurs ad he variace covariace marix are ime varyig ad kow. The radom vecors of asse reurs i wo disic periods are supposed o be idepede E[(r - μ )(r -j - μ -j ) ] = 0 for j > 0. We separae he ivesor s period complee porfolio allocaio problem io he choice of he opimal period risky porfolio ad he choice of he opimal period capial allocaio bewee he (opimal) risky porfolio ad he risk-free asse. We use he oaio w o represe he k vecor of risky porfolio weighs for period. The risky porfolio mea reur ad volailiy for period are give by µ = w µ = w Σ w P P subjec o w = () where is he vecor of oes (he las equaliy is he budge cosrai). Cosequely he period reur o he risky porfolio ca be wrie as r P P P P = µ + ε (3) where ε P are idepede sadard ormal radom variables such ha ε P ~ N(0 ) ad E[ε P ε P-j ] = 0 for j > 0. Noe ha hese properies appear as a cosequece of our assumpio of idepede ad ormally disribued vecors of asse reurs. The ivesor s period capial allocaio cosiss of ivesig proporio y i he risky porfolio ad cosequely proporio - y i he risk-free asse. Thus he period reur o he ivesor s complee porfolio equals r = y ( r r ) + r. (4) C P f f The mea reur ad he variace of he complee porfolio are give by µ = Er [ ] = y ( µ r ) + r C C P f f = Var[ r ] = y. (5) C C P I our seup he ivesor s opimizaio problem is wofold. The firs problem is he choice of he period weighs w of he opimal risky porfolio; he secod problem is he choice of he period opimal fracio of wealh y o be ivesed i he risky porfolio. To obai he soluios we eed o formulae he ivesor s opimizaio problem. I paricular we eed o defie he objecive fucio of he ivesor. PROBLEM FORMULATION AND ANALYTICAL SOLUTIONS A Brief Overview of Problem-Solvig Approaches I a sigle-period model he mea variace formulaio of Markowiz [95] is by far he mos commo. Fall 06 The Joural of Porfolio Maageme 87

4 I his formulaio here is o separaio bewee he opimal risky porfolio choice problem ad he opimal capial allocaio problem. Isead oe fids he opimal fracios x of he ivesor s wealh allocaed o he risky asses; he remaider ( - x ) is allocaed o he riskfree asse. Cosequely he period reur o he ivesor s porfolio i his seig is give by r = x ( r r ) + r. (6) p f f Mahemaically he sigle-period mea variace formulaio for porfolio selecio ca be posed i several forms (Rachev e al. [008]). Oe of possible forms is he maximizaio of he mea variace uiliy γ max Er [ p ] Var[ rp ] x γ = x ( µ rf ) + rf x Σx (7) where γ is he ivesor s level of risk aversio. Afer he ivesor s opimal allocaio o he risky asses is ideified he composiio of he opimal risky porfolio is foud usig he codiio ha x =. The soluio for he weighs of he opimal risky porfolio is give by Σ w Σ ( µ r f ) = ( ) (8) Σ µ r f where is he iverse of he variace covariace marix Σ (Mero [97]). The properies of he opimal Markowiz porfolio are widely kow. The composiio of he opimal risky porfolio does o deped o he risk prefereces of he ivesor; herefore all ivesors regardless of heir levels of risk aversio choose he same risky porfolio. The opimal risky porfolio has he highes rewardo-variabiliy raio beer kow as he Sharpe raio. Therefore he composiio of he period opimal risky porfolio ca also be foud by maximizig he porfolio s period Sharpe raio µ p r f max SR = subjec o x = (9) x p where µ p ad p deoe he mea reur ad volailiy of he risky porfolio. Whereas he composiio of he opimal risky porfolio does o deped o he ivesor s risk aversio he opimal capial allocaio does. Give he opimal risky porfolio ad he ivesor s coefficie of risk aversio he opimal sigle-period capial allocaio is foud by solvig he followig problem: γ max Er [ C ] Var[ rc ] y γ = y( µ P rf ) + rf y P. (0) The soluio o his problem (see Bodie e al. [009 Chaper 6]) is give by µ P r f y =. γ P () This soluio shows ha he opimal posiio i he risky porfolio is iversely proporioal o he level of risk aversio ad he level of risk (as measured by he variace) ad direcly proporioal o he risk premium offered by he risky porfolio. A sigle-period Markowiz porfolio choice model is urealisic for medium- ad log-erm ivesors who ca rebalace heir porfolios may imes before he ed of he ivesme horizo. Naurally a more realisic muliperiod formulaio of he opimal porfolio choice problem has araced he aeio of fiacial researchers. Oe of he firs coribuios o his opic was made by Mossi [968]. He formulaed he muliperiod problem wihi he framework of expeced uiliy heory ad cocluded ha eve i a sigle-period formulaio here are o closed-form soluios o his problem for a geeral uiliy fucio. The coiuous-ime formulaio of he opimal porfolio choice problem cosidered i Mossi [968] ofe allows aalyical racabiliy. Mero [97] provided he soluio o he coiuous-ime porfolio choice model usig opimal sochasic corol heory. He showed ha whe ivesme opporuiies are cosa over ime or ime varyig wih a kow vecor of mea reurs ad variace covariace marix he ivesor behaves myopically; ad he soluio o he ime composiio of he opimal risky porfolio is give by he same formula used i a sigle-period model (Equaio 8). 88 Opimal Dyamic Porfolio Risk Maageme fall 06

5 Relaively recely Li ad Ng [000] obaied a closed-form soluio o he muliperiod porfolio choice problem i which ivesors seek o maximize he mea variace uiliy of heir ermial wealh. As i our seig hose auhors assume a ime-varyig vecor of mea reurs ad variace covariace marix. As i he previous formulaios he auhors fid he opimal fracios x of he ivesor s wealh ha are allocaed o he risky asses wihou discrimiaio bewee he opimal risky porfolio problem ad he opimal capial allocaio problem. The obaied soluio is give by a complex se of equaios wih obscured ecoomic iuiio. Our Formulaio ad Aalyical Soluio I he previous formulaios he mai focus was o fid he opimal fracios x of a ivesor s wealh allocaed o he risky asses. I coras o he previous formulaios our formulaio of he muliperiod porfolio choice problem simulaeously ideifies he composiio of he opimal risky porfolio w ad he opimal capial allocaio y bewee he risky porfolio ad he risk-free asse. The opimal fracios ca be foud subsequely usig x = y w. The opimal muliperiod porfolio choice problem ca be formulaed eiher as he maximizaio of he mea variace uiliy of he cumulaive reur o he ivesor s complee porfolio over periods or as he maximizaio of he (muliperiod) Sharpe raio. The secod formulaio is more compellig for praciioers because he performace of boh passive ad acive porfolios is ypically measured usig he Sharpe raio. Therefore he aural goal of ay porfolio maager is o creae a porfolio wih he highes Sharpe raio. I pracice o esimae a porfolio s Sharpe raio oe eeds a series of observaios of porfolio reurs. Tha is by aure he esimaed Sharpe raio is a muliperiod performace measure. Specifically accordig o Sharpe [994] he Sharpe raio of a porfolio over [0 T] is compued as he raio of he esimaed mea excess reur o he esimaed sadard deviaio of excess reurs. Mˆ ad Vˆ deoe he saisical mea ad variace respecively of he reur o he complee porfolio i excess of he risk-free rae of reur (give by r C - r f = y (r P - r f )) observed over periods. These quaiies are esimaed by ˆ M = y ( r r ) = P f ˆ ( ( ) ˆ ). V = y rp rf M = () I his formulaio he ivesor s goal is o maximize he followig raio: Mˆ max SR = ˆ. (3) y w V To esure aalyical racabiliy we assume ha is raher large. This assumpio is equivale o he assumpio ha he period legh Δ is raher small relaive o he ui legh. 3 The deails of he derivaio of he aalyical soluio ca be foud i he Appedix. Our aalyical soluio reveals ha uder his simplified assumpio he ivesor behaves myopically. Specifically he composiio of he opimal period risky porfolio give by Equaio 8 is he same as ha of he sigle-period Markowiz model ad he coiuous-ime Mero model. The opimal period capial allocaio is give by y µ = A r P P f (4) where A is a arbirary posiive cosa. Noe ha he expressio for he opimal muliperiod capial allocaio is similar o ha i a sigle-period model (give by Equaio ). I boh cases he opimal period posiio i he risky porfolio y is iversely proporioal o he variace of he risky porfolio ad direcly proporioal o he risk premium offered by he risky porfolio. I ca be show ha he maximizaio of a paricular mea variace uiliy fucio is aaied by choosig A = /γ. Our soluio implies he followig. Whe he ivesme opporuiy se is cosa he opimal risky porfolio is he same for all periods ad he opimal capial allocaio is he same for all periods. Wih ime-varyig ad kow ivesme opporuiies (kow vecor of mea reurs ad variace covariace marix) he composiio of he opimal risky porfolio is o cosa bu he soluio for he weighs of he opimal risky porfolio is give by he same well-kow formula. The mai ew implicaio of our soluio is ha wih ime-varyig kow ivesme opporuiies ivesors mus adjus heir posiios i risky porfolios Fall 06 The Joural of Porfolio Maageme 89

6 over ime i order o maximize he expeced uiliy or he muliperiod Sharpe raio. I oher words whe ivesme opporuiies are ime varyig holdig a cosa posiio i he risky porfolio is subopimal. PRACTICAL IMPLEMENTATION OF ANALYTICAL SOLUTION Our aalyical soluio shows ha he ivesor s muliperiod porfolio choice problem is reduced o a sequece of myopic sigle-period porfolio choice problems. This kowledge grealy simplifies he ivesor s decisio because he opimal period risky porfolio ad capial allocaio (bewee he risky porfolio ad he risk-free asse) deped oly o he period vecor of mea reurs µ ad he variace covariace marix Σ. I oher words o make he opimal period porfolio choice he ivesor does o eed o kow he vecors of mea reurs ad variace covariace marixes for he subseque periods beyod period µ +j Σ +j j. The problem is ha he ivesor s opimal dyamic policy give by Equaios 8 ad is difficul o impleme i pracice because of he challeges associaed wih forecasig he ex-period vecor of mea reurs µ. I coras he variace covariace marix Σ is forecasable a leas i he shor ru. Whe we ackowledge ha forecasig he ex-period vecor of mea reurs is subjec o eormous esimaio risk we ca shif our focus from he opimal dyamic mea variace porfolio choice o opimal dyamic porfolio risk maageme. Specifically we shif our focus from he opimal (mea variace) porfolio of risky asses o he opimal porfolio risk corol across asses. I addiio we shif our focus from he opimal (i he mea variace sese) capial allocaio bewee he risky porfolio ad he risk-free asse o he opimal porfolio risk corol over ime. Our aalyical soluios moivae he risk corol echiques. Firs we cosider he opimal porfolio risk corol across asses. To decrease he esimaio risk as argued by Kirby ad Osdiek [0] oe ca make several simplified assumpios. If oe has o iformaio o he period mea asse reurs for he sake of simpliciy oe ca assume ha all period mea asse reurs are alike; hus µ i = µ where µ i is he period mea reur of asse i. Uder his assumpio Equaio 8 becomes w Σ = (5) Σ which is he soluio for he weighs of he miimumvariace porfolio (Mero [97]). I addiio oe ca assume ha all pairwise correlaios bewee he risky asse reurs are zero. I his case he iverse of he variace covariace marix becomes he diagoal marix Σ = diag(/ / k ) where i is he period volailiy of asse i ad he soluio o he period opimal porfolio weighs becomes i w i k. i = = k i= i (6) Kirby ad Osdiek [0] proposed he followig modificaio of he sraegy: η i wi( η) = η i = k k i= i (7) where he uig parameer η is a measure ha deermies how aggressively he ivesor adjuss he porfolio weighs i respose o volailiy chages. They demosraed ha i he presece of errors i esimaig fuure volailiy i is opimal o decrease he sesiiviy of risk exposure o he chages i volailiy. I oher words he opimal (empirically ideified) sraegy is give by some η 0. Noe ha whe η = 0.5 he opimal period porfolio becomes he risk-pariy porfolio i wi(0.5) = i = k k. i= i (8) Furher as η 0 he opimal period porfolio becomes he equally weighed porfolio. We ow ur o he opimal porfolio risk corol over ime. Agai he dyamic capial allocaio give by Equaio 4 is difficul o impleme i pracice because of he challeges associaed wih forecasig he porfolio mea reur. If oe has o iformaio o he period porfolio mea reur for he sake of simpliciy oe ca assume ha all period mea reurs are alike µ p = µ; addiioally µ > r f. Uder his assumpio 90 Opimal Dyamic Porfolio Risk Maageme fall 06

7 Equaio 4 reduces o he followig (recall ha A is a arbirary posiive cosa): y A =. (9) I his form he sraegy ha opimally corols he porfolio risk over ime cosiss of iverse variace weighig over ime. I his sraegy he complee porfolio risk is adjused accordig o he level of forecased variace of he risky porfolio. The pracical implemeaio of opimal porfolio risk corol over ime ca be achieved as follows. Equaio 9 moivaes us o cosider A as he value of volailiy a which he ivesor chooses o ives 00% i he risky porfolio. Therefore we deoe he value of P by Bchmrk whe y =. The Equaio 9 ca be rewrie as y P Bchmrk =. (0) P We ierpre his Bchmrk as he bechmark volailiy. Agai as argued by Kirby ad Osdiek [0] i he presece of errors i esimaig fuure volailiy i is opimal o decrease he sesiiviy of risk exposure o he chages i volailiy. For his purpose we sugges employig he followig modificaio of he sraegy wih opimal porfolio risk corol over ime: y η η = Bchmrk ( ) () P where he expoe η 0 deermies how aggressively we adjus he ivesor s posiio i he risky porfolio i respose o volailiy chages. Noe ha whe η = 0.5 he modified sraegy becomes he familiar volailiyargeig sraegy y Targe (0.5) = P () where Bchmrk is ierpreed as he volailiy arge Targe. Tha is uder some codiios he volailiy-argeig sraegy migh represe he opimal implemeaio of porfolio risk corol over ime. Whe η = 0 he acive sraegy reduces o he buy-ad-hold sraegy y (0) =. EMPIRICAL ILLUSTRATION Daa I our sudy we use four empirical daases which are lised i Exhibi. These daases which come from he daa library of Keeh Frech 4 are similar o hose used i sudies by DeMiguel e al. [009] Krizma e al. [00] ad Kirby ad Osdiek [0]. The daases represe value-weighed porfolios formed usig differe crieria. For example he daase of porfolios formed o size ad book-o-marke raio cosiss of reurs o 5 sock porfolios ha are he iersecios of 5 porfolios formed o marke equiy ad 5 porfolios formed o he raio of book equiy o marke equiy. The daa for all daases are a a daily frequecy ad cover he period from Jauary 97 o December I addiio we use he 90-day omial U.S. Treasury bill rae as a proxy for he risk-free rae of reur. The daily T-bill rae is also obaied from he daa library of Keeh Frech. Covariace Marix Forecasig Mehod To impleme opimal dyamic porfolio risk maageme we eed o forecas he fuure covariace marix. For his purpose we employ he expoeially weighed movig average (EWMA) mehod. I he EWMA mehod he expoeially weighed covariace marix o day is esimaed usig he followig recursive form: Σˆ = ( λ ) r r +λσˆ (3) where r - is he vecor or reurs o day - ad 0 < λ < is he decay cosa. We follow he recommedaios of E x h i b i Lis of Daases Noes: This exhibi shows he lis of daases cosidered i he empirical sudy. N deoes he umber of porfolios i each daase. Fall 06 The Joural of Porfolio Maageme 9

8 he RiskMerics group ad esimae he daily EWMA covariace marix wih λ = The EWMA covariace marix forecas for he subseque period of K days [ + K - ] is obaied by muliplyig he day esimaed covariace marix by he umber of days: Σ ˆ + K = K Σ ˆ. (4) Whe we eed o forecas oly he variace of he risky porfolio he EWMA mehod reduces o ˆ = ( λ ) r +λ ˆ (5) where ˆ is he expoeially weighed variace o day ad r - is he reur o day -. Superioriy of Porfolio Risk Corol over Time The goal of he empirical sudy i his secio is o demosrae he superioriy of porfolio risk corol over ime. For his purpose for each porfolio i a daase we compare he performace of he passive porfolio o he ou-of-sample performace of he acive porfolio i which oe corols he porfolio risk level over ime. The acive porfolio cosiss of ivesig y i he passive (risky) porfolio ad - y i he risk-free asse. I he acive porfolio he period posiio i he risky porfolio is give by y η = Bchmrk ( ) ˆ (6) where ˆ is he period volailiy forecas made a he ed of period - ad η 0 is a uig parameer. Whe volailiy is fully predicable i is opimal o employ η =. However i real markes because of he presece of errors i esimaig fuure volailiy usig η = migh be subopimal because of is aggressiveess. For he purpose of compariso we also impleme he sraegy wih a decreased sesiiviy of risk exposure o he chages i volailiy. Specifically we use η = 0.5 which is equivale o usig he volailiy-argeig sraegy. I our simulaios we se Bchmrk = 0% (i aualized erms) bu he performace of acive porfolios is o affeced by he choice of he bechmark volailiy. η We also wish o deermie how he porfolio revisio frequecy affecs he performace of a sraegy wih risk corol over ime. For his purpose we cosider hree frequecies a which he composiio of he acive porfolio is revised: daily weekly ad mohly. Our coveio is ha a week cosiss of 5 (radig) days ad a moh cosiss of days. The ou-of-sample simulaios proceed as follows. The reurs for he firs year are used as he iiial widow o esimae he variace of daily reurs usig he EWMA mehod. Thus he reurs o he acive porfolios are simulaed over he period from Jauary 973 o December Afer simulaig he reurs o he acive sraegies for every porfolio i a daase we compue he Sharpe raios of boh he acive porfolios ad heir passive couerpars. To repor he resuls of simulaios we aualize he Sharpe raios ad average hem across each daase. For he sake of illusraio cosider for isace he daase of 5 porfolios formed o size ad booko-marke raio. For each porfolio i he daase we compue he Sharpe raio of he passive buy-ad-hold sraegy over he period from Jauary 973 o December 04. Nex we calculae he average Sharpe raio of all 5 passive porfolios i he daase. The for each porfolio i he daase we simulae ou-of-sample he acive sraegy wih daily rebalacig from Jauary 973 o December 04 ad compue is Sharpe raio. Fially we calculae he average Sharpe raio of all 5 acive porfolios wih daily rebalacig. Thereafer we repea he simulaios of he acive sraegies wih weekly ad mohly rebalacig. The resuls of our simulaios are repored i Exhibi. For every daase i our sudy Pael A repors he average Sharpe raio of he passive porfolio versus he average Sharpe raio of he acive porfolio wih risk corol over ime. I he acive porfolio he period Bchmrk posiio i he risky porfolio is give by y =. ˆ Similarly for every daase i our sudy Pael B repors he average Sharpe raio of he passive porfolio versus he average Sharpe raio of he acive volailiy-argeig porfolio. I he volailiy-argeig porfolio he period Targe posiio i he risky porfolio is give by y =. ˆ Our firs observaio is ha a virually ay revisio frequecy he acive porfolio wih risk corol over ime ouperforms is passive couerpar. Our secod 9 Opimal Dyamic Porfolio Risk Maageme fall 06

9 E x h i b i Average Aualized Porfolio Sharpe Raios across Each Daase (//973 /3/04) Noe: The reurs o all acive porfolios are simulaed ou of sample usig hree differe revisio frequecies: daily weekly ad mohly. observaio is ha he performace of he porfolio wih risk corol over ime is greaes whe he composiio of he acive porfolio is revised a he daily frequecy. Apparely he shorer he ime ierval bewee rebalacig pois he beer he volailiy forecas accuracy for he subseque period (a similar observaio is made by Flemig e al. [003]). A he daily frequecy he average Sharpe raio of he acive porfolio is higher ha he average Sharpe raio of is passive couerpar by 0% 50%. Our hird ad fial observaio is ha i real markes here are margial differeces bewee he performace of porfolios wih heoreically opimal risk corol over ime ad ha of he volailiy-argeig porfolios. Specifically a virually all revisio frequecies he performace of he volailiy-argeig porfolios is margially beer ha ha of he acive porfolios wih heoreically opimal risk corol over ime (ye hese differeces are oable maily whe he acive porfolio is revised eiher a he weekly or mohly frequecies). This empirical resul is i lie wih he argumes preseed by Kirby ad Osdiek [0] who advaced he idea ha i is opimal o smooh he respose o chages i volailiy whe forecas accuracy decreases. Superioriy of Porfolio Risk Corol across Asses ad over Time The goal of he empirical sudy i his secio is o demosrae he superioriy of simulaeously corollig porfolio risk boh across asses ad over ime. For his purpose usig he se of porfolios i each daase as he available uiverse of risky asses we compare he ouof-sample performace of he sraegy wih porfolio risk corol oly across asses versus he ou-of-sample performace of he sraegy wih porfolio risk corol boh across asses ad over ime. We cosider four compeig mehods of porfolio risk corol across asses: he equally weighed (or aive) sraegy he risk-pariy sraegy he volailiy-imig sraegy ad he miimum-variace porfolio. All hese mehods of porfolio risk corol excep for he las oe ca be cosidered as a specific realizaio of he geeralized volailiy-imig sraegy proposed by Kirby ad Osdiek [0]. I his sraegy he period asse i weigh is give by w i ( η) = ˆ k i= i η ˆ i η (7) where ˆ i is he period volailiy forecas of asse i. We remid he reader ha he aive sraegy is obaied whe η = 0. The risk-pariy sraegy is give by η = 0.5. The heoreically opimal volailiy imig is give by η =. Kirby ad Osdiek [0] foud ha i he presece of errors i esimaig fuure volailiy i is opimal o se η >. I heir sudy he bes performace is usually aaied a η = 4. Therefore for he purpose of comparig our resuls wih hose of Kirby ad Osdiek [0] we impleme he volailiy-imig sraegy wih boh η = ad η = 4. The commo feaures of all hese sraegies which belog o he geeralized volailiy-imig sraegy are as follows: They do o require esimaig covariaces ad hey do o geerae egaive weighs. I coras he miimum-variace porfolio requires esimaig he full covariace marix. To make our resuls comparable wih hose of DeMiguel e al. [009] Krizma e al. [00] ad Kirby ad Osdiek [0] we impose shorsale resricios o he weighs of he miimum-variace porfolio. Tha is our miimum-variace porfolio is a log-oly porfolio. Fall 06 The Joural of Porfolio Maageme 93

10 I he sraegy wih risk corol across asses oly he ivesor allocaes 00% of his wealh o he risky porfolio; ha is y = for all (equivalely we ca assume ha he ivesor maiais a cosa posiio i he risky porfolio as ime passes). To also impleme porfolio risk corol over ime we compue he period vecor of weighs of he risky porfolio w ad he forecas porfolio volailiy as ˆ = w Σˆ w. (8) P The period weigh of he risky porfolio i he capial allocaio is he compued as y = (9) Bchmrk ˆ P where as before we se Bchmrk = 0% (i aualized erms). The composiio of he acive porfolios is revised a he daily frequecy. We he coduc saisical iferece o he relaive performace of aleraive sraegies. For his purpose we es he ull hypohesis ha he Sharpe raio of sraegy j deoed SR j is o greaer ha he Sharpe raio of bechmark sraegy b deoed SR b. To perform he es (as i DeMiguel e al. [009]) we apply he Jobso ad Korkie [98] es wih he Memmel [003] correcio. Specifically give SR j SR b ad ρ as he esimaed Sharpe raios ad correlaio coefficie respecively over a sample of size T he es of he ull hypohesis H 0 : SR j - SR b 0 is obaied via he es saisic z = SR j SRb ( ) SR j SRb SR jsrb T ( ρ + + ρ ) (30) which is asympoically disribued as a sadard ormal. I our sudy we cosider wo differe bechmark sraegies. Firs we compue p-value which deoes he p-value of he hypohesis es i which he bechmark is he sraegy wih aive diversificaio across asses ad o risk corol over ime. We deoe his sraegy /N. The oucome of his es will coribue o he debae o wheher opimized porfolios ca ouperform aive diversificaio. We compue he p-value for all sraegies wih ad wihou risk corol over ime. Secod for each sraegy ha has risk corol boh across asses ad over ime we compue p-value which deoes he p-value of he hypohesis es i which he bechmark is he correspodig sraegy wih risk corol across asses oly. The oucome of his es will shed ligh o wheher risk corol over ime adds value o risk corol across asses. As i he previous secio we perform ou-ofsample simulaios of acive sraegies over he period from Jauary 973 o December The resuls of our simulaios are repored i Exhibi 3. Our firs observaio is ha for all four daases he performace of he opimized porfolios wih risk corol oly across asses is higher ha ha of he /N sraegy. I addiio for hree ou of four daases he performace of opimized porfolios is saisically sigificaly higher ha ha of he /N sraegy. Alhough for some opimized porfolios he Sharpe raio is oly margially higher ha ha of he /N sraegy he performace is saisically sigificaly higher a he % level. Therefore our resuls cofirm he fidigs preseed i he sudies by Krizma e al. [00] ad Kirby ad Osdiek [0]. Specifically our resuls also sugges ha he risk corol across asses adds value. Our secod observaio is ha for hree ou of four daases he performace of he porfolios wih risk corol boh across asses ad over ime is higher ha ha of he /N sraegy ad ha of he correspodig porfolios wih risk corol across asses oly. The performace is saisically sigificaly higher a he 0% level or beer for he volailiy-imig porfolio wih η = 4 ad he miimum-variace porfolio. Alhough for some porfolios wih risk corol boh across asses ad over ime he Sharpe raio is sigificaly higher ha ha of he /N sraegy ad ha of he correspodig porfolios wih risk corol across asses oly he p-value of he es remais relaively high ad we cao rejec he ull hypohesis. This is because he reurs of porfolios wih risk corol oly across asses are highly correlaed wih he reurs of he /N sraegy whereas he reurs of he porfolios wih risk corol boh across asses ad over ime exhibi a subsaially lower correlaio wih he reurs of he /N sraegy. The higher (he absolue value of) he correlaio coefficie he larger he es saisics (see Equaio 30); hus i is more difficul o rejec he ull hypohesis whe he reurs o wo porfolios exhibi low or o correlaio. 94 Opimal Dyamic Porfolio Risk Maageme fall 06

11 E x h i b i 3 Resuls of Simulaios wih Risk Corol Across Asses Oly ad wih Risk Corol Across Asses ad Over Time (//973 /3/04) Noes: Sharpe raios are aualized. The p-values of esig he ull hypohesis H 0 : SR j - SR b 0 where SR j is he Sharpe raio of a sraegy j ad SR b is he Sharpe raio of he bechmark sraegy are i brackes; p-value deoes he p-value of he hypohesis es i which he bechmark is he /N sraegy. For he sraegy wih risk corol boh across asses ad over ime p-value deoes he p-value of he hypohesis es i which he bechmark is he correspodig sraegy wih risk corol across asses oly. For he sake of illusraio cosider for isace he daase of 5 porfolios formed o size ad booko-marke raio. The Sharpe raio of he risk-pariy sraegy wih risk corol oly across asses amous o 0.54 whereas he Sharpe raio of he /N sraegy amous o The Sharpe raio of he risk-pariy sraegy is oly margially higher ha ha of he /N sraegy bu because he correlaio coefficie bewee he reurs o hese wo sraegies amous o 0.99 he differece i he Sharpe raios is highly saisically sigifica. I coras he Sharpe raio of he risk-pariy sraegy wih risk corol boh across asses ad over ime amous o The differece wih he Sharpe raio of he /N sraegy is ow sigificaly larger bu because he correlaio coefficie bewee he reurs o he wo sraegies amous o 0.65 he differece i he Sharpe raios is o saisically sigifica a coveioal levels. Our hird observaio is ha for he fourh daase of idusry porfolios we do o obai ay saisically sigifica resuls. I addiio for his daase we fid ha he performace of some porfolios wih risk corol boh across asses ad over ime is worse ha ha of he correspodig porfolios wih risk corol across Fall 06 The Joural of Porfolio Maageme 95

12 asses oly. However his daase is oorious for beig very difficul o use i porfolio opimizaio (Duchi ad Levy [009] ad Kirby ad Osdiek [0]); ad moreover o dae o asse pricig model ca explai he cross-secio of reurs o idusry-sored porfolios (see amog ohers Asgharia ad Hasso [005] Lewelle e al. [00] ad Daiel ad Tima [0]). Overall our resuls sugges ha risk corol over ime adds value ad improves he performace of porfolios wih risk corol across asses oly. Implemeig risk corol over ime i addiio o risk corol across asses migh icrease he Sharpe raio of a sraegy by up o 40%. Compared wih he /N sraegy risk corol boh across asses ad over ime allows a porfolio maager o improve he Sharpe raio by up o 00%. CONCLUDING REMARKS I his aricle we formulaed ad solved he muliperiod opimal porfolio choice problem. The ovely of our formulaio was i separaig he opimal risky porfolio choice problem from he opimal capial allocaio problem. We showed ha i our seig he ivesor behaves myopically. As a resul he muliperiod porfolio choice problem reduces o a sequece of sigleperiod problems wih familiar soluios. Specifically he Markowiz soluio provides a sigle-period soluio o he opimal risky porfolio problem. The opimal capial allocaio cosiss i holdig a posiio i he risky porfolio ha is iversely proporioal o he risky porfolio variace. The mai message of our soluios is ha i a world i which he mea reurs ad covariace marix are ime varyig he ivesor has o revise o a regular basis boh he composiio of he risky porfolio ad he capial allocaio bewee he risky porfolio ad he risk-free asse. I pracice because of he challeges associaed wih forecasig he mea reurs porfolio maagers ca use he explici soluios o he opimal dyamic porfolio choice problem as a moivaio for opimal dyamic porfolio risk maageme. We coduced wo empirical sudies o demosrae he beefis promised by opimal porfolio risk corol boh across asses ad over ime. Our resuls sugges ha porfolio risk corol across asses adds value ad ha complemeig risk corol across asses wih risk corol over ime adds addiioal value. As i Krizma e al. [00] we fid ha he miimum-variace porfolio wih shor-sale resricios performs bes amog all compeig models wih dyamic risk corol. The secod-bes performace is geeraed by he volailiy-imig porfolio of Kirby ad Osdiek [0]. Our research showed ha o reap all beefis from opimal risk corol across asses ad over ime he porfolio maager eeds o obai a accurae covariace marix forecas ad revise he porfolio composiio as ofe as possible. This is because he forecas accuracy deerioraes as he legh of he forecasig horizo icreases. To forecas he covariace marix wih high precisio praciioers are advised o use eiher he mulivariae GARCH forecas or a leas he EWMA forecas. Compared wih he radiioal rollig-sample covariace marix forecas he mulivariae GARCH ad EWMA models allow oe o decrease he covariace marix forecasig error by 50% (Zakamuli [05a]). I coclusio i is worh emphasizig ha we demosraed he superioriy of dyamic porfolio risk corol i he absece of rasacio coss. I oher words we showed he poeial beefis of opimal dyamic porfolio risk maageme. The challege ow is o deermie how o opimally impleme dyamic porfolio risk maageme i he presece of rasacio coss. Zakamuli [05b] preseed a possible approach for implemeig opimal porfolio risk corol over ime i real markes a simple modificaio of he opimal sraegy for risk corol over ime ha allows a porfolio maager o dramaically reduce he amou of radig coss. Previously Lelad [999] cosidered a simplified implemeaio of he opimal rebalacig of he risky porfolio i he presece of rasacio coss. The implemeaio of opimal dyamic risk corol across asses i he presece of rasacio coss ca be achieved alog similar lies. A p p e d i x I his echical appedix we derive he soluio o he opimizaio problem give by max y M SR = ˆ Vˆ (A-) where Mˆ ad Vˆ deoe he saisical mea ad variace respecively of y (r P - r f ) over periods. Because he rigorous derivaio of he soluio is raher leghy i his 96 Opimal Dyamic Porfolio Risk Maageme fall 06

13 echical appedix we prese oly a skech of he derivaio. The deailed derivaio is available from he auhor upo reques. To shore he oaio i he subseque exposiio we replace μ p ad p wih μ ad respecively. Uder he assumpio ha Δ is raher small (he umber of periods is raher large) i ca be prove ha he saisical mea ad variace coverge o he probabilisic mea ad variace: Mˆ M ad Vˆ V. Therefore uder our simplifyig assumpio he ivesor s objecive fucio give by Equaio A- ca be resaed as max y M SR = = V = y( µ rf ). y = (A-) The firs-order codiio for he opimaliy of y i Equaio A- ca be saed as MV y MVy SR y = V V = 0 (A-3) where M = y µ r ad V y f y = are he derivaives of M ad V wih respec o y. The soluio wih respec o y yields M µ r f y =. V (A-4) We ow derive he expressio for he muliperiod Sharpe raio of he sraegy wih opimal proporio y. Iserig he soluio for y (give by Equaio A-4) io he expressio for he Sharpe raio (give by Equaio A-) we arrive a he followig afer a sraighforward ad simple derivaio: SR = µ r f = = = SR (A-5) where SR * deoes he Sharpe raio over periods of he sraegy wih opimal capial allocaio ad SR deoes he square of he period Sharpe raio. Equaio A-5 shows ha he muliperiod Sharpe raio SR * equals he square roo of he mea of he period squared Sharpe raios. Now we ur o he problem of he opimal choice of he period composiio of he risky porfolio. Specifically we eed o solve he followig problem: max SR. (A-6) w Noe ha Equaio A-5 ells us ha o maximize SR * we eed o maximize all period Sharpe raios. Tha is o maximize SR * we eed o solve max SR subjec o w = =. (A-7) w This problem has a well-kow soluio. Fially we demosrae ha ay sraegy of he form µ r f y = A (A-8) where A > 0 is a arbirary cosa also maximizes he Sharpe raio give by Equaio A-. To demosrae his le us rewrie he sraegy give by Equaio A-8 as A V y = a y where a =. (A-9) M Obviously i his form he sraegy y represes a rescaled versio of sraegy y. where a ca be cosidered a scalig facor. To see ha he Sharpe raio of sraegy y = a y is he same as he Sharpe raio of sraegy y iser y = a y io Equaio A- ad see he resul. ENDNOTES Noe ha whereas vecor w saisfies he cosrai w = here are o cosrais o vecor x. If he ivesor s muliperiod soluio o he porfolio choice problem is obaied as a series of sigle-period soluios where each period is reaed as if i were he las oe he we say ha he ivesor behaves myopically. I oher words log-erm ivesors behave myopically whe hey make he same porfolio choices as do shor-erm ivesors. 3 I fiace he ime ierval is measured i years. Our assumpio is jusified whe he ivesor rebalaces his porfolio from daily o mohly frequecies. I his case Δ «. 4 hp://mba.uck.darmouh.edu/pages/faculy/ke.frech/daa_library.hml. REFERENCES Albeverio S. V. Seblovskaya ad K. Wallbaum. Ivesme Isrumes wih Volailiy Targe Mechaism. Quaiaive Fiace Vol. 3 No. 0 (03) pp Asgharia H. ad B. Hasso. Evaluaig he Imporace of Missig Risk Facors Usig he Opimal Orhogoal Porfolio Approach. Joural of Empirical Fiace Vol. No. 4 (005) pp Fall 06 The Joural of Porfolio Maageme 97

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15 Perche R. R.L. de Carvalho T. Heckel ad P. Mouli. Predicig he Success of Volailiy Targeig Sraegies: Applicaio o Equiies ad Oher Asse Classes. The Joural of Aleraive Ivesmes Vol. 8 No. 3 (06) pp Rachev S.T. S.V. Soyaov ad F.J. Fabozzi. Advaced Sochasic Models Risk Assessme ad Porfolio Opimizaio. New York NY: Joh Wiley ad Sos 008. Sharpe W.F. The Sharpe Raio. The Joural of Porfolio Maageme Vol. No. (994) pp Zakamuli V. A Tes of Covariace Marix Forecasig Mehods. The Joural of Porfolio Maageme Vol. 4 No. 3 (05a) pp Volailiy Weighig over Time i he Presece of Trasacio Coss. Workig paper Uiversiy of Agder 05b. To order repris of his aricle please coac Dewey Palmieri a dpalmieri@iijourals.com or Fall 06 The Joural of Porfolio Maageme 99