The Markovian portfolio selection model with GARCH volatility dynamics

Transcription

1 Procdings of h 5h WSEAS Inrnaional Confrnc on Economy and Managmn Transformaion (Volum I Th Markovian orfolio slcion modl wih GARCH volailiy dynamics GAETANO IAUINTA, SERGIO ORTOBELLI LOZZA, ENRICO ANGELELLI Darmn MSIA Univrsiy of Brgamo 47-Via di Caniana, ; Brgamo - Ialy; -mail addrsss: sol@unibgi; gaano,iauina@unibgi Diarimno di Modi uaniaivi Univrsiy of Brscia 5 -Cda S Chiara, 5; Brscia - Ialy; -mail addrss: angl@counibsi Absrac: - This ar rooss and comar orfolio slcion modls undr h assumion ha h orfolios of rurns follow a GARCH y rocss W comu h ric/rurn disribuion a som fuur im aroximaing h GARCH rocss wih a Markov chain W rsn an x-os comarison of orfolio slcion sragis alid o som asss of h US Mark Sinc h oimizaion roblms rsn mor local oima, w imlmn an hurisic algorihm for h global oimum in ordr o ovrcom h inrinsic comuaional comlxiy of h modls Ky-Words: - GARCH modls, Markov chains, orfolio slcion, rformanc sragis, x-os analysis Inroducion In his ar, w modl h rurn orfolios wih a Markov chain ha accoun h GARCH voluion of h rurns In aricular, w us h Duan and Simonao s aroximaion of h rurns voluion (Duan, Simonao in orfolios slcion roblms Undr his disribuional hyohsis w comar h x-os rformanc of som orfolio slcion sragis Thr is a gnral consnsus on h imoranc o modl h im varying volailiy (Engl [98], Bollrslv [986] and h lvrag ffc (Black [976] Svral mirical sudis hav showd ha hs saisical ascs srv o solv many biass bwn horical and mirical rics (s Bakshi, Cao and Chn [997], Engl and Musafa [99], and Hson and Nandi [] Bcaus hr is wid consnsus ha h varianc of h financial ass rurns is im varian, a gra amoun of ffors ar dircing o raliz mahmaical modls which, by choosing h varianc dynamics as h modl cornr-son, should b ffcivly abl o modl financial rics Surly h GARCH modl is a rfrnc insrumn o sudy h volailiy dynamics, and among is advanags hr is is high flxibiliy o b suiabl o caur h mos imoran faurs of h financial variabls In his work w analyz h imac of choics basd on h GARCH aramric characrizaion of financial ass sris I is o no ha h assag from h GARCH aramric characrizaion of financial ass sris o h comuaion of h ric/rurn disribuion a som fuur im is no immdia In ordr o build orfolio walh disribuion w us Duan and Simonao's GARCH aroximaion (Duan, Simonao Morovr, w xnd h Duan and Simonao's idas o ohr ossibl GARCH y modls (s Glosn, al(993, Nlson (99As hs auhors xlain many GARCH modls and in aricular h GARCH(, modls can b rrsnd as a bivaria Markovian sysm (i, h sa of h rocss is uniuly rrsnd by ric and varianc sas This faur allows o aroxima GARCH modls by a discr Markov chain To build h ransiion marixs w us h mhod discussd by Duan and Simonao (, Duan al (3 for aramric Markovian rocsss Wih aramric orfolio slcion modls h ransiion marix dnds on h aramrs of h undrlying mulivaria Markov rocss and h aramrs ar funcions of h orfolio wighs Thrfor w should chck for a global oimum for mos of h orfolio slcion roblms In h ar w us an oimizaion hurisic algorihm Angllli and Oroblli (9 ha rducs normously h comuaional comlxiy wih rsc o ohr global oimizaion aroachs lik simulad annaling In h following mirical comarison, w rsn som orfolio slcion sragis ha us diffrn GARCH modls All of hm ar basd on h simaion of h disribuion of h rurns a fuur ims undr h assumion ha h rsiduals of log rurns orfolios follow a GARCH rocss Th ar is organizd as follows In Scion w show h modls imlmnd In Scion 3 w discuss h Markovian aroximaion of orfolio valu ISSN: ISBN:

2 Procdings of h 5h WSEAS Inrnaional Confrnc on Economy and Managmn Transformaion (Volum I and w formaliz h orfolio slcion modl In scion 4 w rform an mirical comarison among diffrn orfolio slcion modls Finally, w brifly summariz h ar Porfolio Valu Wih GARCH Volailiy Dynamics L us considr a discr-im conomy and n risky asss wih log rurns r = r,, K, r n, ' If w dno by x = [ x,, K x ] ' n h vcor of h osiions akn in h n risky asss, hn h orfolio log rurn during h riod, is givn by [ ] n ri, ( x, = i i= r x ( In aricular, w assum ha invsors wan o maximiz h rformanc of hir choics a a givn fuur da T Now w inroduc h alrnaiv GARCH volailiy dynamics modls imlmnd in his work Suos ha undr h hisorical masur P h daily orfolio log-rurn is dscribd by h following rlaion: W ( x r = ln = μ σ ε W ( x whr W ( osiion x, and ε φ ( ( x is h orfolio valu a im, wih ass, undr P For convnion w considr h iniial orfolio walh ual o (i W = In his work w us h sandard GARCH(, (s Bollrslv, T(986 and som wllknown xnsion of h sandard GARCH(, Each modl can b rrsnd as: ( ρσ ε σ = f,, whr h rlaion xrsss ha h condiional varianc a im is funcion of h laggd valu of h varianc ( σ, h laggd shock ( ε and a s of aramrs ( ρ Th varianc dynamics modls w considr ar: Modl I: GARCH (, (G: (s Bollrslv, T(986 σ = ω βσ σε (3 whr ω, > and < β < Modl II: GJR-GARCH (GJR-G: (s Glosn, al(993 σ = ω βσ σε γσε I (4 whr ω, > and < β < and Gnrally, w assum h sandard dfiniion of log rurn bwn im and im of ass i, as Si, di,[, ] ri, =log, whr S i, is h ric of S i, h i-h ass a im and d i,[, ] is h oal amoun of cash dividnds aid by h ass bwn and ; ε < I = ;ohrwis Modl III: E-GARCH (E-G: (s Nlson (99 ln ( σ = ω βln ( σ ( ε γε (5 whr > and < β < Th aramrs of h modls ar θ ( μρ, h consan drif rm and ρ ( ω,, βγ, = whr μ is = is h aramr vcor rlad o h varianc dynamics In h modl II and III h aramr γ allows o modl h asymmric bhavior of h varianc, somim calld Black's ffc I consiss of a grar rsons of h varianc whn h nws arrivd in h mark ar ngaiv ( ε < han whn h nws ar osiiv ( ε > All condiions on h aramrs ω, ar usd o avoid horical inconsisnc on h valu for σ (i, ω < should man a onial ngaiv valu for h varianc, whil < should man ha grar shock movmns induc a dcrasing varianc, whil h condiions on β allow h varianc rocss o b covarianc-saionary L us considr h valu of h saionary varianc lvl in ach modl, suosing h wakly saionariy onσ hn w obain 3 : ω h = in h G P β E ε ( ω h = in h GJR P β E ε γe ε I P ( ( ( P ω E ε γε h = x β in h E-G Sinc w hav o avoid h asymoic divrgnc and h ngaiviy of h varianc rocss w nd h following addiional condiions: P β E ε < in h G ( P P β E ( ε γe ( ε I < in h GJR No ha in h normal innovaion cas (i, P ( ε, φ N h = ω β in G, h i-, E( σ = h (and E( ln ( 3 W us h GARCH rory ha: ( ( σ ( ε ( ( σ ( ( ε ω = in GJR, β γ / σ = for h E-G E f g = E f E g whr f and g ar som masurabl funcion, sinc σ, and hus ach is funcion, is masurabl wih rsc o h sigma-algbra a im ISSN: ISBN:

3 Procdings of h 5h WSEAS Inrnaional Confrnc on Economy and Managmn Transformaion (Volum I ω / π h = x in E-G β and h condiions ar: β < in h G, β γ /< in h GJR-G Th GARCH(, modl is a bnchmark modl and i is usd for modl comarisons Th main characrisics of ohr GARCH modls usd can b summarizd in: Parsimonious modl, only 4 aramrs o modl h varianc dynamics Two sa variabls: ric and varianc 3 Tim varying varianc: GARCH modls driv h varianc rocss 4 Modls ar onially abl o xlain h wllknown sylisd-facs as h "Lvrag ffc" ( γ aramr and h "Clusring ffc" in h sochasic volailiy ( β aramr 3 Porfolio Valu Dynamics As Markovian Evoluion Procss Duan and Simonao hav shown ha h GARCH(, modl can b rrsnd as a bivaria Markovian sysm (i, h sa of h rocss is uniuly rrsnd by ( W, σ, so h rocss is markovian of h firs ordr This faur allows o aroxima GARCH modls by a discr Markov chain Duan and Simonao s analysis can b xndd o GJR-GARCH and E-GARCH modls as w show hr in h following In aricular, w rsn h Markov chain aroximaion of a GARCH (, rocss (Duan, Simonao adad o h work modls L us considr an undrlying orfolio log-rurn modld by h uaion ( or uivalnly l us considr lnw( x = lnw ( x μ σ z whr W dno h orfolio valu a day L b som robabiliy masur and σ b h varianc modld by G, GJR or E-G L z a sandardizd random variabl indndnly disribud wih rsc o h informaion u o im, i, ε φ (, Following Duan and Simonao's suggsions, w form h ariions by using h logarihm of adjusd walh and log varianc for h wo sa variabls considrd Th adjusd walh is usd o rduc h dimnsion of h ransiion marix by a walh convrsion Th logarihms of h valus usd ar jusifid mainly for is br convrgnc bhavior Th adjusd walh is comud by W μ % = W whr % μ = μ h / and h is h saionary varianc, h radjusd walh can b asily rcovr lar Also h uncondiional varianc can b comud in all h GARCH modl mniond No ha in rm of log-adjusd walh h log rurn dynamics bcoms: W W ln ln = % μ = ( h σ σε W W Th uncondiional xcaion of h coninuously comoundd rurn on h adjusd walh is zro, sinc E ( σ = h and E ( σε = L and b h logarihm of h adjusd walh (l us say log walh and h logarihm of h varianc rscivly (i, = ln ( W and = ln( σ hn h modls can b rwrin wih: = ( h ε = ln ω β ε in h G cas, ( ( ω β ε γ ε ( = ln I in h GJR cas or = ω β ε γε in h E-G cas To find a sas ariion o aroxima h GARCH rocss w us: A log walh ariion cnrd on h logarihmic of h iniial orfolio walh: [ I, I ], whr I is drmind by sudying h condiional bhavior of h logarihm of h adjusd orfolio walh ovr h invsor im horizont : T ( ( I = δ m E σ φ (6 = An analyical formula of h condiional varianc of h log walh can b drivd for many GARCH rocsss 3 Log varianc ariion: o form h ariion w would sudy h condiional bhavior of h logarihm of h varianc T = ln ( σ T From h GARCH rocss faurs w know ha hr ar wo noabl valus of h varianc: a h iniial varianc, which h rocss sars from, b h uncondiional varianc( h o whom h rocss asymoically is aracd Boh hs valus hav o b considrd in h varianc ariion, bu h scond has incrasing imoranc as w ar far from h bgin insan Th ariion cnr can b τ min ( τ, T min ( τ, T comud as: = ln σ h τ τ Th valu of τ is a moral indx usd o form h wighs As i incrass as h rlaiv wigh of h uncondiional varianc rsc o h iniial varianc incrass Thn in h sudy of long-rm horizon τ has o b small Anyway i is imoran o nsur ha blongs o h ariion Th log varianc ariion is [ I, I ] In ordr o comu h widh I of h ariion i should b nough o sudy Var ( T φ, bu in-g and GJR-G i could rsul analyically comlx W know by h Jnsn inualiy ha ISSN: ISBN:

4 Procdings of h 5h WSEAS Inrnaional Confrnc on Economy and Managmn Transformaion (Volum I ( T φ ln ( ( σt φ Var Var roos o us a widh: so Duan and Simonao ( ( ( δ σ φ I = ln n Var (7 T Only in h E-G cas w hav o no ha h log varianc ariion can b consrucd dircly by h E-GARCH uaion, bcaus i xrsss h varianc in logarihmic rms: [ I, I] whr: τ min ( τ, T min ( τ, T = ln ( σ ln ( h and τ τ T ( ( φ I = δ n Var = (8 In h E-G h sum of h condiional varianc u o T is givn by: T Var ( T φ = T γ = π 4 Duan and Simonao showd ha δ ( m m δ ( m m and ar sufficin ariion condiions for h m aroximaing Markov chain o convrg o is arg GARCH rocss Th logarihmic adjusd walh ariion and h logarihmic varianc ariion ar ually dividd in m and n odd ars rscivly in ordr o drmin h sa of h bivaria rocss: i m i = I ( m C( i [ c( i, c( i ( i ( i ci ( = for i,, m and h corrsonding clls ar =, whr ( c =, = for i,, m c m = j n j = I and h corrsonding clls ar n D( j = [ d( j, d( j for j =,, n, whr ( i ( i d ( =, d( j = for j =,, n and ( = and ( d( n = Th Markov ransiion robabiliy from sa ( i, j a im o sa ( kl, a im is dfind as ( i j k l = C( k D( l = ( i = ( j π, ;, Pr {,, } for =,, T I is yical in h GARCH(, modls ha h varianc a im is a drminisic funcion of h informaion s 4 No ha in E-G cas: ( ( / Var φ E ε γε π γ ε N, = = whr ( π a im In aricular in h modls invsigad w can wri h varianc as funcion of is laggd valu, and wo laggd walh, i, : =Φ (,, Firs w rcovr ε from h log ric uaion wrin on im forward: ( h ε = and subsiuing in h log varianc uaion w obain: G Φ,, = ( = ln ω β ( h γ GJR Φ,, = ( ln ω β ( γi ( h E G Φ (,, = ( h ( h ω β γ This imlis a sourc of sarsiy in h markovian ransiion marix: for ach combinaion of ( i, j, k i xiss only an indx l whr h ransiion robabiliy can b non zro Thus w can rwri h Markov ransiion robabiliy as: Pr { C( k = ( i, = ( j } π ( i, j, k, l = if Φ( ( j, ( k, ( i D( l,ohrwis Th condiional robabiliy can b comud as: ( ( ( Pr { C k = i, = j } = ( j ( j = Pr ck ( i ( ( h ε < ck ( ( j ( j ck ( i ( ( h ck ( i ( ( h Pr ε ( j < ( j Clarly hs ransiion robabiliis can b asily comud for any classical disribuional assumion on h innovaions In his ar w us innovaions Gaussian disribud 3 Th dynamic orfolio slcion roblm Onc w hav comud h ransiion marix M w obain h disribuion a im T considring h owr of h ransiion marix M T Thrfor, givn h sa (i,j of h bivaria rocss (rurn, varianc corrsonding o h k-h raw of h ransiion marix, h disribuion of h bivaria ISSN: ISBN:

5 Procdings of h 5h WSEAS Inrnaional Confrnc on Economy and Managmn Transformaion (Volum I rocss a im T condiiond o sar by (i,j sa is givn by h k-h raw of h marix M T Thus, o g h robabiliy h log walh is in h sa s afr T ss saring by (i,j sa w hav o sum h robabiliis for h diffrn varianc sas, i, whr is h robabiliy corrsonding o h k-h raw of h marix M T o go in h sa (s,l afr T ss Thrfor, doing so w obain h robabiliy of h forcasd final walh for any GARCH y modl dscribd In h orfolio slcion roblm w assum h iniial walh W = and all admissibl walh rocsss W (x = { W ( x} dnding on an iniial orfolio x S ar dfind on a filrd robabiliy sac ( ΩI,, ( I,Pr Th orfolio slcion roblm whn no shor sals ar allowd, can b rrsnd as h maximizaion of a funcional f :( ΩI,,Pr a alid o h random final walh W T (x obaind wih h orfolio wighs blonging o h n -dimnsional simlx n n S = { x Ρ xi = ; xi }, i= i, max f ( W ( x x S T Tyical xamls ar h rformanc masur y ρ( X funcionals f ( X = whr ρ (, ρ( ar wo ρ( X osiiv incrasing funcions of cohrn risk masurs (s Rachv al (8 and h rfrnc hrin Ths funcionals ar isoon wih h monoony ordr, bu ρ ( and ρ ( ar consisn wih risk avrs rfrncs Thus, ρ( X h funcional f ( X = is no isoon nihr wih ρ( X risk lovr nor wih risk avrs rfrncs W rfr o Rachv al (8 for furhr xamls of h abov masurs Hr in h following w inroduc h rformanc y masur usd in h choic roblm Rachv raio This rformanc funcional is dfind as ETL ( WT ( rb z( x OA RR (, β ( WT ( x = ETLβ ( WT ( z( x rb Whn h bnchmark r b is h riskfr ra and z ( x is h chosn orfolio gross rurn (i z( x = x( r( and x W T ( z( x r b is h final walh a im T w obain invsing in h xcss rurn z( x rb ETL is h Excd Tail Loss or Avrag Valu a Risk (AVaR which is a cohrn masur dfind as ETL ( = F ( u du whr F ( u = inf{ Ρ / Pr( u} is h lf invrs of h disribuion funcion Rcall ha h classic consisn simaor of xcd ail loss is givn by T ETL ( = I [ T F ( ] if F ( whr I [ = F Whn h bnchmark ( ] ohrwis r b is h riskfr ra, X is h orfolio rurn, and h numraor and h dnominaor ar osiiv (ngaiv, hn h Rachv raio is isoonic (consisn wih non-saiabl rfrncs of invsors who ar nihr risk avrs nor risk lovr (s Rachv al (8 4 An Ex-Pos Emirical Comarison among GARCH y modls In his scion, w comar orfolio slcion sragis basd on h GARCH modls inroducd in h rvious scions W us 3 asss uod on h US marks (NSE and NASDA from //97 ill 6/4/ for a oal of 3384 daily obsrvaions W comar h rformanc of: Rachv raio undr h hyohsis h log walh follows or a GARCH(,, or a GJR-GARCH or an E-GARCH Shar raio (s Shar 994 undr h assumion w considr hisorical iid rurns W rcalibra daily h orfolio and for h dynamic sragis w us a moral horizon T= working days Figur Ex-os final walh rocss whn Euroan sragis ar alid wih daily rcalibraion and moral horizon T= days W forcas h fuur walh using 8 sas for h orfolio of rurns and 6 sas for is varianc As cofficns of AVaR in h Rachv raio w us ISSN: ISBN:

6 Procdings of h 5h WSEAS Inrnaional Confrnc on Economy and Managmn Transformaion (Volum I Th comarison consiss in h x os valuaion of h walh roducd by h sragis For ach sragy, w considr an iniial walh W = a h da 4/3/9, and a h k -h rcalibraion ( k =,,,, hr main ss ar rformd o comu h x-os final walh: (k S Drmin h mark orfolio x M ha maximizs h rformanc raio ρ ( W ( x, i h soluion of h following oimizaion roblm: max ρ( W ( x ( k x ( k s ( k ' ( x =, ( k xi ; i =, K, n As shown by Angllli and Oroblli (9 his y of roblms could rsn mor local oimum hn w us h hurisic dvlod from hm o aroxima h global oimum S Drmin h x-os final walh givn by: (( ( ' k ( x os W = W x z, k k M (xos whr z is h vcor of obsrvd gross rurns bwn k and k (k S 3 Th oimal orfolio x M is h nw saring oin for h ( k -h oimizaion roblm Ss, and 3 ar rad unil h obsrvaions ar availabl and for ach rformanc raio Th ouu of his analysis is rrsnd in Figur Figur rors h x -os walh rocss using diffrn GARCH modls In aricular hs rsuls mhasiz h good rformanc of h classic GARCH(, modl ha in h las yar rsn arnings of abou h % Insad h ohr GARCH modls ar almos nvr comarabl o h classic on Howvr h comarison wih saic classic sragy is amazing and i suggss us ha w should nvr us h classic sragis in orfolio choics Thus, h mirical rsuls show ha volailiy GARCH modls could b vry imoran in orfolio hory 5 Conclusion This ar xamins h imac of GARCH y rurn voluion in orfolio slcion roblms W dscrib how o aroxima GARCH y rocsss wih Markov chains and w dal h orfolio slcion roblm undr hs disribuional assumions Thus w roos algorihms ha rmi o solv comuaionally comlx roblms in accabl comuaional ims Finally, w roos an mirical comarison among h myoic orfolio slcion modls and hos basd on h GARCH aroximaion Th x-os mirical comarison among classic aroachs and hos basd on Markovian rs shows h grar rdicabl caaciy of h lar Acknowldgmns Th auhors hank for grans COFIN 6% Rfrncs: Angllli, E and Oroblli, S (9 "Amrican and Euroan Porfolio Slcion Sragis: h Markovian Aroach" Char 5 in Financial Hdging did By Parick N Calr, 9-5 Bakshi, G, C Cao, and Z Chn, (997 "Emirical Prformanc of Alrnaiv Oion Pricing Modls," Journal of Financ, 5, Black, Fishr, (976 "Sudis of Sock Pric Volailiy Changs," in: Procdings of h 976 Mings of Businss and Economic Saisics Scion, Amrican Saisical Associaion, Bollrslv, Tim, (986 Gnralizd Auorgrssiv Condiional Hroskdasiciy, Journal of Economrics Aril, 3:3, Duan, J and J Simonao, ( Amrican oion ricing undr GARCH by a Markov chain aroximaion, Journal of Economic Dynamics and Conrol 5, Duan, J, E Dudly, G Gauhir, and J Simonao, (3 Pricing discrly moniord barrir oions by a Markov chain, Journal of Drivaivs, Engl, Robr F, (98 Auorgrssiv Condiional Hroskdasiciy wih Esimas of h Varianc of Unid Kingdom Inflaion, Economrica 5:4, Engl, RF and C Musafa, (99 Imlid ARCH Modls from Oions Prics, Journal of Economrics, 5, Glosn, Lawrnc R, R Jagannahan and D E Runkl, (993 On h Rlaion Bwn h Excd Valu and h Volailiy of h Nominal Excss Rurns on Socks, Journal of Financ 48:5, Hson, S, and SNandi, ( A closd-form GARCH oion valuaion modl, Rviw of Financial Sudis, 3, Nlson, Danil B, (99 Condiional Hroskdasiciy in Ass Rurns: A Nw Aroach, Economrica 59:, Rachv, S,S Oroblli, S Soyanov, F Fabozzi, and A Biglova (8 Dsirabl roris of an idal risk masur in orfolio hory, Inrnaional Journal of Thorical and Alid Financ (, Shar, WF, (994 Th Shar raio, Journal of Porfolio Managmn, Fall, ISSN: ISBN: