Stochastic Programming Models for International Asset Allocation Problems

Transcription

1 Stohast Programmg Models or teratoal Asset Alloato Problems Herules Vladmrou Nolas Topaloglou, Stavros Zeos HERMES eter o omputatoal Fae & Eooms Shool o Eooms & Maagemet Uversty o yprus RsLab Meetg Madrd, De., 23 Pots o Dsusso: Problem ssues Problem ramewor & rs ators (Maret & urrey Exhage Rs Dversato & Hedgg poles Rs maagemet metrs Modelg Approahes Searo Geerato Optmzato Models (Stohast Programs (jotly determe portolo omposto ad hedgg levels eah maret seletve hedgg va orwards ad optos orporato o Optos Portolo Empral Assessmet o Models & vestmet Strateges Rs/Retur Proles o Portolos (stat tests Out-o-sample Perormae (osstey Batestg (Ex-post perormae

2 teratoal Portolo Maagemet: The Problem: Alloato o uds to teratoal assets Dyam maagemet o portolo The Objetves: Eetve Maagemet o Rs/Retur Tradeos (parametr programs Dversato & Rs Hedgg The Needs: Represetato o uertaty apturg maret & exhage rate radomess oststet prg o Optos Portolo Optmzato Models utlzg sutable rs measures to otrol total rs exposure teratoal Dversato t pays to dversy teratoally Postve empral evdee holds or portolos o equtes ad bods tl. dversato etals addtoal rss (urrey exhage lutuatos Eu & Res, J. o Fae, 988. Hgher orrelatos o teratoal vestmets bear marets 2

3 Eets o teratoal dversato? They deped o the volatlty ad orrelato strutures o the teratoal marets ad urrey exhage rates. teratoal dversato etals exposure to urrey rs. Eu, Res, Joural o Fae, 988. Observatos: Geeral rease loal retur orrelatos Volatlty s otagous aross marets Hgher orrelato bear marets Majorty o peso uds vested abroad, are maaged as overlays portolos (Peso ad vestmet Age, 993 urrey rs (partly hedged wth orward urrey exhages Dervatve seurtes - alteratve rs maagemet meas (to hedge ether maret rs, or exhage rs, or both Holst rs maagemet tools employed. Maret Syhrozato & terdepedees To Hedge or Not to Hedge urrey Rs? Perold ad Shulma, Faal Aalysts Joural, 988: Yes! Free luh urrey hedgg! Kaplas ad Shaeer, J. o Eooms ad Busess, 99: Some tmes Yes ad some tmes No, else we do t ow! P. Joro, J. o Portolo Maagemet, 989: Some tmes Yes, some tmes No, else we eed to determe a hedge rato! F. Bla, J. o Fae, 99: Uversal hedge rato or all vestors ad all oreg holdgs. Flatov ad Rappaport, Faal Aalysts Joural, 992: Some tmes Yes, some tmes No, else we have a hedge rato! Abe ad Shrhade, Federal Reserve Ba o Atlata Eoom Revew, 997: ourse o ato lueed by varous ators. Beltratt, Lauret ad Zeos, Searo Modelg o Seletve Hedgg Strateges, JED, 23: Seletve Hedgg s the Preerred Strategy! We also ormulate mplemetable hedgg poles. Sgle perod MAD model Hstoral observatos used as searos 3

4 Abe ad Shrhade, Federal Reserve Ba o Atlata Eoom Revew, 997. Abe ad Shrhade, Federal Reserve Ba o Atlata Eoom Revew,

5 Fators oud emprally to aet the perormae o alteratve hedgg poles The lterature presets deret vews as to the optmal ourse o (urrey hedgg ato or teratoal portolo maagemet depedg o ators suh as: vestmet opportuty set vestor s reeree urrey deomato Represetato o uertaty Tmerame o study (albrato data vestor s tme horzo ad rs tag rtera vestmet strategy (stat vs dyam Edogeze urrey hedgg desos portolo ostruto model Seletve Hedgg: tegratve ramewor Edogeze hedgg desos portolo seleto proedure Searos o dex domest returs ad exhage rates apturg orrelatos betwee them (dee searos o holdg perod returs urrey hedgg va orward exhages ad/or optos (alteratves or otrollg hedgg desos Portolo optmzato models determe portolo ompostos ad urrey hedgg levels Extesos/otrbutos: VaR rs measure (more approprate or sewed dstrbutos, oheret Searo geerato proedures (& Stablty vestgato Operatoalzato o hedgg desos (speato o orward otrats troduto o optos portolo optmzato models 5

6 Value-at-Rs (VaR ad odtoal Value-at-Rs (VaR Portolo Value Dstrbuto at horzo T VaR VaR The Problem: teratoal asset-alloato problem (sgle- ad two-stage SP models; mothly tme steps Assets: Sto des varous outres (USD, GBP, DEM, JPY Govermet Bod des varous outres Short-term bods -3 years termedate-term bods (3-7 years Log-term bods (7- years Optos: Sto dex Optos, Quatos, urrey Optos Data Soures: Morga-Staley MS Data (Sto des Datastream Salomo Brothers Govermet Bod des Spot & Forward Exhage Rates 6

7 Desrptve Statsts o Hstoral Data Mea St.Dev. Sewess Kurtoss USS.59% 3.9% UKS.64% 4.66% GRS.23% 5.773% JPS -.33% 6.336% US.537%.473% US7.688%.646% UK.723%.7% UK7.93%.932% GR.537%.458% GR7.67%.39% JP.327%.522% JP7.68%.73% UStoUK -.74%.8% UStoGR -.67%.88% UStoJP.33%.33% Perod: Ja. 99 Aug mothly observatos Geerally, asset returs are ot ormal; exhbt asymmetres ad at tals. Motvato or: - alteratve searo geerato proedures - rs metrs sutable or asymmetr dstrbutos (.e.,var - alteratve opto prg proedure ssues: Developmet o VaR models or teratoal asset alloato that jotly determe the level o urrey hedgg ad selet the approprate vestmets Prg o optos o assets/urrees osstetly wth postulated searos orporatg optos (urrey optos, sto optos teratoal portolos o stos ad bods Hedgg urrey ad maret rs jotly usg Quatos Assessmet o alteratve tradg strateges volvg ombatos o optos (strp, strap, straddle, stragle. 7

8 Searo Geerato:. Prpal ompoet Aalyss (PA albrated usg hstoral maret data Dreted seletve samplg rom empral dstrbutos o Ps Bayes-Ste estmato orretos Dult exteso to multstage searo trees Do ot math all statstal haratersts 2. Momet-Mathg Searo Geerato Methods (Hoylad, Wallae Maagemet See, 22 (Hoylad, Wallae, Kaut omp. Optm. & Appl., 23 Frst our margal momets & orrelatos math target values Targets estmated usg hstoral data Searo tree ostrutos Model albrato: 2 past years (rollg horzo Searo Trees Root ode P( 8

9 9 Asset vetory ostrats: { } L N y x w w y x h w p(, \,,, = = ash balae (base urrey: = = b s v P x v P y \ ς ζ ς ζ = = b p s L N v P x v v P y } \{,, \ ( ς ζ ς ζ Geer Multstage SP Formulato or teratoal Portolo Maagemet Problem root ode remag odes but ot leaves Geer Multstage SP Formulato or teratoal Portolo Maagemet Problem root ode remag odes but ot leaves ash balae (oreg urrees: = s b v e P x v e P y ς ζ ς ζ = p p s b L N v v P x v e P y } \{,, ( ( ς ζ ς ζ Asset Sale Lmts } \{, ( L N h y p

10 Geer Multstage SP Formulato or teratoal Portolo Maagemet Problem tal Portolo Value = e P h V = p p p p p L v P w e v P w V \, ( ( ( ( ( Fal Portolo Value Portolo Retur L V V R = Parametr boud o Expeted Portolo Retur L R µ π Geer Multstage SP Formulato or teratoal Portolo Maagemet Problem VaR deto L R z y L y Objetve Futo: L z π y β Maxmze z : the VaR o portolo retur (at -β peretle the objetve value s the respetve VaR Developmet o VaR models: S. Uryasev ad T. Roaellar (2-22, Joural o Rs, Faal Egeerg News, Joural o Bag ad Fae, et.

11 Potetal beets rom teratoal dversato Expeted Retur (mothly Eet Froters or VaR Optmzato Model (Sept. 2.3%.2%.%.%.9%.8%.7%.6%.5%.4% -8.% -7.% -6.% -5.% -4.% -3.% -2.% -.%.% VaR (β=95% US. Portolos tl. Portolos (Seletve Ηedgg tl. Portolos (No Hedgg mprovemet rs-retur proles regardless o rs metr (preerable strategy: seletve hedgg Beets more evdet or termedate rs-tolerae levels. Expeted Retur (mothly.3%.2%.%.%.9%.8%.7%.6%.5% Eet Froters or MAD Optmzato Model (Sept. 2.4%.%.5%.%.5% 2.% 2.5% 3.% 3.5% MAD US. Portolos tl. Portolos (Seletve Hedgg tl. Portolos (No Hedgg Ex-post beets rom teratoal dversato..25 Realzed Returs or US ad tl. Seletvely Hedged Portolos o Stos & Bods (VaR Model w th param eters µ=.%, β=95%.2 Wealth Level.5..5 teratoal Portolos US Portolos. Ja- 98 Apr- 98 Jul- 98 Ot- 98 Ja- 99 Apr- 99 Jul- 99 Ot- 99 Tme Perod Ja- Apr- Jul- Ot- Ja- Apr-

12 Ex-post Realzed Returs (Seletve Hedgg 95%-VaR Model wth varous values o target mothly retur (µ µ=.% µ=.8% µ=.% µ=.2% Wealth Level Ja- 98 Apr- 98 Jul- 98 Ot- 98 Ja- 99 Apr- 99 Jul- 99 Ot- 99 Ja- Tme Perod Apr- Jul- Ot- Ja- Apr- omparso o Hedgg Poles (VaR Model.3% Eet Froters or VaR Model (Stos oly Expeted Retur.3%.2%.2%.%.%.% -7.5% -7.4% -7.3% -7.2% -7.% -7.% -6.9% VaR (β =95% om plete Hedgg No Hedgg Seletv e Hedgg.% Eet Froters or VaR Model (Bods oly Expeted Retur.9%.8%.7%.6%.5%.4% -7.% -6.% -5.% -4.% -3.% -2.% -.%.% VaR (β =95% om plete Hedgg No Hedgg Seletv e Hedgg 2

13 omparso o Hedgg Strateges (VaR Model Expeted Retur.3%.2%.%.%.9%.8%.7%.6%.5%.4% Eet Froters or VaR Model (Stos ad Bods -7.5% -6.5% -5.5% -4.5% -3.5% -2.5% -.5% -.5% VaR (β=95% omplete Hedgg No Hedgg Seletve Hedgg Seletve Hedgg s the more eetve (lexble strategy. Sgle- or Two-Stage SP Model? (omparso stat tests (Ex-ate: Two-stage SP model s learly superor to sgle-stage (Domat eet roters batestg expermets (ex-post: The models exhbt smlar perormae/behavor No domatg model a be dsputably deted -stage model aords er represetato o short-term uertaty, whle 2-stage model aptures eets o subsequet perod(s 3

14 Ex-ate omparso o Sgle- & Two- Stage Models Ex-post omparso o Sgle- & Two-Stage SP Models (m rs ase 4

15 Ex-post omparso o Sgle- & Two-Stage SP Models (more aggressve ase Portolo ompostos o Sgle- & Two-Stage SP Models (Mmum Rs ase 5

16 Portolo ompostos o Sgle- & Two-Stage SP Models (More Aggressve ase orporatg Optos Portolos: troduto o optos portolo (Europea optos wth maturty mathg rebalag requey Optos o sto des Quatos o oreg sto des urrey optos Optos pred osstetly wth postulated searo sets ad satsyg arbtrage-ree odtos vestgato o alteratve rs maagemet (hedgg strateges 6

17 Methodologes or prg Optos Expaso methods: Start wth a bas dstrbuto ad add orreto terms (orrado ad Su (J. Faal Researh, 996, Jarrow ad Radd (JFE, 982 (urrey Optos Dervato o the rs-eutral measure usg utlty utos: Rs-Neutral Prob = Atual Prob Prg Kerel (Roseberg ad Egle, WP-23, Bash et al, RFS 23, Jawerth, J. Dervatves, 999, JF 2 (Sto Optos orporatg Optos teratoal Portolos vestmets deret lasses o optos: A. Quatos: Fxed exhage rate oreg equty optos. Relevat or jotly maagg oreg maret rs ad exhage rate rs. Payo o a all Quato reeree urrey: = Max (S X K, B. Smple Optos: Relevat or maagg oreg maret rs, but uoered about exhage rate rs. Payo o a all opto oreg urrey: = Max (S K,. urrey Optos: Rghts or urrey exhages at prespeed rates at opto s exprato. 7

18 orporatg Optos teratoal Portolos Prg o Optos (osstetly wth postulated searo sets Determe a ew (rs eutral probablty measure o postulated searo set based o Rado-Nodym prple; satsy martgale property. The pre o a opto s the expeted value (uder the rs eutral probablty measure o dsouted (wth rsless rate payos at maturty. urrey optos pred usg proedure o orrado & Su. No-arbtrage odtos vered. Prg urrey Optos = e x rt t log log = Ete Ε = Et Et Et The or the opto o E we have Q ( E x s ot ormal the we use Gram-harler expaso. t geerates a approxmate desty uto or a s.r.v. wth ozero sewess ad exess urtoss: x t rt x = t K e ( E ( te K x log( K / E ω x = t µ t σ t The Gram-harler expaso osders approxmato desty: 3 ( ω = ( ω γ D ( ω γ 3! D 4! ( 4 2 ω dx 8

19 Prg urrey Optos = E e r T log( E d = N( d Ke / K ( r r T r T γ Ee ( d σt[ (2σ T 3! where 3 γ = ( 2 3/2 κ4..., γ 2 = ( κ σ 2 r T σ 2 T T Aouts or sewess ad urtoss, devatos rom ormalty o exhage rates d d N( d σ T γ 2 d d 4! 2..., ( d = (2π /2 2 3dσ e T 2 /2 d /2 3σ 2 T ] Prg Quatos ad Smple optos Rado-Nodym theorem: Statemet about two equvalet probablty measures: The atual measure P The Rs-Neutral measure Q o some measurable spae Ω. Rado-Nodym theorem asserto: Q(dω=ξ(ωP(dω Where ξ(ω s a measurable uto wth respet to the uderlyg sgma eld. 9

20 Prg Quatos ad Smple Optos Hypothess o power utlty uto By ormalzg ad hagg varables (Bash et al, Revew o Faal Studes, 23 The rs-eutral probabltes or eah searo are q ( R = γr γr e Where γ s the relatve rs averso e p p ( R ( R Prg Quatos ad Smple Optos Rs-eutral dex desty obtaed by expoetally tltg the physal desty. To avod arbtrage, the dex must be a martgale Mea o rs-eutral desty must satsy martgale odto: S e ( rδ Τ = q S 2

21 orporatg Dervatves teratoal Portolos Rs Neutral Valuato: The pre o the opto o asset S s the expeted payo o the opto uder all searos, Rs- Neutral Measure, dsouted at the rs-ree rate. Thus, or a Quato all: = e rdt N p = max( XS K, urrey optos or hedgg purposes Ex Post Realzed Returs. urrey Hedgg through urrey optos or Forward otrats, Sgle vs Mult-stage models, put at-the-moey optos (TG=%, 2% Realzed Retur May-98 Aug-98 Nov-98 Feb-99 May-99 Aug-99 Nov-99 Feb- May- Aug- Nov- Feb- May- Aug- Nov- Tme perod Forw ard otrats Tw o-stage Sgle Tw o-stage Forw ard otrats Sgle 2

22 urrey optos or hedgg purposes Ex Post Realzed Returs. urrey Hedgg through urrey optos or Forward otrats, Sgle vs Mult-stage models, BearSpread (TG=%, 2% Realzed Retur May-98 Aug-98 Nov-98 Feb-99 May-99 Aug-99 Nov-99 Feb- May- Aug- Nov- Feb- May- Aug- Nov- Tme perod Forw ard otrats Tw o-stage Sgle Tw o-stage Forw ard otrats Sgle orporatg Dervatves teratoal Portolos Payo Tradg Strateges volvg Optos Straddle Payo Stragle X X X 2 S T S T Log all ad Log Put Log all (exerse pre X 2 wth the same exerse pre X Log Put (exerse pre X ad the same maturty wth the same maturty 22

23 orporatg Dervatves teratoal Portolos Payo Strp Payo Strap X S T X S T Log all ad 2 Log Puts 2 Log all ad Log put Wth the same exerse pre X ad the same maturty wth the same exerse pre X ad the same maturty Shapg portolo rs usg optos.35 Probablty Wthout Stragle Straddle Strp Retur 23

24 Rs/Retur Eet Froters O Portolos wth varous tradg strateges o Quato Optos vs Portolos Wthout Optos 2.5% 2.% Expeted Retur.5%.%.5% Wthout Optos Straddle Stragle Strp Strap.% -% % 3% 5% 7% 9% % 3% 5% 7% 95%-VaR Ex post Realzed Returs o Portolos wth deret strateges o Optos ( µ=%.35.3 Realzed Retur Stragle Straddle Strp Strap NoOptos.95 May-98 Aug-98 Nov-98 Feb-99 May-99 Aug-99 Nov-99 Feb- May- Aug- Nov- Feb- May- Aug- Nov- Tme perod 24

25 Rs/Retur Eet Froters O Portolos wth ad wthout Optos (Stragle 2.5% 2.% Expeted Retur.5%.% QuatoForw ards.5% SmpleForw ards Wthout Optos w th H Wthout Optos No H.% -% % % 2% 3% 4% 5% 6% 7% 8% 95%-VaR Ex post Realzed Returs o Portolos wth ad wthout Optos (Straddle Strategy, µ=% Realzed Retur..5 Quatosorw ards SmpleForw ards WthoutOptos.95 May-98 Aug-98 Nov-98 Feb-99 May-99 Aug-99 Nov-99 Feb- May- Aug- Nov- Feb- May- Aug- Nov- Tme perod 25

26 Ex post Realzed Returs o Portolos wth Quatos ad Smple optos (Stragle Strategy, µ=% Realzed Retur Quatosorw ards SmpleForw ards WthoutOptos.95 May-98 Aug-98 Nov-98 Feb-99 May-99 Aug-99 Nov-99 Feb- May- Aug- Nov- Feb- Tme perod oludg Remars: SPs potetally eetve tools or teratoal portolo maagemet Searo geerato methods provde eetve meas or represetg uertaty VaR models osttute eetve rs maagemet tool teralzg urrey hedgg desos (va orward otrats the models mproves ex-ate ad ex-post results Postve value o tegratve ramewor troduto o dervatves leads to urther perormae mprovemets Partularly the tegrated hadlg o maret ad urrey rss (use o quatos 26

27 HERMES eter o Exellee o omputatoal Fae & Eooms Worg Paper Seres: (WP -23 Joural o Bag ad Fae, 26(7, 22 27