UniSuper s Approach to Risk Budgeting

Transcription

1 UnSuper s Apprch Rs Budgeng Prepred by Dvd H. Schneder Denns Sms Presened he Insue f Acures f Ausrl 009 Bennl Cnvenn 9- Aprl 009 Perh Wesern Ausrl Ths pper hs been prepred fr he Insue f Acures f Ausrl s (Insue) 009 Bennl Cnvenn The Insue Cuncl shes be undersd h pnns pu frrd heren re n necessrly hse f he Insue nd he Cuncl s n respnsble fr hse pnns. Dvd H. Schneder Denns Sms The Insue ll ensure h ll reprducns f he pper cnledge he Auhr/s s he uhr/s nd nclude he bve cpyrgh semen: The Insue f Acures f Ausrl Level 7 Chlls Huse 4 Mrn Plce Sydney SW Ausrl 000 Telephne: Fcsmle: Eml: cures@cures.sn.u Webse:.cures.sn.u

2 UnSuper s Apprch Rs Budgeng Dvd H. Schneder; Denns Sms ABSTRACT UnSuper () hs develped rs budgeng sysem h mesures devns frm he Fund s sregc sse llcn. The pper presens he mhemcs used cse sudy nd brd cnclusns penlly pplcble nsunl nvesrs. Keyrds: rs budgeng; fcr nlyss; mrgnl nd prprnl cnrbun rs; reurn rbun; ex-ne lph; cllnery; reverse pmsn. ACKWLEDGEMETS The uhrs uld le hn UnSuper s Chef Invesmen ffcer (Dvd S. Jhn) fr hs encurgemen nd suppr n reln hs reserch nd UnSuper fr permng he fndngs be publshed. The ves expressed n hs pper ere frmuled n cnsuln h ur cllegues n UnSuper s Invesmen Deprmen bu he uhrs ccep respnsbly fr ny errrs n hs pper.. ITRDUCTI A s smples rs budgeng s he prcess f seng rge level f rs be cceped he prfl level nd llcng hs rs crss number f nvesmens n he ms effcen mnner n rder mxmse reurns hls cnnng rs hn greed rges. A he secr r sse clss level rs budges re ypclly se n erms f rcng errr rge fr he verll secr (.e. h much vlly rund n pprpre secr benchmr uld he Fund be prepred ccep n rder ry nd ncrese reurns). I s mprn ne h rs budgeng prmrly prvdes nvesrs h frmer fr dscussn nd nlyss. I des n hve be (nd ndeed n he uhrs pnn shuld n be) ppled n prescrpve fshn. Ths pper fcuses n sngle cmpnen f verll fund level rs mngemen. UnSuper hs derved frml Invesmen Rs Mngemen Plcy (IRMP) h ses u he Fund s rs mngemen phlsphy denfes ey nernl nd exernl rss fced by he Fund nd he mnner h hch hese rss re mnged (UnSuper 005). The Fund s rs mngemen nves spn he publc nd prve mre res s ell s he Fund s sregc lng nd gvernnce rrngemens. The rs clssfcn frmer nd rs mngemen nves underpnnng UnSuper s IRMP re grphclly represened n he chr verlef. UnSuper s rs budgeng pprch hs been develped ver number f yers nd fcuses n mesurng nd mngng he cve rs cceped by he Fund (.e. em. n he chr verlef). Dels relng UnSuper s mngemen f her cmpnens f rs hn he Fund re usde he scpe f hs pper. Much f he r underen by UnSuper s bul up frm prr r n rs budgeng (fr exmple Mn (007) prvdes n ulne f rs budgeng frmer hch frmed he bss f much f he r n hs pper; Lermn (003b) descrbes prccl mehdlgy fr nsunl nvesrs; hle Scherer (000) Kzun (00) Shrpe (00) Bnz (003) de Bever (003) nd Berelr (006) prvde n verve f rs budgeng fr nsunl nvesrs). Hever he uhrs exended prr fndngs by: remvng he smplfyng ssumpn h excess reurns beeen mngers re uncrreled ( s cmmn fr mngers emplyng smlr nvesmen syles perfrm n crreled mnner); nrducng he de h usfy cve rs ne needs exceed hurdle reurn n excess f 0% (descrbed n mre del n he Appendx secn A.7); nd develped mehd help vercme he dffcules nheren h mulple cllnery (descrbed n mre del n he Appendx secn 4. nd A..4). UnSuper s Apprch Rs Budgeng

3 CHART. Schemc represenn f he rss fced by UnSuper Surce: UnSuper 005 The bve chr summrses he rss fced by UnSuper. The Fund s rs budgeng pprch mesures cve rs cceped by he Fund s hghlghed n bx.. UnSuper s Apprch Rs Budgeng

4 Due he cmpunl cmplexy f he Fund s rs budgeng prcess UnSuper hs develped n n-huse rs budgeng nd fcr nlyss prgrm brnded The UnSuper Rs Budgeng nd pmsn Sysem (TURBs). Ths pper ulnes he becves nd lgc underlyng TURBs prvdes prccl cmmenry s ell s red exmple fr he Fund s 30 June MATHEMATICAL VERVIEW TURBs mnrs he exen hch he Fund deves frm s Sregc Asse Allcn (SAA). UnSuper cn nves pssvely nd brdly mch he be expsures expeced frm ech sse clss. The exen hch he Fund emplys cve mngemen nd he mun by hch he Fund deves frm he pssve benchmr psn represens surce f rs he Fund. The becve behnd TURBs s denfy nd qunfy hs surce f rs. UnSuper predmnnly nvess v fund mngers bu he Fund des hld sme nvesmens drecly. As such fr he purpse f hs pper e use he erm mnger nd nvesmen nerchngebly. TURBs hs been desgned s s : Deermne h ech nvesmen r mnger generes her reurns; Idenfy he mre fcr expsures fr ech mnger nd ggrege hese deermne verll pn fcr expsures; Assess he expeced fuure lph (r excess reurn bve benchmr) fr ech mnger; nd Ensure h he Fund s cve rs s llced pprprely beeen mngers (.e. such h ms f he Fund s cve rs les h mngers h re expeced uperfrm her benchmrs). In rder mee he becves sx prcesses need be cmpued. In prculr TURBs:. Arbues ech mnger s reurns beeen seres f mre fcr expsures (.e. be cmpnen) nd n bserved ex-ps (r hsrc) lph cmpnen. Ths sep s reslved usng fcr nlyss nd mulple regressn;. Deermnes he ex-ps l rs (r vlly) nd rcng errr fr ech pn nd ssess he mrgnl nd prprnl cnrbun h rs frm ech mnger; 3. Uses Byesn echnques deermne n ex-ne (r frecs) esme f ech mnger s lph; 4. Assesses he exen hch ech pn s be expsure dffers he Fund s SAA Benchmrs; 5. Ses mnmum cve rs rge fr ech pn nd ssess he exen hch he hurdle s expeced be cheved; nd 6. Emplys reverse pmsn cnfrm heher he egh ssgned ech f he Fund s mngers s cnssen h he expeced perfrmnce f he mnger. Ech f hese sx prcesses re descrbed mhemclly bel. The equns presened re develped n Appendx. T sss he reder e prvde glssry f nn n Appendx. e h equn references prvded n prenhess fer ech equn rele he rder n hch he equn s develped n Appendx. UnSuper s Apprch Rs Budgeng 3

5 . Fcr Anlyss Regressn echnques re ulsed derve ex-ps esmes f he surce f ech mnger s l reurn. Ech mnger s reurn cnsss f be cmpnen lng h n lph (r reurn n excess f mre fcrs) cmpnen. Specfclly: Where µ β K K µ β F ε...(..5) Denes he bserved reurn (befre x ne f fees) frm mnger me here mnger nvess n sse clss. Denes he esmed ex-ps reurn n excess f mre fcrs fr mnger me here mnger nvess n sse clss. Denes mnger s sse clss. Represens he number f ll pplcble fcrs. Typcl fcrs re reurns n sc ndces neres res vlly he rs free re f reurn ec. Denes he esmed be fcr descrbng he sensvy beeen mnger s expsure fcr me. F Denes he bserved reurns frm fcr me. ε Denes he mnger s resdul errr erm r unexplned reurns h zer men. The lph nd be prmeers frm equn..5 re esmed usng he echnques ulned n secn f Appendx (refer equn.3.). nce ech mnger s lph esme nd be fcrs re bned sndrd sscl echnques re used es he gdness f f.. Mrgnl nd Prprnl Cnrbun Rs Ech pn s rs (r vlly) cn be esmed by cnsderng hsrc d. The l ex-ps vrnce f reurns fr ech pn s defned s nd s derved n equn 3.: Where...(3.) Denes he mnger s egh n pn me (.e. percenge hldng eghed by Funds Under Mngemen). Denes he esmed cvrnce f reurns (grss f x ne f fees) beeen mnger nd mnger me h n dusmen ll fr mnger nd s u-crreln. Denes he l number f mngers spnnng ll sse clsses. The cvrnce beeen ech Mnger s reurns explcly lls fr u-crreln nd he frmul fr he cvrnce s presened n equn 3.. Ech mnger s mrgnl cnrbun rs muns : hch s derved n equn 3.4. UnSuper s Apprch Rs Budgeng 4

6 .3 Ex-Ane Alph Esmn Ech mnger s ex-ps lph s esmed durng he fcr nlyss clculn. Hever ex-ps d s n relble gude esme fuure lph expeced be genered by mngers. T ccun fr UnSuper s nernl ve s ech mnger s sll Byesn pprch s requred. The uhrs dped he Blc-Lermn Mdel (BLM) s s derve ex-ne lph esmes fr ech mnger. The rngs behnd he dpn f he BLM re dscussed n Secn 4 f Appendx nd derved n equn 4..9: ( ) ( ) Ω τ Α Ω Q Where: τ A A A...(4..9) Denes he vecr f ech mnger s expeced ex-ne lph reurns. Ech elemen f he vecr s gven he symbl E[ ]. E[ ] Denes he ex-ne expeced lph frm mnger h peres n sse clss me. Α τ Q A Ω A Denes he vecr f bserved ex-ps lph r excess reurns fr ech mnger derved usng fcr nlyss frm equn..5. e h hs vecr cn lernvely represen he equlbrum excess reurn fr ech mnger (hch uld hen be null vecr nd he erm ( ) τ Α uld be remved frm equn 4..9.) Denes he cvrnce mrx f ech mnger s ex-ps reurns n excess f her fcr expsures. Represens he sclng fcr ppled hch mesures he uncerny f he hsrc d. Represens vecr cnnng he nvesr s ve f he expeced lph genered frm ech mnger lng he dgnl elemens. Represens squre mrx cnnng he nvesr s cnfdence n s ve f he expeced lph genered frm ech mnger lng he dgnl elemens..4 Reurn Arbun By cmbnng nd eghng he Fund s expsure ll mngers e cn derve he ex-ne expeced reurn fr pn () nd rbue hs reurn beeen vrey f surces. Whn he se f vlble be fcrs { F } K here s sub-se f M fcrs h rele he Fund s benchmr fr ech sse clss (s n exmple fr he Ausrln Equy sse clss UnSuper s curren benchmr s he ASX 300 Accumuln ndex hch n urn s penl be fcr fr ll Ausrln equy mngers). Le { BM m} M m dene he se f sregc benchmr fcrs fr ech f he Fund s M sse clsses. Then ne cn deermne he mnner n hch ech be fcr mps n he Fund s sregc benchmr fcrs s flls: M F RP γ m BM m ν m Where RP (5.) Denes he rs premum vlble frm Fcr hch cnn be rbued ne f he Fund s benchmrs. m γ Denes he esmed sensvy beeen fcr nd he m h sse clss benchmr. ν Denes he resduls f he regressn slun h zer men. UnSuper s Apprch Rs Budgeng 5

7 The rs prem nd he m γ ceffcens re esmed n he sme mnner h s used derve he fcrs frm equn..5. By cmbnng equns nd 5. e cn derve he expeced reurn fr ech pn expressed s funcn f: ) The eghed verge f ech mnger s expeced ex-ne lph; b) The eghed verge expsure he Fund s SAA benchmrs; c) Exrneus be rs prem frm fcrs h dffer he Fund s SAA benchmr; nd d) An errr erm. If R denes he ex-ps (r hsrc) reurn frm pn me hen: K M R γ β K m BM m β RP ε m K Where ε β ν ε...(5.5) The bve equn s cenrl he Fund s fcr nlyss. In prculr: The erm denes he eghed verge ex-ps lph bserved frm he Fund s mngers. The erm K M γ m β BM m denes he eghed verge expsure he pn s m SAA benchmr hch ll be cnrsed he pn s cul SAA; Whle K β RP denes he pn s expsure her nn be surces (such s cred rs vlue smll cp bses secr bes ec); nd K The fnl erm β ν ε represens he errr r resduls f he vrus regressn esmes (h zer men). The dsrbun s crclly exmned ensure h s suffcenly nrmlly dsrbued. Evdence f exreme urss r seness culd be cuse fr cncern..5 Rs Budgeng Rs budges hve rdnlly been derved by ms prcners h reference mxmum permssble rcng errr. Ths pprch s pprpre fr sse mngers hse mndes re fen specfed n erms f rcng errr lmns. Hever he cncern h hs pprch fr nsuns h gurneed lbles s h here s n drec nercn beeen he mxmum rcng errr nd he Fund s lbles. In ddn he chce f n pprpre rcng errr budge s subecve. As resul he uhrs hve derved n lernve pprch rs budgeng. The Fund ses s SAA s s bes mee he nvesmen becves (fr he Accumuln pns) nd py lbles s hey fll due (fr he Defned Benef Dvsn). Hence ny devn frm he Fund s SAA represens surce f rs he Fund. Specfclly nrducng cve mngemen dds rs he Fund. The mrgnl ncrese rs s nly usfble f he Fund s expeced (r ex-ne) lph exceeds he benef h culd be bned by chngng he Fund s SAA benchmrs nd mvng lng UnSuper s Apprch Rs Budgeng 6

8 he Fund s cnsrned effcen frner. Ths de (dscussed n mre del n Secn 7 f Appendx nd grphclly n he chr bel) prvdes n nequly h s used n ur rs budgeng frmuln nmely h ech pn s ex-ne lph needs exceed mnmum hurdle usfy deprure frm be llcns. Gemerc Averge Reurns % 0% 9% 8% 7% 6% 5% 4% Cmprsn f The Accumuln pns Rs nd Reurn Prfle Relve he Effcen Frner - Rs eurl Cndns Cs h Cp Sble pml SAA Cns Bl Blnced SRI Hgh Gr h Hgh Gr h Gr h Aus. Eques Innl Eques SRI Blnced p P Ls ed Prpery 0% % 4% 6% 8% 0% % 4% 6% 8% 0% Sndrd Devn f Annul Reurns Effcen Frner - n cnsrns Effcen Frner h UnSuper cnsrns Dversfed Accumuln pns Sngle Asse Clss pns Ε[] Uly Funcn d EF( p ) d Hence usfy deprure frm he pml (Be cnsruced) prfl cve mngemen mus be such h: E[ ] d Aus. Bnds EF( ) p d p P In secn 7 f he Appendx e derve n nequly lnng he ex-ne lph he Fund s SAA nd s cnsrned effcen frner: EF( )...(7.) Where represens pn s vrnce bsed n he SAA lng erm eghs nvesed n he EF( EF benchmr fr ech sse clss nd ) equls he dervve r slpe f he cnsrned effcen frner h respec he vlly f he frner slved hen. e ls h n hs equn denes he cvrnce mrx f ech mnger s reurns n excess f he benchmr. The ble bel summrses he mnmum hurdle fr ech pn lng h he mnmum excess requred lph usfy usng cve mngemen s ppsed ncresng rs by chngng he Fund s SAA. UnSuper s Apprch Rs Budgeng 7

9 Tble : Mnmum requred lph fr dfferng Accumuln pns Accumuln pn () Dervve f cnsrned effcen frner EF( ) Mnmum requred lph ( ) under dfferng pn rcng errr levels.00%.50%.00% Csh % 5.7% 7.6% Cpl Sble % 0.68% 0.90% Cnservve Blnced % 0.5% 0.68% Blnced % 0.4% 0.56% Grh % 0.39% 0.5% Hgh Grh % 0.% 0.8% As cn be seen n he bve ble ech pn s requred mnmum ex-ne lph vres ccrdng he curvure f he effcen frner s ell s he rcng errr fr he pn relve he pn s benchmr. Hence f he bserved pn rs s % (sy) hgher hn h hch uld hve ccurred hd UnSuper nvesed pssvely fr he Blnced pn hen he mnmum requred excess reurn fr he Blnced pn ( usfy deprure frm he pn s SAA) uld mun 0.56%. TURBs cmpues he ex-ne lphs usng equn 4..9 geher h he ex-ps rcng errr fr ech pn nd cnrss hese he mnmum hurdle (prvded by equn 7.) ensure h he Fund remns cmfrble h s mnger lne-up..6 Reverse pmsn Equn 7. cn be dped bn n pml mnger lne-up (dened by ). Such prfl uld hve he hghes rs-dused reurn hls smulneusly meeng he mnmum hurdles derved n secn 3.5 bve. The dpn s derved n secn 7 f he Appendx. EF( ) (7.4) EF( ) nce hs been clculed UnSuper cnrss he Fund s cul mnger eghs per pn h derved by TURBs deermne heher he Fund remns cmfrble h he curren mnger lne up. e hugh h s clculed seprely fr ech sse clss nd n fr he verll fund nd h denes he cvrnce mrx f ech mnger s reurns n excess f her benchmr. 3. CASE STUDY In hs secn e presen he fndngs frm n nlyss underen usng UnSuper s mnger lne up s 30 June 008 lng h he Fund s hsrc d. The cse sudy prvdes del n he Fund s Ausrln shres mngers lng h summrsed fndngs fr her sse clsses. The nlyss cnsders mnhly hsrc reurns ver he hree yers endng 30 June 008. Cnsderns relng prccl dusmens nd llnces requred fr lernve sses s ell s fr dels pernng he clbrn f he BLM re dscussed n secn 4 f hs pper. UnSuper s Apprch Rs Budgeng 8

10 3.. Fcr Reurn Arbun The frs prcess ulsed by TURBs s rbue ech mngers reurns beeen mre fcr expsures (.e. he be cmpnen) nd n bserved ex-ps (r hsrc) lph cmpnen. The esmed fcr expsures nd lph re deermned usng mulple regressn. Fr ech mnger s lph esme nd be fcrs seleced sscl mesures re prvded es he gdness f f. TURBs ssesses ech mnger n urn ulsng bcrd elmnn lgrhm fnd n pml f he mre fcrs. The lgrhm evenully seleced by he uhrs s descrbed n Appendx. The fcrs used nclude ndex reurns by mre cplsn syle (vlue nd grh) mmenum nd secrs bu remve he mpc f hgher rder fcr crrelns s descrbed n Appendx. The ble verlef dsplys ech f UnSuper s Ausrln shres mnger s expsures he vrus be fcrs. A psve llcn ndces bs he fcr hls negve llcn suggess bs y frm he fcr. The ls clumn prvdes he ggreged expsures fr UnSuper s Ausrln shres prfl s 30 June 008. The prmry be used s he ASX 300 (he benchmr fr he prfl). UnSuper s verll Ausrln Shres prfl hd be f The prfl s currenly srucured h lrge be expsure verld by menngful lph. The sgnfcn be expsure s cnssen h he lrgely lng-nly pprch dped by he Fund nd he requremen n ncumben mngers remn s clse fully nvesed s pssble. Plesngly here s mnml resdul urss ndcng h ver he perd nlysed here hve been fe exreme mves relve benchmr. A n verll prfl level TURBs ndces h he prfl hs mdere bs smll cps. Ths psnng s cnssen h UnSuper s expecns. Frm syle perspecve he nlyss dsplys negve expsures bh vlue nd grh (hus brdly syle neurl). Hever he uhrs beleve h he expsure vlue s beng prly undersed nd rher beng ndrecly refleced hrugh n mpled secrl bs fnncls. hhsndng hs he nlyss cnfrms h he prfl des n hve ny unnended syle ls. verll he nlyss suggess h he prfl s cnsruced hu sgnfcn ggrege fcr expsures nd s ell dversfed n h he prfl hs cpured he bul f he ASX 300 be. Plesngly here s sld level f underlyng ex-ps lph (.e. lph remnng fer ll be fcrs re remved) f.6% p... UnSuper s Apprch Rs Budgeng 9

11 Asse Sub Clss Syle Syle Syle 3 Syle 4 Syle 5 Syle 6 Tl Mnger Cde Mnger A Mnger B Mnger C Mnger D Mnger E Mnger F Mnger G Mnger H Wegh 7.6% 6.6% 7.3% 6.% 9.5% 5.3% 7.% 7.0%.9% 6.3%.7%.3% 0.7% 6.% 6.8% 7.5% 5.3% 00% Mnger ex-ps Alph p.. 4.4% -.5% -0.%.6% -0.6%.6% 0.5%.0% -.8% 0.5%.4% 5.6% -4.% 3.9% 7.3% 4.6%.6%.6% ASX300 Accum. Index 94.3%.5% 99.8% 0.9% 03.%.7% 94.0% 83.% 08.9% 96.6% 85.6% 0.% 4.6% 86.% 8.7% 84.6%.7% 96.7% -Mnh Mmenum % % 3-Mnh Mmenum % % 6-Mnh Mmenum % % Cnsumer Dscrenry 9.8% % % % - 3.% Cnsumer Sples % % % Energy Fnncls 46.9% % % % Helh Cre.6% 8.% -5.% % % 4.4% %.0% Indusrls.6% -0.% 0.0%.%.5% % % IT % % LPT % % Merls % % Telecmmuncn % % % % Ules -7.4% % 8.% -6.% % Vlue % 4.9% -43.6% -.5% % -49.4% % - 5.7% -.5% -3.9% Grh % % 53.8% -.% ASX ex % 47.0% 09.4% % ASX Smll rdnres % % 8.5% 5.% % % Mdel Adused R 93.3% 88.4% 98.8% 93.8% 94.7% 90.% 94.6% 89.3% 9.3% 87.6% 78.5% 88.5% 83.4% 94.3% 94.6% 85.% 90.% /A Mdel Sndrd Devn 0.8%.3% 0.3% 0.8% 0.8%.% 0.7%.%.%.5%.7%.4%.% 0.8% 0.7%.%.% /A Resdul Se /A Resdul Kurss /A Resdul DW Ssc /A Resdul Aucrreln -8.0% 0.8% -3.5% -5.4% 7.8% 8.% -0.3% 6.6% -9.6% -3.9% 3.9% 9.9% 5.% -3.4% -.5% 6.% 8.% /A Mnger I Mnger J e: All ndexes represen expsures fer remvl f he effecs f ll fcrs rned hgher n he ls s descrbed n 4.. Mnger K Mnger L Mnger M Mnger Mnger Mnger P Mnger Q UnSuper s Apprch Rs Budgeng 0

12 Deled bel re llusrve cmmens f he regressn resuls fr ne f he Fund s Ausrln equy mngers (vz. mnger Q). Mnger Q Mnger Q dps qunve nvesmen prcess hch cmbnes seven ey fcrs. The mnger s prfl s srucured be brdly syle neurl ver he lng-erm lhugh bses vlue r grh my be evden ver shrer me perds subec prevlng mre cndns. The ble bel dsplys he ey regressn resuls fr he mnger. Tble : Ausrln Equy Mngers Mnger Q s regressn resuls Regressn Fcr Ceffcens Sndrd Errr -Ssc p-vlue Ler 95% Upper 95% Mngers ex-ps Alph (p..).6%.8% % -4.0% 7.% ASX300A ndex 3% 5.0% % 03% 3% ASX Helh Cre Secr ^ -8% 6.3% % -30% -5% ASX Vlue Index ^ -% 0.3% % -4% -% ASX Grh Index ^ 54% 4.4%. 3.7% 5% 03% e: Fcrs mred ^ represen fcr expsures fer remvl f he effecs f ll fcrs rned hgher n he ls. Mdel F Mdel Adused R 90.% Mdel Sndrd Devn.3% F-Sgnfcnce 33. Sgnfcn he 99.95% level Resdul Anlyss Resdul Se 0. Resdul Kurss Resdul Au-Crreln 8% TURBs ssessed h he mnger s reurns culd be descrbed s generng n lph reurn f.6% p.. plus 3% f he ASX 300 Accumuln Index h negve Helh Cre expsure nd sld grh bs. All fcrs her hn he mnger s ex-ps lph re hghly sgnfcn. The verll mdel f s gd h 90.% dused R nd n exremely hgh F-ssc. In ddn he resduls pprxme nrml dsrbun h lms n evdence f seness r urss nd nly mdere u-crreln. The lrge expsure he ASX 300 Accumuln Index s smeh hgher hn expeced gven h he mnger hs cnsruced her prfl be brdly be neurl. The negve expsure he Helh Cre secr s cnsdered lrgely perd specfc ucme nd n n nheren bs n he mnger s nvesmen prcess nd my ls be he cnr psn he hgher ASX 300 be (.e. TURBs s veng he mnger s hgher mre be bu bel mre expsure Helh Cre hch s hgh be secr). TURBs ls dsplys he resuls f he f grphclly s shn n he fllng chrs. The resduls f he regressn fs fr ech mnger s crefully nlysed by UnSuper s hese prvde nsgh s he hgher mmens f ech mnger s lph dsrbun. UnSuper s Apprch Rs Budgeng

13 Chr : Cnrsng mnger Q s bserved reurns h he bes f bned The chr bve cnrss he mdelled fcrs (gven by he hn blc lne) gns he mnger s cul reurns (gven by he red curve) demnsrng h gd f s ned. The nex chr prvdes scer pl f he errr erms (r resduls) hch effecvely represens he dsrbun f he mnger s lph. Chr 3: Scer pl f errr erms fr Mnger Q The scer pl demnsres h he resduls re dsrbued n brdly rndm mnner rund men f zer. Anher mnner f esng he gdness f f s cnsder hsgrm f resduls relve nrml dsrbun s ell s nrml prbbly pl s shn n he nex chr. UnSuper s Apprch Rs Budgeng

14 Chr 4: Mnger Q s resdul dsrbun The bve hsgrm f resduls nd nrml prbbly pl re resnbly clsely lgned nrml dsrbun (gven by he red lne nd he 45 0 lne fr he nrml prbbly pl). 3.3 Expeced vs. Acul Be Expsure by Accumuln pn nce TURBs cmplees he nlyss presened n 3. fr ech mnger nd nvesmen (sme f he prve equy nfrsrucure nd drec prpery mnger reurn seres re gruped geher due he hgh ncdence f sle prces) TURBs ssesses he exen hch ech Be fcr s crreled he Fund s benchmrs. The sme mulple regressn echnques re used mp ech fcr he Fund s benchmrs s s used reve ech f he Fund s mngers. By ggregng he resuls fr ech Accumuln pn ne cn cnrs he pn s derved be expsure h expeced frm ech pn s SAA s shn n he ble bel: Tble 3: Expeced vs. Acul Be Expsure Fr Ech pn Be Arbun Hgh Grh (%) Grh pn (%) Blnced pn (%) Cns. Blnced (%) Cpl Sble (%) Ausrln Lsed nd Prve equy Inernnl Lsed nd Prve Equy Dmesc nd Inernnl Bnds SAA Fed Be Fcrs SAA Fed Be Fcrs SAA Fed Be Fcrs SAA Fed Be Fcrs SAA Fed Be Fcrs Drec Prpery Index Lned Bnds Infrsrucure Lsed Prpery Truss Csh Exrneus rs prem Tl The bve ble ssesses he exen hch ech pn s be expsure dffers he Fund s SAA Benchmrs. Fr exmple he Blnced pn hs slghly ler equy nd ndex Lned Bnd be UnSuper s Apprch Rs Budgeng 3

15 hn expeced hch s ffse by hgher hn expeced fxed neres nd csh llcn. verll he bserved be expsure fr ech pn s brdly n lne h he pn s SAA. 3.4 MARGIAL AD PRPRTIAL CTRIBUTI T RISK Hvng ssessed ech mnger s ex-ps excess reurns TURBs cmpues he mrgnl nd prprnl cnrbun l pn rs frm ech mnger/nvesmen. Fr he Blnced pn (hch represens UnSuper s deful pn nd hence s he ms ppulr pn) he ex-ps l pn vlly muned 8.6% p.. ver he hree yers June 008. The l pn rs s bel he SAA lng-erm ssumpns fr he Blnced pn (9.8%) prmrly s eques exhbed unusully l vlly unl md-007. Clerly he uhrs expec he pn s 3-yer rllng bserved vlly ncrese ver he cmng mnhs. By cnsderng he cvrnce mrx f ech mnger s ex-ps lph reurns TURBs s ble derve he pn s rcng errr hch muned.% fr he Blnced pn. The bles bel demnsre h ech sse clss cnrbued he bserved pn rs nd rcng errr fr he Blnced pn. Tble 4: Blnced pn -Prprnl Cnrbun Rs nd Trcng Errr Asse Clss Asse Clss Wegh Blnced pn Prprnl Cnrbun pn Vlly Prprnl Cnrbun Trcng Errr (%) (%) (%) Ausrln Eques Enhnced Pssve Grh Lng/Shr eurl Smll Cps Vlue Inernnl Eques Drec Prpery Lsed Prpery Infrsrucure Prve Equy TTAL GRWTH ASSETS Dmesc Fxed Ineres Inernnl Fxed Ineres Dmesc Indexed Bnds Csh TTAL DEFESIVE ASSETS pn Vlly & Trcng Errr The bve bles demnsre h ms f ech pn s l vlly sems frm he lsed equy prfls. surprsngly drec prpery nd defensve sses reduced he pn s rs. When cnsderng rcng errr he uhrs ne h he lernve sses (n prculr nfrsrucure sses) genere ms f he pn s rcng errr hever hs resul prly rses s UnSuper s sll n he prcess f bnng pprpre rs fcrs fr hese sse clsses. Ths shrcmng generes n verly l be nd hgh lph cnrbun reurns h n excessve rcng errr fr lernve sses. UnSuper s Apprch Rs Budgeng 4

16 In ddn he sse clss cnrbun l rs TURBs cmpues he mrgnl nd prprn rs frm ech mnger. The ble bel summrses he p 0 mngers cnrbun l rs nd rcng errr fr he Blnced pn. Tble 5: Blnced pn Tp 0 Cnrburs l rs Rn Asse Sub Clss Mnger pns Wegh n Mnger Mrgnl Cnrbun Rs Prprnl Cnrbun Rs (%) (%) (%) Lsed Prperes LPT Mnger A Lsed Prperes LPT Mnger B As Ex-Jpn IEQ Mnger A As Ex-Jpn IEQ Mnger B Ausrln Eques AEQ Mnger D Ausrln Eques AEQ Mnger A Vlue IEQ Mnger C Ausrln Eques AEQ Mnger I Ausrln Eques AEQ Mnger L Ausrln Eques AEQ Mnger J Ausrln Eques AEQ Mnger P Ausrln Eques AEQ Mnger Ausrln Eques AEQ Mnger Q Ausrln Eques AEQ Mnger K Emergng Mre IEQ Mnger D Grh IEQ Mnger E Grh IEQ Mnger F Ausrln Eques AEQ Mnger C Ausrln Eques AEQ Mnger G Ausrln Eques AEQ Mnger B Tl Alhugh he eny mngers buled bve ccun fr 36% f he Blnced pn s sses hey ccun fr 7% f he pn s l rs. As expeced he p 0 mngers/nvesmens resde hn he lsed equy sse clsses. verll he uhrs cnclude he Blnced pn s ell dversfed h he pssble excepn f he lsed prpery mndes hch ccun fr 5.5% f he pn s l rs bu nly 3% f he sses. The recen exreme vlly exhbed by he lsed prpery secr hs ffeced he cnrbun pn rs frm he Lsed Prpery Trus (LPT) mngers. ne pssble fndng s h n ddnl mnger (pssbly h l rcng errr mnde) culd be nrduced he LPT mnger lne-up help reduce he sse clss s rs. The nex sep n he Fund s rs budgeng nlyss s cnsder he mrgnl cnrbun rcng errr. The p 0 cnrburs re presened n he ble verlef. UnSuper s Apprch Rs Budgeng 5

17 Tble 6: Blnced pn Tp 0 Cnrburs rcng errr Rn Asse Sub Clss Mnger pns Wegh n Mnger Mrgnl Cnrbun Rs Prprnl Cnrbun Rs (%) (%) (%) Infrsrucure Dversfed Infrsrucure Arprs Prve Equy Buy us Infrsrucure Uly Vlue IEQ Mnger A Lng/Shr IEQ Mnger B Drec Drec Prpery Ausrln Equy AEQ Mnger K Vlue IEQ Mnger C Lng/Shr IEQ Mnger D Enhnced Pssve IEQ Mnger E Infrsrucure Rds Grh IEQ Mnger F Enhnced Pssve IEQ Mnger G Prve Equy Venure Cpl As Ex-Jpn IEQ Mnger H Ausrln Equy AEQ Mnger A Ausrln Equy AEQ Mnger B Grh IEQ Mnger I Ausrln Equy AEQ Mnger D Tl Trcng errr s cmpued by cnsderng he sndrd devn f ech mnger s excess reurns. When cnsderng rcng errr cre s requred n he nerpren f he resuls fr he lernve sse clsses due he sle smhed nd serlly crreled nure f her reurn srem. neheless he bve ble demnsres h he bul f he Blnced pn s rcng errr resuls frm lernve sses. These fndngs ere expeced UnSuper selecs lernve sses bsed n her rs-reurn chrcerscs nd hese sses ffer fvurble rs dused reurns. 3.5 DERIVATI F EACH PTI S EX-ATE ALPHA The nex sep n he rs budgeng prcess h TURBs cmpues s he esme f ech mnger s ex-ne (r frecs) lph. The uhrs ulsed n dpn f he Blc-Lermn Mdel (BLM) derve he esmes presened n he ble verlef: UnSuper s Apprch Rs Budgeng 6

18 Tble 7: ex-ane Alph Frecss fr he Blnced pn Asse subclss Mnger Blnced pn Mnger Weghs bserved ex- Ps Alph Trcng Errr UnSuper s Ve f Ech Mnger s ex- Ane Alph Cnfdence (Th Alph Les Whn % f Ve) TURBs ex- Ane Alph (%) (%) (%) (%) (%) (%) Ausrln Eques AEQ Mnger A AEQ Mnger B AEQ Mnger C AEQ Mnger D AEQ Mnger E AEQ Mnger G AEQ Mnger H AEQ Mnger I AEQ Mnger J AEQ Mnger K AEQ Mnger L AEQ Mnger M AEQ Mnger AEQ Mnger AEQ Mnger P AEQ Mnger Q Inernnl Eques Drec Prpery Lsed Prpery Infrsrucure Prve Equy Dmesc Fxed neres Inernnl Fxed Ineres (Hedged) Index Lned Bnds Tl/ Weghed Averge There s nsuffcen hsrc reurn d fr Ausrln Equy Mnger F fr hs nlyss s he de f he nvesgn. In ddn he TURBs ex-ne lphs genered re sensve he vlue f u. The bve ble cmbnes he nvesr s ve h he ech mnger s ex-ps lph ssess he exne lph fr h mnger. TURBs clcules he expeced ex-ne lph fr ech mnger eghs he resuls nd generes n verll ex-ne lph esme f 0.9% fr he Blnced pn. Smlr clculns ere perfrmed fr he Hgh Grh pn (ex ne lph esme muns.0%); Grh pn (0.9%); Cnservve Blnced pn (0.7%); nd he Cpl Sble pn (0.5%); 3.6 ASSESSIG WHETHER ACTIVE MAAGEMET IS EXPECTED T ADD VALUE Hvng derved n pn level ex-ne lph esme TURBs s hen ble ssess heher ech pn s expeced genere suffcen lph usfy deprure frm pssve replcn f he Fund s SAA Benchmrs. Hurdles fr ech pn re derved n secn 7 f Appendx. In essence he r f he pn s ex-ne lph dvded by he ncrese n vlly relve he benchmr mus exceed he derve f he Fund s cnsrned effcen frner usfy he use f cve mngemen. The dervve f he Fund s cnsrned effcen frner usng he Fund s nrmve lng-erm ssumpns he Blnced pn s vlly level muns 0.8. If he Fund hd nvesed UnSuper s Apprch Rs Budgeng 7

19 pssvely nd precsely mched s benchmrs hen he pn uld hve genered vlly f 7.%. TURBs esmed h he pn s l vlly muned 8.6%. Hence usfy n cve prgrm he ex-ne lph dvded by he ncrese n vlly (f.6%) mus exceed 0.4% fr he Blnced pn. The resuls fr ll nn-sri dversfed pns re presened n he ble bel: Tble 8: Requred ex-ane lph Jusfy Deprure Frm he Blnced pns SAA pn Hgh Grh Grh Blnced Cns. Blnced Dervve f he cnsrned effcen frner Cpl Sble Acul vlly hd he fund rced benchmr 0.4% 8.7% 7.% 5.0%.8% bserved l vlly.7% 0.% 8.6% 6.8% 4.% bserved ncrese n rs s resul f cve mngemen nd sregc lng.4%.5%.6%.8%.3% Mnmum requred ex-ne lph (ne f fees) 0.% 0.4% 0.4% 0.6% 0.6% Hence usfy he use f devng frm pssve nvesmen phlsphy he Fund hs exceed mnmum excess reurn f 0.% fr he Hgh Grh pn 0.4% fr he Grh pn 0.4% fr he Blnced pn 0.6% fr he Cnservve Blnced pn nd 0.6% fr he Cpl Sble pn. By cmbnng he fndngs frm Tble (he expeced vs. bserved be frm ech pn) h ble 5 (he ex-ne lph frecss fr ech mnger) nd ble 6 (he mnmum lph hurdle fr ech mnger) he nvesr s ble ssess he exen hch ech pn s benchmrs re expeced be me nd heher he Fund expecs exceed he mnmum hurdle requred usfy cve mngemen. The resuls f hs nlyss re prvded n he ble verlef. UnSuper s Apprch Rs Budgeng 8

20 Tble 9: Rs Budgeng nd Fcr Anlyss Summry Fndngs Reurn Arbun Hgh Grh Grh pn Blnced pn Cnservve Blnced Cpl Sble SAA bserved Be Fcrs SAA bserved Be Fcrs SAA bserved Be Fcrs SAA bserved Be Fcrs SAA bserved Be Fcrs (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) Ausrln Lsed nd Prve equy Inernnl Lsed nd Prve Equy Dmesc nd Inernnl Bnds Drec Prpery Indexed Lned Bnds Infrsrucure Lsed Prpery Truss Csh Exrneus rs prem Tl Expeced reurn frm Be surces Impc f reblncng Ex-Ane lph Tl expeced reurn Expeced excess reurn Mnmum hurdle usfy cve mngemen es: ^ The bserved be fcrs fr drec prpery nd nfrsrucure hve been se he benchmr nd he sse clsses ex-ne frecs hs been se zer. The bve ble demnsres h he Fund s curren mnger lne-up fr ech f he nn-sri dversfed pns generes be expsure h s brdly n lne h he pn s SAA nd h ech pn s ex-ne lph exceeds he mnmum hurdle requred usfy cve mngemen. UnSuper s Apprch Rs Budgeng 9

21 3.7 REVERSE PTIMISATI The fnl clculn perfrmed by TURBs s n ssessmen f n pml mnger lne-up (pml n he sense f generng he grees lelhd f beng he requred lph). Ths s herecl cnsruc nd cnsderble cre s requred n nerpreng he resuls. The ble bel summrses he pmsed prfl genered frm TURBs fr he Blnced pn: Tble 0: Reverse pmsn - Blnce pn Asse sub-clss Mnger Mnger Weghs Curren Mnger Lne-up Trcng errr ex-ane lph Resuls f Reverse-pmsn Revsed Wegh Trcng Errr ex-ane Alph (%) (%) (%) (%) (%) (%) Ausrln Eques Mnger A Mnger B Mnger C Mnger D Mnger E Mnger G Mnger H Mnger I Mnger J Mnger K Mnger L Mnger M Mnger Mnger Mnger P Mnger Q Inernnl Eques Drec Prpery Lsed Prpery Infrsrucure Prve Equy Dmesc Fxed neres Inernnl Fxed Ineres Index Lned Bnds Tl/ Weghed Averge Infrmn R There s nsuffcen hsrc reurn d fr Ausrln Equy Mnger F fr hs nlyss s he de f he nvesgn. As cn be seen n he bve ble TURBs recmmends h greer emphss be plced n hgh nfrmn r mngers (e.g. enhnced pssve mndes). By ulsng he recmmended mnger eghs he Blnced pn s expeced bn slghly hgher nfrmn r. Hever he chnge n nfrmn r s slgh suggesng h he pn s curren mnger lne-up hs been ell cnsdered. UnSuper s Apprch Rs Budgeng 0

22 4 PRACTICAL CSIDERATIS 4. Clbrng he BLM The BLM hs been dscussed n vrey f surces (see fr exmple Blc & Lermn (99) He & Lermn (00) Lermn (003) Wlers (008) nd Meucc (008)). The BLM s dscussed nd dped cnsder cve rs n secn A.4.. Whls he develpmen f he mdel nd he nvesrs ves s resnbly srghfrrd cre s requred n he clbrn f u s ell s he frmuln f meg mrx. 4.. Clbrng u Wlers (008) explns h s cmmn fr users f he BLM be cnfused s he pprpre vlue fr u. He nd Lermn (00) use vlue f 0.05 heres Schell nd Sccrf remr h mny peple use vlue f τ clse. Severl her uhrs (eg. Meucc 008) cmpleely elmne τ. Persnl dscussns h he uhr f he BLM (Lermn 008) sugges h τ shuld be such h he sndrd devns re f smlr scle h f Π. Lermn recmmends vlue rund 0.3 hen ne s cnsderng l reurns. The upu frm he BL mdel s n verly sensve he seleced vlue f τ. ur ve s h hgher vlue f τ (vz. n he rnge) generes mre sble nd nerpreble ex-ne lph esme fr he ex-ne lph esmes. 4.. Clculng meg The meg mrx represens ne s cnfdence n ech mnger r sc s bly genere lph. Specfclly meg represens he vrnce f he ve mrx Q. Wlers (008) prvdes rnge f mehds h cn be used derve he meg mrx. The mehds h ms ppeled us ere he use f cnfdence nervls nd usng he vrnce f he resduls frm he fcr mdels. We seleced he cnfdence nervl pprch. We dd hs by defnng cnfdence nervl h ech mnger r sc uld uperfrm her benchmr hn % rnge. Wlers (008) prvdes he fllng exmple f he mehd e use: Asse hs n esmed 3% men reurn h he expecn h s 67% lely be hn he nervl (.5%3.5%). Knng h 67% f he nrml dsrbun flls hn sndrd devn f he men lls us rnsle hs n vrnce f (0.005). We cmpue he ssced vrnce f ech mnger s uperfrmnce nd hese vlues frm he dgnl f he meg mrx. 4. Hndlng cllnery f rs fcrs Gven he hgh levels f cllnery beeen fcrs (fr exmple he reurn fr lrge cp Ausrln sc ndex (ASX 00) s hghly crreled h he reurn fr ASX 300 scs) e dp sndrd ecnmerc echnque f creng ne fcr reurn seres hch eque he resduls f he gven fcr fer regressng n ll prr fcrs. The srucure s sequenl nd requres he selecn f n rderng f he fcrs. In ms cses e chse h e cnsder be he prmry mre ndcr s he frs fcr nd hen rder he remnng fcrs reflec he sgnfcnce h e expec he fcr reurns hve n explnng he reurns f mngers/nvesmens n ech secr. Fr exmple fr he Ausrln shres secr e chse he ASX300 s he prmry fcr nd clcule sy he resdul reurns f he Vlue ndex reurns fer regressng n he ASX300 ndex reurns. e h hs prvdes bh n esme f he mre be f he Vlue ndex he ASX300 mre nd f he verge excess reurn f he Vlue reurn ver he ASX300 fr he esmn perd (n lph esme) nd he resdul seres hch cn be nerpreed s he reurn seres fr vlue fer dus fr s mre cmpnen. Ler rder fcrs re regressed n he prmry fcr nd ech f he prr UnSuper s Apprch Rs Budgeng

23 resdul reurns seres frm he hgher rder regressns. In he nn bel e re hese resduls s F nd fr ese f expressn e cn re F F nd refer hem s he resdul fcrs. e h ech resulng rs fcr s smply he resdul f he excess reurn f fcr fer llng fr he expsure he her (hgher rder) resdul fcrs nd hese resduls re rhgnl. Ths pprch ls hs he dvnge f prvdng sme nsgh n he nure f he mres fr he perd n hch he esmes re bsed. 4.3 Mngng lernve sses Alernve sses re fen vlued h reference mhemcl mdel hve sle nd nfrequenly qued prces nd s such dsply smhed serlly crreled reurn srem. Cnrsng her reurns lsed ndces s f lmed vlue. T vercme hese cncerns e cnrs he reurns frm hese sse clsses he rllng gemerc verge reurns frm lsed mres ver yers (). 4.4 Inerpreng he dervve f he cnsrned effcen frner The dervve f he cnsrned effcen frner represens he hurdle h mus be cheved usfy usng cve mngemen rher hn lerng he be llcn fr gven sregc sse llcn. Three prccl cnsequences rse frm hs smple bservn:. The greer he slpe f he cnsrned effcen frner he greer he hurdle. As such s mre lely h he hurdle uld be cheved fr hgher rs sreges (such s fr Hgh Grh pn) hn fr Csh pn r her cnservve pns (here he slpe f he cnsrned effcen frner s s mxmum). Fr hese mre defensve sreges my ell be mre pprpre ncrese be rs (e.g. by ncresng durn r bnds r by nrducng cred n prfls) rher hn ncrese he use f cve rs r pr lph frm her surces.. Sreges h reduce verll pn rs bu h he cs f slgh reducn n reurn re mre pprpre hn ler rs pns h hgher slpe f he cnsrned effcen frner. As n exmple currency hedgng f nernnl grh sses generlly reduces pn vlly bu he cs f mplemenng he hedge. I my hus be pprpre mnn hgher currency hedge r fr nernnl grh sses hn cnservve pns hn fr n pn h hgher rs lernce. 3. The greer he number f cnsrns mpsed by he nvesr he fler he slpe f he cnsrned effcen frner nd dpng cve mngemen s mre esly usfed. UnSuper s Apprch Rs Budgeng

24 4.5 Reverse pmsn The pmser fvurs mngers nd scs h ler rcng errrs benchmr nd mpled srng perfrmnce cmpred rs. Hever he pmser des n cnsder rnge f her fcrs h re relevn he srucurng prfl ncludng he cpcy f he mnger nd he mprnce f dversfyng mnger rs. Whls he TURBs pmsed prfl prvdes sme nsghs n penl prfl srucure s nly l nd des n e n ccun rnge f her (predmnnly qulve) prfl cnsrucn cnsderns ncludng: The ype f mnger (.e. develpng r mure buque r nsunl); Cpcy f he mnger; Desred syle becve f he verll prfl; Mnger-specfc pernl rss nd he need dversfy mnger expsure; The verll bs smll cps (nd he smller end f he smll cps mre); nd her qulve cnsderns. TURBs s prculrly helpful n reveng expsures crss mngers nd n denfyng nd ssessng he ey surces f reurn fr mnger s ppsed beng used cnsruc fnl prfl. In hs regrd pmsers end hve dffculy prducng menngful prfls gven he dffculy n ncrprng lrge number f vrbles (ncludng qulve cmpnens) nd he endency cncenre llcns mngers h slghly beer perfrmnce chrcerscs. 5 CCLUSIS Rs budgeng s he prcess f seng rge level f rs be cceped he prfl level nd llcng hs rs crss number f nvesmens n he ms effcen mnner n rder mxmse reurns hls cnnng rs hn he greed rges. UnSuper hs develped n n-huse rs budgeng nd fcr nlyss prgrm h mnrs he exen hch he Fund deves frm s Sregc Asse Allcn. TURBs prvdes he Fund h frml frmer fr dscussn nd nlyss. The resulng nlyss prvdes nsghs h help frmule he Fund s nvesmen rrngemens. UnSuper s Apprch Rs Budgeng 3

25 Appendx - Mhemcl Frmuln T d he reder glssry f nn s prvded n Appendx. Le Le A. Bcgrund Wm dene he Sregc Asse Allcn (SAA) egh fr pn hn sse clss m. dene he egh (s prprn f he l Fund) fr mnger h peres n sse clss Ν me. Hence. Where denes he l number f mngers spnnng ll sse clsses. Le dene mnger s egh hn pn (mnger nvess n sse clss ) me. Then m W...() here m mps n sse clss fr he egh W m. Le dene he vecr f eghs f mnger hldngs me hn pn. W Hence......(.) W Furher: here denes he un vecr.... Fr ese f nn vecrs nd mrces ll exclude he me ( ) nd pn ( ) suffx n hs ppendx lhugh ech mrx s ssumed be me nd pn dependen. If µ denes he ex-ne expeced reurn frm mnger me n sse clss hle µ denes he bserved reurn frm mnger me. Then µ s vecr f ex-ne expeced reurns µ fr ech mnger me nd µ... (.) M µ Furher µ equls he Fund s expeced reurn n yer s me (n he bss h he mnger eghs remn cnsn ver he mefrme cnsdered). The vlue f µ s derved n secns A. nd A.4 bel. UnSuper s Apprch Rs Budgeng 4

26 A.. Esmng Ech Mnger s Fcr Expsures Ech mnger s reurn cn be cnsdered s he sum f be (r mre) fcr expsure geher h n cve mnger reurn (lph). The frs sep n esmng he vlue f µ s deermne ech mnger s fcr expsure r be expsures. The mre dffcul sep n he prcess s esme ech mnger s prspecve ex-ne lph. Ths s dscussed n secn 4. A... Decmpsng Mnger Reurns Shrpe (964) derved he Cpl Asse Prcng Mdel (CAPM) hch ses h shre s expeced perfrmnce me (gven by E ] ) s dependen n he exen hch he shre s crreled [ R s he mre (referred s he shre s sysemc rs r be). Specfclly Shrpe derved he CAPM frmul: ( ~. (..) E [ Rs ] R f β E[ RM ] R f ) ε s M Where s M β nd s denes he esme cvrnce beeen he secury (s) nd n Vr( R M ) pprpre mre ndex (M) me hle Vr( R M ) denes he esmed vrnce f he mre me nd ~ε represens he errr erm. s CAPM s exended by Rss (976) n hs frmuln f Arbrge Prcng Thery (APT). APT ses h he expeced reurn f fnncl sse cn be mdelled s lner funcn f vrus mcr-ecnmc fcrs {F } r herecl mre ndces here sensvy chnges n ech fcr s represened by fcr-specfc be ceffcen ( β ) fr secury s me. K R s R f β s F ~ ε s Hence under APT: (..) here ~ε represens he errr erm h nn-zer verge vlue. K s he se f ll pplcble fcrs s (he rs free re s n elemen f h se) nd s F denes he bserved reurn frm fcr me. Rerng equn.. ensure h he errr erm hs zer men nd seng he rs free re s fcr gves: K R s s β s F ε s (..3) Shrpe (00) dscusses he use f fcr mdels prvde rbus predcns n rs esmn prcedures. We cn frmlly ssess fcr mdel fr mngers s flls. Ech mnger hlds se f secures. Le S be he se f ll secures. S s Then & s & dene mnger s hldng n secury s me. Le s Fr sme mndes UnSuper lls shrng s such he usul cnsrn & s 0 s s.. S des n pply n he frmer h flls. We cn express ech mnger s expeced reurn s: S S K S s s s & s &. β s F. s s s µ. & ε s (..4) Where µ represens mnger s bserved reurns me. UnSuper s Apprch Rs Budgeng 5

27 Equn.4 cn be expressed mre smply s: Where K µ β F ε (..5) s he esmed eghed ex-ps verge f ech sc s dsyncrc rs (eghed crss ll secures held by he mnger) β me nd ε s he mnger s eghed errr erm h zer men. s he mnger s eghed verge expsure fcr Hence he mnger s ex-ps reurn cnsss f n lph cmpnen ( ) be cmpnen K ( { } K β F ) nd n errr erm ( ε ). The be cmpnen s esmed by slvng fr he se f β fr ech mnger usng mulple regressn. The frmuls presened bel ere dscussed n mre del n Srumnn & Grd (007). Equn..5 cn be ren n vecr-mrx nn: vz. µ X. β ε (..6) µ r µ... µ r... F F F... r F F F... r F F F... r β... β ε ε... ε r Where r> nd r denes he number f mnhs d (bh fcr nd mnger d) h s nlysed. r mus be greer hn (he number f fcrs nlysed). Here F denes he bserved reurns frm fcr me. Usng mulple regressn slve equn (..6) gves: ( X X ) X µ β (..7) Equn.7 prvdes n unbsed esme f he be fcrs (nng h he frs elemen n he β vecr represens he ex-ps lph esme fr he mnger). A... Deermnng he gdness f f The gdness f f f he regressn mdel gven by equn..7 cn be ssessed usng sndrd sscl echnques e.g. by cnsderng he vecr f errrs rsng frm he regressn me hch s gven by he vecr: ε µ X β (..) The dsrbun f errrs s crefully exmned ensure h he vecr s pprxmely nrmlly dsrbued nd lcs ucrreln. TURBS ulses hree cmmn sscl echnques deermne he gdness f f:. A ble f sscl vlues s derved (sndrd sscl ess re used such s he vrnce seness urss nd ucrreln f resduls he mdel s F-ssc lng h R nd dused R );. A hsgrm f resduls relve nrml dsrbun s chred; nd 3. A nrml prbbly pl s derved. UnSuper s Apprch Rs Budgeng 6

28 nce resnble f hs been bned s pssble explre he sscl sgnfcnce f ech f he be fcrs h ere fund (s ell s he ex-ps lph esmed frm he mnger) n rder deermne he sscl sgnfcnce f hese fcrs. The frmule h fll fr he remnder f hs secn re presened fr cmpleeness nd re ell-nn sndrd sscs f mulple regressn mdellng. The vrnce f he mulple regressn mdel s gven by: ε ε (..0) r We cn ls derve he sscl sgnfcnce nd 00( )% cnfdence nervl f ech elemen hn he β vecr usng -sscs. In prculr he 00( )% cnfdence nervl fr he ex-ps lph nd ech be fcr s cmpued s flls: (..) nd ± dg( X X ) r β β fr.. (..b) ± ( ) dg X X r Where dg ( X X denes he frs elemen n he dgnl f he nverse mrx X X nd dg( X X ) r ) denes he h elemen n he dgnl f he nverse mrx fr ll..k. Furher bned frm he Sudens -dsrbun h r degrees f freedm. Fnlly ne cn ssess he sscl sgnfcnce f ech be fcr by cnsderng he -ssc frm ech be fcr (gven s dg( X X ) ) nd hen dervng s p-scre (en frm he nverse cumulve suden s -dsrbun h r degrees f freedm). The bserved sgnfcnce level (r p-scre) s he smlles fxed level (usully 5%) hch he fed be fcr s deemed be ssclly sgnfcn. The sndrd regressn pprch ulned n secn A. s bsed n he ssumpn h he fcrs ( F ) re uncrreled nd hve equl nd cnsn uncerny (.e. re hmscedsc). In secn A..3 e derve mehd hndle fcrs h unsble vrnces nd n secns 4. nd A..4 e cle he mre cmplced prblem f crreln beeen fcrs nd he mngemen f cnegrn. A..3. Weghed Les Squres (WLS) esmn WLS regressn cmpenses fr vln f he hmscedscy ssumpn by eghng fcrs (.e. F ) dfferenlly. Under WLS fcrs hch cnrbue lrge vrnces n he regressed mnger s reurns cun less n esmng he β ceffcens. The resul s h he esmed ceffcens re usully very clse h hey uld be n equn..7 bu under WLS regressn her sndrd errrs re smller. Specfclly he eghed sum f squred resduls s mnmsed f ech fcr s egh s equl he recprcl f he vrnce f he fcr. If Φ denes squre mrx f errr esmes fr ech fcr.e. Φ dg(... ) (.3.) UnSuper s Apprch Rs Budgeng 7

29 Then equn..7 cn be reren s: ( ) Φ X Φ µ β X X (.3.) X Φ X And he vrnce f he esmes muns ( ) (.3.3) Hence genere WLS esme f he fcr expsures e run he sndrd mulple regressn mdel prvded n secn A.. T vercme he cncerns ssced h he lc f hmscedscy e re-run he regressn frmuln bu usng eghed les squres pprch ulned n equn.3.. The frmuls presened n hs secn re sndrd sscl echnques. A..4. Adusng he regressn slun vercme cllnery Cllnery s sscl cncern h rses hen r mre fcrs n mulple regressn mdel re hghly crreled. In hs sun he be esmes f ech mnger s reurns he fcrs (gven n equn..b bve) nd he sgnfcnce ess fr he fcrs (gven by her p-scres) re underesmed. Furher be esmes chnge errclly n respnse smll chnges n he d nd he esmes f he ex-ps lph becme unrelble. Unfrunely hn he fnnce envrnmen mny fcrs re hghly crreled (s n exmple mnhly reurns frm he ASX300 Accumuln ndex s 98% crreled he ASX lsed prpery ndex beeen 00 nd 007). T vercme hs cncern lernve pprches regressn nlyss ere explred nd reeced befre e derved n lgrhm h s sble nd cmpunlly effcen. The frs emp s ulse prncpl cmpnen nlyss (PCA). PCA s echnque used reduce muldmensnl dses ler dmensns fr nlyss nd hs he benef h he ne se f fcrs re rhgnl (hence vercmng he dngers f cllnery). Unfrunely he genered fcrs lced nuve nerpren nd ere unsble. The nex pprch h e explred nvlved buldng up he regressn mdel by ddng fcrs unl he mdel becme unsble. Hence ne uld frs run he regressn h ech ndvdul fcr nd hen selec he fcr h he les p- vlue (.e. he ms sgnfcn fcr) nd sysemclly dd fcrs he mdel. Unfrunely hs pprch frequenly cnverged sngle fcr h l verll mdel sgnfcnce (vz. The dused R vlues nd F-ess eren del). We fnlly derved he fllng lgrhm vercme cllnery:. Genere mulple regressn nlyss h ll he fcrs pplcble he sse clss;. Sve he dused R f he regressn nlyss; 3. Fnd he fcr h he hghes p-vlue (.e. he fcr h s les ssclly sgnfcn); 4. Remve hs fcr frm he se f nlysed regressn fcrs; 5. Re-run he regressn h ll he remnng fcrs; nd 6. Cnnue he lp (K-) mes unl nly sngle fcr remns. nce ll (K-) runs ere cmplee he TURBS prgrm scns ll he vlble dused R vrbles fnd he hghes vlue. The se f fcrs h re ssced h he pml dused R ere hen re-run hs me remvng ny fcr hse p-vlue s bel he user-defned ssclly sgnfcn vlue (usully se 5%). The end resul s h he TURBS prgrm s ble cnverge n se f sgnfcn fcrs h (lhugh crreled) genere sble nd nuve fcr expsures fr he mnger. UnSuper s Apprch Rs Budgeng 8

30 UnSuper s Apprch Rs Budgeng 9 A.3. Esmng he mrgnl cnrbun rs frm ech mnger Le V dene he cvrnce mrx fr he mngers ( me ). Hence V Where denes he esmed cvrnce beeen mnger nd mnger me. If r represens mnger s l bserved reurn n mnh (ll-..l-m) nd l s he les bserved me perd hen he cvrnce beeen mnger nd me s esmed by: ( )( ) { } r l l l l r µ µ µ µ Where l µ represens he crude verge mnhly reurn hence r l l l r µ µ e h ssumes h mnger s reurn seres s free f u-crreln. If ρ represens he esmed u-crreln fcr fr mnger me hen ρ cn be rughly esmed usng ne-mnh lg s flls: ( )( ) { } ( ) r l l r l l l l µ µ µ µ µ µ ρ r And mus be dused fr ucrreln by dvdng he bserved cvrnce beeen mnger nd mnger by he fcr ( )( ) ρ ρ. e h f n evdence f u-crreln exss hen ρ 0 nd n dusmen s requred he sndrd cvrnce frmul. The pprch generes prccl nd cnservve dusmen he cvrnce mrx bu my ell verse he cvrnce f reurns fr lsed equy mngers h ulse mmenum sreges. Hence cvrnce f reurns s esmed s: ( )( ) { } ( ) ( )( ) r l l l l r ρ ρ µ µ µ µ (3.) Furher ech pn s vrnce me ( ) cn be esmed frm he ex-ps d: V (3.) Where s defned n equn (.) bve.

31 UnSuper s Apprch Rs Budgeng 30 The mrgnl cnrbun he l fund vrnce fr mnger s derved by ng he prl dervve f he l pn vrnce ( V ) h respec he eghng fr mnger : ( ) V <> <> > V (3.3) Equn 3.3 lls us derve he mrgnl cnrbun he l pn rs (en s he sndrd devn f reurns fr pn r ) V (3.5) Hence he mrgnl cnrbun he l pn rs frm mnger me s he r f he cvrnce f reurns beeen mnger nd he pn dvded by he sndrd devn f reurns f he pn. Ech mnger s prprnl cnrbun rs s smply her mrgnl cnrbun mulpled by her egh n he prfl. Ths s he pprch ulned by Mn (007). Hence prprnl cnrbun rs fr Mnger...(3.6) As chec he sum f he prprnl cnrbun rs per mnger equls he pn s sndrd devn f reurn: (3.7) Equns 3.6 nd 3.7 ere derved by Scherer (000) nd Ilrch (004).

32 A.4. Deermnng Ech Mnger s Ex-Ane Alph We sr by cnsderng uly hery hch cn be ulsed ssess equlbrum expeced reurns nd hen re-ren fnd he pml prfl gven cvrnce mrx nd vecr f expeced reurns. Such n pprch nrduces he need fr Byesn frmer deermne ex-ne reurns. Afer reveng cmmnly used mehds fnd ex-ne reurn esmes e seleced he Blc-Lermn Mdel (BLM) hch s dped cnsder lph rher hn l reurn expecns. The pprch used s smlr h cnsdered by Wnelmn (Lermn 003b) excep h e dd n me he smplfyng ssumpn h lph s ndependen beeen mngers. The ssumpn h lph s ndependen beeen mngers ccurs frequenly hn rs budgeng ppers. Whls lph nd be re ndependen (by desgn) lph beeen smlr mnger syles s frequenly crreled (e.g. hen sngle qunve mngers uperfrms frequenly mny qunve mngers ls uperfrm prly s smlr mehdlges re seleced by mngers). As such e fel necessry cnsder he pssbly h mngers lph culd be crreled. A Mrz effcen prfl s ne h ffers he grees expeced reurn fr gven level f rs. T fnd he se f such prfls cmpunlly effcen pprch s mxmse he Fund s l reurn per un f rs. T d s e me use f uly hery nd clculus. Uly hery (ulned n vrey f ppers e.g. Mn (007)) generes vrey f ey fndngs dependng n he shpe f he nvesr s uly funcn bu ne ey cnclusn ses h he nvesmen becve s mxmse expeced reurn per un f rs. λ Mxmse: µ V (4.) Where λ denes he nvesr s rs versn prmeer. Tng he dervve f 4. h respec he egh fr mnger nd seng hs equn zer lng h mng use f equn 3.3 gves: µ λ V λ µ µ λ 0 (4.) Hence he lcl mxmum f 4. fr mnger s derved hen µ λ (4.3) e h λ represens he h elemen f he vecr λ V. If pml mnger eghs hen he lcl mxmum f 4. s derved hen: µ λv (4.4) Pre-mulplyng (4.4) by he nverse f he cvrnce mrx gves: V λ µ (4.5). A.4.. Reverse pmsn denes he vecr f ne cn use equn 4.4 derve he expeced reurn fr n pn gven he esmed rs versn prmeer fr he pn ( & λ ) he esmed vrnce-cvrnce mrx (V ) nd he curren mnger eghs hn he pn prfl vz. µ & λ V (4.6). UnSuper s Apprch Rs Budgeng 3

33 Equns 4.5 nd 4.6 hve been presened n vrey f ppers e.g. Mn (007). As he rs versn prmeer fr ech Accumuln pn cnn be ccurely scerned equn 4.6 s nly f benef deermne he relve dfferences beeen expeced reurns fr mngers. If ne s ble derve he expeced reurn fr sngle mnger h hgh degree f cnfdence usng prr nledge hen he se f l expeced reurns cn be deermned. An bvus cndde fr he selecn f he bse mnger s csh mnger here reurns re generlly sble nd predcble. The bve prgrph nrduces Byesn cnsdern n he deermnn f ex-ne reurn frecss. In ddn equn 4.6 gnres he uncerny f expeced reurns n equlbrum. T ll fr he nvesr s ves s ell s he need del h he uncerny f equlbrum expeced reurns hve been cnsdered usng he Blc-Lermn Mdel (Lermn 003). A.4.. The Blc-Lermn Mdel (BLM) The BLM dels h he uncerny f expeced reurns by dpng Byesn pprch here he nvesr s ves s subec errr nd cn be mdfed by he mre s nl equlbrum reurn expecn derve blended reurn expecn. BL defne eq s he equlbrum prfl f ll sses n he mre (eghed by mre cplsn). If here re sses n he unverse f ll sses hen eq s x vecr. The cvrnce mrx fr ll sses s defned s Σ (n x mrx). Under he BL Mdel he mpled equlbrum f reurns (n excess f he rs free re) s nrmlly dsrbued h n expeced reurn vecr f Π: E[ Π] Σ (4..) eq Equn 4.. s derved n n nlgus mnner equn 4. (.e. by deermnng he mxmum reurn per un f rs). In equn 4.. represens he rld-de rs versn prmeer nd s usully se he rs-regressn slpe f ll sse clsses. The vrnce f he equlbrum reurn s gven by Σ nd n he BL mdel s scled dn by fcr τ hch mesures he uncerny f he prr. Hence he mre s prr equlbrum dsrbun f reurns ( Π ) s dsrbued rmlly.e. Π ~ ( Σ τσ ) (4..) eq Hvng deermned he prr dsrbun f equlbrum reurns he BL mdel hen lls fr he nvesr s ves. If he nvesr hs K ves (clerly K ) hch re ndependen f ech her nd f he mre s prr equlbrum hen he BL mdel deermnes he nvesr s expeced verll reurns gven her ves nd he cnfdence he nvesr hs n her ves. Le Q dene he mrx f nvesr s ves ( K x mrx). Then Q cn be expressed s he prduc f Ρ (hch s K x mrx) nd he prr equlbrum dsrbun f reurns Π. Furher f Ω denes he nvesr s cnfdence n her ves hen Q s nrmlly dsrbued h men Ρ Π nd vrnce Ω. Specfclly: Q ~ ( ΡΠ Ω ) (4..3) UnSuper s Apprch Rs Budgeng 3

34 UnSuper s Apprch Rs Budgeng 33 We sh fnd he Cmbned reurn dsrbun f reurns gven he nvesr s ves. The resul s presened n equn 4..6 nd s derved by Blc nd Lermn (see fr exmple Lermn (003)). The fllng pge descrbes ne mehd f dervng he BLM frm frs prncples bsed n r by Jng e l (005). If e le Π Q Y Ρ I X nd Ω τσ ε hen he regressn slun ε X Π Y (3) hch mnmses he errr erm ε uld prvde he pml slun equns 4.. nd 4..3 nd uld genere eghed verge f he equlbrum reurns nd he nvesr s ves (r usng Byesn ermnlgy he pserr dsrbun f Π gven Q ). The greer he nvesr s cnfdence n Q (.e. he ler he vlues n Ω ) he hgher he egh f he cmbned ves he nvesr s ve. We ls n h Ω Σ ~ τ ε. If e le Ω Σ Θ 0 0 τ hen usng eghed les squres he slun : ε β µ. X s prvded by β hch hs n unbsed expeced vlue f ( ) µ Φ Φ X X X nd n esmed vrnce f ( ) Φ X X (Refer equn.. nd..3 nd ne h Φ denes squre mrx f errr esmes fr ech fcr). By subsun e herefre hve h he unbsed esme f he cmbned ve s: ( ) Y X X X Θ Θ hle he vrnce f he cmbned ve s ( ) Θ X X. Hence he unbsed esme f he cmbned ve [ ] [ ] Π Ω Σ Ρ Ρ Ω Σ Ρ Q I I I τ τ And he vrnce f he cmbned ve [ ] 0 0 Ρ Ω Σ Ρ I I τ Srng h he slun he vrnce gves: Vrnce [ ] ( ) ( ) [ ] ( ) [ ] 0 0 Ρ Ρ Ω Σ Ρ Ρ Ω Σ Ρ Ω Σ Ρ τ τ τ I I I (4..4) We cn ls slve fr he unbsed esme f he cmbned ve: [ ] [ ] Π Ω Σ Ρ Ρ Ω Σ Ρ Q I I I τ τ ( ) [ ] Ρ Ρ Ω Σ τ [ ] Π Ω Σ Ρ Q I 0 0 τ ( ) [ ] ( ) [ ] Π Ρ Ω Σ Ρ Ρ Ω Σ Q τ τ

35 [ Ρ Ω Ρ] [( τσ) Π Ρ Ω Q ] ( τ Σ) (4..5) Hence he BLM generes cmbned ve h s rmlly dsrbued h n expeced vlue gven n equn 4..5 nd vrnce gven by equn The cmbned ve ( τσ) [ Ρ Ω Ρ] [( τσ) Π Ρ Ω Q ] [( Σ) Ρ Ω Ρ] ~ τ (4..6) Equn 4..6 hs been presened n numerus ppers ncludng Lermn (003) Wlers (008) Meucc (008) ec. A.4.3. Adpng he BLM fr cve rs Equn 4..6 prvdes frmer fr cmbnng equlbrum reurns ( Π Σ ) fr sses K nvesr specfc ves ( Ρ ) he egh he nvesr plces n he equlbrum ve nd he τ cnfdence he nvesr hs n her ves ( Ω ). The BLM cn be exended cnsder cve rs s flls. If E[ ] Denes he ex-ne expeced lph frm mnger me. Q A Ω A Denes he vecr f expeced lphs. Denes he cvrnce mrx f ech mnger s reurns n excess f her fcr expsures. Denes he nvesr s ve f he expeced lph genered frm ech mnger. Denes he cnfdence he nvesr hs n ech mnger s bly genere lph. Then ( ) ( ) Where τ ΡA ΩA ΡA τ Π A ΡA ΩA QA (4..7) Π A denes he vecr f equlbrum cve reurns. When cnsderng l reurns (.e. n equn 4..6) Q s usully expressed fr cnvenence s he prduc f Ρ (hch s K x mrx) nd he prr equlbrum dsrbun f reurns s he nvesr fen hs ve h cern sse clss ll uperfrm nher sse clss bu hs less cnfdence s he bslue reurn genered by bh sse clsses. Hever ne s n requred express Q s prduc f Ρ nd Π nd hen cnsderng cve reurns he nvesr ends hve ve s he bslue level f lph genered by ech sse r mnger. Lermn (003b) derved equn 4..7 nd explns h he equn cn be furher smplfed by cnsderng h n equlbrum: ) Acve reurns fr ll sses equl zer (.e. 0 ); nd Π A b) The nvesr s ves bu expeced lph (gven n Q A ) re frmed ndependenly equlbrum reurns ( Π A ) hence ΡA s n deny mrx. Hence ( ) Ω [ Ω Q ] τ A A A (4..8) eq UnSuper s Apprch Rs Budgeng 34

36 Equn 4..8 s presened n Lermn (003b) nd s cenrl UnSuper s rs-budgeng frmer s reles expeced cve reurns UnSuper s ves bu cve reurns UnSuper s cnfdence n hse ves nd he cvrnce beeen hsrc cve reurns. e h equn 4..7 s ls f use f hsrc lph esmes replce he equlbrum lph esme fr ech mnger. Under hs scenr he nvesr s ves bu expeced lph ( Q A ) re sll frmed ndependenly hsrc reurns ( Π A ) nd Ρ A remns n deny mrx. If e le Α dene he vecr f ex-ps lph esmes hen Α... here s s derved frm ech mnger s regressn nlyss prvded n equn.4 hen e cn recs equn 4..7 s flls: ( ) ( ) τ Ω τ Α Ω Q A A A (4..9) The resuls frm equn 4..9 cn be cnrsed h f 4..8 s reles expeced cve reurns he nvesr s ves bu ech mnger s cve reurns her cnfdence n hse ves he cvrnce beeen hsrc cve reurns he cul bserved ex-ps lph nd he cnfdence he nvesr plces n hse hsrc ex-ps esmes gven by. τ Effecvely equn 4..9 generes en ex-ne lph h s eghed verge f hsrc ex-ps lph esmes cmbned h he nvesr s ves. Whls UnSuper uses equn 4..9 hn TURBs equn 4..8 s n pprpre lernve. A.5. Reurn Arbun By cmbnng nd eghng he Fund s expsure ll mngers e cn derve he ex-ne expeced reurn fr pn () nd rbue hs reurn beeen vrey f surces. In secn A. e rbued he ech mnger s l reurn ( µ ) n lph nd be cmpnen vz. µ β F ε Where K β s he mnger s derved expsure Fcr me nd ε errr erm h zer men. s he mnger s eghed Reurn rbun hs been dscussed by vrey f uhrs (ncludng Mn (007) nd Lermn (003b)). We exend he nlyss cnsder h ech f he mnger s regressed fcrs mp n he Fund s SAA benchmrs. Whn he se f vlble be fcrs { F } K here s sub-se f M fcrs h eque he Fund s benchmr fr ech sse clss (s n exmple fr he Ausrln Equy sse clss he curren benchmr s he ASX 300 Accumuln ndex hch n urn s penl be fcr). Le BM m dene he se f sregc benchmr fcrs fr ech f he Fund s M sse clsses. { } M m Then e cn deermne he mnner n hch ech be fcr mps n he Fund s sregc benchmr fcrs s flls: UnSuper s Apprch Rs Budgeng 35

37 UnSuper s Apprch Rs Budgeng 36 M m m m BM RP F ν γ (5.) here RP denes he rs premum vlble frm Fcr hch cnn be rbued ne f he Fund s benchmrs m γ denes he esmed sensvy beeen fcr nd he m h sse clss benchmr. ν denes he resduls f he regressn equn h zer men. The rs prem nd he m γ fcrs re slved n he sme mnner h s used derve he fcrs frm equn..5 (vz. eghed les squres mulple regressn h he sme pmsn lgrhm mnge cllnery). Frm equn..5 e cn express ech mnger s reurns s funcn f lng h se f fcr expsures K F β. Equn 5. lls us ssess he ln beeen ech fcr expsure h f he Fund s SAA benchmrs. By cmbnng equns nd 5. e cn derve he expeced reurn fr ech pn expressed s funcn f: e) The eghed verge f ech mnger s expeced ex-ne lph; f) The eghed verge expsure he Fund s SAA benchmr; g) Exrneus be rs prem frm fcrs h dffer he Fund s SAA benchmr; nd h) An errr erm. Le R dene he hsrc (ex-ps) reurn frm pn me. Then R µ (5.) Subsung equn..5 n 5. gves: R K F ε β (5.3) Subsung equn 5. n equn 5.3 gves: R K M m m m BM RP ε ν γ β...(5.4) K K K M m m m RP BM R ε ν β β β γ...(5.5) The bve equn s cenrl he Fund s fcr nlyss. In prculr: denes he ex-ps lph expeced frm he fund s mngers fr pn. The erm K M m m m BM β γ denes he eghed verge expsure he pn s SAA benchmr;

38 K Whle β RP denes he pn s expsure her nn be surces (such s cred rs vlue nd smll cp bses secr bes ec); K The fnl erm β ν ε represens he errr r resduls f he vrus regressn esmes (nd hs zer men). The dsrbun s crclly exmned ensure h s suffcenly nrmlly dsrbued. Evdence f exreme urss r seness culd be cuse fr cncern. Recll h n secn e defned Wm s he me-ndependen Sregc Asse Allcn (SAA) egh fr pn hn sse clss m. Hence f he Fund ere pssvely mch s SAA hen he expeced reurn fr pn uld mun : W Where [ BM m ] M m m E[ BM (5.6) E denes he expeced reurn fr he benchmr fcr fr sse clss m me. Le RA dene he Fund s l cul rs llcn fr pn me. Hence RA represens he devn frm he Fund s SAA fr pn me. Then > RA RA M µ W BM (5.7) m m m M K M β γ K m BM m Wm BM m β RP ε m m K Where ε β ν ε...(5.8) Hence m ] RA {Acve Mngemen Rs} {Fcr Rs} {Errr Dsrbun Anlyss} As such ech pn s expsed hree frms f rcng errr r rs. The frs reles cve mngemen rs he secnd nd ms sgnfcn rs reles he exen h he be Fcr expsures fr he pn dffers he pn s SAA hls he hrd rs reles he exsence f her exrneus be fcrs hn he prfl. In ddn rs rbun ne cn clcule he expeced reurn fr ech pn. By ng he expecn f equn 5.8 nd nng h he expeced errr erm s zer (.e E [ ε ] 0 ). E[ R ] K M K E[ ] γ m β E[ BM m ] β E[ RP ] m...(5.9) Where E[ ] denes ech mnger s expeced ex-ne lph (derved n secn 5 f hs ppendx). TURBS deermnes ech pn s ex-ne lph vlue nd hen ess heher he esme exceeds he mnmum requred hurdle. Fr ech pn E [ R ] s cnrsed he pn s nvesmen becves ensure h he Fund remns cnfden h he becves remn chevble h pprpre levels f cnfdence. UnSuper s Apprch Rs Budgeng 37

39 A.6. Rs Allcn If e le dene he vecr f esmed ex-ne lph s fr ech mnger nd esmed vrnce-cvrnce mrx f excess reurns hen dene he AEQ ψ AEQ... ψ nd (5.6) CA ψ... ψ Where denes he esmed cvrnce beeen mnger nd mnger s excess reurns r ψ lph srems. Generlly ne expecs ψ r ssumed hese crss-crrelns be zer nd 0 hen <>. Hever mprnly e hve n frced ψ relve he mnger s bserved r fed be expsures. denes mnger s rcng errr TE fr ech mnger me..e. T E [ ψ ψ... ψ ] As resul cn be reren s reurns hch need n equl he deny mrx.. denes he vecr f rcng errrs T E. C. TE here C denes he crreln mrx f excess The pml cve rs llcn prblem cn be specfed s mxmsng he Fund s expeced cve reurn subec specfed rs budge. Rs llcn ypclly fcuses n seng lm n he Fund rcng errr. Alhugh hs s n he pprch dped by TURBs s useful ssess ech pn s rs llcn usng he smpler ssc f rcng errr. We sr h he rdnl slun rs budgeng nmely mxmse expeced excess reurns subec he cnsrn h he Fund s rcng errr remns hn specfed lm. Algebrclly hs prblem cn be specfed s: Mxmse: subec he cnsrn h pn s cve rcng errr s less hn specfed mxmum (.e. TEmx ) here TEmx represens he pre-deermned mxmum permssble rcng errr. Frm secn 4. e n h he mxmum f λ µ V ccurs hen he pml prfl (gven by ) equls V µ. λ By subsun he funcn λ s mxmsed f (6.) λ We n need slve fr λ such h he cnsrn s bned. Hence λ s such h: λ λ > λ (6.3) TE mx > λ (6.4) TE TE mx (6.) mx cn n be slved by subsung λ n equn 6.. UnSuper s Apprch Rs Budgeng 38

40 As such he prfl h he mxmum expeced reurn (h he grees permssble rcng errr) s gven by: TE mx (6.5) Equn 6.5 hs been derved by Lee nd Lm (00) nd s presened n Scherer (004) s ell s Berelr e l (006). Lee nd Lm derved he frmul usng Lgrngn echnques s ppsed he smpler mehd used here. Inerpreng equn 6.5 Equn 6.5 hs nuve ppel. e h TE mx s cnsn. If he lph srems frm ech mnger s ndependen f every her mnger hen s squre mrx h dgnl cnssng f elemens dg( ψ ψ... ψ ) nd C reduces he deny mrx. As such he nverse f s us dg(... ) nd he vecr ψ ψ TE mx ψ. ψ... ψ Hence he prfl h mxmses reurns such h rcng errr s lmed TEmx s prprnl he r f ech mnger s expeced u-perfrmnce he esmed vrnce f her excess reurns. Equn 6.5 s n exensn f ms rs llcn mdels excep h crrelns f excess reurns beeen mngers re n ssumed be zer. A.7. Incrprng he Fund s Sregc Asse Allcn (SAA) Ms rs budgeng frmers re deermned gnrng he Fund s lbles (nd hence SAA) ypclly by budgeng rcng errr. UnSuper s rs budgeng frmer dffers h used by ms prcners s TURBs mnrs he exen hch ech pn deves frm s Sregc Asse Allcn (SAA). UnSuper s SAA fr ech pn s se by pplyng se f nvesmen becves lng h nvesmen cnsrns (such s remvng he bly shr scs lmng expsure lernve sse clsses ec) nd hen ssessng here n he cnsrned effcen frner he Trusee uld le he pn le by cnsderng he blnce beeen rs nd rerd. Asse Lbly Mdels generlly cnsder nly be rs frm ech sse clss. The hn dr blue curve n he chr bel demnsres he Fund s uncnsrned frner hle he hcer lgh blue curve represens he Fund s cnsrned effcen frner. verlyng he Fund s nvesmen becves (pssbly v he use f uly funcn) lls he Fund deermne he pml sse llcn fr ech pn s shn n he chr bel fr he Fund s Hgh Grh pn (4). UnSuper s Apprch Rs Budgeng 39

41 % 0% Cmprsn f The Accumuln pns Rs nd Reurn Prfle Relve he Effcen Frner - Rs eurl Cndns pml SAA Uly Funcn Gemerc Averge Reurns 9% 8% 7% 6% Cp Sble Cns Bl Blnced SRI Hgh Gr h Hgh Gr h Gr h Aus. Eques Innl Eques SRI Blnced Ls ed Prpery 5% Aus. Bnds Cs h 4% 0% % 4% 6% 8% 0% % 4% 6% 8% 0% Sndrd Devn f Annul Reurns Effcen Frner - n cnsrns Effcen Frner h UnSuper cnsrns Dversfed Accumuln pns Sngle Asse Clss pns The Fund hs he bly nves pssvely nd precsely mch he be expsures expeced frm ech sse clss. The exen hch he Fund emplys cve mngemen represens surce f rs he Fund. Cnsder he cnsrned effcen frner derved by nvesng pssvely (gven by he hc blue curve bve). Insed f nvesng pssvely he Fund cn nves cvely hch dds he penl mprve reurns (r dmpen reurns f mngers re prly seleced). Addnlly such lph surces cn dd rs he prfl r pssbly reduce rs f he lph surce s uncrreled he be reurns. Grphclly he ddn f lph culd be hugh f s ncresng he hcness f he effcen frner. UnSuper s Apprch Rs Budgeng 40

42 Frm h e cn ell Brs e l (006) ere he frs represen cve reurns s crcle n menvrnce spce e hve exended her r cnsder he mpc f ddng cve rs he effcen frner. Effecvely he pml rs-reurn blnce h s derved mee he lbles hs been lered h he ncrprn f cve rs. T vercme hs cncern he Fund culd frs remve be rs (eg. by reducng he llcn eques) nd hen dd cve rs s s reesblsh he pml SAA s shn n he chr bel. UnSuper s Apprch Rs Budgeng 4