Quantitative Portfolio Theory & Performance Analysis
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1 Quaave Porfolo heory & Performace Aalyss Week February Coceps. Assgme For February 4 (hs Week) ead: A&L Chaper Iroduco & Chaper (PF Maageme Evrome) Chaper 2 ( Coceps) Seco (Basc eur Calculaos) For February 6: NO CLASS! For Feb (Nex Week) ead: A&L Chaper 2 ( Coceps) Seco 2-3 (elave eur ad sk) Chaper 3 (Basc Elemes of Moder PF heory).2 Coceps Performace Measureme Frs Sage Performace Aalyss ad Maageme Quafy eur o a Asse/Porfolo AIM Sadard Frs Elemes of Performace Evaluao efereces o be used o Compare eurs: Bechmarks ad Peer Groups sk.3 Calculao over -perod Ca be as large as ay erval over whch he asse s held whou beg modfed Le P : prce (marke value) of asse a me P : prce (marke value) of asse a me D : cash flow (dvded coupo...) of asse a me he exac value of he -perod reur o asse (hrough a arhmec calculao): he relave varao of he prce over he perod adjused for ay cash flow applcable he -perod arhmec reur o asse.4
2 Calculao over -perod he Arhmec eur o he asse s gve by P P D P P D prce erm + CF erm P P P Is oal reur aoale s ha for oe perod he oe perod chage value s P P P D P.5 Calculao over -perod A lear combao allows a exac PF reur Sub-perod reurs do o add o mul-perod reurs Arhmec mea provdes a ubased esmae for he reur of he ex perod E he expeced reur o a asse ca be used o forecas s fuure performace HW: Show hs resul (a ubased esmae for he followg perod).6 Calculao over -perod As a exeso we may defe he Logarhmc eur o asse as P D l P he raoale s ha for oe perod (of ay legh) he -perod chage value s e P P D he logarhmc reur s addve for mul-perods HW: Show he mul-perod reur s addve.7 Calculao over mulple perods We ca use he reurs from successve perods o calculae he mul-perod reur Arhmec Mea (of arhmec reur): Wh sub-perods for asse a hs overesmaes he mul-perod reur Cosder P0 00; P 200; P2 00 he 2-perod reur s zero he arhmec mea of he wo -perod reurs: P P0 P2 P P0 P 2.8 2
3 Calculao over mulple perods Geomerc Mea (of arhmec reur): Wh sub-perods for asse g hs eables a exac calculao of he mul-perod reur Cosder P0 00; P 200; P2 00 he 2-perod reur s zero as before he geomerc mea of he wo -perod reurs: 2 2 g g Calculao over mulple perods Geomerc Mea (of arhmec reur): So whle he arhmec mea gves a esmae of he reur for he followg perod he geomerc mea gves he reur for he log erm o jus he ex perod a beer more useful resul HW: For example show wha happes wh logarhmc reur.0 Porfolo eur Smlar o sgle asse excep for he varao capal assocaed wh vesor add/lqudao Basc Formula V V D P where V V s he value of he PF a he ed of perod V s he value of he PF a he begg of perod D s he cash flow geeraed by he PF durg perod Oly vald for PF w/fxed composo durg perod. Porfolo eur Basc Formula Ad sce a PF s a lear combao of asses P x where s he reur o asse durg perod x s he wegh of asse he PF a he begg of perod s he umber of asses he PF durg perod Exac for arhmec reurs; approxmae for logarhmc If successve PF valuaos are kow use prevous resul Always beer o work wh prces vs. reurs.2 3
4 Porfolo eur Basc Formula Smlarly (o Basc Formula) for a marke dex I I I where I I s he value of he Idex a he ed of perod I s he value of he Idex a he begg of perod Porfolo eur Basc Formula w/capal Flow hree mehods Ieral ae of eur (I) Capal-Weghed ae of eur (CW) me-weghed ae of eur (W).3.4 Porfolo eur Basc Formula w/capal Flow Capal-Weghed ae of eur (CW) CW V V0 C CW where V0 C V s he value of he PF a he ed of he perod V0 s he value of he PF a he begg of he perod C s he h capal/cash flow for he PF a (pos f a corbuo/flow & eg f a whdrawal/ouflow).5 Porfolo eur Basc Formula w/capal Flow aoale CW V0 C CWV0 CW C V V0 C Ad CW V 0 CW C V.6 4
5 Porfolo eur Basc Formula w/capal Flow Ieral ae of eur (I) I C V V0 I I aoale I I V0 C V I Feaures Ierave (a challege o deerme o paper) Iflows vesed he PF gog forward.7 Porfolo eur Basc Formula w/capal Flow me Weghed ae of eur (W) W Idea: Break dow perod o elemeary subperods (cash flows); usg he oao as before For each sub-perod V V C V C he reur for he whole perod s geomerc mea V W V C.8 Porfolo eur Basc Formula w/capal Flow me Weghed ae of eur (W) W aoale W Feaures o mpleme eed o kow amou ad mg of cash as well as value of PF a each dae I pracce cash s assumed o occur a moh ed sead of o exac daes; couous verso helps.9 Porfolo eur Basc Formula w/capal Flow r Couous W r W e W W V V V rw l l l V C V 0 V C aoale r W V V e exp l l V 0 V C V V V V 0 V C V C.20 5
6 Porfolo eur Basc Formula w/capal Flow hree mehods: I CW W Comparso W allows maager o be evaluaed separaely from moveme of capal bes for evaluag maager CW allows for oal performace of fud o be measured I more precse ha CW whe here s a sgfca umber of capal flows of dffere szes Freque evaluaos of PF reduce mpac of capal flow Daly s bes.2 HW A Example (pg 32) CW V V C =45.28% V0 C 0 3 CW Coverg o a sadard referece perod rae 3 3 CW CW CW 3.26% I = 3.7% V W W 4.26% V C W couous me = 3.33%.22 Ieraoal Ivesme Up o ow assumpo of sgle currecy Exchage rae defos Forward premum (or dscou) f : fwd premum ( dscou) ( fwd x - rae - spo x - rae) / spo x -rae fwd x - rae ( fwd premum) spo x -rae emember: - f F0 S0e S ( + fwd premum) where 0 S : spo prce of currecy referece ($) also d F : fwd prce of currecy referece ($) also F Q / 0 0 Q/ 0 0 rr : ref ($) rsk free rae currecy rsk free rae f Ieraoal Ivesme Evoluo of he spo exchage rae perceage erms s called he currecy reur he varable C s used o refer o hs quay I a eraoal porfolo he currecy reur eeds o be bfurcaed from he asse reur (whe he asse s prced currecy) eur s expressed as he sum of he reur o he asse ad he currecy reur 6
7 Ieraoal Ivesme Exeso of reur models (arhmec & logarhmc) provdes (asse currecy) bfurcao of reur Sarg wh he prce represeao he eferece currecy of asse a me erms of he Quoao currecy: Q Q/ Q / P P d & d : Prce of quoe currecy he referece currecy Allows he developme of he arhmec reur: Q Q/ Q Q/ P d P d Q Q/ Q Q/ Q Q/ C P d P d Q Q/ Q Q/ Q Q/ C C C ad Q Q/ C.25 Ieraoal Ivesme Smlarly for he Logarhmc eur we have he exac equaly Q Q/ Q Q/ P d P d Q Q/ l l l Q Q/ Q Q/ C Pd P d For he porfolo we also have he expeced ref resul Q Q/ Q/ V P P d & f quoe = ref currecy d P Q Q/ Q Q/ P Q Q/ P : logarhmc x x x C x C x x x C : arhmec &.26 FC Ieraoal Ivesme For bfurcao whe hedge srumes are used Noe he reur FC for a forward exchage corac S F 0 dff bewee currecy reur & fwd premum reur S0 Q / C f where (+ f) s he forward premum o 0 Q / C f hs reur s somemes referred o as he forward surprse.27 Ieraoal Ivesme So reur o a asse hedged w/fwd for currecy rsk s Q Q / Q / C hc f where he hedge rao h 0 Expadg Q Q/ Q Q/ Q/ C C h C f so Q Q/ Q Q/ Q/ C C hc f Q Q/ f h C f he sum of he compleely hedged poso plus he uhedged proporo of he exchage exposure.28 7
8 Ieraoal Ivesme Smlarly for he hedged porfolo of asses m Q Q/ Q/ x x xc h C f P j j j j where h s he hedge rao for each of m curreces j.29 Usg Dervaves Cosder he smple case of a call opo o apprecae wha ca be doe Smply he corbuo of he dervave s he performace of he uderlyg aloe dffered from he performace cludg he dervave; we ca say more abou whece he dervave performace comes he prce performace of he call ca be wre C C0 Now suppose he heorecal value of he call a 0 s V 0 he s possble o break he performace o 2 erms C C V C C V Usg Dervaves Cosder he smple case of a call opo o apprecae wha ca be doe C C V C C V he frs erm measures he heorecal vs. quoed prce based o he spo prce of he uderlyg o dae of purchase 2 d erm measures he dffereal bewee he curre quoe ad he al equlbrum prce he frs ca measure he ably of he maager o pck udervalued opos he secod measures he ably o selec opos wh udervalued uderlyg asses (as we shall see) Are he oucomes from luck or skll?.3 Usg Dervaves Cosder he smple case of a call opo o apprecae wha ca be doe C C V C C V Each erm ca be broke dow aga he frs ca be broke dow o a volaly prof ad a formula prof () () V C C s C V C s Noe 3 formulas: he rue he marke ad he bechmark We use he bechmark (marke) wh he mpled vol. a 0 vs. he real vol. (s) o deerme he vol. prof he formula prof s foud usg he bechmark (rue) wh.32 he real vol. 8
9 Usg Dervaves Cosder he smple case of a call opo o apprecae wha ca be doe C C V C C V Each erm ca be broke dow aga he secod erm ca be broke dow o he prof from asse udervaluao ad oe due oly o he opo overlay he asse prof s measured relave o a bechmark sraegy a forward corac o he uderlyg r d S S0e Assumg he marke s effce ad rsk eural hs prof s zero.33 Usg Dervaves Cosder he smple case of a call opo o apprecae wha ca be doe C C0 V0 C0 C V0 Each erm ca be broke dow aga he secod erm ca be broke dow o he prof from asse udervaluao ad oe due oly o he opo overlay Fally he prof from usg a opo vs. a forward r d C V0 S S0e So we have C C0 C0() s C0 V0 C0() s rd S S0e C V0 S S0e rd.34 Usg Dervaves Cosder he smple case of a call opo o apprecae wha ca be doe C C C () s C V C () s rd rd S S0e C V S S0e Vol. erm + formula erm + asse erm + opo aloe If opos are valued effcely he sum of he frs wo s zero (o average) rrespecve of he bechmark formula beg close o he oe used by he marke If opos are valued effcely ad he bechmark cocdes w/ marke formula boh erms are zero.35 Usg Dervaves Cosder he smple case of a call opo o apprecae wha ca be doe C C C () s C V C () s rd rd S S0e C V S S0e If he asses are valued effcely erm 3 s zero If o he marke s o rsk eural ad s value measures he compesao correspodg o he rsk ake If asses are o valued effcely he erm 3 s o zero ad s value measures he maagers ably o selec opos ha have a correcly valued uderlyg.36 I a effce marke coex erm 4 s also zero 9
10 he GIPS Performace Mus be o a oal reur bass usg me weghed rae of reur Bes s o value o a daly bass ad combe resuls geomercally Calculaed a leas quarerly bu recommeds mohly (e of rasaco coss ad cludg reur o cash).37 0
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