Calibration of factor models with equity data: parade of correlations

Transcription

1 MPRA Much Peroal RePEc Archve Calbrao of facor model wh equy daa: parade of correlao Alexader L. Baraovk WeLB AG 30. Jauary 0 Ole a hp://mpra.ub.u-mueche.de/36300/ MPRA Paper No , poed 30. Jauary 0 7:0 UC

2 Calbrao of facor model wh equy daa: parade of correlao Alexader L. Baraovk WeLB AG, Germay h paper decrbe he proce of ML-emag of he equy correlao whch ca be ued a proxe for ae correlao. I a Gaua framework he ML-emaor are gve cloed form. O h ba he mpac of he Lehma collape o he dyamc of correlao vegaed: afer he Lehma falure Sepember 008 he re correlao ook place acro all ecoomc ecor. Keyword: ra/er ae correlao, maxmum lkelhood emao, gle rk facor model, ormal mxure, VAR of equy porfolo. Emal: Alexader.Baraovk@gmal.com he opo expreed h paper are hoe of he auhor ad do o ecearly reflec vew of he WeLB AG.

3 Emprcal aaly of equy me ere A aural way o emae cred qualy correlao ug horcal daa o exame prce hore of ock a he equy reur are oe fudameal ad very obervable ource of frm-pecfc correlao formao. Our ma daa ource he Bloomberg daa feed. A of coa 765 Norh-Amerca uer from 9 dc ICB dury ecor led able. ID ICB ecor ame # of frm Ol & Ga 495 Chemcal 44 3 Bac Reource 46 4 Coruco & Maeral 6 5 Idural Good & Servce Auomoble & Par 75 7 Food & Beverage 7 8 Peroal & Houehold Good 38 9 Healh Care Real 363 Meda 86 ravel & Leure 49 3 elecommucao 3 4 Ule 5 5 Bak Iurace 4 7 Real Eae 65 8 Facal Servce echology 070 able : Idury ecor clafcao For each uer we rereve he 90 weekly log-reur V coverg perod: Kolmogorov-Smrov e ad mxure of drbuo We coruc a emprcal drbuo fuco for obervao ( ) V of -h ock F ( ) ( x) = IV ( ) = x ad calculae he Kolmogorov-Smrov ac ( ) ( D F ) ( x) F( x) drbuo. Here we aume he ormaly of he daa,.e. F( x) Φ( x) = up, where F a heorecal cumulave x. O he ex ep we compare KS-ac for every oblgor from ecor wh he crcal value of Kolmogorov drbuo for a 5%-gfcace level ad cou oblgor, f KS-e accep he ormaly hypohe. able how he drbuo of a umber of frm havg ormal daa acro ecor for wo group of frm amely belogg o DJ SOXX Amer 600 or o.

4 "ooxx" frm "oxx" frm #of frm # of "ormal" frm k k/ #of frm # of "ormal" frm k k/ Ol & Ga 460 0, ,86 Chemcal ,34 5 0,80 Bac Reource , 4 4,00 Coruco & Maeral ,39 9 9,00 Idural Good & Servce , ,9 Auomoble & Par , ,80 Food & Beverage , ,83 Peroal & Houehold Good ,9 3 0,96 Healh Care , ,65 Real 3 8 0, ,95 Meda ,7 5 0,73 ravel & Leure , ,7 elecommucao 0 0 0,0 3 0,5 Ule , ,85 Bak , ,37 Iurace ,4 6 0,7 Real Eae ,0 0 0 N/A Facal Servce , 4 0,64 echology , ,89 able : Drbuo of umber of ooxx ad oxx frm v KS-e he calculao of KS-ac for every ecor pecfc emprcal drbuo fuco lead o he followg able F ( ( x) = F ) ( x) = ooxx frm oxx frm Ol & Ga 0 Chemcal 0 Bac Reource 0 Coruco & Maeral 0 Idural Good & Servce 0 Auomoble & Par 0 Food & Beverage 0 Peroal & Houehold Good 0 Healh Care 0 Real 0 Meda 0 ravel & Leure 0 elecommucao 0 Ule Bak 0 0 Iurace 0 0 Real Eae 0 0 Facal Servce 0 echology 0 able 3: KS-e reul acro ecor hu he Kolmogorov-Smrov e wh he 5%-gfcace level reec ( 0 ) boh he ormaly ad -hypohe for he daa of NO SOXX frm (excludg Ule ) ad accep ( ) he ull hypohe o ormaly of daa of SOXX frm excludg hree cae for ecor Bak, Iurace ad Real Eae. Qualy of daa Here we roduce he followg rao θ := d

5 where a umber of all he dc eleme ha appear a me ere of log-reur. he parameer 0 θ d reflec a lqudy or radably of hare. Clear for he large ock θ. A he ame me he ock of low lqud ame havg repeaed quoe are characered by mall θ <<. able 4 coa he hogram of θ for o oxx frm before ad afer KS - adume wh a 5% - gfcace level hea # of "o oxx" frm # of frm afer KS-e 0<θ<0% <θ<0% <θ<30% <θ<40% <θ<50% <θ<60% <θ<70% <θ<80% <θ<90% <θ<=00% oal able 4: drbuo of umber of o oxx frm w.r.. he parameer θ hu he more radable a ock he more lkely prce follow a geomerc Browa moo a well a he Kolmogorov-Smrov e a uable ool o fler he daa accordg o he aumpo of a ormaly. I equel for purpoe of he calbrao of a gle facor model we wll ue he daae of he ormal me ere of log-reur of 063 frm (ee rgh colum of able 4). Here we plo her mea emprcal drbuo fuco acro ecor Fg. : he mea EDF (blue curve) acro ecor v ormal (gree) ad Sude (red) -drbuo wh 3 degree of freedom ad calculae he average F 9 9 ( x) = F ( x) for he e of 9 emprc CDF from Fg. a well a fd lea-quare f a famly of ormal drbuo wh zero mea ad ukow varace. Fg. depc hee wo curve =

6 Fg.. Mea CDF (blue) ad f (red) hu he adardzed log-reur o a whole perod are urprgly decrbed by a ormal drbuo fuco wh a ou varace: x σ Φ, σ = F x. A uch pheomeo ca be explaed by a mxure of ormal drbuo gve o he uberval where λ =, ad λ = a = l = l = drbued wh mea m ad varace σ. F x x m, l λφ = σ. he log-reur a obervao perod are aumed o be ormally So, f we dvde our daa o wo perod =90 week ( ) ad =00 week ( ) ad he emae he mea CDF for boh perod we come o x ( ) x x F x λ Φ + λ Φ x Φ Φ, λ ( λ) where 90 λ = =. 90 A aural choce of a a po of regme chage behavour a perod Sepember/Ocober 008 ca be mahemacally cofrmed by a chage po aaly of a kuro a well a KS-ac κ 4 4 σ + σ = 3 σ + σ x x KS = up F ( x) Φ Φ x σ σ

7 where he f parameer {, } σ σ o he obervable perod (0, ) ad ( +, ), repecvely, are how Fg. 3 gma Fg. 3 Evoluo of he d. devao σ ( )(blue), ( ) We depc boh KS-ac ad kuro σ (red) ad her weghed um σ (gree) KS-ac Kuro KS-ac [%] Kuro Dae Fg.4 Kuro ad KS-ac of mxure of ormal drbuo over me ad oe ha he kuro alway greaer ha 3 ce he mxure of wo zero-mea ormal dee alway ha a hgher peak ad heaver al ha he ormal dey of he ame varace o he oe had. O he oher had he kuro a well a KS-ac ocllae aroud wo dffere value. Here we alo defe a perod ( = 87 (.09.08), = 98 (8..08)) of a rao from oe magude of ocllao of he ac o aoher oe. Wha wa remarkable durg h perod wa he Lehma Broher collape.

8 Correlao emao.. Oe Facor Model Aume we have a e of oblgor (ock) belogg o a dury ecor. Aocaed wh a oblgor a lae varable ( ) V, whch repree he ormalzed log-reur o a oblgor ae a. ( ) V gve by V ( ) ( ) f ε =, +, () where f a yemac rk facor (eg, duy pecfc dce) a me. ( ) ε repree -h oblgor-pecfc rk., Baed o above emprcal evdece for he kuro for log-reur (Fg. 4) whch ca be approxmaed by a coa of 3 a well a accordg o he KS-e boh f ad ( ) ε are here aumed o have a adard ormal drbuo ad, are oly depede ad ( ) ε depede acro oblgor. We alo aume ha oblgor a gve dury have a gle commo rk facor ad meaure he evy of each oblgor o f, by a facor loadg,. For wo dure ad, he correpodg facor f ad f are aumed o be correlaed ad o poe a correlao coeffce., he correlao emao procedure ue he wo-ep MLE mehod decrbed [Kalkbreer, Owua 009]. Fr he correlao of frm wh each of he dury ecor are calculaed (ra-ecor correlao). Ug hee reul, he correlao of frm wh dffere dury ecor (er-ecor correlao) are calculaed... Emao of he ra-ecor correlao Gve a daae of 063 ormal me ere of log-reur ( ) V we defe he maxmum lkelhood emaor o he oe-facor model () or more precely o he model parameer he followg hree ep:. By coruco () for a oblgor from ecor we ge: ( ) V f, ~ N ( 0, ) erm of a lkelhood fuco a a me ( ) L, ( f, ) = e π ( ) ( ) ( V ) f ( ). he margal lkelhood for V durg a obervao perod hu: or mmedaely. () f ( ) / ( ),,, (, ) Λ = L f dφ f π e L f df = = = = (3) 3. Emae of ca be obaed by maxmzg he margal lkelhood for each ecor

9 = arg max Λ. (4) { ( ) 0 < We oe ha boh he egrao cheme calculao of lkelhood (3) (e.g. Gau-Herme cheme) ad umercal mehod earchg of exreme (4) ca lead o gfca error. Foruaely he lkelhood (3) ca be boh egraed ad maxmzed aalycally. I Appedx we derve from (3) where a ma of a correlao marx ( ) ( ) ( ) ( ) ( + ) / Exp / µ + + Λ ( ), (5) + P wh eleme µ = Σ (6) ( P ) r, ( ), = V V. (7) = r a Pearo correlao of weekly log-reur for a par (,) of frm from ecor., r Followg Düllma e al. (008) ae correlao are emaed by µ, he mea of he par-we correlao of all frm. I referred a drec emao mehod. For he parameer µ hold µ <. A fr dervave of he lkelhood (5) facorzed o produc of cubc polyomal ad expoeal fuco. Hece he maxmzg of he MLE (5) lead o earchg of he roo of a cubc equao. Omg echcal deal we pree a Cardao formula for opmum (4): where q q a = arg max Λ = + +,, µ, (8) { 3 D 3 D ( ) 0 < 3 ( )( 3 ) ( ) ( ) ( ) 3 3 p q a a ab + + µ D = + > 0, p = b ; q = + c; a = ; µ b = ; c = 3 able 5 ad 6 collec he reul of (6)-(8) calculao for hree group of frm.

10 ICB ecor/norh Amerca all frm DJ SOXX compae # of frm rho # of frm rho Ol & Ga Chemcal Bac Reource Coruco & Maeral Idural Good & Servce Auomoble & Par Food & Beverage Peroal & Houehold Good Healh Care Real Meda ravel & Leure elecommucao Ule Bak Iurace Real Eae Facal Servce echology Average able 5: Ira-ecor correlao % Aume for every ecor # of frm ma mu rho Ol & Ga Chemcal Bac Reource Coruco & Maeral Idural Good & Servce Auomoble & Par Food & Beverage Peroal & Houehold Good Healh Care Real Meda ravel & Leure elecommucao Ule Bak Iurace Real Eae Facal Servce echology Average able 6: ra-ecor correlao [%] of o oxx frm all ν oxx + ν = (9) where he wegh are ormalzed o um up o a parameer ν uch ha oxx ooxx ν = ν, ν = ν. From he reul able 6 7 we derve a almo uform drbuo of he wegh coeffce ν acro ecor a how able 7. ooxx all all

11 ICB ecor/norh Amerca wegh coeffce Ol & Ga 0,944 Chemcal 0,948 Bac Reource 0,94 Coruco & Maeral 0,960 Idural Good & Servce 0,97 Auomoble & Par 0,853 Food & Beverage 0,878 Peroal & Houehold Good 0,963 Healh Care 0,940 Real 0,96 Meda 0,9 ravel & Leure 0,963 elecommucao 0,86 Ule 0,960 Bak 0,98 Iurace 0,935 Real Eae N/A Facal Servce 0,959 echology 0,964 able 7: drbuo of he wegh parameer acro ecor Deoe oxx γ = ad rewre (9) he followg form all all ( oxx ooxx) + ν ooxx = γ ν. (0) Hece a commo ra-ecor correlao learly creae wh γ (or a umber of oxx frm ) o he erval oxx ν ooxx ν all oxx. hu e.g. o emae a qule-baed cred/marke rk meaure for a porfolo coag boh lqud ad llqud ame wo compoe of he ra-ecor correlao are o be ued accordg above weghed rule (9) - (0). I Appedx: able A ad A3 we alo collec he MLE-reul for dffere aggregao of o oxx frm acro ecor ad oe ha he ra-ecor correlao for compae wh greaer marke capalzao / hgh cred qualy / umber of employee are bgger oe for compae wh maller capalzao / low cred qualy / umber of employee. Fg. 6 gve a geomerc erpreao of depedecy of he MLE (8) o he ma µ (6) acro ICB ecor a gve able 7. Fg. 6. MLE v µ

12 A cloud of he ra-ecor correlao bouded by wo curve ( = 3, 90,µ ) ooxx 7 = (red curve) ad ( = 43, 90 ),µ ooxx 5 = (gree curve) wh mmal ad maxmal umber of frm (ock) ecor =7 ( Real * Eae ) ad =5 ( Idural Good & Servce ), repecvely. Noe alo ha for he ame fxed umber of frm every ecor he all ra-correlao le o a uque Cardao curve ( *, 90, µ ) he Cardao curve approache le (,, ) µ µ., Proof. Due o L'Hôpal' rule we have he followg cha of he lm: =. 3 + µ µ µ b 0; c 0; a µ p ad q, 3, 7 leadg fr o zero dcrma D ad he o a double zero roo ad a mple roo 9c 4ab + a 3b a µ 0 µ µ a a 3b µ ,.e., 3 0 ( µ ) I mea ha aympocally = µ () or by oher word a aympoc MLE of he ra-ecor correlao gve by a ma of a marx wh he Pearo correlao (7). If we adm he heavy al for he rk facor drbuo for each of he 9 dury ecor he emae for racorrelao ca be alo calculaed above hree ep wh he lkelhood ()-(3) modfed accordg o he drbuo aumpo: e.g. he yemac facor follow a ormal mxure drbuo ad he doycrac facor are ormally drbued ad hece a lae varable ( ) V by () ha a ormal mxure drbuo...3 Impac of Lehma Broher collape o he correlao akg o accou he acal aaly of log-reur me-ere he ra-ecor correlao ca be decompoed o hree compoe for (.0.07, ), ( , 8..08) ad (4..08, ) perod. Applyg he mehodology ()-(5) eparaely o every perod oe ca oba he followg reul:

13 # of frm ma mu rho ma mu rho ma mu rho Ol & Ga Chemcal Bac Reource Coruco & Maera Idural Good & Ser Auomoble & Par Food & Beverage Peroal & Houehold Healh Care Real Meda ravel & Leure elecommucao Ule Bak Iurace Real Eae Facal Servce echology able 8: Varao of ra-ecor correlao over me Afer he Lehma falure Sepember 008 he re correlao ook place acro all ecoomc ecor. I order o vegae a mpac of he Lehma perurbao o he dyamc of correlao we ubue o he margal lkelhood (3) he me ere wh a varable umber of he log-reur {, =,Kτ } V coverg perod (.0.07, τ week ) ad calculae he MLE replacg (8) by τ. Fg. 7 how a perurbed dyamc of he emae of correlao acro ecor for a perod ( , ),.e. τ = 9, K, Ol & Ga Chemcal Bac Reource Coruco & Maeral Idural Good & Servce Auomoble & Par Food & Beverage R-quared[%] Peroal & Houehold Good Healh Care Real Meda ravel & Leure elecommucao Ule Bak 0.00 Iurace Real Eae Facal Servce echology Dae Fg. 7. Evoluo of he ra-ecor correlao acro ICB ecor a well a Fg.8 depc her average curve (blue) comparo o a dyamc of he correlao derved from he log-reur daa a po-lehma epode.

14 Dyamc of R-quared/USA R-quared (cumulave) po-lehma R-quared (cleaed) R-quared [%] Dae Fg. 8. Varao of average R-quared over me Our reearch dcae ha he emae moohly vary over me before a well a afer Lehma falure. A he ame me he creae correlao durg he Lehma epode goe expoeally. I a po-lehma epode he correlao do reur o he value before he Lehma Broher collape ad ablze aroud a ew magude 3%...4 Ba adume : parade of correlao he ML-emae (8) ad he ma (6) have he followg bae [ee deal appedx] ba , 0 00 f 0.3; , = , 0 00 f , µ µ + µ 0.39µ 0.09, 0 00 f µ 0.3; 4.68µ µ µ µ + µ µ , ba = µ 0.903µ.47µ µ µ µ. 64µ +.73, 0 00 f 0.3 µ µ 0.49µ µ 6.605µ µ µ , whch ca be fed from yhec adard ormally drbued me ere f ad e geeraed by () wh a gve rue parameer r. Noe alo ha a adard devao goe o zero a ~ /.

15 he ba adume of boh parameer lead o a mple approxmae ˆ ba ˆ µ µ ba Fg. 9 ad able 9. = =, a how µ Fg. 9. Parade of ra-ecor correlao # of frm ma mu rho Ol & Ga Chemcal Bac Reource Coruco & Maeral Idural Good & Servce Auomoble & Par Food & Beverage Peroal & Houehold Good Healh Care Real Meda ravel & Leure elecommucao Ule Bak Iurace Real Eae Facal Servce echology Average able 9: Nooxx frm: adued ra-ecor correlao ad mae of correlao marce acro ecor..5 Emao of he er-ecor correlao he above mehodology ca be exeded o cro- correlao of oblgor dffere ecor, ay ad. We aume ha all oblgor ecor deped oly o he yemac facor f ad have he ame R. he yemac facor f ad f follow a bvarae ormal drbuo wh correlao. hu we oba he followg lkelhood fuco er, cro = ( ) L ( x, y,, ) d ( x, y, ) Λ = Φ () where ( k ) ( k L,,, ) cro x y = L, x, L, ( y, ) wh he pre-calculaed ecor-pecfc codoal k = k= lkelhood () a me. I Appedx we derve from ()

16 er Q Λ, ( ) Exp Χ + 4 p % ( ) ( ) ( 4 p p ( ) ). (3) Maxmzg (3) lead o = arg max Λ. (4) er { ( ),, 0 < We apply (4) ad calculae he er-correlao for each par of he 9 dure a how able 0. Ol & Ga Chemcal Bac ReouCoruco Idural GAuomoble Food & BevPeroal & Healh CareReal Meda ravel & LeelecommuUle Bak Iurace Real EaeFacal S echology Ol & Ga 00,00% ,63% 69,06% ,69% ,8% Chemcal - 00,00% 8,5% 90,63% ,47% 7,97% -,09% 93,75% 5,8% - 68,9% 4,06% - - 7,50% - Bac Reource - 8,5% 00,00% ,9% 0,63% -,66% 97,34% 59,84% - 73,8% 4,06% - - 3,50% - Coruco & Maeral - 90,63% - 00,00% ,38% 0,63% -,8% - 59,84% - 68,3% 4,53% ,63% - Idural Good & Servce ,00% Auomoble & Par ,00% 6,88% 3,9% -,34%,7% 0,47% 90,3%,7% 0,78%,03% - 5,63% - Food & Beverage - 80,47% 83,9% 84,38% - 6,88% 00,00% 3,44% - 4,53% - 68,9% - 89,69% 5,3% ,7% - Peroal & Houehold Good - 7,97% 0,63% 0,63% - 3,9% 3,44% 00,00% - 69,69% ,8% Healh Care 0,63% ,00% Real 69,06%,09%,66%,8% -,34% 4,53% 69,69% - 00,00% ,3% Meda - 93,75% 97,34% - -,7% ,00% 56,4% - 67,03% 4,06% - - 8,75% - ravel & Leure - 5,8% 59,84% 59,84% - 0,47% 68,9% ,4% 00,00% - - 7,8% ,6% - elecommucao ,3% ,00% 3,8% 0,78% 3,9% - 5,3% - Ule - 68,9% 73,8% 68,3% -,7% 89,69% ,03% - 3,8% 00,00% 4,84% - - 3,5% - Bak 4,69% 4,06% 4,06% 4,53% - 0,78% 5,3%,8% - 35,3% 4,06% 7,8% 0,78% 4,84% 00,00% Iurace ,03% ,9% ,00% -,5% - Real Eae ,00%,66% - Facal Servce - 7,50% 3,50% 30,63% - 5,63% 36,7% ,75% 55,6% 5,3% 3,5% -,5%,66% 00,00% - echology 3,8% ,00% able 0: er-ecor correlao of ooxx frm he average er-ecor correlao he equal o 7.8%. 3 Regreo Aaly A alerave way o emae he ra/er correlao for oxx frm o reolve he followg adard oxx lear regreo equao ( ) ( ) V = β f + ε (5), wh repec o bea aalogy o he capal ae prcg model (CAPM). hu he correlao wll be calculaed o he daa of he large Norh Amerca ock from SOXX Amerca 600 Idex. he reur ( ) V are excluvely correlaed by mea of her compoe facor f whch are modelled by dury pecfc dce. We deoe he -h week reur o he -h dex by f, ad for each of he dce, we coder he la 90 weekly reur ad compue he Pearo correlao of weekly reur for all par of dce by R, = f, f, = Mmzg redual ( ) ε (5) lead fr o a opmal bea for -h ock (oblgor) ecor : β ( ) arg m { ( ) = V β f, β

17 V 4 β f, - -4 Fg. 0. he log-reur of Exxo Mobl Corp v DJS Amer 600 Ol & Ga ad he o he aural emae of he ra-ecor correlao ( ) β = β (6) = ad fally o he er-ecor covarae = β β R. (7),, Drec comparo of wo model () ad (5) gve a obvou lkg equao: β = whch ca be ued o e boh model. able ad collec he ra- ad er-ecor correlao compued by (6) ad (7), repecvely.

18 ID ICB ecor ame DJ SOXX compae: bea Ol & Ga 7,68 74,45 Chemcal 75,3 75,6 3 Bac Reource 6,59 64,47 4 Coruco & Maeral 70,03 76,44 5 Idural Good & Servce 60,6 7,95 6 Auomoble & Par 73,87 77,5 7 Food & Beverage 55,3 58,94 8 Peroal & Houehold Go 5,34 63,03 9 Healh Care 56,64 64,74 0 Real 57,74 63,40 Meda 67,85 7,07 ravel & Leure 6,74 68,83 3 elecommucao 59,7 6,04 4 Ule 74,6 77,89 5 Bak 68,5 76,7 6 Iurace 63 67,97 7 Real Eae 87,04 86,07 8 Facal Servce 68,45 7,94 9 echology 63,06 66,85 Average 65,69 70,64 able : bea -vero of he ra-ecor correlao Ol & Ga Chemcal Bac ReoCorucoIdural GAuomobleFood & BevPeroal & Healh CareReal Meda ravel & LeelecommuUle Bak Iurace Real EaeFacal Sechology Ol & Ga 00,00 78,63 8,85 77,6 7,5 63,4 70,5 66,90 63,3 63,39 77,3 60,40 69,84 84,43 56,75 7,06 N/A 66,63 73,7 Chemcal 78,63 00,00 78,38 86,44 8,65 7,94 67,5 63,3 58, 70,46 74,97 7,00 6,67 67,03 59,6 70,55 N/A 7,39 77,4 Bac Reource 8,85 78,38 00,00 74,6 64,65 57,64 5,4 47,74 45,5 50,75 6,30 5,77 49,90 63,59 39,89 54, N/A 5,85 63,49 Coruco & Maeral 77,6 86,44 74,6 00,00 88, 80,9 67,06 69,74 63,03 78,7 8,9 80,9 69,0 66,94 7,64 80,05 N/A 83,0 80,8 Idural Good & Servce 7,5 8,65 64,65 88, 00,00 86, 75,80 78,67 70,56 85,43 89,86 87,88 7,97 68,94 77,0 84,5 N/A 87,56 85,89 Auomoble & Par 63,4 7,94 57,64 80,9 86, 00,00 63,0 67,93 55,4 79,99 8,0 8,4 64,55 6,57 73,8 77,9 N/A 8,07 80,46 Food & Beverage 70,5 67,5 5,4 67,06 75,80 63,0 00,00 89,00 83,54 77,05 8,37 7,09 75,73 76,87 59,5 75,59 N/A 7,80 7,7 Peroal & Houehold Good 66,90 63,3 47,74 69,74 78,67 67,93 89,00 00,00 8,55 84,0 8,83 79,57 76,6 73,4 6, 76,35 N/A 75,86 74,95 Healh Care 63,3 58, 45,5 63,03 70,56 55,4 83,54 8,55 00,00 74,43 76,88 69,67 73,4 73,44 59,4 77,7 N/A 7,83 69,46 Real 63,39 70,46 50,75 78,7 85,43 79,99 77,05 84,0 74,43 00,00 86,80 88,38 76,5 66,98 68,97 78,6 N/A 86,5 84,06 Meda 77,3 74,97 6,30 8,9 89,86 8,0 8,37 8,83 76,88 86,80 00,00 84,34 80,0 74,79 76,65 86,60 N/A 88,04 84,43 ravel & Leure 60,40 7,00 5,77 80,9 87,88 8,4 7,09 79,57 69,67 88,38 84,34 00,00 69,04 60,40 73, 78,9 N/A 84,70 80,9 elecommucao 69,84 6,67 49,90 69,0 7,97 64,55 75,73 76,6 73,4 76,5 80,0 69,04 00,00 73,35 6,98 74,38 N/A 74,56 74,94 Ule 84,43 67,03 63,59 66,94 68,94 6,57 76,87 73,4 73,44 66,98 74,79 60,40 73,35 00,00 5,5 73,5 N/A 64,7 7,3 Bak 56,75 59,6 39,89 7,64 77,0 73,8 59,5 6, 59,4 68,97 76,65 73, 6,98 5,5 00,00 85,99 N/A 88,96 6,4 Iurace 7,06 70,55 54, 80,05 84,5 77,9 75,59 76,35 77,7 78,6 86,60 78,9 74,38 73,5 85,99 00,00 N/A 89,67 73,8 Real Eae N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 00,00 N/A N/A Facal Servce 66,63 7,39 5,85 83,0 87,56 8,07 7,80 75,86 7,83 86,5 88,04 84,70 74,56 64,7 88,96 89,67 N/A 00,00 80,9 echology 73,7 77,4 63,49 80,8 85,89 80,46 7,7 74,95 69,46 84,06 84,43 80,9 74,94 7,3 6,4 73,8 N/A 80,9 00,00 able : bea -vero of he er-ecor correlao o a perod: Value a Rk (VaR) of equy porfolo wh ormal mxure I kow ha f all k eque of a porfolo are mapped o he ame gle rk facor (e.g. he marke dex, ee deal eco 3) he ormal VaR a he (-α )-cofdece level mply equal o k σ N VaRα = Φ ( α ) P ωβ where P a oal porfolo value, y ω =, y marke value of -h equy. A he ame me he ormal aumpo P could lead o he uderemao of VaR a wll be how equel. We have already ee ha he degree of exce kuro he ock reur me ere coderably hgher before ha afer cr (Lehma falure). Here we qualavely vegae a mpac of al behavour of ock reur drbuo o VaR of a lear equy porfolo wh a porfolo mea µ ad a volaly of σ over -day horzo. N N

19 We aume he hree drbuo of he porfolo P&L: ormal, ormal mxure ad ormal wh he weghed mea ad devao uch ha a -day VaR a a gfcace level α ha hree poble value a roo of he followg olear algebrac equao: VaR α / P µ N pl VaRα Φ, V ~ N( 0,) σ N VaR α / P µ NM + VaRα / P µ N α = Pr P & L < VaR = Φ ( ) Φ α pl VaRα p p, V ~ NM λ σ NM σ N VaR [ + ( ) ] α / P pµ NM p µ N pl Φ 3 VaRα, V ~ N 0, σ λ pσ NM + p σ N (8) where p he probably of regme (before cr), µ a -day porfolo reur ad NM σ NM a -day adard devao regme. Regme characerze ordary marke crcumace wh a par µ ad N σ. N For fxed parameer: P = 00 Mo; µ NM = 0.3; σ NM = ; µ N = ; σ N = ; = 0 we reolve (8) w.r.. 50/ 50/ 50/ VaR for wo gfcace level α = 5% (dahed) ad 0%(old) ad arbrary probably of crah p a how o Fg.. VaR@MoD VaR@MoD prob_of_crah prob_of_crah Fg.. VaR v probably p for ormal (gree), NM (red) ad equvale ormal (blue) α hu gorg he pobly of a crah ca erouly uderemae he VaR. For low gfcace level (e.g. 0% or 0%), he ormal aumpo ( pl 3 ( VaR α ) ) ca overemae VaR f p α. he parameer of a ormal mxure dey fuco ca be emaed from horcal daa by ue of he expecao maxmzao (EM) algorhm [7]. hu we would be able o quafy he probably p. Aoher cae of udy o fx he probably p, e.g. % ad 0%. he we ge VaR@MoD 00 p = % p = 0 % VaR@MoD a a

20 We ee ha for hgher gfcace level NM VaR coderably bgger boh ormal VaR eve for mall probably of crah. 5 Cocluo I h work we fr carred ou a emprcal aaly of he equy me ere coverg a 4y perod from 007 o 00. he we coder a ormal drbuo aumpo for he rk facor wh a wo-ae vero of he CredMerc framework ad derve he maxmum lkelhood emaor cloed form. Cocurre o MLE ae correlao are emaed by ma µ or he mea of par we equy ample correlao. We how ha he ample correlao are le baed ha he ML-emae ad aympocally boh mehod lead o he ame correlao. Baed o he ICB dury clafcao we compued he ba adued ra- ad er - correlao for he 9 dury ecor wh a average value of 9.45% ad 7.8%, repecvely. We alo vegaed a dyamc of he correlao ad correlao chage uder reed marke codo (Lehma falure Sepember 008) ad uded a mpac of ormal mxure aumpo o he VaR of mple equy porfolo. 6 Referece [] M.Kalkbreer ad A. Owua, Valdag rucural cred porfolo model. I Model Rk - Challege ud Soluo for Facal Rk Model, Rk Book, 00 [] Duellma, M., J. Küll ad M. Kuch, Emag Ae Correlao From Sock Prce or Defaul Rae Whch Mehod Superor?, Joural of Ecoomc Dyamc ad Corol, 34(),00 [3] Gradey ad Ryzhk, able of Iegral, ere, ad produc, Seveh Edo, 007 [4] A. L. Baraovk, Dyamcal yem forced by ho oe a a ew paradgm he ere rae modellg, SFB 649, Humbold-Uverä zu Berl, 00 hp://fb649.ww.huberl.de/paper/pdf/sfb649dp pdf [5]. Alma,. Schmd ad W. Sue. "A Sho Noe Model for Facal Ae ", 008. Ieraoal Joural of heorecal ad Appled Face, Vol, No., p [6] A.P.Demper, N.M.Lard, ad D.B. Rub. Maxmum lkelhood from compleedaa va he EM algorhm. Joural of he Royal Sacal Socey B, 39():-,977 Appedx. Proof of Eq. (5): he egrad (3) ( ) ( ) f ( V ) f f V + f f V / = = p f + q f e e = e ( π ( )) e A( ) e, π ( )

21 ( ) where we deoed ( V ) / = A( ) = ( π ( )) e ad + ( ) p = ; q = V. = Sce q p x + q x 4 p e dx = e π p [Gradey&Ryzhk, p.337] ( ) V = / π ( ) ( )( + ) Λ ( ) = ( π ) A( ) e = + we ge a lkelhood a form:. ( ) ( ) / ( V ) V / / π = = = = = ( π ) ( π ( )) Exp / + ( ) ( ) ( )( ) + + hu / ` = = = = Λ ( ) Exp / ( + ) ( )( ) + ( ) ( + ) V + V (A.) ( ) akg o accou ha he log-reur me ere are adardzed,.e. V we ge: `=. ( ) ` = =. ( V ) ( ) ( ) ( ) ( ) ( ) ( + ) V = V V = ( V ) + V V ( ) + ( ) r, + = = = = = = = = = = = = ( ) = + Σ P = Σ P he (A.) raform o (5). MLE-reul for dffere aggregao of o oxx frm acro ecor

22 Ol & Ga Chemcal Bac Reource Coruco & Maeral Idural Good & Servce Auomoble & Par Food & Beverage Peroal & Houehold Good Healh Care Real Meda ravel & Leure elecommucao Ule Bak Iurace Real Eae Facal Servce echology Average able A: Ira-ecor correlao for dffere aggregao of o oxx frm acro ecor employee < <employee < <employee M<ae<Mrd ae >Mrd Ol & Ga Chemcal Bac Reource Coruco & Maeral Idural Good & Servce Auomoble & Par Food & Beverage Peroal & Houehold Good Healh Care Real Meda ravel & Leure elecommucao 7 6 Ule Bak Iurace 7 8 Real Eae Facal Servce echology oal able A: umber of frm bucke 639 # of frm raed < BBB ub-veme grade veme grade # of frm raed >=BBB Ol & Ga Chemcal Bac Reource Coruco & Maeral Idural Good & Servce Auomoble & Par Food & Beverage Peroal & Houehold Good Healh Care Real Meda ravel & Leure elecommucao Ule Bak Iurace Real Eae Facal Servce echology oal Average able A3: Ira-ecor correlao v umber of frm bucke

23 Comme o ba for MLE (8) ad for a ma μ (6):. Se a rple {,,} ad mulae muually depede adard ormally drbued me ere for ε { = = } ad a me ere for a commo facor f, {,..., } doycrac facor ( ),,..., ;,.., { } Geerae he log-reur V, =,..., ; =,..., by (). =.. For he gve e of me ere fd boh a opmal r from (4) ad a ma μ from (6)-(7), keep hem ad he repea () N me. 3. Calculae he mea value ( ), µ ad adard devao σ ( ) {, =,..., N} of r ad ( ) { µ, =,..., N} of μ, repecvely. 4. Se a ew value for ad repea ()-(3) K me. 5. Se a ew value for ad repea ()-(4) L me. A a reul we ge he followg able σ, from he ample µ umber of frm\rho 5,00% 0,00% 5,00% 0,00% 5,00% 00 9,8% 4,09% 8,37%,69% 7,03% 00 8,48%,75% 7,7%,77% 6,4% 300 7,84%,9% 6,80%,40% 5,94% 400 7,46%,8% 6,49%,7% 5,73% 500 7,3%,59% 6,30%,% 5,58% 600 7,03%,46% 6,0%,00% 5,53% 700 6,87%,34% 6,% 0,93% 5,48% 800 6,75%,5% 6,07% 0,87% 5,47% 900 6,67%,8% 6,00% 0,83% 5,46% 000 6,57%,% 5,96% 0,8% 5,4% able A4. Emprcal drbuo of a MLE (8) v umber of frm ad gve rho umber of frm\mu 5,00% 0,00% 5,00% 0,00% 5,00% 00 6,00% 0,89% 5,87% 0,78% 5,58% 00 5,56% 0,45% 5,44% 0,47% 5,4% 300 5,38% 0,3% 5,4% 0,35% 5,% 400 5,30% 0,% 5,30% 0,6% 5,00% 500 5,9% 0,5% 5,3% 0,9% 4,9% 600 5,4% 0,6% 5,% 0,4% 4,9% 700 5,0% 0,3% 5,% 0,% 4,89% 800 5,9% 0,% 5,% 0,9% 4,90% 900 5,9% 0,% 5,9% 0,7% 4,9% 000 5,6% 0,% 5,8% 0,9% 4,89% able A5. Emprcal drbuo of ma μ (6) v umber of frm ad gve rho We fr fd he be f o hee daa a form b( ) / a( ) + +. he from he f we ge he ere B = { b( 5% ), b( 0% ), b( 5% ), b( 0% ), b( 5% )} a well a { a( 5% ), a( 0% ), a( 5% ), a( 0% ), a( 5% )} A = whch ca be ealy approxmaed by uable polyomal a how o Fg. A ad Fg. A for MLE (8) ad Fg. A3 ad Fg. A4 for MLE ad ma μ, repecvely. 3 b ( ) a

24 Fg.A Drf B of MLE ad polyomal f Fg.A Shf A of MLE ad polyomal f able A6: Plo of he MLE-f compoe ad her approxmae 4 3 b ( ) a ( ) µ µ Fg.A3 Drf B of ma μ ad polyomal f Fg.A4 Shf A of ma μ ad polyomal f able A7: Plo of he ma-f compoe ad her approxmae Proof of Eq. (3): Fr we derve ( k) ( V m, x) ( V, y) ( ) ( ) e e L x, y,, = L x, y,, / / k = π m= π ( ) %, cro cro where k k m m L% cro x, y,, = Exp V, + x x V, V, + y y V,. k ( = ) m= he a eral egral () calculae a x x y+ y α α + y y β y p x + q x I = L% cro ( x, y,, ) e dx = Exp e dx ( ) ( ) ( ) π q α α + y y β y = p where Exp 4 p ( ) ( ) ( )

25 α ( k ) ( k ) = ( V, ) ; β = V, k= k = = + y ; q = β +. ad p ( ) ( ) A exeral egral he gve by q% 4 p% ξ π π II = I dy = e e, p% p where β β p% = p ; q% = + ad 4 p ( ) p ( )( ) ( ) β α α ξ = 4 ( ) ( ) ( ) p. Sce er, ( ) Λ = Deog we ge / / ( ) ( ) II we come o Λ, ( ) ( ) ( ) ( ) = π, α = α ; β = β ; β = β β ; = = = β Χ = p 4 α ( ) ( ) ( ) q% / + er 4 p% β β Q = = + + α, β q % 4 = 4 p ( ) ( ) ( ) ( ) p ( )( )( ) ( ). er Q. Λ, ( ) Exp Χ + 4 p 4 p p % A he log-reur me ere are adardzed we have where µ, = rk, l k = l= ( ) α = α ; = = ( ) = β β µ ; ( ) β = β β µ,, = p% p e ξ. a ma of a marx of he Pearo cro-correlao ( k ) ( l) rk, l = V, V for a par, (k,l) of frm from ecor ad, repecvely. Collecg he above formula we fally come o (3). = =