ON OPTIONED PORTFOLIO SELECTION UNDER OPTION STRATEGIES

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1 ON OIOND OROLIO LCION UNDR OION RAGI Wee Hüla chöholzweg 4 CH-8409 Wehu wzela el al whula@bluew.ch Absac. A cobuo o he ooous usolve ooe ofolo seleco oble s ae. We oose a class of fe coss oo saeges fo whch ooe ofolo seleco ue vaous cea coul be leee fo accal uoses. Isea of ea-vaace ofolo seleco we oose ea-car ofolo seleco whee CaR s he cooal value-a-s easue cosee he ece s aagee leaue. Keywos : ofolo seleco oo saeges shofall s cooal value-a-s. Iouco. ea-vaace ofolo seleco oeee by aowz95/59/87/94 s oe of he coesoes of oe ofolo heoy. A ae o ee he eho o ofolo seleco ue ofolo suace has bee oose by he auho996. hs lae eho eues vesg ehe a log u holg sulaeously fu o a log call holg sulaeously cash. I geeal howeve ooe ofolos ca be bul usg ves oo saeges cossg of abay lea cobaos of log a sho osos of us a calls. ce he esulg ofolo eu sbuo of a ooe ofolo ay be ahe asyec a ffcul o calculae elcly e.g. Boosabe a Clae983 ea-vaace aalyss s usually o ecoee hs suao e.g. cheuesuhl a Zags996. he cosuco of a geeal ooe ofolo seleco eho s aog he ooous usolve s aagee obles boh fo a heoecal a accal o of vew. Dese he ay aoaches o hs oble e.g. Boosabe a Clae984/85 Lee993 Albech aue a el995 Aa aue a ölle996 De a Olea996 cheuesuhl a Zags996 Jasse a ab998 Aa a aue000 o sasfacoy soluo has bee oose whch has a uvesal oeal fo face acce le ea-vaace ofolo seleco. I he ese suy we oose a class of fe coss sucue oo saeges fo whch ooe ofolo seleco ue vaous cea coul be leee fo accal use. Isea of ea-vaace ofolo seleco we oose ea-car ofolo seleco whee CaR cooal value-a-s s a ecse cohee easue of s cosee he ece s aagee leaue.

2 . A secal class of oo saeges. he lag hozo. uose a veso has bough shaes of soc a e 0 hs vese aou beg 0. he veso wshes o have a leas soees eacly > shaes of he sae soc a e. o hs he veso ca buy ecly shaes a ay e [ 0 ]. He s face wh he ecso of choosg he oal e of buyg he shaes a a ossbly low ce. o avo hs ffcul ecso a veso ca aleavely ves oos o he soc o acheve he sae goal a e. eveal uesos ca be ase : Cose a ae e o sgle soc wh ce ocess [ 0 ] Q Whe s a oo saegy efeable o he ec-buy-soc saegy? Q Does hee es a bes oo saegy o acheve he above goal? Q3 Gve a sle class of oo saeges s hee a bes oo saegy o acheve he above goal? o aswe he ffcul uesos Q a Q ay sou scefc aoach wll eue fs sasfacoy soluos o he oe acable ueso Q3 whch clues he cosuco of sle classes of oo saeges whch ae aoae fo ec vesgao. I oe ha ou oo saeges ae accally feasble we assue ha he caal ae s lu acula evey oo oso o he gve soc a soe saaze eecse ces...s s always avalable he e eval [ 0 ]. Cosae by coo vese aagee escos sellg sho us ay eue ha soe cash s eose boowe o ohewse secue a sellg sho calls ay eue eos of shaes cash boowg o ay ohe secuy e.g. Jaow a Ru983 Cha. 3. uheoe vese bas usually eue a u volue o vese o ae sho osos whch les ha ou oo saeges wll be feasble fo facal suos bu geeal o fo he oay vae veso. Whou loss of geealy we assue ha all oos ae of Aeca ye a ay be eecse a ay e u o he auy ae. Cose a elavely sle oo saegy cossg of he followg vese oos all wh he sae auy ae : sho us wh eecse ce each a he oo ce sho calls wh eecse ce each a he oo ce C log calls wh eecse ce each a he oo ce LC log us wh eecse ce each a he oo ce L Iuvely s aual o assue ha euce coss whe sho us ae eecse a euce coss whe sho calls ae eecse a. he o-egave aual ubes ae assue o sasfy he followg cosas :.. hs choce s easoable bu o coellg fo he followg easos. If > a oly he sho us ae eecse he he goal of shaes wll be suasse wh a ossble au loss of aou whch s a uecessay wase of caal hece a seculave vese saegy. Choosg allows oe o eos all of he

3 3 avalable shaes as ogage whou gog o a heoecally ule loss shae ces ay cease abaly of aou a 0 C case > a he call oos ae eecse a e. he fac ha oe buys eacly log calls s eve. If he sho calls ae eecse all log calls us be eecse o ge a leas shaes. Neglecg ees ae effecs ue o he fac ha hese oo osos ay be sele a ffee es he vese value say e of asaco coss of hs oo saegy euals :. W C LC L. A auy ae he oos have bee ehe eecse o o. Assug fo slcy hs s o he case acce ha fo all sho osos of a gve ye eacly oe of hese wo ossbles occus hee ae fou a so obablsc eves of oace u o fuhe sub-eves eeg o whehe o o he veso eecses soe o all log oo osos. Neglecg ve effecs s aual o ae he assuo ha a Aeca sho call o a o-ve ayg soc s eve eecse ealy Resco 5 Jaow a Ru he he eleva fou oo eves ae escbe as follows : ve : sho us eecse sho calls o eecse { fo soe C fo all } ve : sho us o eecse sho calls o eecse { C fo all } ve 3 : sho us o eecse sho calls eecse 3 { fo all C } ve 4 : sho us eecse sho calls eecse 4 { fo soe C } ae ow wh ceay ha s obably oe oly a e. heefoe ealy eecse of he log osos s usually o ossble. If 4 occus eale eecse ay be ossble as soo as s ow ha he sho osos have bee eecse. Accog o whch eve acually occus s ow ecessay o escbe how ay log osos wll be eecse. uch a esco eees he vese coss K a he vese value a auy ae of he ooe soc o ae e. Obseve ha he occuece of he eves {...4} ve ce he sho us have bee eecse a s aoal o eecse he log us because hs aco euces he vese coss of he oo saegy. he ooe soc cosss of shaes a auy ae. wo sub-cases ae ossble.

4 4 ub-case : o have eacly shaes a ae oe us eecse log calls a hee eas log call osos. Oe has [ ] W K ub-case : > hee ae oe ha he eue shaes a auy ae a hee eas log call osos. Oe obas K W ae ogehe he foulas ca be suaze as follows :.3 K a 0 W a.4 ve o have eacly shaes a auy ae oe us eecse log calls a hee eas log call a log u osos a auy ae. Oe has.5 K W.6 ve 3 o have eacly shaes a ae oe us eecse log calls a hee eas log call a log u osos. heefoe oe obas.7 K 3 W.8 3 [ ] ve 4 wo sub-cases us be sgushe. ub-case 4 : 0 o have eacly shaes a ae oe us eecse 0 log calls a hee eas log call a log u osos. Oe has

5 5.9 4 W K.0 4 ub-case 4 : 0 o ge shaes a ae eecse 0 > log us. hee eas log call a log u osos. Oe has. 4 W K. 4 ae ogehe he foulas ae suaze as follows :.3 { } { } 0 a 0 a 4 W K.4 { } { } 0 a 0 a 4 Rea.. If oe allows > fo he sole goal of a seculave vese saegy hee s by occuece of eve 4 a fuhe sub-case : ub-case 43 : 0 ecse he log us o have a ae a ube of > shaes. hee eas log call a log u osos. Oe has.5 4 W K.6 4 he hee sub-cases of eve 4 ca be fuhe suaze o he foulas :.7 { } { } 00 a a 0 a 4 W K.8 { } { } 00 a a 0 a a 4 I s eesg o as whehe hee ess a subclass of he efe class of oo saeges fo whch he vese coss K of he ooe soc ae cosa each sae of he wol { } 34. hese so-calle fe coss oo saeges ae chaaceze by he followg esul.

6 6 heoe.. e coss oo saeges Gve s he class of oo saeges sasfyg he cosas. whose vese value s gve by.. he oe has K cos. fo all { 34 } f a oly f oe of he followg wo cases s fulflle : Case : he eecse ces of all oos ae eual ha a fo each feasble choce hee es s ffee oo saeges wh fe coss.9 K W {... s} fo all { 34 } Case : > ub-case : 0 he eecse ces ae eual a ecessaly.0 wh >. o each feasble choce hee es s ffee oo saeges wh fe coss. K W {... s} fo all { 34 } ub-case : 0 he eecse ces ae eual a ecessaly.. o each feasble choce hee es s ffee oo saeges wh fe coss.3 K W {... s} fo all { 34 } oof. I oe ha K cos. Case he followg uaes us be eual : { } : [ ] { } : : { 3} { 4} : fo sub-case 4 fo sub-case 4.

7 7 I sub-case 4 coae { } wh { 3} a { 4}. Coag ow { } wh { } o ge eaely oe us have 0. If hee ae o log u osos a ca be chose abaly say. If > he ecessaly. hs seles sub-case 4. I sub-case 4 coae { } wh { 3} o ge. Coag ow { } wh { 4} oe us have 0. ce a oe has ecessaly hece. Coaso of { } a { } a Case s show. I Case he above uay fo { } { } : les. hs seles sub-case 4 us be elace by I sub-case 4 oceeg as above oe us have wh { }. Coag ow { } yels he coo > a hus.0 hols. I sub-case 4 coae { }. Coag { } wh { 4} wh { }. ce > oe us have wh { 3} o ge oe ges ha s he fs a of. a coag { } yels. Iseg he fs elao o he seco oe yels he seco a of.. 3. obables of he oo eves. o calculae he eece eu a elae s easues of a ooe soc o ae e vese ue he class of oo saeges cosee eco whch wll be oe eco 3 s fs ecessay o evaluae he obables of he eves { 34 }. Cose a sho u o he soc wh eecse ce a a oo ce a a a sho call wh eecse ce b a a oo ce C b. As show below Lea. suffces o cose he followg hee obables :. a { a a fo all [ 0 ]} he obably ha a sho u 0 wh eecse ce a s o eecse he e eval. b { > b C b } he obably ha a sho call wh eecse ce b s eecse ue he ae assuo ha Aeca sho calls ae eve eecse befoe auy 3 a b a a fo all > b C b he obably ha sulaeously a sho call wh eecse ce b s eecse bu a sho u wh eecse ce a b s o eecse.3 { [ 0 ] } o abay soc ce ocesses he evaluao of. a.3 belogs o he so-calle oc of cossg obables. I he ubuous secal case ha follows a geoec

8 8 Bowa oo e.g. Blac-choles oel hese obables follow fo classcal esuls o absoo obles. I he followg assue ha has a log-oal sbuo such ha he accuulae logahc eu a e euals :.4 R l W wh W he saa Wee ocess. 0 I accal calculaos s useful o se l. he elao [ ] 0 efes he he aaee as he accuulae eu e u of e. o coveece se he followg a ' l b' l. a a b C b I s eae ha b'.6 b Φ whee Φ z s he saa oal sbuo. o calculae. a.3 cose he sog e.7 f { R a' } τ. I s o ffcul o see ha a τ > a a a 3 a b τ a > R b'. Ue he assuo.4 s well-ow ha.8 a' a' a' a Φ e Φ.9 b' a' b' a' a b Φ e Φ 3. I geeal he obables of he fou eleva oo eves ae sly eee fo.-.3 as follows. Lea.. Gve s a sho u wh eecse ce a oo ce a a sho call wh eecse ce a oo ce C as eco. he he obables of he eves {...4} sasfy he followg elaos :

9 9 oof. By efo of he eves eco a eleeay obably oe has eaely ce 3 3 a he eves ae so he elaos.0-.3 follow whou ffculy. 4. he eece eu of a ooe soc vese. Ue he assuo.4 of a geoec Bowa ce ocess he eece eu of ou sucue ooe soc vese ca be obae as follows. s oe calculaes he cooal eece eu a auy ae gve a oo eve whch s eoe a efe by 3. [ ] R {...4} K whee a K have bee efe eco. he eece eu a auy ae of a ooe soc vese s he gve by 4 3. R R whee he obables of he oo eves ae evaluae as eco. Of elae ees fo ecso uoses ae he ea absolue evao of he cooal eece eus fo he eece eu efe by 3.3 ADR R R 0 4 o slaly he vaace of he cooal eece eus efe by R R ar. ale 3. : he ufo eecse ce oo saeges lc eessos fo he cooal eece eus 3. ae geeal ahe colcae. I he oa secal case elavely sle foulas ca be gve. I he oaos of eco le C he obably ha he sho calls ae eecse a cose he uvaae fucos calle so-loss a cougae so-loss asfos of whch ae gve by

10 u u Φ Φ 3.6. e 0 l l u hough saghfowa calculaos oe obas fo he ueaos 3. he foulas 3.7 a a C C C C C 3.8 C C C C C 3.9 { } 3 3 C C C > 3.0 { } 4 4 C C C > o ge he eece eu of a ufo eecse ce ooe soc vese se a use he eessos fo he obables of he oo eves gve Lea.. I ou secal case hese obables ae gve by whee accog o a.9 oe has 3. Φ 3.3 Φ Φ e 3.4 Φ Φ 3 e 3.5 l l 0 0 C.

11 4. Ooe ofolo seleco wh a shofall s ceo. o a facal acoe he a ees wll o be he eece eu of a sgle ooe soc vese bu ha of a whole ooe ofolo vese. o slfy he aalyss we esc ouselves o he secal fe coss oos saeges obae seg heoe.. A sla bu oe cole aalyss coul also be oe fo ou geeal sucue oo saeges. Gve s a ofolo aage whch vess ffee socs a oos o hese socs all wh he sae auy ae. he basc aa of ou ooe ofolo cosss of he followg es : : ce a e of he -h soc... : -h eecse ce of a oo o he -h soc... s : ce of a sho u o he -h soc wh eecse ce C : ce of a sho call o he -h soc wh eecse ce LC : ce of a log call o he -h soc wh eecse ce L : ce of a log u o he -h soc wh eecse ce : ube of shaes a e 0 of he -h soc > : ube of ese shaes a e of he -h soc : ube of sho call osos o he -h soc : ube of sho u osos o he -h soc o he -h soc he ofolo aage chooses a fe cos oo saegy wh ufo eecse ce {... s} whose vese value s by. eual o 4. W C LC L. Iclug he ce of he shaes he fe cos of he -h ooe soc s hus 4. K W. 0 o each a hee ae fou eleva oo eves efe as follows see eco : 4.3 { fo soe C fo all } { C fo all } 3 C } 4 { fo soe C { fo all he whole ooe ofolo s chaaceze by he chose veco... fo whch hee ae s ossble choces. o each ooe ofolo eacly 4 ffee ooe veco eves ay occu each chaaceze by a veco whch ae eoe a efe by }

12 Deog by...4 whch ca be calculae alyg he eho of eco as wll be oe lae eco 5 he obably of a veco eve 4.4 wll ee soe ulvaae oel o he obables of he sgle eves. o eale f he eves ae eee a assuo whch us be ese he obables ae gve by he obably of he eve { } 4.5. Noe ha ue soe oe geeal assuo of osve eeece bewee he eves he lef-ha se wll always be a leas eual o he gh-ha se. oeove ue easoable assuos hee es ue sle ue bous e.g. e a La998 see also Chow a Lu968. As a ec coseuece he obably ca always be wo-se boue ways whch esuably wll suffce fo os accal suaos. o ge he vese value of he ooe ofolo a e gve ha occus s ecessay o slay fs he vese value of a -h ooe soc gve ha occus. o he aalyss ae eco oe has [ ] [ ] 4 [ ] hough ao he vese value of he ooe ofolo gve ha occus s eual o 4.7. Usg 4. he fe cos of he ooe ofolo euals 4.8 K K. laly o eco 3 cose ow he cooal eece eu gve a veco eve of he ooe ofolo a ae eoe a efe by 4.9 [ ] R. K he he eece eu of he ooe ofolo a ae euals 4.0 R R.

13 3 A elae ea absolue evao a vaace of he cooal eece eus sla o 3.3 a 3.4 ca also be efe. o accal calculaos of hese uaes hee eas he ueso of he evaluao of he obables see eco 5. Of a oace ofolo seleco s he ae-off bewee eece eu a s. ce he vaace of he eu shoul esuably o be a aoae s easue ue o he asyec ooe ofolo eu sbuo a aleave s easue of he ooe ofolo us be use. ollowg he shofall eho whch s a slgh eeso of he well-ow value-a-s eho we oose o use a secal shofall s easue o he vese value of he ooe ofolo ow as CaR easue cooal value-as easue e.g. Hüla00a/b003 Rocafella a Uyasev00. he CaR easue ay be efe as follows. Le be he ao vaable of he egave eu of he ooe ofolo whch ees o he eves gve by 4. f occus. K Gve a sall loss oleace level ε usually ε o 0. 0 he CaR easue of o he level ε s efe by he uay CaR ε ε 4. [ ] Q ε [ Q ε ] whee u s he so-loss asfo of. he CaR easue of s sasfes ueous esable oees. o eale s a cohee s easue he sese of Aze e al.997/99. I geeaes a oeg of s calle CaR oe whch s euvale o he so-loss oe o euvalely he ceasg cove oe Hüla00a heoe. Hüla003 ooso.. heefoe he CaR oe s coable wh he coo efeeces of s avese ecso aes whch use cocave o-eceasg uly fucos. laly o he classcal ea-vaace ofolo seleco oe ca cose a ea-car ofolo seleco acula a ea-car ooe ofolo seleco. o accal calculaos of he CaR easue 4. he obables ea o be evaluae. Iee he sbuo fuco a he so-loss asfo of whch us be ow fo he evaluao of 4. ae obae fo he cooal sbuos of gve usg he heoe o oal obably : Q s he uale fuco of a [ ]. 4 he eue cooal sbuos a so-loss asfos ca be obae fo usg a he secfcao of a o sbuo of... usg he soc ce yac escbe eco 5. Uless e.g. he sgle ufo eecse ce ooe soc vese of ale 3. he aalycal evaluao of he eue cooal uaes 4.3 a 4.4 gh be ahe cubesoe a uecal ehos shoul be cosee.

14 4 5. obables of he ooe veco eves. he eece eu 4.0 a he CaR-easue 4. eue fo a ea-car ooe ofolo seleco ee o he obables 4.5. Usg he esuls of eco a geoec Bowa oo assuo fo he behavou of soc ces a he couous Caal Asse cg oel by eo97 we show how he obables of he eleeay oo eves 4.3 ca be evaluae. he vual soc ces ae le o he geeal e of he facal ae whch s escbe by a soc ae e whose yac s suose o follow a Ioocess 5. W whee s he saaeous ae of eu of he ae e s he saaeous vaace ae a W s a saa Wee ocess. he e le couous ce ocesses of he vual socs ae escbe by 5. η W η W whee s he saaeous ae of eu of he -h soc c η s he elaosh bewee he saaeous covaace-ae c of a a he volaly of he ae e a W s a saa Wee ocess. Oe assues ha he aaees η a η ae cosa ove he e hozo a ha he ocesses W a W ae eee fo all.... I follows ha 5.3 W ' ' whee η η s he saaeous vaace ae of he -h soc a W s aohe saa Wee ocess. uheoe oe has c ρ whee ρ s he coelao coeffce bewee a. he ae of eu of he -h soc s le o he ae ae of eu followg he couous CA elaosh : 5.4 f f whee f s he saaeous s-fee ae of eu a 5.5 ρ s he bea coeffce of he -h soc. he owlege of ρ f a suffces o eee usg 5.4 a 5.5 a hus he soc ce ocess 5.3. I acula he

15 5 ce ocess follows a logoal sbuo such ha he accuulae logahc eu a e s gve by 5.6 ' 0 l W R. he above aalyss has euce he ofolo suao o he saa suao.4 of a sgle soc whch allows oe o calculae he obables of he eves. o each s se slaly o l l 0 0 C. I aalogy o.6.8 a.9 efe 5.8 Φ 5.9 Φ Φ e 5.0 Φ Φ e 3. Alyg Lea. oe obas fally he ese obables : CaR fo a ooe soc vese wh ufo eecse ces. I s useful o llusae ou heoecal aalyss fo he secal case of a ooe soc vese wh ufo eecse ces whch clues acula fe cos oo saeges Case of heoe.. he eece eu of such a vese ca be calculae wh he foulas usg o coue CaR accog o foula 4. s ecessay o oba fs he sbuo of he egave eu 6. 4

16 6 a s so-loss asfo 4 6. whee he egave eu s escbe by 6.3 f occus wh 0 K he al cos of he ooe soc vese. We sgush bewee wo cases. he sbuo of a eoes he suvval fuco. s eoe by Case : eves 0 a a alyg he ules fo cooal obables oe obas usg he foulas of eco afe soe calculaos Decoosg he eve { C } o he wo so eves { } { C } { } { 0 } { } { C } C C a C [ ] [ ] [ ] a { 0} C a C { } C laly oe obas he ecooso

17 7 6.7 { } { } 0 C I I C α α α α wh 6.8 a α α α α he as above a {} I he cao fuco of he eve {}. Afe soe saghfowa bu eous calculaos oe ges he eessos C C C C C C Case : eves 3 I s o ffcul o see ha 6. { } { } C C C C wh { } a C laly oe ges

18 8 6.4 wh α C [ I{ C }] α 6.5 α 3 α4 3 4 a 6.6 [ ] [ C ] C C. Now o ge accal values of CaR ε [ ] fo gve ε say f a accuae aoao o Q ε f{ : ε} ε ε a/o 0. 0 fs hough uecal couao of 6. usg 6.4 a 6.. he base o 6.7 a 6.4 calculae ε accog o 6.. he obae value ε ε ε wll be a accuae aoao o CaR ε [ ] Q ε [ Q ε ]. ε Refeeces. Aa. a R. aue 000. Aalysche valuao es Rso-Chace-ofls obee Ae- u Oossaege. Bläe e Deusche Gesellschaf fü escheugsahea I Aa. aue R. a. ölle 996. valuao of cobe soc a oo saeges usg ecess-chace a shofall-s-easues. I : Albech.. Auaelle Asäze fü az-rse AIR 996 vol. II elag escheugswschaf Kalsuhe. Albech. aue R. a. el 995. A shofall aoach o he evaluao of s a eu of osos wh oos. oceegs 5 h AIR Ieaoal Collouu Büssels vol Aze. Delbae. be J.-. a D. Heah 997. hg coheely. RIK 0 o. Novebe Aze. Delbae. be J.-. a D. Heah 999. Cohee easues of s. aheacal ace Boosabe R. a R. Clae 983. A algoh o calculae he eu sbuo of ofolos wh oo osos. aagee cece Boosabe R. a R. Clae 984. Oo ofolo saeges : easuee a evaluao. Joual of Busess Boosabe R. a R. Clae 985. obles evaluag he efoace of ofolos wh oos. acal Aalyss Joual Chow C.K. a C.N. Lu 968. Aoag scee obably sbuos wh eeece ees. I asacos o Ifoao heoy vol. I De C. a B. Olea 996. Ooe ofolos : he ae-off bewee eece a guaaee eus. I : Albech.. Auaelle Asäze fü az-rse AIR 996 vol. II elag escheugswschaf Kalsuhe.

19 9 Hüla W ea-vaace ofolo seleco ue ofolo suace. I : Albech.. Auaelle Asäze fü az-rse AIR 996 vol. I elag escheugswschaf Kalsuhe. Hüla W. 00a. Aalycal evaluao of ecooc s caal fo ofolos of Gaa ss. AIN Bulle Hüla W. 00b. A ufe aoach o cooal value-a-s wh alcao o RAROC a RARAROC. ausc avalable fo he auho. Hüla W Cooal value-a-s bous fo coou osso ss a a oal aoao. Joual of Ale aheacs Jasse J. a.-. ab 998. he ae-off bewee eece a guaaee eus fo ooe ofolos. oceegs s uo-jaaese Wosho o ochasc Rs oellg fo ace Isuace ouco a Relably Bussels vol. I. Jaow R.A. a A. Ru 983. Oo cg. R.D. Iw Hoewoo Illos. Lee D.J ofolo seleco he esece of oos a he sbuo of eu of ofolos coag oos. oceegs 3 Ieaoal Collouu AIR Roa vol aowz H ofolo eleco. he Joual of ace aowz H ofolo eleco ffce Dvesfcao of Iveses. Joh Wley. eco o 99. Basl Blacwell. aowz H ea-vaace aalyss ofolo choce a caal aes. Basl Blacwell. aowz H he geeal ea-vaace ofolo seleco oble. I : Howso.D. Kelly.. a. Wlo.. aheacal oels face. hlosohcal asacos of he Royal ocey of Loo e. A eo R. 97. A aalyc evao of he effce ofolo foe. Joual of acal a Quaave Aalyss Rocafella R.. a. Uyasev 00. Cooal value-a-s fo geeal loss sbuos. Aeas Joual of Bag a ace. cheuesuhl G. a R. Zags 996. Oal ooe ofolos wh cofece ls o shofall cosas. I : Albech Auaelle Asäze fü az- Rse AIR 996 vol. II elag escheugswschaf Kalsuhe. e. a C.D. La 998. O elably bous va cooal euales. Joual of Ale obably