A Mean-Variance Portfolio Optimal Under Utility Pricing

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1 Jural f Mathematcs ad Statstcs (4): , 6 ISSN Scece Publcats Mea-Varace Prtfl Optmal Uder Utlty Prcg Hürlma Werer Feldstrasse 45, CH-84 Zürch, Stzerlad bstract: expected utlty mdel f asset chce, hch taes t accut asset prcg, s csdered. he btaed prtfl select prblem uder utlty prcg s slved uder several assumpts cludg quadratc utlty, expetal utlty ad multvarate symmetrc ellptcal returs. he btaed uque slut, called ptmal utlty prtfl, s sh mea-varace effcet the classcal sese. Varus quests, cludg cdts fr cmplete dversfcat ad the behavr f the ptmal prtfl uder uvarate ad multvarate rderg f rss as ell as rsadusted perfrmace measuremet, are dscussed. Key rds: Prtfl select, utlty prcg, mea-varace effcecy, ellptcal dstrbuts INRODUCION he mea-varace mdel f asset chce by Martz [] s e f the crerste f mder face. Despte ts shrtcmgs ad restrcts (multvarate rmal returs fr arbtrary prefereces r quadratc utlty fr arbtrary dstrbuts) ths fudametal mdel remas ppular, maly due t ts aalytcal tractablty ad ts rch emprcal mplcats. I the preset study, e csder a mdel f prtfl select, hch taes t accut asset prcg. s usual, the gal s maxmzat f expected utlty f the vestr s termal radm ealth. Hever, as sde cstrat, e suppse that asset prces are determed by the utlty prcple, hch expresses the dfferece betee the future radm value ad ts certaty equvalet. he aalyss s restrcted t the case here all assets are rsy. hs assumpt s mst useful fr strategc asset allcat, here eve mey marets are csdered rsy vestmets. Our prtfl select prblem uder utlty prcg s defed Sect ad slved uder several alteratve assumpts. I Sect., e assume quadratc utlty fucts but arbtrary returs ad bta the ptmal quadratc utlty prtfl. ssumg multvarate rmally dstrbuted returs ad a expetal utlty fuct Sect., the slut s called ptmal expetal utlty prtfl. he slut Sect.3, btaed fr multvarate symmetrc ellptcal returs but arbtrary left trucated utlty fucts, s called ptmal ellptcal prtfl. s a remarable ma result, e sh herem 3. that the three derved ptmal utlty prtfls defe a uque ptmal utlty prtfl, hch s mea-varace effcet the classcal sese. I Sect 4, e study the behavr f ths prtfl th respect t sme typcal stuats. he cdts fr cmplete dversfcat are stated. he, e dscuss the behavr f the ptmal prtfl uder uvarate ad multvarate rderg f rss as ell as uder rsadusted perfrmace measuremet.. Prtfl select uder utlty prcg: Csder the prtfl select ver a statc e-perd a ecmy th a fte set f rsy assets, hch has the fllg characterstcs: S : tal vestmet amut R : radm retur the -th rsy asset, =,..., W : prprt f tal amut vested the -th rsy asset, =,..., P : tal prce f the -th rsy asset, =,..., Let S = S be the tal amut vested the -th rsy asset such that = S = S. Defe the prprt λ f shares f the -th rsy asset by the cdt λ P = S, =,...,. (.) We suppse that the tal prces f the rsy assets are utlty prces. Fr ths, let be gve -decreasg ad ccave utlty fucts u (x), =,...,, e fr each rsy asset. he utlty prcple determes prces such a ay that there s dfferece betee recevg the future radm value P (+ R ) ad ts crrespdg certaty equvalet P, that s prces slve the system f equats u ( P) = E u ( P ( R ) ) +, =,...,. (.) Usg these prces, the e-perd crease ealth r radm ga f a prtfl chce λ = ( λ,..., λ ) s gve by λ = G( λ) = P R. (.3) Crrespdg uthr: Hürlma Werer, Feldstrasse 45, CH-84 Zürch, Stzerlad 445

2 Suppse a vestr uses a -decreasg ad ccave utlty fuct u(x). he, the prtfl select prblem f the vestr cssts t maxmze the expected utlty f the termal radm ealth ϕ( λ) = E u( S+ G( λ) ) uder the cdts (.). I the ext three Subsects three typcal stuats are csdered. We beg th the t ell mea-varace legtmated cases f quadratc utlty ad multvarate rmal returs ad exted the latter t the case f multvarate symmetrc ellptcal returs... Quadratc utlty fucts: Csder frst quadratc utlty fucts u (x) = u(x) = x γ x, =,...,. he margal utlty u'(x) = γ x s -egatve prvded x γ, hece satat s reached he x γ. avd ths, e assumes that rates f retur ad tal ealth are such that satat ll t be attaed []. Slvg (.) yelds the quadratc utlty prces P =, =,...,, (.4) γ + (+ ) here, are the meas ad ceffcets f varat f R, =,...,. maxmze ϕ( λ ) t suffces t slve the frst rder cdts ϕλ ( ) = P E R u '( S G( λ) ),,...,. λ + = = (.5) Uder the assumpt f a quadratc utlty fuct, e btas the equvalet cdts γ λ P ( Cv R,R ),,..., γ S + = = (.6) = Let e = (,..., ) be the vectr f expected returs ad let V% = ( Cv R,R + ) = ( E RR ) be the matrx f secd rder mmets, hse verse s assumed t exst. he (.6) has the slut γ λp = ( V% e ), =,...,. (.7) γ S It flls that the ptmal eght vectr = (,..., ), th λp =, =,...,, (.8) λp satsfes the smple relatshp V e = %, % th % = ( V% e), (.9) Where = (,...,) detes a vectr f es. he slut (.9) defes the ptmal quadratc utlty prtfl. Furthermre, e has usg (.7) that γs γ S S = λp = %, that s γ γ % γ = + % S J. Math. & Stat. (4): , 6 (.) 446 Isertg (.) t (.4) e btas the quadratc utlty prces + % P = S, =,...,. (.) + (+ ) % he ptmal quadratc utlty prprts defed (.) are the gve by + (+ ) λ = S= ( V% e ), =,..., (.) P + %.. Multvarate rmal returs ad expetal utlty: ther mprtat stuat, hch alls fr explct aalytcal frmulas, s a multvarate rmal dstrbut f returs ad expetal utlty fucts γ x u (x) = u(x) = ( e ), =,...,, th γ the γ ceffcet f rs avers. ga, dete by, the meas ad ceffcets f varat f R, =,...,. Slvg (.) yelds the expetal utlty prces P =, =,...,. (.3) γ Sce ( R,S + G( λ) ) s bvarate rmal, the applcat f Ste s Lemma yelds Cv R,u '( S + G( λ) ) = E u'' ( S+ G( λ) ) Cv[ R,G( λ) ], =,...,. (as (.6) ppedx). It flls that (.5) s equvalet t the cdts P Cv R,R,,...,. (.4) = γ λ = = Dete by V ( Cv R,R ) = the cvarace matrx, hse verse s assumed t exst. he (.4) has the slut γ λp = ( V e ), =,...,. (.5) he ptmal eght vectr = (,..., ) s gve by V e =, th = ( V e), (.6) ad defes the ptmal expetal utlty prtfl. Wth (.5) e has γs= γ λp =, hece γ = S. Iserted t (.3) yelds the expetal utlty prces S P =,,...,. = (.7) he ptmal expetal utlty prprts (.) are the gve by S ( V λ = = e ), =,..., (.8) P.3. Multvarate ellptcal returs ad left trucated utlty: Suppse that the radm vectr f returs (R,...,R ) flls a multvarate symmetrc ellptcal desty f returs th mea e ad pstve defte cvarace matrx V gve by

3 = det(v) (.9) f(x) g (x e) V (x e) Where g :[, ) [, ) s sme apprprate geeratr fuct. he -decreasg almst everyhere dfferetable ccave utlty fuct f the vestr s assumed t tae -egatve values such that u(x) = fr all x λ. he cstat l s chse such that the utlty asscated t udesrable egatve prtfl returs vashes (see the examples bel fr llustrat). pplyg Fag et al. [3], herem.6, p. 43 ad herem.8, p. 45, e tes that ( R,S+ G( λ) ) s bvarate ellptcal ad satsfes the lear regress prperty requred herem. f the ppedx. Sce u(x)= fr x l, the varat (.) f Ste s Lemma yelds the relatshp Cv R,u '( S + G( λ) ) Cv G( λ),u '( S + G( λ) ) (.) = Cv[ R,G( λ) ], =,...,. Var G( λ) [ ] It flls that (.5) s equvalet t the system f equats Cv G( λ),u '( S + G( λ) ) Var[ G( λ) ] E[ u '(S + G( λ)) ] (.) λ PCv R,R =, =,..., = hch has the slut Var[ G( λ) ] E u '( S + G( λ) ) λ = ( ) = Cv G( λ),u ' ( S + G( λ) ) he ptmal eght vectr (.8) s gve by P V e,,..., V e, = th ( V e) J. Math. & Stat. (4): , 6 (.) =, (.3) ad defes the ptmal ellptcal prtfl. By (.6) ths ccdes th the ptmal expetal utlty prtfl. llustrate, e assume the examples bel left trucated expetal utlty fucts such that fr =,..., e sets, x P ( + ), u(x) = (.4) γ ( exp{ γ x P( + ) } ),x P( + ), ad, x S( + ), u(x) = (.5) γ ( exp{ γ [ x S( + )]}), x S( + ), here = E S G( λ) s the mea expected retur f the prtfl. he specal chce f the left trucat mdels the fact that returs bel average are asscated a zer utlty. hs mght be apprprate fr the desg f ptmal prtfls, hse gal s t beat the maret. I ths stuat, the prcg equats (.) are equvalet t E exp( γ PR ) { R } (.6) = exp( γ P ), =,..., 447 here use has bee made f Pr(R ) = (symmetrc margs). Example.: Nrmal verted gamma returs It appears structve t llustrate ur results th a -trval but tractable multvarate ellptcal dstrbut, hch fds de terest bth Isurace ad Face. he mxture f a rmal th verted gamma varace yelds the Pears type VII dstrbut r geeralsed Studet t [4-6]. It has bee prpsed t mdel facal returs by Praetz [7], Blattberg ad Gedes [8], K [9], aylr [], Hürlma [,]. actuaral applcat s fud Hürlma [3]. he multvarate desty ( β + ) + (xe) C (xe) f(x) =, B ( β, ) det(c) (.7) π Γ( β) B ( β, ) =, β >, Γ ( β + ) has lcat-scaled trasfrmed Pears VII margal destes f(x) = B ( β, ) c ( β + ) (.8) x +, =,..., c If ß> the varace σ = Var[ R ] exsts ad e has υ c = ( β ) σ. If β =, υ =,,3,..., e recvers a lcat-scale trasfrmed Studet t th υ degrees f freedm. I partcular β = s a Cauchy ad ß= s a Bers dstrbut. If β the radm varable ( β ) ( R ) c cverges t a stadard rmal radm varable. O the ther had, ay lear R = R,...,R has desty cmbat ( ) ( β + ) x R = ( β ) + f (x) B,, (.9) c c th = e ad c= C [3], herem.6). he aalytcal frmulas bel use the specal tegral fuct a bx I(a,b) = ( + x ) e dx, (.3) hse cmputatal evaluat s dscussed the ppedx B. Uder the assumpt ß> (varaces are fte), e btas after sme calculat the frmulas [ ] + λ = β β + γ [ λ λ ] γ [ λ ] Eu'(S G( )) B( ;) I(, Sc), Cv G( ),u '(S + G( )) = Var G( ) (.3) β B( β; ) I( β, γsc), β hch serted t (.) yeld the relatshps

4 β I( β +, γsc) λp = γ β I( β, γsc) V e, =,..., ( ) J. Math. & Stat. (4): , 6 (.3) smlar calculat shs that the utlty prcg equats (.6) are equvalet t B( β; ) exp( γ P ) I( β +, γsc) (.33) = exp( γ P ), =,..., Usg (.3) ad S= λpe sees that β I( β +, γsc) γ S= (.34) β I( β, γsc) Whch s a mplct equat fr the parameter γ. Isertg ts umercal value t (.33) t s pssble t determe umercally the crrespdg trucated expetal utlty prces ad the ptmal prprts defed (.). 3. Mea-varace effcecy f the ptmal utlty prtfls: It s rthhle ad structve t cmpare prtfl select uder utlty prcg th the classcal mea-varace prtfl select. Let the set W = = descrbe all prtfl chces. = Recall that a prtfl W belgs t the meavarace budary f ad ly f, fr sme, the prtfl slves the prblem m{ σ = V} W subect t = e=. prtfl W belgs t the mea-varace effcet frter f ad ly f prtfl v W exsts such that v ad σ σ, here at least e f the equaltes s strct. Frm Mert [5] r Huag ad Ltzeberger [], e s that a prtfl belgs t the mea-varace Example.: Symmetrc Ktz type returs budary f ad ly f ther tractable famly f symmetrc ellptcal = g + h, (3.) dstrbuts has the desty here g ad h are the vectrs defed by K N s f ( x) = [( x e) C ( x e) ] exp{ r[ ( x e) C ( x e) ] } (.35), det( C) g = B( V e ) ( V ), D r, s >, N + >, (3.) h = C( V ) ( V e ) D th the rmalzg cstat th s Γ( ) N+ s K = r = (V e), B= e (V e), (.36) (3.3) N+ π Γ( s ) C= (V ), D = BC. hs famly, trduced by Ktz [4], s called he mea-varace effcet frter cssts f symmetrc Ktz type dstrbut Fag et al. [3]. It thse eght vectrs (3.) fr hch. It s reduces t the multvarate rmal dstrbut case C N = s =, r =. he margal destes tae the frm remarable that the ptmal prtfls (.9), (.6) ad (N) s (.3) are all equal ad belg t the mea-varace K x x f(x) = exp r (.37) effcet frter. c c c herem 3.: he ptmal quadratc utlty prtfl, ad the prtfl retur s aga f symmetrc Ktz type the ptmal expetal utlty prtfl ad the ptmal such that ellptcal prtfl ccde ad are mea-varace (N) s K x x effcet. f R (x) = exp r (.38) c c c Prf: he expected retur f the ptmal expetal alytcal evaluat requres the fllg specal tegral fuct e (V e) B utlty prtfl s = e = =. s a rx bx J(a, b,r,s) = x e dx b (.39) Usg (3.) ad (3.) e btas mmedately V e hse cmputatal evaluat s prvded the that g+ h =, hch by (3.) shs that ppedx B. straghtfrard calculat yelds the belgs t the mea-varace budary. Sce frmulas E[ u '(S + G( λ)) ] = KγSc J((N ), γsc,r,s), = B > C (because D = BC > ) the prtfl Cv[ G( λ),u '(S + G( λ)) ] = K γ(sc) J((N ) (.4) s mea-varace effcet. Sce (.3) s detcal t +, γ Sc, r, s). (.6), t remas t sh that (.9) ad (.6) yeld the Prceedg as Example., t s pssble t same ptmal prtfls. Let % ad dete the determe umercally the crrespdg trucated ptmal prtfls defed by (.9) ad (.6). expetal utlty prces ad the ptmal prprts bta them, e slves the lear systems Vx %% = e, (.). 448 v

5 J. Math. & Stat. (4): , 6 Vx = e ad set x% % = x %, x =. Sce x V % = V+ M th M = ( ), e sees mmedately that the equat V ~~ x = e s equvalet t the equat Vx% = ( e x% ) e. But ex% = e ( V% e) = B% s a cstat. It flls that ( B)( V e x% % ) V e % = = = =, hch shs x% ( B% )( V e) the assert. I ve f the preset ufcat result, the V e prtfl defed by the eght vectr = ll smply be called ptmal utlty prtfl. s a addtal prperty, t s mprtat t bserve that ay mea-varace effcet prtfl s ptmal the fllg sese. Fr ay such, vestg the amut S yelds by cstruct a termal radm ealth, hse crrespdg expected utlty satsfes the equalty E u( S+ G( λ) ) E u( S+ G( λ )), here λ s the ptmal prprt vectr defed (.). I the very arr sese f maxmzg ths expected utlty, the ptmal utlty prtfl s the preferred mea-varace effcet prtfl chce. 4. Sme prpertes f the ptmal utlty prtfl: Gve the ptmal utlty prtfl V e =, hch maxmzes the expected utlty f the termal radm ealth uder varus assumpts, t appears useful t study the behavr f ths partcular slut th respect t sme typcal stuats. I the specal case =, fte used as llustrat, e use the tats R,R fr the radm returs stead f R,R. ssume that the bvarate dstrbuted returs have meas,, stadard devats σ, σ ad crrelat ceffcet ρ. he ceffcets f varat are deted σ σ =, =. Quest : Whe s cmplete dversfcat ptmal? I case sme f each rsy asset s purchased, that s > fr all {,...,}, e speas f cmplete dversfcat [6-8]. hs stuat ccurs f ad ly f V e > (each vectr cmpet s strctly pstve). Fr example, f = e has σ ρσσ V e= > ( ρ ) σ σ σ ρσσ (4.) f ad ly f ρ < m, r equvaletly R R R R Cv, < m Var, Var. (4.) I partcular, ths stuat alls fr pstvely crrelated returs, hch are typcally bserved facal marets ad hse facal rs cat be elmated thrugh dversfcat [9]. Quest : H des the ptmal prtfl behave uder uvarate rderg f rss? () () () I geeral, let R = (R,..., R ), =,, be t vectrs f multvarate rmally dstrbuted returs ad () let, =,, the crrespdg ptmal prtfls. he radm returs f the ptmal prtfls are () () deted here R = R, =,. Cmparg ther expected expetal utltes, e s frm stadard rderg f rss thery [] that Eu(R) [ ] Eu(R) [ ] (4.3) f ad ly f R sl R, (4.4) here sl detes the stp-lss rder r equvaletly the creasg cvex rder. Uder rmally dstrbuted returs, the relat (4.4) hlds f ad ly f the meas ad stadard devats are rdered as flls:, σ σ. (4.5) Frm the prf f herem 3., e s B that =, =,, here the cstats,b are defed as (3.3). Usg Huag ad Ltzeberger [], frmula (3..b), e btas further B σ = ( C + B ) =, =,. (4.6) D hese results all fr a umercal evaluat f the crter (4.5), hch s left t the terested reader. Quest 3: H des the ptmal prtfl behave uder multvarate rderg f rss? Gve sme multvarate rderg f relatve rsess betee the t vectrs R () ad R (), h d the crrespdg ptmal eght vectrs W () ad W () behave? hs dffcult quest has ly bee scarcely dscussed the lterature. Fr smlar but dfferet prtfl select prblems, e fds sme results the studes by Fshbur ad Prter [], Ladsberger ad Mels [], Eechudt ad Gller [4], Hürlma [8]. llustrate, let us restrct ur attet t the bvarate stuat = ad let us assume that the rmally () () () () dstrbuted pars (R,R ) ad (R,R ) have equal margs, partcular equal meas,, equal 449

6 J. Math. & Stat. (4): , 6 stadard devats σ, σ, but dfferet crrelat ceffcets p () ad p (). We assume that the secd retur s mre rsy tha the frst the sese that the ceffcets f varat satsfy the equalty >. What happes uder the crrelat rder f rsess () () () () (R,R ) c (R,R )? hs ma bvarate rderg f rss has bee csdered amg thers aagmt ad Oamt [3], Cambas et al. [5], che [6], Dhaee ad Gvaerts [7]. I ur specal bvarate stuat, the abve relat hlds exactly () () he ρ ρ. N, the eghts the mre rsy asset are gve by () =, =,. (4.7) () ρ + () ρ () () It s mmedate that ρ ρ ad > mples () (). hs meas that a decs maer ll prprtally vest mre the rser asset f the less crrelated rsy par. hs duces a preferece relat fr l depedece betee returs, hch seems t be accrdace th the usual stadards mder face. Quest 4: H des the ptmal prtfl behave uder rs-adusted perfrmace measuremet? mder vestr decdes up vestmet by lg at the tradeff betee expected retur ad rs, here rs s measured usg a s-called cheret measure f rs [8]. smple ad ppular cheret measure f rs, at least sce Rcafellar ad Uryasev [9], s cdtal value-at-rs t sme cfdece level α, hch fr a rmally dstrbuted radm retur R equals CVaRα [ R] = E R R VaR [ R] > α (4.8) = ϕ Φ ( α) σ, α here VaRα [ R] = f { x Pr(R x) α} s the value-atrs, Φ(x) s the stadard rmal dstrbut, ϕ (x) =Φ '(x) ad, σ are the mea ad stadard devat f R. Fr rs-adusted perfrmace measuremet, e ls at the radm retur per ut f cdtal value-at-rs t a fxed cfdece level α, called CVaR retur rat, hch s defed by R CVaR [ (4.9) α R] he expected value f the CVaR retur rat measures the rs-adusted retur captal. hs ay f cmputg the retur s cmmly called RROC [3] ad s defed by ER [ ] RROCα [ R] = (4.) CVaR R α [ ] N, f a vestr has t decde up the mre prftable f t ptmal prtfls th radm returs R ad R, a decs favr f the secd retur s tae f ad ly f e has RROCα [ R ] RROCα [ R] at gve cfdece levels a. hs preferece crter tells us that a retur s preferred t ather f ts expected value per ut f ecmc rs captal s greater. By (4.8) ad the results fr Quest, e btas the cha f equvalet equaltes RROCα [ R ] RROCα [ R] σ σ = = = = (4.) B B B B. ppedx : varat f Ste s Lemma herem.: Let (,) be a bvarate real radm vectr th dstrbut F (,) (x,y) ad let g(x) be a real dfferetable fuct such that lm g(x) =. x Suppse (, ) satsfes the lear regress prperty (LR) E = y = E[ ] + β ( ye[ ]), Where ß=ß[,] s sme cstat depedg (,). he e has the detty Cv,g() = β Cv,g() (.) [ ] [ ] Prf: Frst, tegrate bth sdes f (LR) th respect t the cdtal dstrbut F( y) t get the detty E [ ] E y = β { E [ ] E y } (.) he cvarace frmula by Heffdg [3] (r Lehma [3], Lemma ) yelds Cv,g() = F (x,y) F (x)f (y) g'(y)dxdy [ ] {, } { } = F(x y) F(x)F(y)g'(y)dxdy F (x) dx F (x)dx = F(y)g'(y)dy F (x y) dx + F (x y)dx E E y F(y)g'(y)dy. [ ] = { [ ] } Isertg (.) e gets Cv[,g() ] y = β I(y)g'(y)dy, th (.3) { [ ] } I(y) = E E y F (y) = E tdy= te dy ( [ ] ) ( [ ]) y (.4) Isertg (.4) t (.3) ad applyg Fub s therem as ell as the assumpt lm g(x) =, e x btas furthermre 45

7 J. Math. & Stat. (4): , 6 I(y)g '(y)dy = ( t E[ ] ) df (t) g '(y)dy y t = g'(y)dy ( t E [ ]) df(t) = ( te [ ]) g(t)df(t) = E ( E[ ] ) g() = Cv[,g() ], Whch shs the desred detty. I the specal case f a bvarate rmal radm vectr, e has Cv[,] β = Var[ ] ad the detty [33] Lemma. Cv[,g() ] = E[ g'() ] Var[ ], (.5) hch mples the relatshp Cv[,g() ] = E[ g'() ] Cv[,], (.6) I the lterature ths s attrbuted t Ste [34,35], Huag ad Ltzeberger []. Sce a lt f bvarate radm mdels satsfy the requred lear regress prperty, the dsplayed cvarace detty has a de applcat. mg the may multvarate mdels satsfyg lear regress prpertes, let us met the fllg fe but mprtat classes ad famles f multvarate dstrbuts: * he class f symmetrc ellptcal dstrbuts [3] * Bvarate ad multvarate dstrbuts f Pears type [36,37] * Bvarate ad multvarate Paret dstrbuts f the frst d [38] * Bvarate ad multvarate dstrbuts cstructed frm lear Spearma r Fréchet cpulas th margs frm lcat-scale famles [,,37,39] ppedx B: Numercal evaluat f t specal tegral fucts Frst, e sh h t cmpute the tegral (.3), that s a bx I = I(a,b) = ( + x ) e dx (B.) Dvde the tegral t t parts such that I= I+ I th a bx I = (+ x ) e dx (B.) a bx I = (+ x ) e dx (B.3) Recall the bmal seres expas α α α ( + z) = z, z <, (B.4) αα ( )...( α + ) =! hch s vald fr all real umbers α. Iserted t (B.) e btas a bx I = x e dx = (B.5) a Γ ( + ) = + G(b; + ) b x α t here G(x; α) =Γ( α) t e dt detes the cmplete gamma fuct. Fr (B.3) te that a a a ( + x ) = x ( + x ) ad apply (B.4) the secd term. Oe btas a a bx I = J, J = x e dx, (B.6) here the tegrals ca be calculated recursvely as flls (use partal tegrat) : b b b J = ( bj e ) e, a + a + (B.7) Γ (a + ),J = a [ G(b;a+ ) ]. + b he evaluat f the secd specal tegral (.39) s smpler. Usg the expetal seres ( ) r s s exp( rz ) = z, e btas the Gamma! fuct seres expas a rx s J(a,b,r,s) = x e bx dx b ( ) r Γ (a+ s+ ) = a+ s+.! b REFERENCES (B.8). Martz, H.M., 95. Prtfl select. he J. Face, pp: Huag, C. ad R.H. Ltzeberger, 988. Fudats fr Facal Ecmcs. Elsever Scece Publshg Cmpay. 3. Fag, K.-., S. Ktz ad K.-W. Ng, 99. Symmetrc Multvarate ad Related Dstrbuts. Chapma ad Hall. 4. Hgg, R. ad S. Klugma, 984. Lss Dstrbuts. Jh Wley. Ne r. 5. Helma, W.-R., 989. Decs theretc fudats f credblty thery. Isurace: Mathematcs ad Ecmcs, 8: Jhs, N.L.,. Ktz ad N. Balarsha, 995. Ctuus Uvarate Dstrbuts. (d Ed.). Jh Wley, Ne r. 7. Praetz, P.D., 97. he dstrbut f share prce chages. J. Busess, 45: Blattberg, R.C. ad N.J. Gedes, 974. cmpars f the stable ad Studet dstrbuts as statstcal mdels fr stc prces. J. Busess, 47: K, S.J., 984. Mdels f stc returs - cmpars. J. Face, 39: aylr, S.J., 99. Mdelg Facal me Seres. (3rd reprt). Jh Wley.. Hürlma, W., a. Facal data aalyss th t symmetrc dstrbuts. SIN Bullet, 3:

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