Jean Fernand Nguema LAMETA UFR Sciences Economiques Montpellier. Abstract

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1 Stochstc domnnce on optml portfolo wth one rsk less nd two rsky ssets Jen Fernnd Nguem LAMETA UFR Scences Economques Montpeller Abstrct The pper provdes restrctons on the nvestor's utlty functon whch re suffcent for domntng shft no decrese n the nvestment n the respectve sset f there re one rsk free sset nd two rsky ssets n the portfolo. The nlyss s then confned to portfolo n whch the dstrbutons of ssets dffer by frst degree stochstc domnnce shft. Ctton: Nguem, Jen Fernnd, (5) "Stochstc domnnce on optml portfolo wth one rsk less nd two rsky ssets." Economcs Bulletn, Vol. 7, No. 7 pp. 7 Submtted: June 9, 5. Accepted: July 5, 5. URL: 5G7A.pdf

2 . Introducton The queston of whether rsk-verse nvestors should or should not choose dversfed portfolo hs been nlysed qute extensvely n the lterture; for exemple Fshburn nd Porter (976), Kr nd Zemb (98), Meyer nd Ormnston (989) nd recently Meyer nd Ormnston (994) extend nd generlze ths lterture by consderng portfolos contnng more thn one rsky sset. The frst reserch ssumes tht the rsky returns re ndependently dstrbuted nd the lst ones, focuses on removng ths ndependence restrcton, llowng the rsky returns to be stochstclly dependent. All of these studes were lmted to one decson vrble, so they were not ble to nlyze the effects of generlzed FDS on optml portfolo wth three ssets or more. We extend these erler works by consderng portfolos wth one rsk free sset nd two rsky ssets, nd we nlyze the effects of shft n the sense of frstdegree stochstc domnnce (FDS). The nlyss of comprtve sttc s for portfolos wth two or more sset s rther dffcult. Followng Hrt (975), n order to derve the desred effects of chnges n the nvestor s welth, the utlty functon must possess prtculr type of seprton property, nd, s t turn out, not mny clsses of utlty functons stsfy ths property. For the cse of portfolo wth one rsk free sset nd two rsky ssets we were ble to determne the effects of shfts mentoned bove. Our pper provdes very smple condtons on the utlty functon whch re necessry nd suffcent for domntng shft n the dstrbuton of sset not to cuse decrese n the nvestment n tht sset. Two mplctons of ndependence restrcton ply n mportnt role n our nlyss. Frst, ndependence llows one rsky return to be ltered wthout chngng the mrgnl or the condtonl cumultve dstrbuton functons (CDF) for the others. Second, when the return to rsky sset s ndependent of other rsky returns, the mrgnl nd ech of the condtonl CDF s for tht return cn be chnged n exctly the sme wy. The pper s orgnzed s follows. In the next secton, the notton nd ssumptons of one rsk free sset nd two rsky ssets portfolo decson model re presented nd one mn comprtve sttc result s vewed. Followng ths, comprtve sttc ssues tht only rse wth stochstc ndependence re dscussed n secton 3. In secton 4, for Frst stochstc domnnce chnge, we determne the condtons on the decson mker s preferences tht re necessry nd suffcent for the chnge to cuse n ncrese n the proporton of welth nvested n the sset whose return s ltered. Fnlly, concludng remrks pper n the end.. The model nd ssumptons We consder rsk-verse ndvdul endowed wth non-rndom ntl welth who lloctes hs welth normlzed to one between one rsk free sset (wth return x ) nd two rsky ssets wth returns ~ y nd z ~. The portfolo shre of rsky ssets ~ y nd z ~ re α nd α, respectvely. The ndvdul s end of perod welth W ~ s then equl to ~ W α, ~ y, ~ z x ~ y x ~ + + α + α z x () ( ) ( ) ( ) Relztons of W ~ depend on relztons of ~ y, z ~ nd the selected vlues for α nd α. The rndom returns ~ y nd z ~ re ssume to tke vlues n the ntervl [, ]. H y, z. The The jont cumultve dstrbuton functon (CDF) for ~ y nd z ~ s denoted ( ) condtonl nd mrgnl CDF s for ~ y re denoted F ( y nd F ( y), respectvely, nd for Hdr nd Seo (99) mke ths ther ceters prbus ssumpton. Hdr nd Seo (99) ssume ths when they requre the rsky returns to be ndependent both before nd fter one return s ltered.

3 z~ there re G ( z y) nd G (. If ~ y nd z ~ re ndependently dstrbuted then F ( y F( y) for ll z, ( z y) G( H y, z F y G z. G for ll y, nd ( ) ( ) ( ) The decson mker s ssumed to choose ( α, α ) the termnl welth tkng rndom vrble ~ y nd z ~ s gven. Formlly, ( ) to mxmze the expected utlty (EU) from α, α s selected to mxmze EU,.e. optml portfolo solves the followng progrm (P): mx u α, α ( W( α, ~ y, ~ z )) d H( y, ( y dydz H, To smplfy notton, the symbol d H( y, s used to denoted nd y z u s the von Neumnn Morgenstern utlty functon whch s ssumed to be three tmes contnuously dfferentble, non decresng nd concve n W, wth u' ( ) > nd u ''( ) <. Now, ssume we hve nteror solutons, the frst order condtons ssocted to the bove progrm re: ) (, ~ y x u' W, y z )) d H( y, ), α z (3) ) (, ~ z x u' W, y z ), α z (4) In the cse of ndependence, condton (3) nd (4) becomes, respectvely nd y x )) df ( y dg( z ) y x ( W (, ~ y, ~ z )) df( y) dg( z ) () α (5) z x )) df ( z y) df ( y ) z x ( W (, ~ y, ~ z ) df( y) dg( z ) Evluted t α, nd α (5) nd (6) cn be wrtten, respectvely s ( y) x ( + x + z x )) dg( z ) α (6) E α (7) ( z ) x ( + x + y x )) df( y ) E α (8) Whch hve the sgn of the expected excess return ( x ) x ) nd only f ( ( x ) ) E. It follows tht α s postve f E ~ x s lso postve, tht s f x ~ offers postve rsk premum. 3. Effects of welth on optml portfolo We nvestgte the effect on the demnd for rsky sset when there re chnges n welth. We desgn effects of welth on optml portfolo by dervng the frst order condtons wth respect to the portfolo shre of rsky sset α nd the gent s end of perod welth W. We frst hve the next result Proposton. Defne the functon Φ ( u' W(, y, R nd Φ ( ) ) If ' ( )( y x ) df( y α z x, then (9)

4 ) If ' < where dw dα R nd Φ ( ) < dw ( α, ~ y, ~ z ) dα, then ( α, ~ y, ~ z ) >, for, (), for, () R' s the dervtve of the coeffcent of bsolute rsk verson Proof. Assume we hve nteror solutons, snce u s concve, the frst order condtons ssocte to the bove progrm (P) gves the optml llocton α stsfyng the followng equtons: ) (, ~ y x u' W, y z ), α z () ) (, ~ z x u' W, y z ), α z (3) Frst, dfferenttng () nd () wth respect to αnd α, respectvely, second snce u s concve we hve Defnes nd ) (, ~ y x u' ' W, y z ) d H( y, ), α z (4) < ) (, ~ z x u'' W, y z <, α s n mplct functon of W ( α, ~ y, ~ z ). We thus obtn y x ' ( W( α ~ ~, y, z dα dw ( α ~ ~, y, z ) y x ' dw dα ( α, ~ y, ~ z ) α (5) z x ' z x ' ) d H( y, The denomntor s unmbguously negtve nd clerly the sgn of (6) nd (7) depends upon the sgn of the numertor. Now, strtng from (6) where Note tht ( y x ' ( W ~, y, z ) d H ( y ψ ( dg( z ) (6) (7) α (8), ( ( y x ' ) df ( y ψ ~ B (9) ψ s nonnegtve f B ( ) ( ( ) ( ) ( ~ ~ ) ~ ~ u'' W α, y, z y x u' ' W α, y, z df y z u' ) ( W ( α, ~ y, ~ z )) y x ) df ( y B u' 3

5 R x R x R x ( W y, ~ z R ( )( y x ) df ( y ) ~ α, z () ( W (, ~ y, ~ z ( )( x y ) df ( y ~ α () ( ) y x ) df( y + R x z [ x, ] nd y [, x ] we hve ~ ~ ~ y, ~ z x y~ x ~ α, + z x x z~ + α + α + + α x W α R ' > mples tht R ( W ( y~, ~ α, z ) < R( W ( α, z~ ). Thus W x x ( ) ( ) ( ) ( ) (, ~ z ) R R ( )( x y) df ( y > ~ ( W (, ~ z ( )( y x ) df( y ~. Hence α () nd z [ x, ] nd y [ x, ] we hve W ~ ~ y, ~ z x y~ x ~ α, + z x x z~ + α + α + + α x W α R ' > mples tht R ( W y, ~ α, z ) > R ( W ( α, z~ ). Thus ( ) ( ) ( ) ( ) (, ~ z ) ~ W( α, ~ y, ~ z ) ( W(, ~ z Hence ( )( x ~ y) df( y > ( ) y x ) df ( y. Hence α (3) ( W, z ( ) y x ) df ( y ) > B > R α z (4) snce Φ ( u' W ( α, y, ( )( y x ) df( y ' > Smlrly B f R, nd f R < we hve tht ' ( ) (, ~ )) W, z u' W α, y z y x ) df ( y < R ( W ( α, ~ z ) Φ( ) < B < R α z (5) Hence B < Anlysng the results bove, we shll descrbe the behvour of the portfolo n term of the behvour of smple chrcterstc of the utlty functon. Proposton revels tht, there re mny lterntve sttstcs by whch portfolo mght be chrcterzed. We focus on one nd we cn ssert tht, f there re two rsky 4

6 ssets nd money, then the welth elstcty of the demnd for one rsky sset s greter thn, equl to, or less thn unty s bsolute rsk verson s n ncresng, constnt or decresng functon of welth. The ntuton of proposton s s follow. Optml demnd of one rsky sset depends upon the gent s tttude n the presence of rsk. 4. Shfts n returns dstrbuton of one rsky sset We desgn shfts n one dstrbuton, we consder chnges n α when ~ y undergoes n FDS shft from F to F. And we nvestgte whether or not frst degree domnnce shft n one rsky sset wll result n hgher demnd for ths rsky sset. For the cse of ndependent ~ y nd z ~, followng Hdr nd Seo, we crry out the comprtve sttc nlyss by replcng H ( y~ z~, ) F ( y~ ) G z ) wth H ( y~ z~, ) F ( y~ ) G z ). Tht s, we ssume tht ~ y nd z ~ re ndependently dstrbuted both before nd fter the chnge n ~ y occurs. Tht mples tht the sme chnge occurs [ ] for ll condtonl CDFs for ~ y ; tht s, F ( y F ( y s the sme for ech z ~. For ths ndependent cse, t s lso true tht the mrgnl nd condtonl CDFs for z ~, G z ) nd G ( z y), remn unchnged s ~ y s ltered. FDS mprovement n the CDF for ~ y re represented by F ( y) F ( y) for ll y n [, ]. Proposton. Assume frst tht the utlty functon u stsfes u '>, u '' <, second j y~ nd z ~ re stochstclly ndependent,, j mxmzed t α. Then α α for ny u' W α, y, z X x ( X ~ y or X z~ ) s nondecresng. ( ( ))( ) Proof. Let ( ) ~ j ; thrd, ( ( ) u W, ~ y, ~ z d H ( y, z ) α s ~ y FSD y f nd only f ξ α denote the dfference between the dervtves wth respect to utlty under the two dstrbuton of X ; tht s, ( ) dg( ) ( α ) ( ) ( ~ y x u' W α, y, z )) d F ( y) F ( y) α of expected ξ ~ z, (6) the ssumptons of the proposton n conjuncton of the bsc FDS theorem (Fshburn nd ( ) dg( Vckson, 978) mply tht, ) (, ~ y x u' W, y z ) d F ( y) F ( y) nd z ~, so tht ( ) ξ α for ll α for ll α α. Evlutng ( ) ( α ) y x W ( α, ~ y, ~ z ) ξ α t α ( ) α ( F y ) dg( ( α ) we hve ξ ( ) d ( ) (7) snce the frs order condton gves ) ( ) ~ ~, y x u' W, y z d ( F ( y) ) dg( α α (8) The sme comment pples for 5

7 ξ ( α ) z x W ( α, ~ y, ~ z ) ( ) dg( ( ) d F ( y) ( α ) α, (9) then the concvty of u mples α α Fgure : represents effects of FSD on optml portfolo wth one rsk-free sset nd two rsky ssets. u' ( W( α, ~ y, ~ z ))( X x ) d H( y, j ( α ) ( α ) ( α ) It s worth notng tht the condtons on utlty n proposton re smlr s those found n Fshburn nd Porter (976) for FSD chnges, nd Rothschld nd Stgltz (97). Thus, wth ndependence, extenson to portfolo wth one rsk free sset nd two rsky ssets preserves the fndngs from the one rsky nd one rskless sset cse. Concluson The queston of whether rsk-verse gents should or should not choose dversfed portfolo (s opposed to speclzed ones) hs been nlysed qute extensvely n lterture. In ths pper, we hve ddressed the queston of the optml proporton of one rsky sset n dversfed portfolo. For smplcty, the nlyss n ths pper s confned to rndom vrbles whose dstrbutons re confned on the unt ntervl. Agents re ssumed to mxmze the expected utlty of termnl welth whch s the end-of-perod vlue of 6

8 portfolo. Ths pper s concerned wth condtons under whch the proporton of gven rsky sset n the optml portfolo of rsk verse gent s t lest s lrge s some gven proporton. We then n secton 3 looked t portfolo n whch the optml demnd of one rsky sset depends upon the decresng, constnt or ncresng rsk verson. Ths frst result s consstent wth those obtned n the portfolo wth one rskless nd one rsky sset. Ths nlyss s fcltted by the use of specl subset of rsk-verter. The object of our secton 4 s to provde brefly condtons tht re suffcent for rsk verter to nvest more n the domntng sset. Note tht our study could be nterpreted n terms of mutul fund seprton. References Fshburn, P nd W. B. Porter, (976). Optml portfolos wth one sfe nd one rsky sset: Effects on chnges n rte of return nd rsk, Mngement Scence,, Hdr, J. nd T. K. Seo, (99). The effects of shfts n return dstrbuton on optml portfolos, Interntonl Economc Revew, 3, Hrt, O, (975). Some negtves results on the exstence of comprtve sttc results n portfolo theory, Revew of Economc Studes, 4, Kr, D nd W. T. Zemb, (98). The demnd for rsky sset, Mngement Scence, 6, Meyer, J. nd M. B. Ormston, (989). Determnstc trnsformtons of rndom vrbles nd the comprtve sttcs of rsk, Journl of Rsk nd Uncertnty,, Meyer, J. nd M. B. Ormston, (994). The effect on optml portfolos of chngng the return to rsky sset: the cse of dependent rsky return, Interntonl Economc Revew, 35, 3, Rothschld, M. nd J. Stgltz, (97). Incresng rsk II: Its economc consequence, Journl of Economc Theory, 3,