Credit portfolio optimization with replacement in defaultable asset

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1 SPECIAL SECION: MAHEMAICAL FINANCE Credi porfolio opimizaio wih replaceme i defaulable asse K. Suresh Kumar* ad Chada Pal Deparme of Mahemaics, Idia Isiue of echology Bombay, Mumbai 4 76, Idia I his aricle, we propose a ew model for credi porfolios. We prove he exisece of opimal porfolios for he associaed power uiliy maximizaio problem ad he bechmarked opimizaio problem. Keywords: Bech-marked opimizaio, credi porfolio, defaulable asse, power uiliy. Iroducio IN his aricle, we iroduce a ew porfolio model of moey marke accou, a sock ad a defaulable asse. Credi porfolio models cosisig of moey marke accou, sock ad defaulable asse have already bee sudied by various auhors,. Bu all hese models assume ha upo defaul (i.e. of he defaulable asse), he porfolio coiues wih he moey marke accou ad he sock ill mauriy. Oe exisig mehod o circumve his siuaio is o icorporae large umber of defaulable asses i he porfolio ad look a he large porfolio asympoics; see Sircar ad Zariphopoulou 3 ad he refereces herei. his approach i geeral leads o a ifiie dimesioal value process for he porfolio wealh. We propose a differe approach. Our mehod is o replace he defaulable asse wih aoher defaulable asse upo each ime of defaul. By doig his, we arrive a a porfolio opimizaio model, where he dyamics of he value process is give by a Markov swichig jump diffusio model (see eq. (3) laer), which is fiie dimesioal. here exiss lieraure o porfolio opimizaio usig jump diffusio models 4,5. Noe of hese models icorporaes swichig i he diffusio. I geeral here are hree approaches o porfolio maageme, uiliy maximizaio of porfolio wealh; bechmark opimizaio ad mea variace opimizaio. Usig our porfolio model, we sudy uiliy maximizaio usig power uiliy ad bechmark opimizaio problem. his aricle is srucured as follows. Firs, we describe he porfolio model ad he porfolio opimizaio problems. he uiliy maximizaio problem is addressed ex. he, we prove he exisece of he opimal porfolio. I he ex secio, we prove he exisece of opimal bechmarked porfolio, followed by some umerical resuls. *For correspodece. ( suresh@mah.iib.ac.i) 65 Porfolio model ad problem descripio We cosider a fiacial marke wih a baske of defaulable asses, oe sock ad a risk-free securiy called moey marke accou. A ay give ime he baske of defaulable asses coais M-ype of asses wih ih ype beig ideified by a pair of parameers (b(i), σ (i)), where b(i) deoes he expeced rae of reur ad σ (i) deoes he volailiy vecor, i =,,, M. he asse price evoluio uil defaul of ay ih ype defaulable asse from he baske is give by he sochasic differeial equaio (SDE i shor), d S( ) = Si( ) b( i)d + σ ( i, j)d Wj( ), i =,,..., M, where σ (i) = (σ(i, ), σ(i, ),, σ(i, )) ad W( ) = (W ( ), W ( ),, W ( )) is a sadard R -valued Wieer process o he complee probabiliy space (Ω, F, P). he evoluio of he sock price is give by d S ( ) = S ( ) bˆ d + ˆ σ ( j)d Wj ( ). he moey marke accou has a price S ( ) which evolves accordig o he equaio, ds () = S ()r d, where r is he ieres rae of he moey marke accou. Our porfolio model a ay give ime cosiss of a sigle defaulable asse chose from he baske, he sock ad he moey marke accou. he evoluio of he porfolio over ime is as follows. Upo defaul of he defaulable asse i he porfolio, ivesors choose aoher defaulable asse from he baske as a replaceme ad he procedure coiues up o he ermial ime. he defaul ime ad he replaceme of defauled asse i he porfolio is accordig o a coiuous ime Markov chai Y( ) wih sae space S = {,, 3,, M} ad rae marix Π = (π ij ) M M, i.e. he defaul ime is modelled by he swich ime of Y( ), ad whe he Markov chai is i he sae i he porfolio coais he ih ype defaulable asse. CURREN SCIENCE, VOL. 3, NO. 6, 5 SEPEMBER

2 SPECIAL SECION: MAHEMAICAL FINANCE he he evoluio of Y( ) is give by πijδ+ ο( δ) if i j PY ( )( + δ) = jy ( ) = i) = + πδ ii + οδ ( ) if i = j, where () i.e. X () φ S() + φ () S () + φ () S () if τi foray i, = φ S() + φ () S () for= τi forsome i. λipij if i j πij : = λi oherwise. Remark. he baske of asses coais M-ype of defaulable asses wih muliple umber of each ype of asse i he baske. Hece eve if he ih ype defaulable asse is defauled a a ime, he baske of asses will coai he ih ype defaulable asse. he model as i sads ow will o preve keepig a asse i he porfolio eve if he asse of he same ype was defauled i he pas. he model oly preves he same ype of asse i he porfolio for immediae replaceme a defaul. Recifyig his issue is a ieresig fuure work. We assume ha Markov chai Y( ) ad he Wieer process W( ) are idepede. Le F be he righ coiuous augmeaio of σ(w(s), Y(s), s ). Le τ deoe he ime of he h swich of he Markov chai Y( ), i.e. he ime of he h defaul i he porfolio is τ,. Se ad I = τ τ,, I =, τ =, N() = if{ I, I + > }. he N( ) is a Cox process wih iesiy λ(y()). For more deails, oe ca refer o Las ad Brad 6 Cosider he process S () give by SY() () if τi foray i, S () = for = τi for some i. for all i. he process S () deoes he combied dyamics of he price process of defaulable asses used i he porfolio. Noe ha S () is eiher a RCLL (righ coiuous wih lef limis) or a LCRL (lef coiuous wih righ limis) process. Also oe ha he righ coiuous versio of he process S () is S Y ( ). Cosider a porfolio process φ( ) = (φ ( ), φ ( ), φ ( )), which is predicable ad self-fiacig. Here φ (), φ () ad φ () deoe he umber of shares of he moey marke, sock ad he defaulable asse a ime respecively. he value a ime of he porfolio φ ( ) is give by X() = φ ()S () + φ ()S() + φ () S (), () Uder he self-fiacig codiio, he wealh process X( ) is give by he soluio of he followig swichig jump diffusio d X ( ) = [ r+ u ˆ ()( b r)d + ( b( Y()) r)]d X ( ) + ( u () ˆ σ( j) + u () σ( Y(), j))d W () u ()d N (), (3) where u( ) = (u ( ), u ( )) deoes he predicable process represeig fracio of wealh ivesed i he sock ad he defaulable asse a ime respecively. We assume ha shor sellig of he defaulable asses ad he sock is o allowed. Hece we resric our porfolios o he followig. A porfolio process u( ) is said o be admissible ad wrie u( ) A if: (i) he process u( ) is A = [, ] [, ε]-valued ad is predicable wih respec o {F }, where ε > is a fixed cosa. (ii) Equaio (3) has a uique weak soluio correspodig o he process u( ). Noe ha he admissibiliy codiio demads he exisece of a uique, weak soluio o eq. (3). Codiio (ii) is o commo whe he corol does o appear i he drivig oise erms. Bu whe he drivig oises are corolled, i is geerally imposed 7,8. Furher oe ha, whe he corol is a prescribed corol, he from Proer 9 (heorem 37, p. 84), i follows ha eq. (3) has a uique srog soluio. herefore, he se A of all admissible corols is o empy, because, all prescribed corols are admissible. Also, i is ieresig o oe ha we i fac prove he exisece of opimal corol which is prescribed; see heorems ad 3 below. Now we will show ha he wealh process X( ) correspodig o u( ) A is posiive. For a RCLL process Z( ), le ΔZ(s) deoe he jump Z(s) Z(s ) of he process Z( ) a ime s. We sae he followig heorem from Proer 9. heorem. Le Z( ) be a semimarigale, Z() =. he here exiss a uique semimarigale X( ) ha saisfies he equaio X() = + X ( s )d Z( s). Moreover X( ) is give by j CURREN SCIENCE, VOL. 3, NO. 6, 5 SEPEMBER 65

3 SPECIAL SECION: MAHEMAICAL FINANCE 65 X () = exp Z() [ Z, Z]() ( +ΔZ()) s s exp ΔZ ( ) ( ΔZ ( )), where he ifiie produc coverges. Le u( ) be a admissible corol ad X( ) be he correspodig wealh process. he, from heorem, we have X () = exp Z() [ Z, Z]() ( +ΔZ()) s where s exp ΔZ ( ) ( Δ Z ( )), >, d Z( ) = ( r+ u ( )( bˆ r)d + u ( )( b( Y( )) r))d + ( u ( ) ˆ σ( j) + u ( ) σ( Y( ), j))d W ( ) u ( )d N( ). j From he above equaio i is clear ha ΔZ(s) = u (s) > + ε a he ime of jump; oherwise ΔZ(s) =. herefore, our wealh process is sricly posiive for ay admissible corol. Remark. Porfolio u( ) akig values i A implies o shor sellig of he sock ad he defaulable asse. Also u ( ), u ( ) pus a resricio o borrowig from he moey marke accou. Moreover, furher resricig u ( ) o [, ε], preves puig all moey i he defaulable asse. We sudy wo ypes of porfolio opimizaio problems usig our porfolio model described above. I he firs problem we look a porfolio opimizaio usig power uiliy ad i he secod, we sudy he bechmark opimizaio problem. We assume ha he ime horizo is [, ]. Power uiliy opimizaio I his porfolio opimizaio, he objecive of he ivesor is o maximize he ermial expeced uiliy: Jθ, ( xiu,, ( )) : = E[( X ( )) X() = xy, () = i], (4) θ where X( ) is give by eq. (3) correspodig o u( ) A. A ivesme sraegy u*( ) is said o be opimal if J θ, (x, i, u( )) J θ, (x, i, u*( )), u( ) A. Noe ha i geeral u*( ) may deped o he iiial wealh x >. he fucio f defied by f( x, i) : = sup J ( x, i, u( )), θ, is called he value fucio of he opimizaio problem. Our mehod o prove he exisece of a opimal ivesme sraegy relies o characerizig he value fucio as a uique soluio o cerai Hamilo Jacobi Bellma (HJB) equaios. i.e. we use dyamic programmig approach. Bechmarked opimizaio I his porfolio opimizaio, he objecive of he ivesor is o selec a porfolio which ouperforms a bechmarked porfolio. Le L( ) deoe he price evoluio of he bechmark porfolio ad is give by he soluio of he SDE (5) d L ( ) = L ( ) α d+ γ j d Wj( ), L() = l, (6) where α ad γ = ( γ, γ,..., γ ) are cosa. he ivesor s objecive is o maximize he riskadjused growh of his porfolio relaive o he bechmark. his objecive ca be modelled hrough defiig a ew opimizaio crierio, represeig he logarihm of he excess reur of he asse porfolio over is bechmark, F() give by X() F( ) = l = l X( ) l L( ). L () See, for example, Davis ad Lleo. By Io s formula, i follows ha F( ) is give by he soluio of he SDE d F( ) = ( r+ u ˆ ( )( b r)d + u( )( b( Y( )) r)) α ( ( ) ˆ ( ) ( ( )) γ u σ + u σ Y + u( ) u( ) ˆ σ σ( Y( ))) d + ( u () σ( j) + u () σ( Y(), j) γ )d W () + l( u ())d N(), (7) where ˆ σ = ( ˆ σ(), ˆ σ(),..., ˆ σ( )). he ivesor s objecive is o maximize he ermial expeced uiliy J θ, (z, i, u( )) := E[ exp( θ(f())) F() = z, Y() = i], (8) over all admissible ivesme sraegies u( ) subjec o eq. (7). he defiiio of value fucio is as i eq. (5). A porfolio u*( ) is said o be bechmarked opimal if J θ, (z, i, u( )) J θ, (z, i, u*( )), u( ) A. CURREN SCIENCE, VOL. 3, NO. 6, 5 SEPEMBER j j

4 SPECIAL SECION: MAHEMAICAL FINANCE Uiliy maximizaio I his secio we sudy he porfolio opimizaio wih cosa risk aversio parameer θ >, described earlier. For his i suffices o cosider he followig payoff crierio give by E[(X()) θ X() = x, Y() = i], (9) where X( ) is he soluio of eq. (3), sice he maximizer of J θ, (x, i, u( )) is he miimizer of above crierio ad vice versa. Le ad I θ, (, x, i, u( )) := E[(X()) θ X() = x, Y() = i], fxi (,, ): = if I (, xiu,, ()), θ, () be he correspodig value fucio. he HJB equaio correspodig o he value fucio f is give by (, x, i) + if x( r+ u ˆ ( b r) + u( b() i r)) u A (, x, i) + x ( u + u () i ˆ σ σ f ˆ σ σ πij + uu ()) i (, x, i) + f(,( u ) x, j) + λ i { f(,( u) x, i) f(, x, i)} =, wih ermial codiio () f (, x, i) = x θ. () We look for a soluio of eqs () ad () of he form f (, x, i) = x θ g(, i). (3) I is easy o see ha f give i eq. (3) is a soluio i C, ((, ) R S) o eqs () ad () iff g i C ((, ) S) is a soluio o dg (, i ) + if θ ( r u ˆ ( b r ) u ( b () i r )) g (, i ) d + + u A θθ ( + ) + ( u ˆ σ + u σ( i) + uu ˆ σ σ( i)) g(, i) + λi(( u) ) g(, i) + πij( u) g(, j) =. (4) i =,, 3,..., M wih ermial codiio. g(, i) =. (5) Now we rewrie eq. (4) i he vecor form. Le g () = (g(, ), g(, ),, g(, M)), I = I M M, b = diag(b() r, b() r,, b(m) r), λ = diag(λ, u u u λ,,λ M ) ad σu = diag( σ, σ,..., σm), where σ u i = u ˆ σ + u σ ( i ) + u u ˆ σ σ ( i ). he eq. (4) reduces o he form d g ˆ θθ ( + ) () + if θ{ ub [ r u ( b r )]} Ig () σu g () d u A λ + (( u ) ) g( ) + ( u ) Π g( ) =. his implies (6) d g () + if B( u) g() =, (7) d u A wih ermial codiio M imes g ( ) = (,,...,), (8) where B(u) := (b ij (u)) M M, bij ( u ) = ˆ θθ ( + ) u θ( r + u( b r) + u( b( i) r)) + σi λi, if i = j, λi( u) pij, if i j. heorem. he value fucio defied by eq. () is he uique posiive C, ((, ) R S) soluio of he eqs () ad (). Moreover, if u*(, i) miimizig selecor i eq. (4), he u*(, Y ( )) is a opimal corol. Proof. Cosider he fucio M F( y) = if B( u) y, y R +. u A he F( ) is Lipschiz coiuous because A is compac. Hece eqs (7) ad (8) have a uique soluio i C ((, ) S). herefore, i follows ha he eqs () ad () have a uique soluio i C, ((, ) R S). Le u( ) be a admissible corol ad X( ) be he correspodig wealh process eq. (3). Applyig Io s formula (see ref. 9, heorem 33, pp. 8 8) o f (, X(), Y()), we ge f(, X( ), Y( )) f(, X(), Y()) = f (, X( ), Y( ))d + f (, X( ), Y( ))d X () x c CURREN SCIENCE, VOL. 3, NO. 6, 5 SEPEMBER 653

5 SPECIAL SECION: MAHEMAICAL FINANCE 654 c + fxx (, X( ), Y( ))d[ X, X]() + { f (, X ( ), Y ( )) f (, X ( ), Y ( ))}, (9) < where [X, X] c () deoes he coiuous par of [X, X]() ad X c () deoes he coiuous par of X(). akig expecaio we ge E{ f(, X( ), Y( )) f(, X(), Y())} = E [ f (, X( ), Y( )) + X( ) f (, X( ), Y( )) { r+ u ( )( bˆ r) + u ( )( b( Y( )) r)} + X ( ) fxx (, X( ), Y( )) ˆ σ σ ( u ( ) + u ( ) ( Y( )) + u ( ) u ( ) σ σ ( Y( )))]d ˆ + E { f(,( u ( )) X( ), Y( ) + h( Y( ), z)) R f (, ( u ( )) X( ), Y( ))} m(d z)d + E { f(,( u ( )) X( ), Y( )) f (, X( ), Y( ))} λ( Y( ))d. () akig codiioal expecaio codiio o X() = x, Y() = i we ge f (, x, i) E u( ) [(X()) θ X() = x, Y() = i], which is rue for all u( ). his implies ha f (, xi, ) if E [( X ( )) X() = xy, () = i], x () Now, we are goig o prove ha if u*(, i) is he miimizig selecor i eq. (4), he u*(, Y ( )) is a opimal corol. Observe ha û () = u*(, Y( )) is a prescribed corol. Le g*(, ) be he uique soluio of eqs (4) ad (5) ad u*(, ) = (u* (, ), u* (, )) is a miimizig selecor i eq. (4), he g*(, ) is he uique soluio o d g (, i ) θ ( r + u * * (, i )( b ˆ r ) + u (, i )( b () i r )) g (, i ) d θθ ( + ) + (( u* (, i)) + ( u* (, i)) ( i) ˆ σ σ + u* (, i) u* (,) i ˆ σ σ()) i g(, i) + λ (( *(, )) ) (, ) i u i g i * πij ( u (, i)) g(, j), () + = wih ermial codiio g(, i) =. (3) Now we cosider he equaio (, x, i) + x( r+ u * (, i)( b ˆ r) + u*(, i)( b( i) r)) (, x, i) + * * x u i + u i i (( (, )) ˆ σ ( (, )) σ ( ) * * f + u (, i) u(, i) ˆ σ σ()) i (, x, i) + π f (,( u (, i)) x, j) i, j * + λ { (,( * i f u (, i)) x, i) f(, x, i)} =, (4) wih ermial codiio f (, x, i) = x θ. (5) Le f ˆ(, x, i ) be he uique soluio of eqs (4) ad (5). he, i is easy o check ha ˆ( f, x, i) = x g*(, i), where g*(, i) is he uique soluio of eqs () ad (3). We already kow ha f (, x, i) = x θ g*(, i), where f (, x, i) is he uique soluio of eqs () ad (). Hece fˆ = f. (6) Usig Io s formula ad eq. (4), we have fˆ (, x, i) = I (, x, i, uˆ ( )), (7) θ, where u ˆ( ) = u*(, Y ( )),. From eqs (), (6) ad (7), we ge i.e. if I (, x, i, u( )) I (, x, i, uˆ ( )), θ, θ, if I (, x, i, u( )) = I (, x, i, uˆ ( )). θ, θ, Hece, uˆ() give by u ˆ( ) = u*(, Y ( )), is a opimal corol. CURREN SCIENCE, VOL. 3, NO. 6, 5 SEPEMBER

6 SPECIAL SECION: MAHEMAICAL FINANCE Bechmarked asse maageme I his secio, we prove he exisece of opimal sraegy for he bechmarked problem described earlier. I suffices o cosider he followig payoff crierio give by Le ad E[exp( θf()) F() = z, Y() = i]. (8) I θ, (, z, i, u( )) := E[exp( θf()) F() = z, Y() = i], ψ (, z, i): = if (,,, ()). I θ, z i u (9) he HJB equaio correspodig o he value fucio ψ is give by ψ (, z, i) + if ( r+ u ˆ ( b r)d + u( b() i r)) u A ψ u ˆ σ + uσ( i) α γ (, z, i) z ˆ σ σ γ ( ) (,, ) u u i ψ + + z z i + πψ(, z+ l( u ), j) ij + λi{ ψ(, z+ l( u), i) ψ(, z, i)} =, wih ermial codiio (3) ψ(, z, i) = exp( zθ). (3) We look for a soluio for eqs (3) ad (3) of he form ψ(, z, i) = exp( zθ)ϕ(, i). (3) he subsiuig eq. (3) i eq. (3), we obai ϕ (, i) + if θ ( r+ u ˆ ( b r)d + u( b() i r)) u A uˆ σ + uσ( i) α γ ϕ(, i) θ + uˆ σ + uσ( i) γ ϕ(, i) + λi(( u) ) ϕ(, i) + πij( u) ϕ(, j) =, (33) i =,, 3,, M wih ermial codiio ϕ (, i) =. (34) As i he proof of heorem, we ca show he followig. heorem 3. he value fucio defied by eq. (9) is he uique posiive C, ([, ] R) soluio of eqs (3) ad (3). Moreover, if u*(, i) is he miimizig selecor i eq. (33), he u*(, Y ( )) is a opimal corol. Numerical compuaio of opimal sraegy I his secio we give a umerical mehod for compuig opimal expeced ermial uiliy fucio ad correspodig opimal corol for fiie horizo problem described earlier. We have already see ha he sysem of ODEs (4) ad (5) has a uique soluio. From he eqs (7) ad (8) i is clear ha we ca compue value fucio ad opimal sraegy by usig umerical mehod for ODE. Here we use he Euler mehod. Defie fˆ (, gˆ()) = if B( u) gˆ(). (35) u A he we wa o solve he sysem of ODEs give by dgˆ () = f ˆ(, g ˆ()). (36) d Le Δ be he sep size. A each sep, wih gˆ( Δ ) available from he previous sep, he explici compuaioal scheme is give by gˆ(( + ) Δ ) = gˆ( Δ ) +Δ fˆ ( Δ, gˆ( Δ )), wih iiial codiio M imes g ˆ() = (,,...,). A each sep, whe we compue f ˆ ( Δ, gˆ ( Δ )), we have o firs fid he u* for which miimum will aai for eq. (35) ad ha is he opimal sraegy. o illusrae he resuls we ex cosider a example of Markov modulaed marke wih hree regimes for he ime ierval [, ]. We ake mauriy ime = moh. he ime sep legh Δ is ake as. moh. he sae CURREN SCIENCE, VOL. 3, NO. 6, 5 SEPEMBER 655

7 SPECIAL SECION: MAHEMAICAL FINANCE space is S = {,, 3}. he drif coefficie ad volailiy rae a each regime are chose as follows: (.65,.4), if i =, ( bi ( ), σ ( i)) = (.5,.35), if i=, (.4,.3), if i = 3. he rasiio probabiliy marix is assumed o be give by ( p ij ) =. he drif coefficie ad volailiy rae of he sock are b ˆ =.3 ad ˆ σ =. respecively, bak ieres rae r =.5, risk aversio parameer θ = ad defaul iesiy (λ, λ, λ 3 ) = (,.,.). For his Markov modulaed marke we compue he value fucio ad opimal sraegy usig he umerical echique described above. Figure describes he effec of differe values of defaul iesiy i he opimal porfolio sraegy. I is clear from he figure ha a ivesor ivess a small porio of his wealh io ha defaulable asse which has large defaul iesiy ad ivess a very large porio his wealh io sock. For very small defaul iesiy of he defaulable asse, he ivesor ivess a sigifica amou of his wealh io defaulable asse. ables ad describe he value of opimal sraegy agais ime wih differe defaul iesiies. ad. respecively. Here, ad deoe he fracio of wealh ivesed i he sock ad he defaulable asse respecively. Sice he sep size Δ =. is very small, acual daa give he values of ad for, ime-pois. Due o lack of space, we oly display ad a some specific ime-pois. Figure. Opimal porfolio sraegy fucios:, Wealh porio ivesed io sock., Wealh porio ivesed io defaulable asse. able. Opimal porfolio sraegy for λ =. ime able. Opimal porfolio sraegy for λ =. ime Bielecki,. ad Jag, I., Porfolio opimizaio wih a defaulable securiy. Asia-Pac. Fiac. Markes, 6, 3, Kor, R. ad Kraf, H., Opimal porfolios wih defaulable securiies a firm value approach. I. J. heor. Appl. Fiace, 3, 6, Sircar, R. ad Zariphopoulou,., Uiliy valuaio of credi derivaives ad applicaio o CDOs. Qua. Fiace,,, Davis, M. H. A. ad Lleo, S., Jump-diffusio risk-sesiive asse maageme I: diffusio facor model. SIAM J. Fiac. Mah.,,, Davis, M. H. A. ad Lleo, S., Jump-diffusio risk-sesiive asse maageme II: jump-diffusio facor model,, arxiv:.56v. 6. Las, G. ad Brad, A., Marked Poi Processes o he Real Lie: he Dyamical Approach (Probabiliy ad is Applicaios), Spriger, Berli, Framsad, N. C., Oksedal, B. ad Sulem, A., Sufficie sochasic maximum priciple for he opimal corol of jump diffusios ad applicaios o fiace. J. Opim. heory Appl., 4,, Oksedal, B. ad Sulem, A., Applied Sochasic Corol of Jump Diffusios, Uiversiex, Spriger, Berli, Proer, P., Sochasic Iegraio ad Differeial Equaios, Spriger, Berli, 99.. Davis, M. H. A. ad Lleo, S., Risk-sesiive bechmarked asse maageme. Qua. Fiace, 8, 8, ACKNOWLEDGEMENS. We hak he referees for heir useful commes, which helped improve he mauscrip. C.P. ackowledges fiacial suppor from he Coucil of Scieific ad Idusrial Research, New Delhi. 656 CURREN SCIENCE, VOL. 3, NO. 6, 5 SEPEMBER