A simple SSD-efficiency test

Transcription

1 A simple SSD-efficiecy es Bogda Grechuk Deparme of Mahemaics, Uiversiy of Leiceser, UK Absrac A liear programmig SSD-efficiecy es capable of ideifyig a domiaig porfolio is proposed. I has T + variables ad a mos 2T +1 cosrais, whereas he exisig SSD-efficiecy ess are eiher uable o ideify a domiaig porfolio, or require solvig a liear program wih a leas OT 2 + variables ad/or cosrais. Key Words: sochasic domiace, porfolio aalysis, liear programmig. 1 Iroducio The cocep of secod-order sochasic domiace SSD, iroduced i ecoomics by Hadar ad Russell 1969 ad Rohschild ad Sigliz 197, has become oe of he ceral coceps i risk modellig. We say ha porfolio rae of reur X, modelled as a radom variable r.v. o some probabiliy space, domiaes r.v. Y by SSD, ad wrie X 2 Y, if X is preferred o Y for ay risk-averse expeced uiliy maximizer ha is, for ay age wih icreasig ad cocave uiliy fucio, see vo Neuma ad Morgeser Thus, he oio of SSD allows oe o compare some of he ivesme opporuiies wihou kowig exac uiliy fucio of a paricular age. This is paricularly impora, because ideifyig a uiliy fucio is a difficul ask, which resors o a exesive quesioaire procedure. Give a covex se V of admissible porfolio rae of reurs, r.v. Y V is called SSD-domiaed wihi V, if X 2 Y for some X V i his case, X will be called a domiaig porfolio, ad SSD-efficie oherwise. The SSD defiiio implies ha a opimal ivesme for a risk-averse expeced-uiliy maximizer belogs o he se of SSD-efficie porfolios. However, his fac goes beyod he expeced uiliy heory: oly SSDefficie porfolios may be opimal for a age who maximizes covex Yaari dual uiliy fucio see Yaari 1987, Theorem 2, miimizes a law-ivaria covex risk measure see Föllmer ad Schied 24, Corollary 4.59, or uses he mea-deviaio model Grechuk e al SSD-efficiecy is also ceral o solvig he iverse porfolio problem for ideifyig ivesor s risk prefereces, see Grechuk ad Zabaraki 213. This moivaes he followig quesio: deermie wheher he give porfolio Y is SSD-efficie wihi a give se V, ad if o, fid a domiaig porfolio. Assumig ha he uderlig probabiliy space is a fiie T -eleme se Ω = {ω 1,...,ω T }, ad V cosiss of liear combiaios of raes of reur of asses, Pos 23 developed a liear program wih OT + variables ad cosrais, which ess wheher a give Y V is SSD-efficie, subjec o he addiioal assumpio Y ω i Y ω j, i j, 1 i.e. ha ies do o occur i he disribuio of Y. This assumpio holds wih probabiliy 1 if he disribuio of Y is a approximaio of a coiuous oe usig T Moe-Carlo simulaios. However, assumpio 1 is rarely ecouered i pracical applicaios. As explaied i Pos 23, Secio II-C, ied reurs may occur, for example, whe aalysig boosrap pseudo-samples or evaluaig a riskless aleraive. Eve if 1 holds 1

2 for base asses, i migh fail for some mixures of hem, or for derivaive securiies. Kopa ad Pos 211 show how he aalysis ca be geeralized usig a weakly icreasig rakig o accou for ies 1. Based o his es, Pos 23, Secio IV made a surprisig coclusio ha some of he popular fiacial idices are SSD-domiaed ad hece cao be opimal ivesmes for a risk-averse age. Pos s es, however, fails o ideify a domiaig porfolio X if Y is SSD-domiaed. I oher words, for a age holdig a porfolio wih rae of reur Y, his es may show he exisece of beer ivesme opporuiies wihi se V, bu does o ideify hem. Assumig 1, Pos 28 shows ha a give porfolio is domiaed by is mixure wih he dual soluio porfolio of Pos 23, provided ha he mixure lies i he local eighborhood wih he same sricly icreasig rakig as he evaluaed porfolio. For a geeral case, several SSD-efficiecy liear programmig ess capable o ideify a domiaig porfolio have bee developed, see e.g. Kuosmae 24, Kopa ad Chovaec 28, Kopa ad Pos 211. Moreover, mehods developed by Decheva ad Ruszczyński 23, 26 ad Kopa ad Chovaec 28 ca be used o fid a opimal domiaig porfolio uder various defiiios of opimaliy. However, all hose ess use OT 2 + variables ad cosrais, which ca be compuaioally iese, because he ypical values of T are above 1 2 or eve 1 3. Fabia, Mira, ad Roma 211 iroduced echique for solvig opimizaio problems wih SSD cosrais which uses O variables, bu wih cus from a expoeial umber of iequaliies added algorihmically. I coras, Luedke 28 suggesed a es wih OT + cosrais bu OT 2 + variables. The exisece of a liear programmig SSD-efficiecy es wih OT + variables ad cosrais, ad o be capable of fidig a domiaig porfolio, was a ope quesio. Such a es is he mai resul of his work. We prese a liear programmig es, wih T + variables ad 2T + 1 cosrais, which, give ay porfolio reur Y V, possibly wih ies, ess wheher Y is SSD-efficie wihi V, ad if o, fids a domiaig porfolio X. A possible limiaio is ha our soluio porfolio X may iself be iefficie, ha is, domiaed by a hird porfolio Z. If his limiaio is of cocer, a addiioal es may be eeded, e.g., he full Kopa ad Pos 211 es. 2 The SSD-efficiecy Tes Le Ω = {ω 1,...,ω T } be a fiie probabiliy space, wih probabiliy measure P such ha P[ω i ] = p i, i = 1,...,T. A radom variable r.v. is ay fucio X : Ω R. F X x = P[X x] ad q X α = if{x F X x > α} will deoe he cumulaive disribuio fucio CDF ad quaile fucio of a r.v. X, respecively. We say, ha r.v. X domiaes r.v. Y by SSD, ad wrie X 2 Y, if EuX EuY for every icreasig cocave fucio u : R R, wih iequaliy beig sric for some u. Equivalely, X 2 Y if ad oly if q X βdβ q Y βdβ, α,1], 2 wih iequaliy beig sric for some α,1], see Theorem 2.58 i Föllmer ad Schied 24. Le V be a covex se of r.v.s, ad Y V be fixed. This paper preses a es which deermies if Y is SSD-efficie wihi V, ad if o, fids a domiaig porfolio. Because Y is fixed ad give, we ca assume wihou loss i geeraliy ha Y ω i Y ω j, i < j. 3 Give ay r.v. U, we deoe s1,...,st a permuaio of se 1,...,T such ha Y ω si Yω s j, i < j, ad Uω si Uω s j wheever Y ω si = Y ω s j, i < j. 1 A mehod for reame of ies has bee oulied already i Pos 23, Secio II-C. However, o SSD-es for he case wih ies has bee explicily formulaed i ha paper. 2

3 Proposiio 1 Y is SSD-domiaed if ad oly if here exiss a o-zero r.v. U V Y such ha u sip si { }] y, = 1,...,T. Moreover, i his case X =Y +εu is a domiaig porfolio for ay ε,mi 1,mi si+1 y si i J u si u si+1 where J {1,...,T } is he se of idices 2 for which y si+1 y si > ad u si u si+1 >. Proof If Y is SSD-domiaed, X 2 Y for some X V. Take U = X Y. The Xω sip si q Xβdβ q Y βdβ = y sip si, where y i =Yω i, i = 1,...,T, ad α = p si, = 1,...,T. Hece u sip si, = 1,...,T. Coversely, le U ad ε be as described. The Xω s1 Xω st, where X = Y + εu. Ideed, Xω si Xω si+1 is equivale o εu si u si+1 y si+1 y si, which holds due o defiiio of ε if i J, ad due o εu si u si+1 y si+1 y si if i J. Thus, q X βdβ = Xω si p si y si p si = q Y βdβ, = 1,...,T 4 Because fucios f X α = q Xβdβ ad f Y α = q Y βdβ are piecewise liear wih verices a α = α, = 1,...,T, 4 implies 2. Le = be he smalles idex such ha u s. The u si p si = u s p s, hece u s >, ad sric iequaliy holds i 4 for =. Thus, X 2 Y. Fially, because r.v.s Y ad Y +U belog o V, ad ε,1], X = Y + εu V due o covexiy of V. Proposiio 1 cao be applied direcly for cosrucig a liear programmig SSD-efficiecy es, because permuaio s depeds o U ad hece ukow i advace. Le I k {1,...,T }, k = 1,...,l be he ses of idices of cardialiy a leas 2 such ha Y ω i = Y ω j if ad oly if i, j I k for some k. Le also J k = {i {1,...,T } i < j, j I k }, k = 1,...,l, ad I = {i {1,...,T 1} Y ω i < Y ω i+1 } {T }. The {1,...,T } = I I 1 I l. Proposiio 2 For a r.v. U, he followig saemes are equivale a u sip si, = 1,...,T ; b u ip i, I ad i Jk u i p i + i Ik u i p i, k = 1,...,l, where x = mix,. Proof a b: Because u si p si = u i p i, I 5 a implies firs saeme of b. For k {1,...,l}, le be he larges idex I k wih u s, ad le be he larges idex i J k if u s >, I k. The u si p si = u i p i +,s 1 i s 1 u i p i = u i p i + u i p i ad b follows. b a: If I, a follows from he firs saeme of b ad 5. If I k for some k {1,...,l}, u si p si = u i p i +,s 1 i s 1 u i p i u i p i +,s 1 i s 1 u i p i u i p i + u i p i. 2 The se of such idices may be a empy se. Throughou he paper, we will use he coveio ha he miimum over a empy se is equal o +. 3

4 I follows from Proposiios 1 ad 2 ha he program l max u i p i + u i p i + u i p i, I s.. k=1 u i p i, I, u i p i + u i p i, k = 1,...,l U = u 1,...,u T V Y 6 has a posiive opimal objecive { value if ad oly if Y is SSD-domiaed, ad i his case a domiaig porfolio y is X = Y +εu wih ε = mi 1,mi si+1 y si i J as i Proposiio 1. The program 6 is o liear because of u si u si+1 } he presece of u i bu ca be liearised i a sadard way by iroducig variables v i ogeher wih cosrais v i u i ad v i. Le { } V = X X = r j x j, x j = 1, x j, j = 1,...,, 7 where r 1,..., r are he raes of reur of asses, x j is he fracio of capial ivesed io asse j, x j = 1 is he budge cosrai, ad x j, j = 1,..., are opioal o-shor-sellig cosrais. Le r i j = r j ω i, i = 1,...,T, j = 1,..., be he reur of asse j uder sceario ω i. The codiio U V Y i 6 becomes u i = r i jx j y i, i = 1,...,T. Because Y V, y i = r i jx j i = 1,...,T, for some x 1,...,x, ad he codiio U V Y becomes u i = r i jx j x j, i = 1,...,T. Hece, for V give by 7, program 6 ca be wrie as max x j,u i,v i I s.. u i = p i u i + l k=1 p i u i + v i p i p i u i, I, p i u i + v i p i, k = 1,...,l r i j x j x j, i = 1,...,T, v i, v i u i, i I 1 I l, x j = 1, x j, j = 1,...,, 8 or, afer excludig u i, max x j,v i I s.. p i r i j x j x j p i + l k=1 p i r i j x j x j, I, p i v i, v i r i j x j x j, i I 1 I l, x j = 1, x j, j = 1,..., r i j x j x j + v i p i, r i j x j x j + v i p i, k = 1,...,l 9 The resulig liear program has T + variables ad a mos 2T + 1 cosrais. 4

5 Example 1 Assume ha here are T = 3 equiprobable scearios, = 2 asses wih reurs r 1 =.24,,.6 ad r 2 =.4,.12,.12, ad he bechmark porfolio has weighs,1. I his case, Y =.4,.12,.12, ad codiio 1 does o hold. I he oaio iroduced before Proposiio 2, l = 1, I 1 = {2,3}, J 1 = {1}, I = {1,3}, ad he liear program 9 akes he form max x 1,x 2,v 2,v x 1 +.4x x x x v v 3, s x 1 +.4x 2 1, x 1 +.4x x x x 2 1, x 1 +.4x v v 3, v 2.12x 2 1, v 3.6x x 2 1, v 2, v 3, x 1 + x 2 = 1, x 1, x 2, 1 which simplifies o max.26x x x 1,x 2,v 2,v 3 3 v v 3, s.. 6x 1 + x 2 1, 15x x 2 1,.24x 1 +.4x v 2 + v 3, v 2.12x 2 1, v 3.6x x 2 1, v 2, v 3, x 1 + x 2 = 1, x 1, x The opimal soluio x 1 = 1, x 2 =, v 2 =.12, v 3 =.6, wih he correspodig objecive value.26x x v v 3 =.8 >, { hece he } bechmark porfolio is o SSD-efficie. Nex, U =.2,.12,.6, si = i, i = 1,2,3, J = {1}, ε = mi 1, y 2 y 1 u 1 u 2 =.25, ad a domiaig porfolio X = Y +.25U has weighs, , 1 =.25,.75. Example 1 is a sligh modificaio of Example 4 i Kopa ad Pos 211, which illusraes ha heir dual es reurs he same domiaig porfolio ad has 14 variables ad 1 cosrais. I coras, es 11 has jus 4 variables ad 6 cosrais. Kopa ad Pos 211 compared he size of differe SSD-efficiecy ess i he case T = 48, = 12. Table 1 preses heir daa for Pos 23 dual es, Kuosmae 24 es, Kopa ad Chovaec 28 es, Kopa ad Pos 211 dual es, ad Kopa ad Pos 211 reduced dual es, ogeher wih he correspodig daa for Pos 28 es, Luedke 28 es ad our proposed es 3. I shows ha he proposed es has subsaially smaller size ha he exisig ess which allow ies ad are able o ideify a domiaig porfolio. 3 The colums AT, DP, ad EP idicae wheher he es allows ies i he reur disribuio, wheher he resuls of he es ca be used o ideify a domiaig porfolio, ad wheher he soluio porfolio is SSD-efficie, correspodigly. 5

6 Table 1: SSD-ess compariso Tes SizeCosrais Variables T = 48, = 12 AT DP EP Pos 23 dual es, Eq. 12 T + 1 T No No No Kuosmae 24 es, Th. 6 T 2 + T + 1 3T No Yes Yes Kopa ad Chovaec 28, Eq. 16 T 2 + T + 1 T 2 + 2T No Yes Yes Pos 28 es, Eq. 5 T + 1 T No Yes No Kopa ad Pos 211 reduced es T + 1 T Yes No No Kopa ad Pos 211 full es T T 2 + T Yes Yes Yes Luedke 28 es, Eq. cssd1 3T T Yes Yes Yes Proposed es 2T + 1 T Yes Yes No 3 Coclusios ad Fuure Research We have cosruced a liear program 8-9 wih OT + variables ad cosrais, such ha is objecive value is sricly posiive if ad oly if he evaluaed porfolio Y is SSD-domiaed wihi admissible se V give by 7. If Y is SSD-domiaed, he oupu of he program ca be used o cosruc a domiaig porfolio. The suggesed SSD es is releva for porfolio maageme: a domiaig porfolio may be suggesed as a aleraive for a ivesor who is currely holdig he bechmark porfolio Y. Oe may argue ha a domiaig porfolio may i geeral be SSD-iefficie, ad, eve if efficie, i is geerally o opimal for he ivesor who holds he porfolio Y. Ideed, if he exac uiliy fucio of he ivesor is kow, he i may be opimised o fid he opimal porfolio wih respec o his/her prefereces. However, i pracice, a ivesor rarely kows his/her uiliy fucio. I his case, deermiig a opimal porfolio is impossible, ad a risk-averse ivesor may be advised o buy a domiaig porfolio, which, i geeral, is o opimal, bu ayway is beer ha he porfolio he/she currely holds, o maer wha his/her uiliy fucio is. A obvious quesio for fuure research is wheher here exiss a liear programmig SSD-efficiecy es wih OT + variables ad cosrais reurig a domiaig porfolio which is i addiio SSD-efficie. Aoher ieresig research direcio would be geeralisig he resuls of his paper o higher order sochasic domiace. We say, ha r.v. X domiaes r.v. Y by N-h order sochasic domiace, or NSD, ad wrie X N Y, if EuX EuY for every fucio u U N, wih iequaliy beig sric for some u, where U N is he se of N imes differeiable fucios u : R R such ha 1 1 u x, x R, = 1,...,N. A r.v. Y V is called NSD-efficie wihi se V, if here are o X V such ha X N Y. I would be ieresig o obai a liear programmig NSD-efficiecy es wih abiliy o ideify a domiaig porfolio. However, his is o eirely sraighforward. Our SSD-efficiecy es relies o he quaile characerizaio 2 of SSD. Theorem 4 i Levy 1992 claims wihou proof ha a similar represeaio holds a leas for N = 3, amely, X 3 Y if ad oly if β q X γdγ dβ β q Y γdγ dβ, α,1]. 12 However, Ng 2 provides a couerexample o his saeme. To he bes of our kowledge, o coveie represeaio of NSD i erms of quaile fucios is kow for N 3. Recely, Pos ad Kopa 213 derived a represeaio of he NSD crierio i erms of piecewise polyomials ad co-lower parial momes, ad used i o develop a efficie liear programmig NSD-efficiecy es for ay N. However, heir es cao ideify a domiaig porfolio. This issue calls for ew ideas. 6

7 Refereces [1] Decheva D., Ruszczyński A.: Opimizaio wih sochasic domiace cosrais. SIAM Joural of Opimizaio, , [2] Decheva D., Ruszczyński A.: Porfolio opimizaio wih sochasic domiace cosrais. Joural of Bakig ad Fiace, 32 26, [3] Fabia C.I., Mira G., ad Roma D.: Secod-order sochasic domiace models usig cuig-plae represeaios. Mahemaical Programmig, , [4] Föllmer, H., Schied, A.: Sochasic fiace, 2d ed.. Berli New York: de Gruyer 24. [5] Grechuk B., Molyboha A., Zabaraki M., Mea-Deviaio Aalysis i he Theory of Choice, Risk Aalysis: A Ieraioal Joural, , [6] Grechuk B., Zabaraki, M., Iverse Porfolio Problem wih Mea-Deviaio Model. Europea Joural of Operaioal Research 213. Acceped. [7] Hadar, J., Russell, W.: Rules for Orderig Ucerai Prospecs, America Ecoomic Review, , [8] Kopa M., Chovaec P.: A secod-order sochasic domiace porfolio efficiecy measure. Kybereika, 44, 2 28, [9] Kopa M., Pos T.: A Geeral Tes for Porfolio Efficiecy. Workig paper 211. hp://papers.ssr. com/sol3/papers.cfm?absrac_id= [1] Kuosmae T.: Efficie diversificaio accordig o sochasic domiace crieria. Maageme Sciece, 5, 1 24, [11] Levy H.: Sochasic domiace ad expeced uiliy: Survey ad aalysis. Maageme Sciece 38, , [12] Luedke J. New formulaios for opimizaio uder sochasic domiace cosrais. SIAM Joural o Opimizaio, 19 28, [13] vo Neuma J., Morgeser O.: Theory of Games ad Ecoomic Behavior, 3rd ed., Priceo, NJ: Priceo Uiversiy Press, [14] Ng, M.C.: A Remark o Third Degree Sochasic Domiace. Maageme Sciece 46, 6 2, [15] Pos T.: Empirical ess for sochasic domiace efficiecy. Joural of Fiace, 58 23, [16] Pos T.: O he dual es for SSD efficiecy wih a applicaio o momeum ivesme sraegies. Europea Joural of Operaio Research, , [17] Pos T., Kopa M.: Geeral liear formulaios of sochasic domiace crieria. Europea Joural of Operaioal Research, , [18] Rohschild M., Sigliz J.: Icreasig risk I: A defiiio. Joural of Ecoomic Theory, , [19] Yaari M.E. The dual heory of choice uder risk. Ecoomerica, ,