On convexity of SD efficiency sets - no short sales case

4. mezinárodní konference Řízení a modelování finančních rizik Osrava VŠB-U Osrava Ekonomická fakula kaedra Financí.-. září 008 On conveiy of SD efficiency ses - no shor sales case Miloš Kopa Absrac his paper deals wih a se of efficien porfolios wih respec o sochasic dominance (SD) crieria. If only quadraic uiliy funcions are considered as a generaor of sochasic dominance relaion he SD porfolio efficiency coincide wih Markowiz porfolio efficiency. We consider scenario approach for reurns of considered asses and we assume no shor sales. On a numerical eample using SD admissibiliy rules based on crossing cumulaive probabiliy disribuion funcions crieria we show ha he SD porfolio efficiency se is no conve for any finie order of SD. o prove he same propery for he infinie order of SD we employ SD opimaliy approach. Keywords Sochasic dominance porfolio efficiency se conveiy Inroducion Eising approaches o esing for he efficiency of a given porfolio make srong parameric assumpions abou invesor preferences and reurn disribuions. he ypical porfolio efficiency classificaion is based on well-known mean-variance crieria (see [6] or [7]). Sochasic dominance based procedures promise a useful non-parameric alernaive especially in he case when no informaion abou risk aiude of he decision maker is known. Assuming only he nonsaiaion of decision maker or increasing uiliy funcion one can use firs-order sochasic dominance (FSD) porfolio efficiency ess inroduced in [4] and []. In all hese ess he scenario approach for choice alernaive reurns is supposed. For risk averse decision makers concave uiliy funcions and second-order sochasic dominance (SSD) porfolio efficiency ess are used. See [9] [4] or [3] for more deails. For an N -h order of sochasic dominance relaion a se of uiliy funcion wih alernae sign of heir derivaives up o N is used (see [5] and references herein). As a sraighforward generalizaion one can employ he se of infiniely differeniable uiliy funcion wih alernae sign of heir derivaives. hese funcions are also called compleely monoonic uiliy funcions and hey form a generaor of infinie-order sochasic dominance (ISD) relaion (see [3]). In general any generaor of sochasic dominance can be considered. If only quadraic uiliy funcions are considered as a generaor of sochasic dominance relaion he SD porfolio efficiency coincide wih Markowiz porfolio efficiency. When esing SD porfolio efficiency one mus disinguish beween efficiency crieria based on admissibiliy" and opimaliy". here is a suble difference beween hese wo conceps. A choice alernaive is SD admissible if and only if no oher alernaive gives higher epeced uiliy for all uiliy funcion from he considered generaor of SD. A choice alernaive is SD opimal if and only if i is he opimal choice for a leas some uiliy RNDr. Ing. Miloš Kopa Ph.D. Charles Universiy Prague Faculy of Mahemaics and Physics Deparmen of Probabiliy and Mahemaical Saisics Sokolovská 83 86 75 Prague Czech Republic. e-mail: kopa@karlin.mff.cuni.cz

4. mezinárodní konference Řízení a modelování finančních rizik Osrava VŠB-U Osrava Ekonomická fakula kaedra Financí.-. září 008 funcion from he considered generaor of SD. For firs-order sochasic dominance FSD admissibiliy is a necessary bu no sufficien condiion for FSD opimaliy. In oher words a choice alernaive may be FSD admissible even if i is no opimal soluion of maimizing epeced uiliy problem for any increasing uiliy funcion. For higher orders of sochasic dominance relaion applying minima heorem [] hese wo conceps are idenical. When analyzing a se of SD efficien porfolios for any SD generaor he conveiy is he mos desirable propery. I is well-known ha mean-variance efficien porfolios always form a conve se. Unforunaely porfolio efficiency ses for FSD opimaliy FSD admissibiliy SSD efficiency crieria are generally no conve. For FSD relaion i was illusraed in [] for SSD relaion i was shown in e.g. [] or [3]. he aim of his paper is o analyze he conveiy of NSD porfolio efficiency ses for all N >. A numerical eample showing non-conveiy of all hese ses is presening. For any finie N we use SD admissibiliy approach based on crieria in [6]. o prove non-conveiy of ISD porfolio efficiency se we employ SD opimaliy approach. Ecluding shor sales and scenario approach for choice alernaive reurns are he crucial assumpions for hese resuls. he remainder of his e is srucured as follows. Secion inroduces preliminary noaion assumpions and definiions. Ne Secion 3 recalls he SSD porfolio efficiency es derived in [3]. he ess for NSD porfolio efficiency based on SD admissible approach are summarizing in Secion 4. he main resul - eample showing non-conveiy of SD porfolio efficiency ses - is presened in Secion 5. Preliminaries Consider a random vecor r = ( r r... r N ) of reurns of N asses and equiprobable scenarios. he reurns of he asses for he various scenarios are colleced in he reurn mari: X = ( ) wih = ( N) =. Wihou loss of generaliy we can assume ha he columns of X are linearly independen. In addiion o he individual choice alernaives he decision maker may also combine he alernaives ino a porfolio. We will use = (... N ) for a vecor of porfolio weighs and he porfolio possibiliies are given by a simple N Λ = { R = n 0 n = N} which arises as he relevan case if we eclude shor sales and impose a budge resricion. he esed porfolio is denoed by τ = ( τ τ... τ N ). Le U be he se of all uiliy funcions. Le U be he se of uiliy funcions such ha: k ( k) ( ) u 0 UN for all k =... N. As he limiing case U U is he se of infiniely differeniable uiliy funcions wih alernaing signs of he derivaives. hese funcions are also called compleely monoonic uiliy funcions. More deails abou he se of compleely monoonic uiliy funcions can be found in [3] and references herein. Definiion (NSD relaion): Le N is an ineger posiive number. A porfolio Λ dominaes porfolio τ Λ wih respec o N h-order sochasic dominance ( r f NSD rτ ) if and only if Eu( r ) Eu( r τ ) 0 for every uiliy funcion u UN U wih a leas one sric inequaliy. he mos favorie is firs-order (FSD) and second-order (SSD) sochasic dominance relaion. A porfolio Λ dominaes porfolio τ Λ by FSD if here eiss no oher beer

4. mezinárodní konference Řízení a modelování finančních rizik Osrava VŠB-U Osrava Ekonomická fakula kaedra Financí.-. září 008 porfolio (wih higher epeced uiliy of reurns) for all decision makers. Similarly he SSD relaion holds if here eiss no beer porfolio for all risk averse individuals. Definiion (ISD relaion): A porfolio Λ dominaes porfolio τ Λ wih respec o infinie-order sochasic dominance ( r f ISD rτ ) if and only if Eu( r ) Eu( r τ ) 0 for every uiliy funcion u U U wih a leas one sric inequaliy. Since U U... U sochasic dominance relaion of N -h order implies sochasic dominance relaion of M -h order for all M = N + N +... hese wo definiions deals wih he pairwise comparison of wo porfolios. Following [9] [4] [] [3] we may generalize hese definiions o he analysis of a given porfolio relaive o all feasible porfolios - SD porfolio efficiency. When analyzing SD porfolio efficiency one mus disinguish beween efficiency crieria based on admissibiliy" and opimaliy". here is a suble difference beween hese wo conceps for FSD efficiency. For higher orders of sochasic dominance relaion applying minima heorem [] hese wo conceps are idenical. Definiion 3 (N(I)SD Porfolio admissibiliy): A given porfolio τ Λ is N(I)SD efficien (N(I)SD admissible) if and only if no porfolio Λ dominaes τ by N(I)SD. Oherwise porfolio τ is N(I)SD inefficien. Definiion 4 (N(I)SD Porfolio opimaliy): A given porfolio τ Λ is N(I)SD efficien (N(I)SD opimal) if and only if i is an opimal soluion of epeced uiliy maimizing problem for a leas some u U N ( I ) SD. Oherwise porfolio τ is N(I)SD inefficien. As a consequence of U U... U NSD porfolio inefficiency implies MSD porfolio inefficiency for all M > N including M = (ISD porfolio inefficiency). Similarly NSD porfolio efficiency implies MSD porfolio inefficiency for all M < N. o es FSD admissibiliy and FSD opimaliy of a given porfolio one can use algorihms suggesed in [4] and []. he SSD efficiency es based on SSD opimaliy approach can be found in [9] while he SSD efficiency ess using SSD admissibiliy approach were inroduced in [4] and [3]. he las es is discussed in more deails in he ne secion. 3 SSD porfolio efficiency ess Before presening a SSD porfolio efficiency es we recall he basic crieria for pairwise comparisons. We define condiional Value a Risk (CVaR) of a given porfolio τ a level α as he opimal value of he following linear program: CVaRα( τ ) = mina+ w aw ( α) = sw.. Xτ a w 0.

4. mezinárodní konference Řízení a modelování finančních rizik Osrava VŠB-U Osrava Ekonomická fakula kaedra Financí.-. září 008 More deails abou properies of CVaR can be found in [] [8] or [3]. Le [] [] [ ] smalles reurn of porfolio τ i.e. ( Xτ ) ( X τ) ( Xτ). heorem (Crieria for SSD relaion): [] ( Xτ ) i be he i -h Le τ Λ and Λ. he following hree saemens are equivalen: (i) Porfolio Λ dominaes porfolio τ Λ by SSD. [] i [] i (ii) ( X) ( Xτ ) i= for all =... wih sric inequaliy for a i= leas one. (iii) CVaRα( ) CVaRα( τ) α {0 } wih sric inequaliy for a leas one α. Proof: ( i) ( ii) was proved in e.g. [5] and ( i) ( iii) was shown in [3]. Applying hese crieria o SSD porfolio efficiency analysis one can derive he following es inroduced in [3]. heorem (SSD porfolio efficiency es): Le τ Λ and D ( τ ) = ma Dk Dk n bk wk k = s.. CVaR k ( τ ) b ( k k w ) D = k k k = w k bk k = w 0 k = k Dk 0 k = Λ. If D ( τ ) > 0 hen τ is SSD inefficien. Oherwise D ( τ ) = 0 and τ is SSD efficien. 4 NSD crieria for pairwise comparisons For a given porfolio Λ he cumulaive disribuion funcion is: ( ) = ( y) = F y he wice cumulaed disribuion funcion is given by: z () ( ) = ( ) = ( ( )) (( ) z) = = ( z ( X) [] ) [ ]. (( X ) z) = F z F y dy z and he N -imes cumulaive probabiliy disribuions funcion is defined recursively as: ( N) ( N ) F () = F ( ) d. he following heorem proved in [5] and references herein gives a necessary and sufficien condiion for NSD relaion for any finie N.

4. mezinárodní konference Řízení a modelování finančních rizik Osrava VŠB-U Osrava Ekonomická fakula kaedra Financí.-. září 008 heorem 3 (NSD es for pairwise comparison): A porfolio τ Λ dominaes porfolio Λ by N -h-order sochasic dominance (NSD) if and only if (i) τ and (ii) = = ( N) ( N) () τ () F F for all R wih a leas one sric inequaliy. For firs and second order sochasic dominance relaion condiion (i) direcly follows from condiion (ii). he following necessary condiion for NSD relaion follows from he definiions of N -imes cumulaive disribuion funcions and condiion in heorem 3(ii). Corollary 4 (Minimal reurn rule): If porfolio τ Λ dominaes porfolio Λ by NSD hen [] [] ( Xτ ) ( X). 5 Numerical eample Consider hree asses wih hree scenarios: 0 0 X = 0 0. 7 5 Using SSD porfolio efficiency es (see heorem ) i is easy o check ha = ( 0 0) and 9 = (0 0) are SSD efficien porfolios. Le τ = ( 0) hen Xτ = ( ) and according o heorem (ii) τ is SSD dominaed by 3 = (0 0 ). Hence porfolio τ is SSD inefficien and herefore he se of SSD efficien porfolios is no conve. Since τ is SSD inefficien i is also NSD inefficien for any finie N and ISD inefficien. Since no shor posiions are allowed porfolio is he only one wih he highes mean reurn. herefore using heorem 3(i) porfolio is NSD efficien for any finie N. he remaining ask o show non-conveiy of NSD porfolio efficiency ses is o prove ha is NSD efficien for any finie N >. For a conradicion assume ha here eiss a porfolio = { 3} Λ ha dominaes by NSD for a leas one finie N. I is easy o see ha 3 because i is fulfilled for all porfolios in Λ. Hence applying Corollary 4 mus be equal o zero. hus for he reurns of we have: X = (0 + 5 3). Since Λ we can rewrie he reurns of in he form: X = (0 5 3 )where0 <. () o find a conradicion we will show ha here eiss no saisfying () and no finie N such ha crierion in heorem 3(ii) is me for =. Firs we compue N -imes cumulaive disribuion funcions of in poin =. Direcly ( N from he definiion of hese funcions we can see ha value ) ( N F () depends on F ) ( ) ( N where (. Moreover F ) ( ) = 0 for all ( 0) and for all N. herefore we ( N compue F ) ( ) only on inerval 0. I is easy o check ha:

4. mezinárodní konference Řízení a modelování finančních rizik Osrava VŠB-U Osrava Ekonomická fakula kaedra Financí.-. září 008 () () F ( ) = F ( ) = 0 ) F () = 3 3 () () F ( ) = ( ) 0 F y dy = 3 (3) () F ( ) = F ( y) dy 0 = 6 M N ( N) ( N ) F ( ) = F ( y) dy 0 = 3( N )! ( N Hence F ) () = / 3( N )! for all N >. Similarly we compue he N -imes cumulaive disribuion funcions for porfolio saisfying (): () () F ( ) = F ( ) = 0 ) F ( ) = 3 3 () () F ( ) = F ( y) dy 0 = 3 () () = F ( y) dy + F ( y) dy ( ) = + 3 3 = + ( ) 3 3 (3) () F ( ) = F ( y) dy 0 = 6 () () = F ( y) dy+ F ( ydy ) = + + ( ) 6 6 6 6 ( = + ) 6 6 M N ( N) ( N ) F ( ) = F ( y) dy 0 = 3( N )! ( N ) ( N ) = F ( y) dy+ F ( y) dy N N y y N = + + ( ) 3( N )! 3( N )! 3( N )! N N N N = + + ( ) dy y dy 3( N )! 3( N )! 3( N )! 3( N )! N N = + ( ) 3( N )! 3( N )! Combining hese wo compuaions we have ( N) N ( N) F () = + ( ) > = F () 3( N )! 3( N )! 3( N )!

4. mezinárodní konference Řízení a modelování finančních rizik Osrava VŠB-U Osrava Ekonomická fakula kaedra Financí.-. září 008 for all 0) and for all N >. Hence no porfolio dominaes by NSD for any N > and herefore is NSD efficien for all N >. Finally consider wo uiliy funcions u( ) u( ) U : u( ) = e u 00 ( ) = e. Solving maimizing epeced uiliy problem: ma Eu( r ). Λ gives as opimal soluion for u = u u = u respecively. Hence using ISD opimal approach are ISD efficien porfolios while NSD inefficiency of τ implies is ISD inefficiency. References [] P. H. DYBVIG AND S. A. ROSS: Porfolio Efficien Ses. Economerica 50 6 (98) 55 546. [] M. KOPA AND. POS: A porfolio opimaliy es based on he firs order sochasic dominance crierion. Forhcoming in Journal of Financial and Quaniaive Analysis. [3] M. KOPA AND P. CHOVANEC: A Second-order Sochasic Dominance Porfolio Efficiency Measure Kyberneika 44 (008) 43-58. [4]. KUOSMANEN: Efficien diversificaion according o sochasic dominance crieria. Managemen Science 50 0 (004) 390 406. [5] H. LEVY: Sochasic dominance: Invesmen decision making under uncerainy. Second ediion. Springer Science New York 006. [6] H. M. MARKOWIZ: Porfolio Selecion. Journal of Finance 7 (95) 77 9. [7] H. M. MARKOWIZ: Porfolio Selecion: Efficien Diversificaion in Invesmens. John Wiley & Sons Inc. New York 959. [8] G. CH. PFLUG: Some remarks on he value a risk and he condiional value-a-risk. In: Probabilisic Consrained Opimizaion: Mehodology and Applicaions (S. Uryasev ed.) Kluwer Academic Publishers Norwell MA 000 pp. 78 87. [9]. POS: Empirical ess for sochasic dominance efficiency. Journal of Finance 58 (003) 905 93. [0] W. R. RUSSELL AND. K. SEO: Represenaive ses for sochasic dominance rules. In: Sudies in he Economics of Uncerainy (. B. Fomby and. K. Seo eds.) Springer Verlag New York 989 pp. 59 76. [] M. SION: On general minima heorems. Pacific Journal of Mahemaics 8 (958) 7 76. [] S. URYASEV AND R.. ROCKAFELLAR: Condiional value-a-risk for general loss disribuions. Journal of Banking & Finance (00) 443 47. [3] G. A. WHIMORE: Sochasic Dominance for he Class of Compleely Monoonic Uiliy Funcions. In: Sudies in he Economics of Uncerainy (. B. Fomby and. K. Seo eds.) Springer Verlag New York 989 pp. 77 88.

4. mezinárodní konference Řízení a modelování finančních rizik Osrava VŠB-U Osrava Ekonomická fakula kaedra Financí.-. září 008 Summary Konveia množín SD eficienných porfólií pri vylúčení predajov nakráko áo práca analyzuje množiny eficienných porfólií vzhľadom ku kriériám sochasickej dominancie (SD). Ak sa zvolí rieda kvadraických úžikových funkcií ako generáor relácie sochasickej dominancie množina SD eficienných porfólií je zhodná s množinou eficienných porfólií Markowizovho modelu. Uvažujeme scenárový prísup pre výnosy akív a nepovoľujeme predaje nakráko. S využiím SD admissibiliy kriérií založených na inegrovaných disribučných funkciách ukazujeme na numerickom príklade že množiny SD eficienných porfólií obecne nie sú konvené pre žiadny konečný rád sochasickej dominancie. K dôkazu nekonvenosi množiny eficienných porfólií pre nekonečný rád SD použijeme prísup SD opimaliy.