POSSIBILISTIC OPTIMIZATION WITH APPLICATION TO PORTFOLIO SELECTION

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume, Number /, pp 88 9 POSSIBILISTIC OPTIMIZATION WITH APPLICATION TO PORTFOLIO SELECTION Costi-Cipria POPESCU, Cristica FULGA Departmet of Mathematics, Academy of Ecoomic Studies, Bucharest, Romaia cipriapopescu@csieasero This paper proposes a type of fuzzy umbers called superior-lr-piecewise (s-lr whose theoretical properties are ivestigated We also propose a chace costrait-type possibilistic programmig model; the coefficiets of the model are s-lr fuzzy umbers ad a Value at Risk (VaR)-type s-lr parameter which cotrols the chaces (i possibilistic terms) of obtaiig usatisfactory value of the obective fuctio The results obtaied have a high degree of geerality, but a immediate applicatio may refer to the portfolio selectio problem for which both modellig ad solvig aspects are discussed Key words: Fuzzy umber; Possibilistic mea value; Liear programmig; Portfolio selectio INTRODUCTION The theory of fuzzy sets, itroduced by Zadeh [7], has bee applied i various fields of sciece Oe of them is Fiacial Mathematics, ad, more precisely, the portfolio selectio issue [6, 8,, 6] Oe of the mai theoretical tool used is the fuzzy mathematical programmig (FMP) [9, 5] I some situatios, the fuzzy approach is used as a alterative to classical probabilistic approach [, 8, 5], i other more complex (multistage) situatios, both methods are used i successive steps [5] I this work, our goal is to create a ew framework for the portfolio optimizatio problem We first propose a special type of fuzzy umbers called superior-lr-piecewise (s-lr) whose theoretical properties are ivestigated By defiig this ew fuzzy cocept, we better capture the two maor aspects i modellig the securities rate of returs i portfolio optimizatio: the statistical data iformatio ad the experts kowledge We also propose a chace costrait-type possibilistic programmig model i which the coefficiets are s-lr fuzzy umbers We use a Value at Risk (VaR )-type s-lr parameter which cotrols the chaces (i possibilistic terms) of obtaiig usatisfactory total retur Sice our approach provides a model that ca be reduced to solvig several liear programmig models, the beefit is ot oly theoretical but also computatioal PRELIMINARY THEORETICAL RESULTS Defiitio Cosider a subset F of the whole set of fuzzy umbers deoted by F A fuzzy umber a is cosidered to belog to the set F if the followig requiremets are met: there exist the sets AB,, = {,, } i = i < i, B { r,, r r + } A l i l l + = = <, l < r5, ad a is characterized by the membership fuctio a : [,], with a ( x) = for ( l ) ( r, ) x ad, 8

2 Possibilistic optimizatio with applicatios i portfolio selectio 89 ( ) a x ( x l)( l l) x [ l l] ( x lk )( lk+ lk ) k x [ lk lk+ ] k x [ l r] ( k+ )( k k+ ) [ k k+ ] ( x r )( r r ) x [ r r ] 5,, 5 +,,, =, =,, 5 x r r r k +, x r, r, k = 5, , 7, 8 We also ote the umber (( l,, l )) writte i the parametrized form a ( a() t, a() t, t) t [,], ( ( ) :[,] a = As i [] ad [7], the fuzzy umber a F ca also be { } t( l l ) + l, t [,5] a () t = t( l l ) + ( l l t [ 5,75], () t t( l l ) + ( l l t [ 75,], a t a t ad ( r7 r8 ) 8, t [,5] ( r6 r7 ) + ( r7 r6 t [ 5,75] ( r r ) + ( r r t [ 75, ] t a = t () t The additio ad the scalar multiplicatio (the scalars are real umbers) ca be expressed as a + b a() t + b( t), a( t) + b ( t), t t,, () {( ) [ ]} {( ka() t, ka( t), t) t [,] } k a, if k, k >, () {( ka() t, ka( t), t) t [,] } k a, if k, k < (5) Usig previous lies, the followig result emerges PROPOSITION If, b F la,, la, ra 5,, r uv,, u>, v<, the a, a = (( ) ( )), b = (( l,, l )(, r r )) a8 (( l + l,, l + l ( r r r )) a b a b a5 b5,, a8 b8 b b b5,, b8 (), ad a + b = +, (6) of u a = (( ul,, ul ( ur ur ) a (( vr,, vr ( vl vl )) a a a5,, a8 v = (7) a8 a5 a,, Example Let a = ((,,7,8 (,5,6,) ) ad b ((,,5,6 ( 9,,,) ) a + b (Fig ) ad a (Fig ) are below = a The graphical represetatios Fig The sum of two umbers from F Fig Scalar multiplicatio Carlsso ad Fullér [] defied the crisp possibilistic mea value of a fuzzy umber a as

3 9 Costi-Cipria Popescu, Cristica Fulga E ( a ) = tdt ta() t dt + ta() t dt = t a() t + a() t dt (8) PROPOSITION Cosider a (( l,, l )) F Proof: From relatio (8 we have J ta() t dt = The followig statemet holds; ( a ) = 96 [ ( l ) + 5( l ) + 8( l ) + ( l r )] E + = ad () O the other had, we have J = ta t dt 8 7 ( a ) J J 6 5 (9) E = +, () 5 75 ( ) ( ) ( ) ( ) ( ) J = t t l l + l dt + t t l l + l l dt + t t l l + l l dt = 5 75 ( l + 5l + 8l + ) = 96 l J is calculated similarly ad usig ( the fial result is obtaied I Dubois ad Prade [] ad Liu [] the possibility of a b (for a, b F ) ad the possibility of a k (for a F, k ) are defied as ( a b ) = { ( a( x) b( y) ) x y x y} Pos sup mi,,,, () ( ) ( ) PROPOSITION If a (( l,, l )) F { } Pos a k = sup a x x, x k () = the, l 5 l( l l), l l Pos( a ) = () 5li ( li+ li) + i, li li+, for i =,, l THEOREM Cosider p i (, ) ad a fuzzy umber a (( l,, l )(, r5 )) F ( a ) p (i) ( p) l + pl, whe l l ; (ii) ( p) l + ( + p) l, whe l l ; (iii) ( p) l + ( + p) l, whe l l Proof: (i) Assume ( a Pos ) p Takig ito accout ( we obtai l ( l l ) p Pos if ad oly if l l >, we have p ( l l ) + l, which leads to ( ) ( p) l pl = The 5 Sice p l + pl Reciprocally, if +, the coclusio is obtaied by reasoig i reverse order Aalogously, we obtai (ii) ad (iii)

4 Possibilistic optimizatio with applicatios i portfolio selectio 9 THEOREM Let a = (( l,, l )) F be a fuzzy umber ad (,) the followig assertios hold (i) If l the Pos ( a ) p ; (ii) If l < the ( a ) p ( i + 5i ) + ( 5i 5 + ) Pos if ad oly if i {,, } such that ( ) p a real umber The ll h l + hl, i i+ i i i i+ h i = p i Proof: (i) From () we get l Pos ( a ) = p (ii) We suppose that Pos ( a ) p If l the Pos ( a ) = > p Thus, l > We have the followig possible situatios: l < l, l < l, or l < l ; l < l, results i l l ad ( a ) = 5l ( l l ) p p l + pl, or ( h ) l + hl Reciprocally, for Pos Hece ( ) i =, we have ll ( h) l+ hl ad l l double iequality, we deduce ( a ) p < O the other had, from the right side of the previous Pos The other cases ( i = ad i = ) are similar THE POSSIBILISTIC PROGRAMMING MODEL Cosider the possibilistic optimizatio model max x Λ = E x a, () Λ = x R Pos = = x a b p, x,α x β,,, (5) = = ad α, β are real umbers which belog to the closed iterval [,] umbers ad (,,, ) T x = x x x, a are basically ay shape fuzzy represets the decisio vector The model () caot be solved i this geeral form I literature, triagular ad trapezoidal umbers are preferred because they are coveiet to use The problem is that this choice leads to limited results i true reflectio of the pheomea studied By modellig the ucertai parameters i the model as s-lr fuzzy umbers, we better capture the two maor aspects: the statistical data iformatio ad the experts kowledge Therefore, i () we cosider a ad s-lr fuzzy umbers, ie, of the form a = ( l,, l,, r8 ) F, for all =,, respectively b = (( b,, b ( b5,, b8 )) Thus, by Theorems -, the equivalet form of model () is max ( l 8) + 5( l 7) + 8( l 6) + ( l 5) x x Γ 96 =, (6) Γ = Γ, the sets Γ i, i =, are disoit ad i= i Γ = x x =, α x β, (7) = Γ i = x ( hi ) l, i + hi li x ( hi ) b i + hi b9 i, x =,α x β, i=, (8) = =

5 9 Costi-Cipria Popescu, Cristica Fulga 5 So, the fial form of the iitial possibilistic model is a liear optimizatio problem At this poit, the solutio ca be obtaied with well established techiques such as the simplex method APPLICATION TO THE PORTFOLIO SELECTION PROBLEM The portfolio selectio model I the probabilistic framework, sice the itroductio of mea-variace portfolio selectio models i the work of Markowitz [, ], variace has bee accepted as a risk measure However, over time, other measures of risk with better properties have bee itroduced The most represetative of them is the Valueat-Risk (VaR ) [, 6] I terms of losses, the Value at Risk is defied as the amout of loss such that the probability of ruig a loss this large or eve larger over a certai period of time, is limited If our model formulatio is cosidered i terms of gais (retur the defiitio of VaR ca be writte aalogously A Mea-VaR efficiet portfolio [8], maximizes the portfolio total retur ad, simultaeously miimizes the VaR at a specified cofidece level Solvig a bi-obective problem is a difficult task ad eve more as VaR is itself the optimal value of a miimizatio problem Therefore, a value V of the miimum accepted retur is specified ad the followig model is cosidered, amely, max x Ψ = E R x, (9) Here P meas probability, Ψ= x P Rx < V ε, x =,α x β, =, () = = R, x is the fractio of the total capital ivested i the security, =,, are the retur rates of security expressed as radom variables, α, β are the lower ad upper limits of the ivestmet i security ad ε is the give cofidece level applied to the probabilistic costrait (chace costrait) The possibilistic portfolio selectio model formulatio correspodig to the previously metioed probabilistic model (9) has the form max E x a, Ω () x = Ω= x Pos x a b ε, x =,α x β, =,, () = = The retur rates a ( =, ad the predetermied acceptable value of the miimum accepted retur b are modeled as s-lr fuzzy umbers ad E represets the possibilistic mea value from Propositio Short cosideratios o the method Whe solvig the possibilistic portfolio optimizatio problem ( the possibilistic costrait compels us to take ito cosideratio the cases preseted i Theorems ad ad cosequetly, the portfolio model () has a similar formulatio to (6) as a liear optimizatio problem with the feasible set beig a reuio of disoit sets like i (7) (8) Sice writig the portfolio model as a liear optimizatio model is

6 6 Possibilistic optimizatio with applicatios i portfolio selectio 9 straightforward (oly some replacemets i the otatio have to be doe, eg, a is replaced by =, ) we omit it ad, i the sequel, we refer to (6)-(8) as the portfolio model The problem (6) (8) ca be solved by direct search followig the steps: ) Iitializatio; ) Checkig the feasibility of the curret poit; ) Checkig if the value of the obective fuctio correspodig to the curret iteratio is superior to the previous oe; ) Testig for the stoppig criteria; 5) Stoppig, or performig a search for fidig the ext curret poit, icremetig the idex ad cotiuig the algorithm from Step O the other had, we ca also cosider four liear programmig models ad use the simplex algorithm for them The simplex algorithm provides always a exact output (a uique optimal solutio, multiple solutios or o solutio) for all four problems, as opposed to the approximate solutio give by the direct search method Therefore, we cosider the four liear programmig models max ( l 8) + 5( l 7) + 8( l 6) + ( l 5) x x Γi 96 = ( i =, each of the feasible sets beig defied i (7 respectively, (8) The performed simulatios o portfolios made of securities listed at the Bucharest Stock Exchage showed the effectiveess of our method R, () 5 CONCLUSIONS I this work we propose a type of fuzzy umbers called superior-lr-piecewise (s-lr we ivestigate their properties ad we preset ad discussed a chace-costrait type optimizatio model with fuzzy costraits ad obective fuctio expressed through the possibilistic mea value This geeralizes some previous models i literature which oly used trapezoidal fuzzy umbers Compared with the classical approach based o trapezoidal umbers, our approach provides substatial beefits: usig the theoretical results obtaied i this paper, the portfolio optimizatio problem ca be trasformed ito a special fuzzy optimizatio model ad moreover, ito a set of crisp liear optimizatio problems which ca be solved by classical meas such as the simplex algorithm Practical applicatios of our model ad method are umerous, particularly i mathematical laguage is ecessary to itroduce a subective poit of view But the fuzzy approach do ot replace but rather complemets probabilistic methods The model discussed is ope to such mixed approaches ad to develop it is oe of our future directios of research ACKNOWLEDGEMENTS This work was supported by CNCSIS-UEFISCSU, proect umber 8 PN II-IDEI, code 778/8 REFERENCES CARLSSON, C, FULLÉR, R, O possibilistic mea value ad variace of fuzzy umbers, Fuzzy Sets ad Systems,, pp 5 6, CARLSSON, C, FULLÉR, R, A fuzzy approach to real optio valuatio, Fuzzy Sets ad Systems, 9, pp 97, DUBOIS, D, PRADE, H, Operatios o fuzzy umbers, Iterat J of Systems Sci, 9, pp 6 66, 978 DUBOIS, D, PRADE, H, Possibility Theory, Pleum, New York, FANG, Y, LAI, K, K, WANG, S, Y, Portfolio rebalacig model with trasactios costs based o fuzzy decisio theory, Europea Joural of Operatioal Research, 75, pp , 6 6 FULGA, C, POP, B, Portfolio selectio with trasactio costs, Bulleti Mathematique de la Societe des Scieces Mathematiques de Roumaie, 5,, pp 7, 7 7 GOETSCHEL, R, VOXMAN, W, Elemetary fuzzy calculus, Fuzzy Sets ad Systems, 8, pp, 98 8 INUIGUCHI, M, RAMIK, J, Possibilistic liear programmig: a brief review of fuzzy mathematical programmig ad a compariso with stochastic programmig i portfolio selectio problem, Fuzzy Sets ad Systems,, pp 8, 9 LAI, I, J, HWANG, C, L, Fuzzy mathematical programmig: theory ad applicatios, Lecture Notes i Ecoomics ad Mathematical Systems, 9, Spriger, Berli, 99 LI, X, QIN, Z, KAR S, Mea-variace-skewess model for portfolio selectio with fuzzy returs, Europea Joural of Operatioal Research,, pp 9 7, LIU, B, Ucertai programmig, Wiley, New York, 999

7 9 Costi-Cipria Popescu, Cristica Fulga 7 MARKOWITZ, H, Portfolio selectio, Joural of Fiace, 7, pp 77 9, 95 MARKOWITZ, H, Portfolio selectio: efficiet diversificatio ad ivestmets, Wiley, New York, 959 McNEIL, A, FREY, R, EMBRECHTS, P, Quatitative risk maagemet: cocepts, techiques ad tools, Priceto Uiversity Press, 5 5 ROMMELFANGER, H, Fuzzy liear programmig et applicatios, Europea J Oper Res, 9, pp 5 57, ZMEŠKAL, Z, Value at risk methodology uder soft coditios approach (fuzzy-stochastic approach Europea Joural of Operatioal Research, 6, pp 7 7, 5 7 ZADEH, L, A, Fuzzy sets, Iformatio ad Cotrol, 8, pp 8 5, 965 Received March,