A Mean- maximum Deviation Portfolio Optimization Model

Transcription

1 A Mea- mamum Devato Portfolo Optmzato Model Wu Jwe Shool of Eoom ad Maagemet, South Cha Normal Uversty Guagzhou 56, Cha Tel: E-mal: Abstrat The essay maes a thorough ad systemat study about a mea- mamum devato portfolo optmzato model Frst, we mae a areful aalyss about the problem ad buld a model about ths d of problem The essay gves two d of dfferet ad haraterst solutos lear programmg soluto ad rtal le soluto Key words: Epeted yeld rate, s, Crtal le, Mamum rs-measuremet Itroduto I the path-breag wor o Portfolo Seleto, Marowtz (95) developed the oept of a effet portfolo terms of the epeted retur ad stadard devato of retur Moder Portfolo Theory (MPT) has beome oe of the most mportat bases moder aptal maret, ad bee appled wdely the prate of vestme I the absee of spef owledge of vestor s preferee, however, t aot be determed whh of ay two effet portfolos s better Baumel (96) replaed the ( E, σ) rtera wth the ( E, E σ) rtera, where stads for the vestor s atttude toward rs Baumol demostrated that hs ( E, E σ) rtera yeld a smaller effet set, whh s a subset of the Marowtz effet set, ad therefore redues the rage of alteratves from whh the vestor has to selet hs portfolo eetly, L () studed the optmal portfolo seleto wth trasato osts Hrosh Koo dsussed the mea-absolute devato portfolo optmzato model ad ts applatos I ths paper we dsuss the mea- mamum devato portfolo optmzato model More presely, we see to defe portfolo that produes mamal yeld ad at the same tme satsfes ostrats o mmum rs Our am s to develop a theory smlar to Marowtz theory for optmal mea-varae portfolos ad provde algorthm tools for omputg suh portfolos Our emphass here s o algorthms beause, ule optmal mea-varae portfolos, the meamamum devato portfolo optmzato model geerally defy aalyss wth smple aalytal tools Let s osder suh a problem; assume a ompay selets ds of seurtes S (,, Λ, ) to ves The epeted yeld rate of S s, ad the rs of S s The ompay defes that the total rs of portfolo s measured by the mamum rs of S Assume the terest rate s r at the same perod ad wthout ay rs How to defe the vestg weght of the a ds of seurtes to mae the epeted yeld rate of the portfolo to possble mamum, ad the total rs to possble mmum Deote the rs of portfolo s, the epeted yeld rate of the portfolo s s the vestg weght of bought assets S (,, Λ, ), s the vestg weght of ba savgs The ( =,,, Λ, ) Aordg to the reuremets of the problem, we a get the followg model: or ( I) m = ma{ } = r + Λ s ( =,,, Λ ) +

2 Aprl, 8 ( II) ma = r = ma{ } s ( =,,, Λ ) The lear programmg soluto of the model Now let's osder the model (I) oly (a) Deote + X = = (,, Λ, ), r + =, = The we ow that the feasble set X s a losed bouded polyhedral ove se (b) = ma { } ( (,),,, Λ ), s the pot supreme of lear futos X So t's a otuous bouded lear ove futo Aordg to (a)(b), m must have the optmal soluto The () X X Soluto: Deote { =,, Λ, ) X, Λ,,, } = ( Λ, = X Solve problems of lear programmg,, Λ, LP separately: m s X Apparetly all these problems have optmal solutos Suppose the optmal soluto of Here, Deote s the th ompoet of = = =,, Λ, m =, the s the optmal soluto of (I), ad = LP s, the optmal value s s the optmal value of (I) Usg ths method, uder a group of gve epeted yeld rate of the portfolo, we a get a relevat group of optmal weghts to mmum ts rs The rtal le soluto of the model Now we osder the model (I) ad (II) Suppose < < < Λ <, ad r < < < Λ < For Λ +, we have Λ So the epeted yeld rate ad rs of portfolo s respetvely: = r = ma + Λ + = r + ( r ) () = () { } Smlarly, we a defe the oepts of so-epeted-yeld-rate super-plae of portfolo, so-rs super-urved-surfae of portfolo, rtal le, e Dfto : I the weght spae (,, Λ, ), gve the epeted yeld rate of portfolo, the super-plae defed by the euato () s alled a so-epeted-yeld-rate super-plae of portfolo All the portfolos o ths 5

3 super-plae have the same epeted yeld rate Wth dfferet, we a get a famly parallel so-epeted-yeld super-plae suh as f or f fgure Wth the reasg of, the yeld rate s preseted by the so-epeted-yeld-rate le rease steadly Dfto : I the weght spae (,, Λ, ), gve the rs of portfolo, the super-urved-surfae defed by the euato () s alled a so-rs super-urved-surfae of portfolo All the portfolos o ths super-urved-surfae have the same rs Wth dfferet, we a get a famly parallel so-rs super-urve suh as b or b 5 fgure Wth the le segmet OB from O to B, the rs preseted by the so-rs le reases steadly Deote a so-epeted-yeld super-plae (suh as f fgure ) Above the segmet OB, the rs o the so-epeted-yeld super-plae derease from the top dow to the pot of terseto (pot M) of ths plae ad the segmet OB Below the segmet OB, the rs o ths plae dereases from the rght to the left, utl to the pot of terseto (pot M) of ths plae ad the segmet OB We all the pot of terseto the frst d of tagey pot of the so-epeted-yeld super-plae ad the so-rs super-urve; we all the lous of these pots of terseto the frst d rtal le Its euato s = = = Λ = I the weght spae,, Λ, ), the frst d rtal le tersets wth the boudary of vestmet area at pot 6 ( H For he same reaso, we a defe the seod rtal le, ad ts euato s = = Λ = Furthermore, we a get the euato of the th rtal le: = + + = Λ = = = Λ = = + Whe the gve so-epeted-yeld super-plae does ot terset the segmet OB OAB, the plae should terset the le BC (suh as f ), whh the dereasg of 6, the rs o the plae derease steadly ad ome to m at the pot of terseto (pot B) of the plae ad the segmet BC We all ths pot the th pot of tagey of the so-epeted-yeld super-plae ad the so-rs super urve, ad all the lous of those pots of tagey the th rtal le Its euato s: = = Λ = = + The frst rtal le, the seod rtal le,, the th rtal le are all alled the rtal le of portfolo Aordg to the defto, the rtal le of portfolo s a otuous spae broe le, we mar the broe pot respetvely as: H, H, Λ, H Aordg the defto of rtal le, gve ay a epeted yeld rate, we a fd the optmal weght of portfolo, whh mae the rs mmum o rtal le At the same tme, gve ay a deoted rs of portfolo, we a fd the optmal weght of portfolo that a mae the epeted yeld rate mamum o rtal le So we eed oly to solve the rtal le euato of portfolo, order to fd the soluto of model (I) or (II) Aordg to the defto of rtal le, the rs, the epeted yeld rate ad the weght at pot H (,, Λ, ) should be respetvely: A Eample = ( ) () = = Λ + () ( ) +,, Λ,, Λ, =, Λ,,,, Λ, + (5)

4 Aprl, 8 Suppose there are four seurtes: S, S, S, S, ther epeted yeld rate ad rss are: = 7, = 8, = 9, = Aordg to euatos (), (), (5), we ow: At pot H, = 869, = 76, (,, Λ, ) = (, 778, 6, 7, 8) At pot H, = 97, = 7, (,, Λ, ) = (,, 8, 9, 8) At pot H, =, = 77, (,, Λ, ) = (,,, 6, ) So the rtal le should be: The frst rtal le euato s: = = = 6 = = 6 The seod rtal le euato s: The thrd rtal le euato s: The forth rtal le euato s: = = = + Wth dfferet or, usg rtal le euato () or (), we a fd respetvely ther optmal weght (See Table or ) Wth dfferet, usg the lear programmg soluto, we a fd ther optmal weght (Table ) 5 Coluso Our essay gves two ways to solve a mea- mamum devato portfolo optmzato model Oe s the lear programmg soluto The other s the rtal le soluto Both of the soluto a fd the optmal soluto of the problem, ad both the smulated umbers results are detal The lear programmg soluto a oly fd the optmal weght of portfolo whh mae the rs to mmum whle gve the epeted yeld rate, but t s helpless to fd the optmal weght of portfolo whh mae the epeted yeld rate to mamum whle gve the rs Ad the lear programmg soluto a oly fd the soluto uder gve data However, the rtal le soluto a resolve both of these problems eferee Harry Marowtz, Portfolo Seleto, Joural of Fae, Marh 95, 77-9 Baumol W J, A Epeted Ga-Cofdee Lmt Crtero for Portfolo Seleto, Maagemet See, 96, 7-8 L Z F, Wag S Y ad Deg X T, A Lear Programmg Algorthm for Optmal Portfolo Seleto wth Trasato Costs, Iteratoal Joural of Systems See,, () 7-7 Hrosh Koo ad Asta Wjayaayae, Mea-Absolute Devato Portfolo Optmzato Model uder Trasato Costs, Joural of the Operatos esearh Soety of Japa, 999, () -5 Table Gve dfferet, the optmal weght of portfolo ad ts m-rs

5 Table Gve dfferet, the optmal weght of portfolo ad ts ma-epeted-yeld-rate Fgure 8