Portfolio selection based on nonparametric estimation and quadric utility maximization framework

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1 Avalable onlne at Proceda Engneerng 3 (0) Power Electroncs and Engneerng Alcaton Portfolo selecton based on nonarametrc estmaton and quadrc utlty maxmzaton framework Ha-xang Yao Scool of Informatcs, Guangdong Unversty of Foregn Studes, Guangzou 50006, Cna Abstract s aer adots te metodology of nonarametrc estmaton and utlty maxmzaton model to exlore a ortfolo selecton roblem under te assumton tat nvestors ave quadrc utlty functon. Frst, we obtan te estmated calculaton formula for te exected utlty by usng te nonarametrc estmaton of ortfolo return s densty functon. en, te otmal nvestment strategy for te utlty maxmzaton model s obtaned. Fnally a numercal examle based on real data of Cnese stock market s gven to sow te usefulness and effectveness of te results. 0 Publsed by Elsever Ltd. Oen access under CC BY-NC-ND lcense. Selecton and/or eer-revew under resonsblty of [name organzer] Keywords: Portfolo selecton; quadrc utlty functon; utlty maxmzaton model; nonarametrc estmaton. Introducton [] establsed te teoretcal bass for usng exected utlty functon to study uncertan decson roblem. [] ntroduced te utlty maxmzaton metod to nvestgate a ortfolo selecton roblem, and consder te roblem of rsk averson and tolerance. [3] studed asset allocaton roblem for statc case by usng utlty maxmzaton model, and obtaned a necessary and suffcent condton for two Fund Searaton eorem to be establsed. Usng utlty maxmzaton model, [4] nvestgated some results on comaratve statcs of otmal strategy under uncertan market arameter. ere are also many oter aers ave studed ortfolo selecton roblems by usng te exected utlty maxmzaton framework, suc as [5,6,7,8,9], and so on. However, most of tese researces are under te assumton tat return on assets obey some secfc robablty dstrbuton tye, suc as normal dstrbuton, or tere are only a el.: ; fax: E-mal address: yaoaxang@mal.gdufs.edu.cn Publsed by Elsever Ltd. Oen access under CC BY-NC-ND lcense. do:0.06/j.roeng.0..59

2 Ha-xang Yao / Proceda Engneerng 3 (0) fnte number of ossble states n te future. Oterwse, wtout tese condton, revous lterature only can dscuss te related roertes for te exected utlty maxmzaton model, but can not solve for te model exlctly or numercally. It s well known tat nonarametrc estmaton metod s aealng n several asects. One of tese s tat lttle or no restrctve ror nformaton on functon s needed. Anoter advantage s tat t allows a wde range of data deendence (see [0]), wc makes t adatable n te context of carcous fnancal market. erefore, te major objectve of ts aer s to set u a framework, so tat, on one and, t can drectly gve out te nonarametrc estmated calculaton formula for te exected utlty under consderng nvestment strategy, on te oter and, t can study te utlty maxmzaton ortfolo selecton roblem at te same tme. e remnder of ts aer s organzed as follows. Frstly, we obtan te estmated calculaton formula of exected utlty by usng nonarametrc estmaton of te ortfolo return s densty functon. Secondly, we obtan closed form exresson for te otmal strategy. Fnally, a numercal examle base on real data of Cnese stock market s resented to sow te valdty and te ractcablty of tese results.. Market and model settng Suose tere are n assets (among tem, tere can be one rsk-free assets) n te fnancal market. Let ξ = ( ξ, ξ, ξ n denote te returns of te assets, W = ( w, w,, w n denote te ortfolo of te n assets. Here A denotes te transose of matrx A. en, return of ortfolo s ξ : = wξ = W ξ, and te utlty maxmzaton ortfolo selecton model can be exressed as max E[ U( W ξ )], st.. W =, () W were = (,,, reresents te vector wt every comonent equal to one, U () denotes te utlty functon corresondng to nvestor s reference. In order to guarantee te otmal soluton exst, U () often needs to satsfy some matematcal roertes, suc as concavty and monotoncty. In ts aer, we suose tat U () s a quadrc utlty functon, tat s say U () as te form as follow U ( x) = x 0.5 bx, b > 0, () were b measures te degree of rsk averson. 3. Exected utlty maxmzaton model base on nonarametrc estmaton Snce n general, we know very lttle nformaton about robablty or densty functon of ortfolo s return ξ or asset s returns vector ξ. And on te oter and, we know tat nonarametrc estmaton metod does not need to make assumton about te dstrbuton tyes, and lttle or no restrctve ror nformaton on functon s needed. erefore, n ts aer, we wll adot te nonarametrc metod to estmate te dstrbuton of ξ or ξ, and ten obtan te estmated formula of exected utlty. And based on ts foundaton, we nvestgate te utlty maxmzaton ortfolo selecton roblem. Snce n general te asset s returns vector ξ s a multdmensonal random vector, f we adot nonarametrc metod to estmate ts densty functon, te convergence rate of nonarametrc estmator would be very slow, wc sometmes referred to as te curse of dmensonalty (see [0]). So n ts aer, we estmate te densty of return of ortfolo, wc s only of one dmenson, overcomng te roblem of curse of dmensonalty, and fnally obtan te nonarametrc estmaton for E[ UWξ ( )].

3 394 Ha-xang Yao / Proceda Engneerng 3 (0) Now we ntroduce some relmnary knowledge of nonarametrc teory (see [0]). e nonarametrc estmaton of robablty densty functon (PDF) x ( ) of unvarate random varable X wt samle set { X, X,, X } s X x x ˆ( ) = k( ), (3) were () k s a kernel functon, wc, for examle, can be cosen as ( ) 0.5v kv () π = e. v G( v) = k() t dt, and = ( ) s a smootng arameter (or alternatvely, bandwdt or wndow wdt). It can be roved tat te kernel estmator x ˆ( ) defned n (3) s a consstent estmator of x ( ) wen kernel functon k() and bandwdt () satsfy te followng condton ) k() s nonnegatve and bounded, k() v dv =, k( v) = kv (), v k() v dv = κ > 0; ) ( ) 0 and ( ) as. So trougout ts aer we always assume tat k() and () satsfy te condtons () and (). In bot teoretcal and ractcal settngs, te nonarametrc kernel estmaton s nsenstve to te coce of kernel functon, but te coce of bandwdt () s a crucal roblem. ere are many metods for selectng (), ere we only ntroduce te common used metods. Rule-of-tumb:.06σ ( ) X X ˆ σ = ( ) 0., were, were σ s te devaton of X. σ can be estmated by te samles: X = X. Gven nvestment strategy W and te samle set { R, R,, R } of ξ, ten return of ortfolo ξ = W ξ as te samle set { { WR, WR,, WR }. Adotng te metod of nonarametrc estmaton, nonarametrc PDF estmaton of ξ s WR x x ˆ( ) = k( ), (5) After avng estmated te PDF of ξ, one can use te luggng-n metod to estmate E[ UWξ ( )] as ˆ WR x E[ U ( W ξ )] = U ( x) ˆ ( x) dx = ( x 0.5 bx ) k( ) dx. WR x Let z = n every ntegral, after ntegral transformaton, and notce tat U x x bx k z dz zk z dz z k z dz κ ( ) = 0.5, ( ) =, ( ) = 0, ( ) =, we can smlfy Ê[ UWξ ( )] as Ê[ U ( W ξ )] = [( W R z) 0.5 b( W R z) ] k( z) dz = [ W R 0.5 b( W R ) + ( bw R ) z 0.5 b z ] k( z) dz κ κ = ( WR 0.5 bwrrw) 0.5b = WR0.5bW ΞW0.5 b, (6)

4 Ha-xang Yao / Proceda Engneerng 3 (0) were,. R = R Ξ= RR We adot Rule-of-tumb to select bandwdt, namely devaton of en ξ, wt ts estmator beng ( ) ˆ σ = ( WR WR ) 0. =.06σ, were σ s te standard Κ= ( WR, WR,, WR = ( R, R,, R) W =R W, M0 = I. ˆ σ can be exressed as ˆ σ = ( ) Κ M Κ= ( ) W R M R W = W Ω W, were. Let 0 0 R= ( R, R,, R, I denotes te dentty matrx of order, ( ) M0 Ω = R R. ereby te 0. selected bandwdt can be wrtten as =.06 W Ω W. So, we ave 0.4 Ê[ U ( W ξ)] = W R 0.5bW ΞW 0.5b.06 W Ω Wκ = W R 0.5 bw ΣW, 0.4 were Σ=Ξ+.06 Ω κ. erefore, utlty maxmzaton model based on nonarametrc estmaton metod can be wrtten as te followng otmzaton roblem max E[ ˆ U ( W ξ )] = W R 0.5 bw Σ W, s. t. W =. (8) W 4. Solvng te model In te followng, we study te solvng of otmzaton roblem (8). Frst, we set u te corresondng Lagrange functon as L = W R 0.5 bw Σ W + λ( W ). e frst-order condton s as follow L L = RbΣ W + λ = 0, = W = W = 0. (9) W λ From te frst equaton of (9), t follow tat W b ( = Σ R+ λσ ). Substtutng t nto te second λ = Σ b Σ R. ereby, te otmal soluton to otmzaton (7) s equaton of (9) gves ( ) ( ) W = b Σ R+ C b ( b A) Σ, (0) were C = Σ, A= Σ R. Substtutng (0) nto (8), te maxmum exected utlty s gotten as Ê[ U ( ξ W )] = 0.5C b b + Ab + D, () were B R R D BC A = Σ, = > Examle analyss ( ) We randomly select sx stocks from Senzen and Sanga stock excange. ese stock codes are 00053, , , 60699, 0007 and 600. Selectng te storcal daly data of tese stocks from June, 008, to May 6, 0, we get = 77 Day returns samles { R, R,, R77}, were te unt of return s / 50. We take te kernel functon as Gauss kernel functon ( ) as κ = v k() v dv =. Substtutng data and troug some calculaton, we obtan 0.5v (7) kv () π = e, ten t

5 396 Ha-xang Yao / Proceda Engneerng 3 (0) Σ=Ξ+.06 Ω κ = Let b = 0., from formula (0)-(), te maxmum utlty s Ê[ U( ξ W )] = , and te otmal ortfolo s W = (0.467, 0.9, 0.458, 0.089, , ). Smlarly, wen b =, te maxmum utlty s Ê[ U( ξ W )] = 0.567, and te otmal ortfolo s W = (0.57, 0.380, 0.09, , 0.00, 0.007). Wen b = 0, te maxmum utlty s Ê[ U( ξ W )] = , and te otmal ortfolo s W = (0.583, 0.840, 0.885, , 0.077, 0.054). Acknowledgements s researc s suorted by te Humanty and Socal Scence Foundaton of Mnstry of Educaton of Cna (No. 0YJC790339), Natural Scence Foundaton of Guangdong Provnce (No. S ), and Plosoy and Socal Scence Foundaton of Guangdong Provnce (No. 09O-9). References [] Neumann JV, Morgenstern O. eory of Games and Economc Beavor. New Jersey: Prnceton Unversty Press; 944. [] Arrow K. Essays n te eory of Rsk-Bearng. Nort-Holland: Amsterdam; 970. [3] Cass D, Stgltz J. e structure of nvestor references and asset returns, and searablty n ortfolo allocaton: A contrbuton to te ure teory of mutual funds. Journal of Economc eory 970;:-60. [4] Ceng HC, Magll MJP, Safe WJ. Some Results on Comaratve Statcs under Uncertanty. Internatonal Economc Revew 987;8(): [5] Hart O. Some negatve results on te exstence of comaratve statcs results n ortfolo teory. Revew of Economc Studes 975;4:65-6. [6] Samuelson PA. Lfetme ortfolo selecton by dynamc stocastc rogrammng. Revew of Economcs and Statstcs 969;5: [7] Merton RC. Otmum consumton and ortfolo rules n a contnuous-me model. Journal of Economc eory 97;3: [8] Muturaman K, Kumar S. Multdmensonal Portfolo Otmzaton Wt Proortonal ransacton Costs. Matematcal Fnance 006;6(): [9] Lu J. Portfolo Selecton n Stocastc Envronments. e Revew of Fnancal Studes 007;0():-39. [0] L Q, Racne JS. Nonarametrc Econometrcs: eory and Practce. New Jersey: Prnceton Unversty Press; 007.