Optimal MinMax Problems Regarding Capitalization of Compound Interest
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1 Proeedg of the 5th WEA Iteratoal Coferee o Eoomy ad Maagemet Traformato (Volume II Optmal MMax Problem Regardg Captalzato of Compoud Iteret ILIE MITRAN Departmet of Mathemat ILIE RĂCOLEAN, IMOLA DRIGĂ Departmet of Eoom Uverty of Petroa tr.uvertat, r.0, 33006, Petroa ROMANIA lemtra@gmal.om, leraolea@yahoo.om, mola.drga@gmal.om Abtrat - The reult of the tudy regardg aptalzato uder ompoud teret are related to the ba properte of polyomal aptalzato. ede the problem of aulmet of th polyomal, there are two optmal type problem whh mply extremely mportat reult eoom ad faal alulato: the mmum devato problem ad the equlbrum problem. The mmum devato problem loely related to the properte of Cebev polyomal whle the equlbrum problem tuded (wth mple ad mxed tratege by ug gfat reult from the game theory. The purpoe of th paper to preet a ew form of the optmal trateg behavour of mmax type ad hee of maxm type, regardg a mportat ue of aptalzato. The form of the effey futo aot determe the equlbrum pot, ad therefore we mut determe oly the optmal guarateed value of maxm ad mmax type. The reult are mportat ad they provde extremely teretg eoom terpretato. Key-Word: - aptalzato polyomal, maxm optmum guarateed value, mmax optmum guarateed value, ut teret, aptalzato fator, equlbrum pot Itroduto The geeral form of aptalzato polyomal a be determed from the followg elemet (Fg.: At tme 0,,,, the followg um are veted or wthdraw 0,,, ; th mared by the followg ymbol, 0,,, ae moey vetmet are made; where 0, ae there are o faal traato; -, ae ah wthdraw.,,, repreet the ut teret harged durg the followg terval 0,,,,...,[, ; 0 3, ad Alway 0 0,, whh mea that at baele there are ah vetmet, ad at the ed there are fud wthdrawal or o faal operato at all. Aordg to the perod of tme 0,,,, the followg aptalzato of ompoud teret wll tae plae: 00 00( (...( ( ( 3...( ( ε 0, 0 ε, ε, ε 3, 3 ε -, - ε, Fg. IN: IN:
2 Proeedg of the 5th WEA Iteratoal Coferee o Eoomy ad Maagemet Traformato (Volume II Uder the rumtae, f P the aptalzato polyomal, the t aalytal expreo the followg: P ( ; ; 00,,..., ( ( 0 If we ue the otato u, a,, for the aptalzato 00 fator, the the aptalzato polyomal a be wrtte a follow: ( ; ; 0 0,,..., 0 0( P u au ( Whe tudyg the properte of ompoud teret aptalzato, the ey ue ad the otext whh they appear are the followg: a The problem of aptalzato polyomal aulmet; th ae otat, ad, 0, are varable, ad the problem that ha to be olved repreet the determato of the oluto of the equato: u a u 0 (3 The ma reult o the aaly of the aulmet of the aptalzato polyomal are the followg: * * *. The ut teret,,..., for whh the equato (3 aulled verfy the followg equalty: * * *,,..., max a, =, (4. If 0..., the the ut teret for whh aptalzato polyomal are ull wll be below. b The mmum devato problem; th ae fxed, ad, 0, are varable, ad the problem that eed to be aalyzed : (P m max P(, The ma reult aroud the aaly of the problem (P are the followg:. The mmum devato of the aptalzato polyomal alway greater or equal to the 0 ze ;. If the aptalzato polyomal of Cebşev type, tha the mmum devato alway 0 equal. Remar Cebşev polyomal of degree mared wth T ad t a be defed a: T :[,] R, T( x o( aro x, N The ma properte of th polyomal are the followg: the degree of the polyomal ad the oeffet of x ; the extreme pot are x o, 0, ; the extreme value are ±. The equlbrum problem (wth mple ad mxed tratege; the ae of mple tratege we may oder a ow quatty, 0,, ad varable. The problem that eed to be olved the followg: (P max m P (, m max P (, For mple tratege, the equlbrum teret * [8], where N the maxmum for the N perod whh faal traato tae plae ( th ae, N. Problem Formulato We hall tart from the followg elemet: At the momet,,, the followg faal operato tae plae: - we depot the um,,,, A, A beg fxed - we wthdraw the um,,,,, beg fxed,,, repreet the ut teret value ued for the terval [0,, [,,, [-,. The problem to determe the amout eeded to be depoted ad wthdraw order to obta the maxmum aptalzed um the momet. eaue the wthdrawal of um,,, mple atually the wthdrawal of,,..., whh repreet the potetal aptalzato ug ompoud teret for,,, we get (table : IN: IN:
3 Proeedg of the 5th WEA Iteratoal Coferee o Eoomy ad Maagemet Traformato (Volume II Table Momet The wthdraw um The aptalzed um ( ( 3...( ( (...( 3 4 t (...( t t t t Remar If we ote ( (...(, the the followg equalte tae plae: ( ( ( (...( We mar wth f the adopted effey futo, f : R R defed by the followg equato: f (, max ;0 (5 where (,,...,, (,,...,,, ( (...( Remar 3 e the effey futo f ot otuou, t obvou that t ot dfferetable ether, o the regular operato related to the dfferetal alulato have o effet. The formulated problem may be terpreted a a uooperatve game problem wth two deo maer where the effey futo W ad W admt the followg aalytal expreo: W (, max ;0 (6 W (, m ;0 (7 We ote that the followg relato verfed: W (, W (, therefore, we are dealg wth a zero um game. Th game doe ot preet equlbrum pot for mple tratege beaue of the effey futo partular form; oequetly the ma problem regardg th ue to determe the guarateed optmal tratege ad value. 3 Problem oluto 3. The geeral oluto For the followg equato: max m W (, m maxw (, m max W (, (8 m max W(, max m( W(, (9 max m W (, t obvou that we eed to determe oly two guarateed optmal value (ad the orrepodg guarateed optmal tratege. Pratally, we eed to aalyze the followg problem: P max mw (, ;( m maxw (, ( P The problem (P olved ug the equalzato prple [6], [7]: * * * * - the optmal oluto (,,..., that we are earhg for, determed by olvg the followg algebra ytem:,, (0 After a mmedate alulato, we get: *,, ( * D therefore, where D * D * D - the mmax guarateed optmal value gve by the followg relatohp: IN: IN:
4 Proeedg of the 5th WEA Iteratoal Coferee o Eoomy ad Maagemet Traformato (Volume II ( max m W (, m A;0 Obvouly, the problem (P m maxw (, equvalet wth the problem (P 3 max mw (, ad t a be alo olved by ug the equalzato prple [9]. Aumg that..., the optmal oluto that we 3 are loog for the followg: - - * * * m - * ;, = m ;, f m - ; 0 = = = 0, otrarwe (3 The guarateed optmal oluto type mmax the followg: m maxw (, m m A ;0 (4 Remar 4 It obvou that max m W (, m m A ;0 (5 Remar 5 e W ovex reported to the varable, t obvou that the game value gve by the guarateed optmal value of W mmax ee. Tag to oderato the equato max mw (, m maxw (,, the followg relato tae plae: m max W (, max A ;0 (6 eaue,,...,,... after a mmedate alulato we hall get: (7 ad, oequetly the optmal tratege that we are earhg for are gve by the followg relato:... *,( , 3. Eoom Iterpretato If = = = ad we mar wth I the ommo value, the the optmal tratege that we are earhg for allow the followg expreo: *...,, (9 hee *,, (0 If I mall eough, the ; oequetly *,, ( We a mmedately oberve that the optmal oluto are gve by the produt betwee the rato ad the power of the aptalzato fator (+. Furthermore, all the optmal oluto are foud o the le: x, f we oder that,, Coequetly, f a, b, we have (fg.: a ab 3 ab a( b Fg. I th ae, the optmum guarateed value are the followg []: ( a m max W (, max A ;0 Th ze mut have a o-zero value; th lead u to the odto that the aptalzato fator ha IN: IN:
5 Proeedg of the 5th WEA Iteratoal Coferee o Eoomy ad Maagemet Traformato (Volume II A to verfy the followg equalty:. I other word, the ut teret ued mut meet the A requremet:. max m W (, m m[ A];0 b Th optmum guarateed value ha to be ozero; order to meet th odto, t eeary that the ut teret hould verfy the equalty: A. ( 3.3 Applato We aume that at the tmet 0 we vet the um 0 ad at tme,,..., o other faal traato tae plae; therefore... 0, For eah momet t poly fud wthdrawal are made. The problem whh are am to determe the ut teret eeded to be ued order to wthdraw the um 0 at tme t p, t p,... ( other word, p p I vew of the pef form of th problem, we a fd t oluto by olvg the followg equato: p p 0( u ( u u... u 0 ( where u repreet the aptalzato fator. Pratally, th equato a be wrtte a follow: u p u 0 u (3 ad, aordgly, we are led to olve the followg equato: u p ( u p p u p u 0 (4 eaue the aptalzato fator u alway o-zero ad uffetly loed to 0 for p u large eough, we are led to olve the followg p p hgher degree equato: ( ( 0 where the varable repreet the ut teret that we are earhg for. It obvou that th equato a be wrtte the followg equvalet form: ( p 0ad admt a gle oluto ( 0,. Remar 6 If p, the the ut teret that we are earhg for the potve oluto of the equato 5 0; therefore 06.. Geerally, the equato of the optmum teret ( p 0 a hghe r degree equato ad t a oly be olved by mea of approxmate method. The mot oveet method mple, pratal ly, the learzato of the expreo ( p. We a mmedately oberve that ( p pwhe mall eough (th amout to the fat that p uffetly large. I th ae, the equato ( p 0tur to p 0ad the potve oluto of th equato the followg: 4p (5 p If we learze the fator ( p a arbtrary hoe pot 0 (, 0, the the optmum teret equato beome:( 0( p0 p0. eaue the watg perod p vere rato to the optmum teret, t very oveet form alulato pot of vew to oder 0 p. I th ae, the prevou equato beome p ad, therefore, the ut teret that we p are earhg for : p (6 p Fg.3 IN: IN:
6 Proeedg of the 5th WEA Iteratoal Coferee o Eoomy ad Maagemet Traformato (Volume II We a mmedately oberve that for p, the optmum teret p ad 4p ode ad th reult p p oordae wth the graphal mage of th teret (fg.3. 4 Coluo The paper approahe a optmum problem regardg the aptalzato of ompoud teret whh dfferet from other ow problem (the problem of aulmet, the mmum devato problem, the equlbrum problem. Eve thee ow problem are aalyzed uffetly at preet ad, therefore, few theoretal reult are mared out. y vrtue of the form of the effey futo, the approahed optmum problem requre a peal mathematal apparatu whh all to requto to ba reult from the game theory (epeally the reult related to mmax optmzat o. The applato performed, extremely gfat from pratal pot of vew, repreet fat, a partular ae of the optmum problem approahed. Atually, the olvg of the problem a be odered a aulmet problem for the aptalzato polyomal, whh ha a partular form. Pratally, the ut teret that we are earhg for a be determed a a oluto of a hgher degree algebra equato; hee the proe of olvg th equato performed by mea of a approxmate method. Therefore, we appealed to learzato method ( fat, a varat of the method, whh mple the developmet to Taylor ere beaue Newto method ad ueve approxmato method requre extremely dffult alulato. Referee: [] Athoy, M., gg, N., Mathemat for Eoom ad Fae. Method ad Modellg, Cambrdge Uverty Pre, 008 [] rgo, D., Meruro, F., Iteret Rate Model: Theory ad Prate, prger 006 [3] Car, A., J., Iteret Rate Model: A Itroduto, Preto Uverty Pre, 004 [4] Drgă, I., Guţă, A., Nţă, D., Iteret Rate R Maagemet ag, The Youg Eoomt Joural, vol. (4, pp. 4-48, 00 [5] Dura, C., C., Mtra, I., Drgă, I., About a Optmum Model of Maret Equlbrum, Proeedg of the d WEA World Multoferee o Appled Eoom, ue ad Developmet (AED '0, Publhed by WEA Pre, pp , Kataou, oue, Tua, 00 [6] Ghermeer, L.., Operato Reearh, Tehal Publhg Houe, uharet, 973 [7] Mtra, I., Co-operatve ad Partal Cooperatve Deoal Model, Moograph, Ed. AME, Frae, 990 [8] Mtra, I., Teho-Eoomal Deo Modellg, AGIR Publhg Houe, uharet, 009 [9] Mtra, I., Dura, C., Magu,., About Equalzato Prple. Applato Rug Problem ad Faal Arbtrato, Proeedg of the d WEA Iteratoal Coferee o Fte Dfferee, Fte Elemet, Fte Volume oudary Elemet (F-ad- '09, Publhed by WEA Pre, pp , Tbl, 009 [0] Mtra, I., Dura, C., Magu,., About oumer optmum dyam model ad maret equlbrum teret, Proeedg of the d WEA Iteratoal Coferee o Fte Dfferee, Fte Elemet, Fte Volume oudary Elemet (F-ad- '09, Publhed by WEA Pre, pp , Tbl, 009 [] Mtra, I., Crtera ad Optmum Traetore for equetal Deo Problem, Lambert Aadem Publhg Houe, aarbrüe, 00 [] ydaeter, K., Hammod, P., Eetal Mathemat for Eoom Aaly, Pearo Eduato, 008 [3] Zemba, W., Vo, R., tohat Optmzato Model Fae, World etf, 006 IN: IN:
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