An Indian Journal FULL PAPER. Trade Science Inc. The principal accumulation value of simple and compound interest ABSTRACT KEYWORDS

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1 [Type ex] [Type ex] [Type ex] ISSN : Volume 0 Issue 8 BioTechnology 04 An Indian Journal FULL PAPER BTAIJ, 08), 04 [ ] The principal accumulaion value of simple and compound ineres Xudong Li*, Mingxing He School of Mahemaics and Compuer Engineering, Lab of Securiy Insurance of Cyberspace, Xihua Universiy, Chengdu, Sichuan Province, 60039, PR, CHINA) lixudong73@63com ABSTRACT The calculaion of he principal accumulaion value is he foundaion of all financial calculaions, and plays criical role in all financial aciviy such as invesmen, and financing The heorem for calculaions of he principal accumulaion value of simple and compound ineres is proposed When a erm is less han a period, he accumulaed value of simple ineres is more han ha of compound ineres; When a erm is equal o a period, he accumulaed value of simple ineres is he same as ha of compound ineres; When a erm is more han a period, he accumulaed value of simple ineres is less han ha of compound ineres, and he growh rae of he accumulaed value of simple ineres is less han ha of compound ineres The heorem enriches he heory of ineres Finally, I sricly prove he heorem wih Rolle s Theorem KEYWORDS Principal accumulaion value; Compound ineres; Simple ineres; Rolle s heorem; Growh rae Trade Science Inc

2 BTAIJ, 08) 04 Zhang Zenglian 0057 INTRODUCTION The principal accumulaion value is a fundamenal concep of he heory of ineres and financial heory, in which he ime value of money is embodied The ime value of money is he relaionship beween ime and money I is o say ha money in hand oday is worh more han money ha is expeced o be received in he fuure if here are no inflaionary and deflaionary rends in an economic sysem The reason is sraighforward: A dollar ha you receive oday can be invesed such ha you will have more han a dollar a some fuure ime This leads o he saying ha we ofen use o summarize he concep of ime value: A dollar oday is worh more han a dollar omorrow This concep is used o choose among alernaive invesmen proposals Undersanding he conceps is crucial o undersanding all sors of soluions o financial problems in personal finance, invesmens, banking, insurance, ec Khan [] raises he issue of ime preference and he ime value of money and heir relevance no only o discouning bu also o wage, ren The ime value of money is especially appreciaed Hudson [], Rober and Murdick [3] have provided he formulae o enable he ime value of money o be refleced in valuaion pracice One of he bigges obsacles o correcly solving ime value of money problems is idenifying he cash flows and heir iming Every ime value of money problem has five variables: Presen value, fuure value, and number of periods, ineres rae, and a paymen amoun The ineres rae is he growh rae of your money over he life of he invesmen I is usually he only percenage value ha is given However, some problems will have differen ineres raes for differen ime frames For example, problems involving reiremen planning will ofen give pre-reiremen and posreiremen ineres raes Frequenly, when you are being asked o solve for he ineres rae, you will be asked o find he compound average annual growh rae On he oher hand, Snowden [4], Xi [5] and Li [6] have given he compuaion issues of compound ineres Compound ineres has imporan applicaions in human sociey based on he benefi of sorage and invesmen in economics [7] The key o compound ineres research primarily lies in he organism s choice of ineracion characerisics on differen emporal and spaial scales [8] Three formulas for compound ineres exis in erms of differen periods of ime: coninuous compound ineres; com-pound ineres of k periods which has received more aenion in evoluionary ecology; compound ineres of a uni period Modeling he erm srucure of ineres raes TSIR) has been he objec of many sudies and he aim of aenion for economiss and financial insiuions Applicaions and analysis of some of TSIR models can be found in he references [9-] However, here are few sudies of he principal accumulaion value when aking ino accoun simple ineres, and he difference beween he principal accumulaion of simple ineres and ha of compound ineres is no given In his paper, he heorem for he principal accumulaions calculaed based on simple ineres and compound ineres is presened and proved In wha follows, Firsly, he essenial noaion and some definiions will be inroduced Secondly, he example for he principal accumulaions calculaed under simple and compound ineres is given Thirdly, he heorem for he principal accumulaions calculaed under simple and compound ineres is presened and proved Finally, he conclusion is given Preliminaries In his secion, we inroduce some definiions and noaions In his paper, ineres raes will be denoed symbolically byi To simplify he formulas and mahemaical calculaions, when i is used i will be convered o decimal form even hough i may sill be referred o as a percenage The iniially deposied amoun which earns he ineres will be called he principal amoun and will be denoed P Accumulaion funcion a ) and amoun funcion A) are defined respecively as follows Define Le be he number of invesmen years 0 ), where a 0) =, he value a ime, denoed by a ) will be called he accumulaion funcion In he case of a posiive rae of reurn, as in he case of ineres, he accumulaion funcion is a coninuously increasing funcion Define Le P be he iniial principal invesed P > 0 ), where A0) = P, he accumulaed value ha amoun P grows o in years, denoed A), is defined as he amoun funcion In he case of a posiive rae of reurn, he accumulaion funcion also is a coninuously increasing funcion Ineres earned is he difference beween he accumulaed value a he end of a period and he accumulaed value a he beginning of he period Therefore, he relaionship beween a ) and A) is A) = Pa) ) In his paper, we focus on he accumulaion funcion Accumulae a single invesmen a a consan rae of ineres under he operaion of simple and compound ineres If an amoun P = is deposied in an accoun which pays simple ineres a he rae of i per annum and he accoun is closed afer years, here being no inervening paymens o or from he accoun, hen he amoun paid o he invesor when he accoun is closed will be a ) = + i )

3 0058 Causes of governmen financial informaion disclosure qualiy BTAIJ, 08) 04 This paymen consiss of a reurn of he iniial deposi P =, ogeher wih ineres of amouni The essenial feaure of simple ineres, as expressed algebraically by expression ), is ha ineres, once credied o an accoun, does no iself earn furher ineres This leads o inconsisencies which are avoided by he applicaion of compound ineres heory The essenial feaure of compound ineres is ha ineres iself earns ineres The operaion of compound ineres may be described as follows Consider a savings accoun, which pays compound ineres a rae i per annum, ino which is placed an iniial deposi P = We assume ha here are no furher paymens o or from he accoun) If he accoun is closed afer one year, he invesor will receive+ i More generally, le a ) be he amoun which will be received by he invesor if he accoun is closed afer years Thus, a ) = + i By definiion, he amoun received by he invesor on closing he accoun a he end of any year is equal o he amoun which would have been received, if he accoun had been closed one year previously, plus furher ineres of i imes his amoun Thus he ineres credied o he accoun up o he sar of he final year iself earns ineres a rae i per annum) over he final year So ha Thus, if he invesor closes he accoun afer years, he amoun received will be a ) = + i) 3) Compound ineres is he ypical compuaion applied in mos ime value applicaions In general, we have implicily assumed ha is an ineger However, he normal commercial pracice in relaion o fracional periods of a year is o pay ineres on a pro raa basis, so ha he expressions in his paper may be considered as applying for all non-negaive values of Examples for accumulaion value of simple and compound ineres In order o inroduce our heorem in nex secion, le us give an example for simple ineres and compound ineres TABLE : Simple and compound ineres principal accumulaion value comparison years) 00 0 o a ) $) a ) $) years) a ) $) a ) $) years) a ) $) a ) $) Figure : Simple and compound ineres principal accumulaion value comparison Le he annual ineres rae i = 07 o illusrae he difference effecs of he principal accumulaion values calculaed wih simple ineres and compound ineres, respecively, he principal amoun P = $ The principal accumulaion values, which are calculaed using expression ) and expression 3) respecively, are given in TABLE or in Figure

4 BTAIJ, 08) 04 Zhang Zenglian 0059 TABLE or Figure shows ha he accumulaion funcions of simple and compound ineres are sricly monoone increasing on[0, + ) Furhermore, Figure shows ha he graph of he accumulaion funcions of simple ineres is a proporional line; however, he graph of accumulaion funcions of compound ineres is he upward sloping one The heorem in he secion can also be clearly seen from comparison wih he graphs The heorem for he principal accumulaion value of simple and compound ineres Form he example above, he accumulaion funcion heorem can be derived as follows Theorem3 Assume ha he ime variable is a coninuous one on he inerval [0, + ) We can denoe he simple and compound ineres accumulaion funcion by he following forms respecively a ) = + i 4) and a ) = + i), 5) where i denoes he ineres rae Then we have a)if = 0 or =, hen a ) = a ) ; 6) b) If 0< <, hen a ) > a ) ; 7) c) If >, hen a ) < a ) and a ) < a ) 8) Proof Le f ) = a ) a ), ie f ) = + i + i) 0) 9) Using he derivaion rules, we have f ) = i + i) ln + i) 0) f and ) = + i) [ln + i)] < 0 Therefore, f ) is sricly monoone decreasing on he inerval[0, + ) Firsly, we shall prove a) Since ) a 0) = a 0), = and a ) = a ) = + i, The saemens of a) follow a once from he wo expressions above

5 0060 Causes of governmen financial informaion disclosure qualiy BTAIJ, 08) 04 Secondly, we shall prove b) Obviously, f ) is differeniable in he inerval 0,) Also, f ) is coninuous on he inerval [ 0,], and f 0) = f ) By Rolle s Theorem we have ha here exis a leas one poin ξ 0,) such ha f ξ ) = 0 Noe ha f ) is sricly monoone decreasing in he inerval 0,), so ha we have ha here exis one and only one poin ξ 0,) such ha f ξ ) = 0 When 0 < < ξ, we have f ) > f ξ ) = 0 ) Hence, we have ha f ) is sricly monoone increasing on he inerval [ 0, ξ ], hen f ξ ) > f ) > f 0) = 0 3) So, we have a ) > a ) On he oher hand, whenξ < <, obviously, we have f ) < f ξ ) = 0 4) Hence, we have ha f ) is sricly monoone decreasing on he inerval [ ξ,], hen f ξ ) > f ) > f ) = 0 5) So, we have a ) > a ) Therefore, his proves b) In he end, we shall prove c) Since f ) is sricly monoone decreasing on he inerval 0, + ), when >, we have f ) < f ) < f ξ ) = 0, 6) So, we have a ) < a ) Obviously, f ) is sricly monoone decreasing on he inerval [, + ), when >, we have f ) < f ) = 0 7) Namely, a ) < a ) Therefore, hese prove c) CONCLUSIONS In his paper, he accumulaion funcion heorem can be derived from he pracical example, and proved When a erm is less han a period, he accumulaed value of simple ineres is more han ha of compound ineres; When a erm is equal o a period, he accumulaed value of simple ineres is he same as ha of compound ineres; When a

6 BTAIJ, 08) 04 Zhang Zenglian 006 erm is more han a period, he accumulaed value of simple ineres is less han ha of compound ineres The heorem enriches he heory of ineres and financial heory ACKNOWLEDGEMENT This work was suppored by he Key Naural Science Fund of Sichuan Educaion Deparmen Gran No 3ZA003), he Key Scienific research fund of Xihua Universiy Gran No R68), Open Research Fund of Key Laboraory of Xihua Universiy Gran No Sjj0-00), Disinguished Young Fund of he School of Mahemaics and Compuer Engineering of Xihua Universiy, he Naional Key Technology R&D program No 0BAH6B00), he Inernaional Cooperaion Projec of Sichuan Province No 009HH0009), he Fund of Key Disciplinary of Sichuan Province No SZD ), he Naional naural Science Foundaion of China NSFC, No / U43330) and he Fund of Lab of Securiy Insurance of Cyberspace, Sichuan Province REFERENCES [] MFKhan; The Value of Money and Discouning in Islamic Perspecive Review of Islamic Economics, ), ) [] MHudson; The Mahemaical Economics of Compound Ineres: A 4,000-year old overview Journal of Economic Sudies, 745), ) [3] Murdick, GRober; Time value of money Engineering Managemen Review, IEEE, ), ) [4] CMSnowden; Compound ineres IEE Review, 48), ) [5] Xi Zhong-En; The calculaion of compound ineres: an oher perspecive Applicaion of Saisics and Managemen, ), ) [6] Li Wei; On calculaion of coninuous compounding of ineres Applicaion of Saisics and Managemen, 76), ) [7] JAlmenberg, CGerdes; Exponenial growh bias and financial lieracy Applied Economics Leers, 9, ) [8] KFischer, KFiedler; Reacion norms for age and size a mauriy in responseo emperaure: a es of he compound ineres hypohesis Evoluionary Ecol-ogy, 64), ) [9] CChiarella, VFanelli, SMusi; Modelling he evoluion of credi spreads using he Cox process wihin he HJM framework: A CDS opion pricing model European Journal of Operaional Research, 8), ) [0] JFalini; Pricing caps wih HJM models: The benefis of humped volailiy European Journal of Operaional Research, 073), ) [] SMira, P Dae, RMamon, ICWang; Pricing and risk managemen of ineres rae swaps European Journal of Operaional Research, 8), 0 03)