TIME VALUE OF MONEY: APPLICATION AND RATIONALITY- AN APPROACH USING DIFFERENTIAL EQUATIONS AND DEFINITE INTEGRALS

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1 MPRA Muich Prsoal RPEc Archiv TIME VALUE OF MONEY: APPLICATION AND RATIONALITY- AN APPROACH USING DIFFERENTIAL EQUATIONS AND DEFINITE INTEGRALS Mahbub Parvz Daffodil Itratioal Uivrsy 6. Dcmbr 26 Oli at MPRA Papr No. 1794, postd 27. Sptmbr 28 14:27 UTC

2 TIME VALUE OF MONEY: APPLICATION AND RATIONALITY- AN APPROACH USING DIFFERENTIAL EQUATIONS AND DEFINITE INTEGRALS Mahbub Parvz, Dpartmt of Busiss Admiistratio, Faculty of Busiss & Ecoomics, Daffodil Itratioal Uivrsy, Dhaka. Abstract Th tim valu of moy is o of th most importat cocpts i fiac. Moy that a firm has i s possssio today is mor valuabl tha futur paymts bcaus today s moy ca b ivstd to ar posiv rturs i futur. Th pricipls of tim valu aalysis hav may applicatios, ragig from sttig up schduls for payig off loas to dcisios about whthr to acquir w quipmt. Problms cocrig Tim Valu of Moy, which ivolvs calculatio of ths cocpts, ar usually solvd by algbraic formula. This papr attmpts to solv such problms usig diffrtial quatios & dfi itgral tchiqus ad maks a compariso wh th rsults obtaid by th tradioal mthod. 1. Itroductio: Th Tim Valu of Moy is o of th most importat cocpts i fiac. Th sam amout of moy of today is mor valuabl tha that i th futur bcaus th valu of moy is always dcrasig. So, th timig of cash outflows ad iflows has importat coomic cosqucs. Th fiacial dcisio makig is basd o th cor cocpts of itrst, prst valu, auis, amortizatio, cratio of sikig fuds, tc. Problms cocrig Tim Valu of Moy which ivolvs calculatio of compoud itrst ad futur valu ar oft solvd by algbraic formula, which ivolvs xpots, logarhms tc. This papr attmpts to solv such problms usig diffrtial quatios & dfi itgral. From that poit of viw this study is somwhat diagostic format ad tris to stablish som formula rlatd to diffrtial quatio ad dfi itgral rgardig th solutios of th problms of tim valu of moy. 1

3 2. Tim Valu of Moy: Moy has tim valu. Th valu of Tk. 1 today is mor worthy tha th valu of Tk. 1 tomorrow. This coomic pricipl rcogizs that th passag of tim affcts th valu of moy. This rlatioship btw tim ad moy is calld th Tim Valu of Moy. Evryo bcoms ivolvd i trasactios whr itrst rats affct th amout to b paid or rcivd. Th valu of th moy is chagd cosidrig that itrst rat togthr wh tim. Modr busiss rlis 1 prct o th tim valu of moy ad o o ca mov a stp whout glctig th chagig rat. 3. Compoud Itrst Ad Th Futur Valu: Compoud itrst is th itrst that is ard o a giv dpos ad has bcom part of th pricipal at th d of a spcifid priod. Th futur valu of a prst amout is foud by applyig compoud itrst ovr a spcific priod of tim. Th cosquc of compoudig is that at ay tim t, th amout i th accout is y( t ) which is icrasig at th rat of th giv itrst. If th itrst rat is K prct ad spcifid priod of tim is vry small, this quatio approximatly ca b wrt as dy ( t) K y ( t)...( i) 1 whos solutio is Kt y A 1...( ii) Now, if th iial amout is y () y wh t, w gt y A, ad quatio (ii) bcoms, y y Kt 1 i.. (itrst rat) (tim) Futur Amout (Pricipal Amout) 2

4 Illustratio 3.1 If Tk. 5 is ivstd at 6 prct compoudd mothly, what will b th futur valu aftr 3 yars? Giv Pricipal Amout Tk. 5, Itrst rat 6 prct.5/moth, Tim 3 yars 36 moths. Accordig to our rul w gt Futur Amout 5.5(36) i.. Futur Amout Tk Usig algbraic formula w gt Futur Amout Tk. 5 (1 +.5) 36 Tk Hr rror i th amout.45 prct, which is ot sigificat. 4. Ordiary Auy: Futur Valu A auy is a sris of qual paymts mad at fixd itrvals for a spcifid umbr of priods. Auis ca b classifid by wh thy bgi ad d, by wh th paymts ar mad, ad by whthr or ot th paymt itrvals coicid wh th itrst itrvals. A ordiary or dfrrd auy cosists of sris of qual paymts mad at th d of ach priod. Now if th itrst rat is i, ad th first paymt is A, th accordig to th prvious sctio th chag i th amout dposd at ay tim t is giv by A ( t ) A If th itrval is small th th total futur valu aftr all paymts will b approximatly th ara udr th icrasig rat curv. Now if th total priod t 3

5 Th A ( ) A which is th formula to calculat th futur valu for ordiary auy. Illustratio 4.1 If Tk. 1 is dposd i a accout at th d of vry moth for th xt 5 yars, how much will b i th accout at th tim of th fial dpos if itrst rat is 8 % compoudd mothly? Giv, A Tk. 1, Itrst rat i 8 prct compoudd mothly 2/3 Total priod 5 yars 6 moths Th w hav from th formula A( ) A A i 1 i Hc th futur amout aftr 5 yars is Tk Usig algbraic formula w gt Futur amout 1 [ ( 1+ 2/3) 6 1] / (2/3) TK ad rror i th amout.4 prct, which is ot sigificat. 5. Ordiary Auy: Sikig Fud To grat th futur valu of a auy, priodic paymts ar mad ito a sikig fud. A sikig fud is a fud ito which priodic paymts ar mad i ordr to accumulat a spcifid amout at som poit i th futur. From th futur valu formula of ordiary auy w hav A ( ) A 4

6 Hc th formula for sikig fud paymt is A A( ) Illustratio 5.1 How much should b dposd i a sikig fud at th d of ach moth for 5 yars to accumulat Tk. 1, if th fud ars itrst 8 prct compoudd mothly? Hr, A ( ) 1,, whr 5 yars 6 moths, Itrst rat i 8 prct compoudd mothly 2/3 So, w hav from th formula for th sikig fud paymt A( ) A 1, 1 ( i 1) i 1 3 (.4 1) Ovr th lif of th sikig fud, th sum of th dposs will b 6 (135.55) TK This sum plus itrst ard will provid th dsird Tk. 1,. Usig algbraic formula w gt 2 Sikig fud Error i th amout.4 prct, which is isigificat. 5

7 6. Ordiary Auy: Prst Valu Th prst valu formula of a auy ca b gratd from th compoudd prst valu formula togthr wh th auy futur valu formula. Now, if w rfr Pricipal amout Prst valu P, Futur amout F, Itrst rat i, ad tim t (o. of yars) x (o. of priod pr yar) th, F P P i..( i ) Also w kow, A ( ) A, whr A ( ) is th futur valu aftr compltio of th priod, Hc w ca wr, Futur Valu aftr priod is F A.( ii ) Combiig quatios ( i ) ad ( ii ) w hav th formula for th prst valu of th ordiary auy as A P i Illustratio 6.1 Th dirctors of a compay hav votd to stablish a fud that will pay a rtirig accoutat or his stat Tk. 1 pr moth for th xt 1 yars, wh th first paymt to b mad a moth from ow. How much should b placd i th fud if ars itrst at 7% compoudd mothly? How much itrst will th fud ar durig s xistc? Giv paymt pr priod A Tk. 1, i 7 prct compoudd mothly.7 / 12, Total priod 1 yars 12 moths. W hav 1 ( i 1) i 6

8 Hc accordig to th formula A P i A i 1 i i A 1 i i Hc P Tk should b placd i th fud. For calculatig itrst, w hav (12 paymts) x (Tk. 1 pr paymt) Tk. 12, Itrst ard Tk. 12, Tk Usig algbraic formula w gt Prst Valu 1 [1- ( 1+.7/12) -12 ] / (.7/12) TK Hc rror i th amout.2 prct, which is a gligibl amout. 7. Ordiary Auy: Amortizatio Loa Amortizatio is th rmiatio of th qual priodic loa paymts cssary to provid a ldr wh a spcifid itrst rtur ad to rpay th loa pricipal ovr a spcifid priod. Promit xampls of amortizatio ar loas tak to buy a flat or a car ad amortizd ovr a priod of 5 to 1 yars. Giv th amout of th loa (th currt pricipal, P), th umbr of priods (), ad th itrst rat ( i ), th quaty to b calculatd is A, th amout of th priod paymt. Th paymts of A Taka ach costut a ordiary auy whos prst valu is P, ad w hav lard that i P A Solvig this for th ukow A, w hav Amortizatio Paymt: P i A 7

9 Illustratio 7.1 If som o borrow Tk. 5. H will amortiz th loa by halfmothly paymts of Tk. A ach ovr a priod of 3 yars. Fid th half-mothly paymt if itrst is 12 prct compoudd half-mothly. Also fid th total amout that th prso will pay. Giv P Tk. 5, Itrst rat i 12 % compoudd half-mothly.12/ 24.5 pr half moth, (3 yars )x ( 24 half moths pr yar) 72 priods. 1 Hr ( i 1) i Hc P i A i P i ( i 1) ip 1 i Th amortizatio paymt is Tk From algbraic formula w gt A 5[.5/ 1- (1+.5) -72 ] Hc thr is a rror of.21 prct which is vry isigificat. Amortizatio Schdul, 12% itrst compoudd half-mothly. Priod Bgiig Amout (a) Paymt (b) pr half moth ( c ) Rpaymt of pricipal d (b-c) Rmaiig balac (a -d)

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11 Rsults Ad Discussios: Th formula w usd i solvig problms as giv i illustratios hav most b wly drivd. Ths formula othr tha usd algbraic procdurs, utiliz solutio tchiqus of diffrtial quatios ad dfi itgrals. Ths mthods ad tchiqus ar w os. This papr attmpts to look at Th Tim Valu Of Moy problms wh mor advacd mathmatical formula which giv rsults wh som rrors. Ths rsults ar show i Amortizatio tabl. This discrpacy could, prsumably i ar futur, b proprly hadld by som rsarchr itrstd i th fild. Th author is thakful to Profssor M. Abdul Malk, Dpartmt of Mathmatics, Jahagiragar Uivrsy, Savar, Dhaka for his assistac i prparig this papr. Rfrcs: Brigham, Eug F, Gapski, Louis, Ehrhar, Michal C, (21), Tim Valu of Moy, Fiacial Maagmt-Thory ad Practic, dio-9. Gma, Lawrc J, (24), Tim Valu of Moy, Pricipls of Maagrial Fiac, dio-1. Prichtt, Gordo D & Sabr, Joh C, (24), Mathmatics of Fiac, Itroductio to calculus,mathmatics wh applicatio i Maagmt ad Ecoomics, dio-7. 1