CAPITALIZATION (PREVENTION) OF PAYMENT PAYMENTS WITH PERIOD OF DIFFERENT MATURITY FROM THE PERIOD OF PAYMENTS

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1 Iteratioal Joural of Ecoomics, Commerce ad Maagemet Uited Kigdom Vol. VI, Issue 9, September ISSN CAPITALIZATION (PREVENTION) OF PAYMENT PAYMENTS WITH PERIOD OF DIFFERENT MATURITY FROM THE PERIOD OF PAYMENTS Shkelqim Fortuzi Professor, Faculty of Busiess, Aleksader Moisiu Uiversity, Durres, Albaia Vladimir Muka Dea of Faculty of Studies Itegrated with Practice, Aleksader Moisiu Uiversity Durres, Albaia Abstract The cash maturity practice at the momet of paymet is applied i all cotracts that bakig istitutios associate with their depositig cliets or borrowers. But this does ot exclude the review of auity cases where the maturity of paymets becomes more frequet or less frequet tha the paymet of paymets. I the dyamics of daily life, there are also auities with a maturity period differet from the paymet period. For illustratio, let's preset the followig situatio: Let's assume that a perso wats to accumulate i his bak accout a fud to be available durig retiremet years. I fulfillmet of his wish he deposits every 3 moth ed the amout of 500 with a iterest rate of 5%. Its deposits cotiue periodically for a certai period of time. Subsequetly, the accumulated fud deposits it with the same iterest rate util the momet of retiremet. After retiremet she withdraws every moth the sum of 100. Let's assume, however, that after makig a certai umber of withdrawals, the perso wats to kow the situatio i his bak accout. To fix this situatio, actios should be made for two fiacial operatios: the deposit operatio ad the withdrawal operatio. The deposit series forms a deposited auity, the future value of which is matched by the formula: (1 i) 1 S R (1 i ) Where, (S) amout of a ordiary auity of () paymets, (R) i periodic ret is the size of each paymet which are made at the ed of each period, (i) the Licesed uder Creative Commo Page 313

2 Fortuzi & Muka iterest rate for a maturity period, () the umber of coversio periods ad ( ) the umber of maturity periods after the last paymet made. The withdrawal series forms aother auity with differet specificatios from the auity formed by deposits. Uder the agreemet draw up at the iitial poit (the agreemet remais i force for all time) the moey matures at every 3 moth ed ad is withdraw each ed moth. As we ca see, durig the secod fiacial operatio we have to do with a auity where the maturity period is differet from the paymet period. Key words: Maturity, Future Value, Preset Value, Paymet period, Auity, Iterest Rate, Capitalizatio, Compoud Iterest, Effective Aual Rates INTRODUCTION For rets with the same maturity as the paymet period, the applicable formulas are: S (1 i) 1 R i (to calculate the Future Value) ad A 1 (1 i) R i (to calculate the Preset Value). We ote with (m) the umber of maturity per year ad (p) the umber of paymets per year (m p). The iterest rate for a maturity period is equalized by the formula: r m i k m Where, (r) is the aual iterest rate. The fractio p idicates the umber of maturity periods that cotais a period of paymet. Auity with maturity differet from the paymet period Auities with maturity periods differet from the paymet period are of two types: - Auities with more frequet maturities tha paymets; - Auities with less frequet maturities tha paymets. Figure 1 shows a more frequet maturity tha the paymets. A paymet period cotais four maturity terms. R R Figure 1 More frequet maturity tha the paymets Licesed uder Creative Commo Page 314

3 Iteratioal Joural of Ecoomics, Commerce ad Maagemet, Uited Kigdom Figure 2 shows a auity with less frequet maturities tha paymets. I this figure, a maturity period cotais four paymet periods. R R R R R Figure 2 Auity with less frequet maturities tha paymets I Figure 1 ad Figure 2 paymets are made at each ed of the paymet period. I these types of auities formulas for future value or preset value of ordiary auity, caot be applied directly. I these formulas there is iterest rate (i). Accordig to this iterest rate, the maturity of the moey is made at the time of paymet. Therefore, the iterest rate (i) is used i ay case whe the maturity of the moey coicides with the momet of paymet ad caot be used i cases whe the maturity date does ot match with the momet of paymet. We ote with: p: Number of paymets made per year; j: iterest rate for a period of paymet; k: the umber of maturity periods that cotais a period of paymet. Meawhile, accordig to symbolism used for capitalizatio of simple or compoud iterest, for capitalizatio or actualizatio of auities the symbol (m) shows the umber of maturity periods i a year ad the symbol (i) idicates the iterest rate for a maturity period. m k The umber (k) is equalized by equatio, p Formulas o Capitalizatio (Actualizatio) Of Auity Sice two ordiary auity formulas become applicable also for auities with maturity periods differet from the iterest rate period (i), the iterest rate (i) should be replaced by the rate (j) for a period of paymet. The iterest rate (j) should be such that it meets two coditios: - mature the moey at the momet of paymet; - be equivalet 1 to the rate (i). 1 Two iterest rates are equivalet to each other if they geerate the same future value over the same period ad for the same pricipal. Licesed uder Creative Commo Page 315

4 Fortuzi & Muka To derive a formula that sets the iterest rate (j) we base o the equatio of their effective 2 aual rates. The effective aual rate of iterest rate (i) is equalized by the formula: i ef (maturity) = (1 + i) m 1 The effective aual rate of iterest rate (j) is equalized by the formula: j ef (of paymets) = (1 + j) p 1 By equatig these two effective aual rates we have: j ef (of paymets) = i ef (matured) (1 + j) p 1 = (1 + i) m 1 (1 + j) p = (1 + i) m 1 j (1 i) m p k j (1 i) 1 (1) The iterest rate (j) has the specificity that achieves the matchig of the paymet period with the maturity period. Accordig to the iterest rate (j) each paymet period is simultaeously a maturity period. Thus, the total umber of maturity periods is ow equal to the total umber of paymets. p t Well, (2) Usig the recociled iterest rate (j) by equatio (1) ad the umber of maturity periods () equalized by the equatio (2) formulas for the future value or the preset value of the auity commoly coverted ito valid auity formulas with maturity dates differet from the paymet period. For a auity with maturity periods differet from the paymet periods The future value is equated by formula: S (1 j) 1 R j (3) 2 The effective aual rate (i ef ) is the aual compoud iterest rate equivalet to the aual iterest rate (r) maturig several times a year Licesed uder Creative Commo Page 316

5 Iteratioal Joural of Ecoomics, Commerce ad Maagemet, Uited Kigdom Let us refer to Figure 3: R R R R R R (1 + i) k R (1 + i) 2k R (1 + i) ( 2)k R (1 + i) ( 1)k S Figure 3 The future value The followig table shows the future value of each paymet as well as the umber of periods durig the paymet geerates iterest. Table 1 Future value of each paymet ad umber of periods durig the paymet geerates iterest Paymet: Paymet geerates iterest for: Future Value Last oe 0 (zero) maturity periods R The peultimate k maturity periods (1 paymet period) R (1 + i) k Third from the ed 2k maturity periods (2 paymet period) R (1 + i) 2k Secod ( 2)k maturities ( 2 paymet period) R (1 + i) ( 2)k First ( 1)k maturities ( 1 paymet period) R (1 + i) ( 1)k Startig with the last paymet, the sum of the future values of all paymets with R value (the future value of the auity with maturity dates differet from the paymet period) is: S = R + R(1 + i) k + R(1 + i) 2k + + R(1 + i) ( 2)k + R(1 + i) ( 1)k. If we write q = (1 + i) k this equatio takes the form: S = R + Rq + Rq Rq 2 + Rq 1. The right side of this sum forms a geometric progressio with the first limit (R), the quotiet (q), ad the umber of the sums (). We have: q 1 (1 q 1) 1 S R R q 1 q 1 Licesed uder Creative Commo Page 317

6 Fortuzi & Muka If we write j = q 1 = (1 + i) k 1 this equatio takes the form: S (1 j) 1 R j. I the same way, the formula for calculatig the preset value is: A 1 (1 j) R j (4) RECOMMENDATIONS 1. To calculate the future value (S ) or the preset value (A ) of a ret with more frequet maturity tha the paymet, we follow two steps: First, the iterest rate (j) is calculated for a period of paymet equivalet to the rate (i) for a maturity period by applyig the formula (2). Secod, depedig o the demad expressed i the problem situatio, is applied formula (1) if the future value (S ) is applied or formula (3) is applied if the preset value (A ) is required. 2. The procedure for calculatig the actual paymet for a more frequet maturity tha the paymet is: First, the iterest rate (j) is calculated for a period of paymet equivalet to the rate (i) for a maturity period by applyig the formula (2). Secod, apply the formula (1) if the future value (S ) is applied or formula (3) is applied if the preset value (A ) is give. 3. To calculate the future value (S ), the preset value (A ) or the periodic paymet (R) of a ret with less maturity tha the paymets ad paymets betwee the two maturity dates is doe with compoud iterest rate formulas of the settlemet of the problem situatio is the same as the procedure applied to rets with more frequet mature payables tha paymets. 4. To calculate the future value (S ) or the preset value (A ) of a ret with less frequet maturity tha the paymets ad paymets betwee the two maturity dates do ot geerate iterest or geerate iterest uder simple iterest formulas, follow as: First, the value of virtual paymet (R v ) is calculated as the arithmetical amout of real paymets betwee maturity dates if these paymets do ot geerate or how may of the matured amouts if the paymets betwee the two maturity dates geerate iterest uder the simple iterest formulas. Secodly, depedig o the demad expressed i the problem situatio, apply formula (4) if the future value (S ) is applied or formula (5) is applied if preset value (A ) is required. Licesed uder Creative Commo Page 318

7 Iteratioal Joural of Ecoomics, Commerce ad Maagemet, Uited Kigdom 5. For the calculatio of the periodic paymet (R), whe the values of the other parameters of the ret with less maturities tha the paymet are give, ad the paymets betwee the two maturity dates are capitalized with simple iterest formulas, apply the formula (4) if it is give (S ) or (5) if it is give (A ). By applyig oe of the formulas we calculate the virtual paymets (R V). Equatig the virtual paymet calculated with the matured amout (the arithmetical amout whe the paymets betwee the two maturity dates do ot geerate iterest) of the real paymets is formed a first-rate equatio with a ukow whose solutio gives the value of the real paymet. REFERENCES A Itroductio to the Mathematics of Fiace, Secod Editio, S. J. Garrett, Copyright 2013 Istitute ad Faculty of Actuaries (RC000243). A Itroductio to the Mathematics of Moey, David Lovelock, Marilou Medel, A. Larry Wright, 2007 Spriger Sciece+Busiess Media, LLC. Elemets of Mathematics for Ecoomics ad Fiace, Vassilis C. Mavro ad Timothy N. Phillips. Copyright Spriger-Verlag Lodo Fiacial Mathematics, J Robert Buchaa, MiNersviile Uiversity, USA. Prited i Sigapore by World Scietific Priters (S) Pte Ltd (2006). Fudametals of Corporate Fiace, Third Editio, Richard A. Brealey (Bak of Eglad ad Lodo Busiess School), Stewart C. Myers (Sloa School of Maagemet Massachusetts Istitute of Techology), Ala J. Marcus (Wallace E. Carroll School of Maagemet Bosto College). Copyright 2001 by The McGraw-Hill Compaies. Mathematics for ecoomics ad busiess, Ia Jacques, botim i pestë i vitit Prited ad boud by Mateu-Cromo Artes Graficas, Spai. Mathematics of Fiace, Eighth Editio, Robert Cissell Formerly of Xavier Uiversity, Copyright 1990 by Houghto Miffli Compay. The Math of Moey, Morto D. Davis, 2001 Spriger Sciece+Busiess Media New York. The Mathematics of Bakig ad Fiace, Deis Cox ad Michael Cox, Califoria. Copyright 2006 Joh Wiley & Sos Ltd, The Atrium, Souther Gate, Chichester, West Sussex PO19 8SQ, Eglad The Mathematics of Moey, Timothy J. Biehler, Copyright 2008 by The McGraw-Hill Compaies, Ic. Licesed uder Creative Commo Page 319

Chapter 3. Compound interest

Chapter 3. Compound interest Chapter 3 Compoud iterest 1 Simple iterest ad compoud amout formula Formula for compoud amout iterest is: S P ( 1 Where : S: the amout at compoud iterest P: the pricipal i: the rate per coversio period