Development of New Loan Payment Models with Piecewise Geometric Gradient Series


 Marjorie Williams
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1 Yayın Geliş Tarihi: Dokz Eylül Üniversitesi Yayın Kabl Tarihi: İktisadi ve İdari Bilimler Fakültesi Dergisi Online Yayın Tarihi: Cilt:29, Sayı:2, Yıl:2014, ss Development of New Loan Payment Models with Piecewise Geometric Gradient Series Erdal AYDEMİR 1 Ramazan EROĞLU 2 Abstract Engineering economics plays an important role in decision making. Also, the cash flows, time vale of money and interest rates are the most important research fields in mathematical finance. Generalized formlae obtained from a variety of models with the time vale of money and cash flows are inadeqate to solve some problems. In this stdy, a new generalized formlae is considered for the first time and derived from a loan payment model which is a certain nmber of payment amont determined by cstomer at the beginning of payment period and the other repayments with piecewise linear gradient series. As a reslt, some nmerical examples with soltions are given for the developed models. Keywords: Annity, Piecewise Geometric Gradient Series, Payments, Time Vale of Money JEL Classification Codes: C0, C02, G21, G12 Parçalı Geometrik Değişimli Seriler ile Yeni Borç Ödeme Modellerinin Geliştirilmesi Öz Karar vermede, mühendislik ekonomisi önemli rol oynamaktadır. Bnnla birlikte, finans matematiği alanında en önemli konlar arasında paranın nakit akışı, zaman değeri ve faiz oranları yer almaktadır. Paranın zaman değeri ve nakit akışı problemlerinden elde edilen formüller bilimsel yazında blnmasına rağmen bazı problemlerin çözümünde b formüller yetersiz kalmaktadır. B çalışmada, başlangıçta belirli sayıda taksit miktarını müşterinin belirlediği, sonraki taksit miktarlarının parçalı aritmetik (miktarsal) değişim gösterdiği bir borç ödeme modeli ilk olarak ele alınmakta ve çözüm için genel formülleri elde edilmektedir. Sonçta, geliştirilen modeller sayısal örneklerle yglamalı olarak gösterilmiştir. Anahtar Kelimeler: Anüite, Parçalı Aritmetik Değişimli Seriler, Amortisman, Paranın Zaman Değeri JEL Sınıflandırma Kodları: C0, C02, G21, G12 1 Yrd. Doç. Dr., Süleyman Demirel Üniversitesi, Mühendislik Fakültesi Endüstri Mühendisliği Bölümü, 2 Süleyman Demirel Üniversitesi, İİBF İşletme Bölümü,
2 E. AYDEMİR R. EROĞLU 1. INTRODUCTION The problems with payments of a debt stand with eqality of the debt s present vale and the sm of the present vales (Shao and Shao, 1998: 502). For the economic analysis, we know the engineering economics also plays an important role in many decision making problems (Blank and Tarqin, 2005, Parvez, 2006). The repayment models of the series (instalments of the series) are fixed, geometric and arithmetic gradients that are located in the literatre for the general formlae (Park, 1997: 55). Another way to describe types of loans from scientific literatre, some papers have been presented abot its contracting (Graham et al., 2008; Hertzel and Officer, 2012), secrity, credit risk and insrance (Hatchett and Kühn, 2009; Mahayni and Schenieder, 2012), pricing options (Gündüz and UhrigHombrg, 2011; Peng et al., 2012; Dai and Chi, 2013), payment instrments and transactions (Hancock and Hmphrey, 1998; Parlor and Rajan, 2003; Maskara and Mllineax, 2011; Goel and Mehrota, 2012). All of this stdy, there is no contribtion abot repayment plans to identify in generalized formlae. The all of above payment models has an assmption that the payments are made at the end of each period. When we look at the scientific literatre in chronological jst last twenty years, some models were developed. According to Formato (1992), the new payment series was developed as skipped instalments that have addressed the problem of repayment of a debt in eqal instalments ronds withot instalments (back payments) in a certain time period at some randomly selected periods in some cases (for example holiday expenditres). Then, instead of the cycliceqal instalments, Moon (1994) developed a new model that was called geometric gradients skip payment. In 2002, an arithmetic gradients skip payment model was developed in loan payment models by Erogl and Karaoz. Firstly, the generalized formlae which has inclded reglar/irreglar piecewise and geometric/arithmetic gradients are developed by Erogl (2000). In addition, advanced models that are piecewise payment models with arithmetic and geometric gradients with arbitrary skips in randomly determined are also developed with the 96
3 Dokz Eylül Üniversitesi İktisadi ve İdari Bilimler Fakültesi Dergisi Cilt:29, Sayı:2, Yıl:2014, ss generalized formlae by Erogl (2001) and Erogl and Ozdemir (2012a). Then, the other developments were abot rhythmic skip models for the loan payments model by Erogl and Ozdemir (2012b) and Erogl et al. (2013). However, the costmers can be reqested with eqal instalments, increasing/decreasing instalments, and nonpayments periods in a payment model jointly and/or separately within some cases which are interest rates, solvency levels, concrrent debt payments etc. In order to provide these sitations, a new mathematical model is developed loan amortization payment models with piecewise geometric gradients in the case of the periods which have a certain nmber of instalments to be determined by the cstomers and the others have eqal annities with inclding nonpayment periods in this stdy. The main prpose of this stdy is to reply the differences of the cstomer reqests. So, the payment models developed by obtaining new two mathematical models are derived from the general formla and these are explained with nmerical examples in crrent convenient financial payment problems. 2. MATHEMATICAL MODEL Fixed payment models are mostly sed model for the loan amortization payments that are given from credit instittions. In other words, the amont of payment is eqal to each other. In fact, after a certain period from the payments was started, the cstomer's payment ability cold be involved from the acqisition of the variability (cases of increasing or decreasing). In this case, the cstomers want to change the instalment level in the amont of a certain nmber of beginning periods itself for considering nder changes in income for the coming periods. In this stdy, a new mathematical model is developed to pay a debt that is taken from a credit instittion in repayments (instalments) circits (monthly, qarterly etc.). The basic assmption of the model is determined to instalment of the pieces that is assmed to be eqal to each other for the periods in the beginning by the cstomer payment ability, the remaining pieces (+1 periods) of the debt are paid in the period. In here, is defined as nonpayment periods. A payment period has the nmber of periods and also a nonpayment period has the nmber of h 97
4 E. AYDEMİR R. EROĞLU periods. The payment and nonpayment periods are followed each other. In addition, another assmption is the amonts eqal instalments dring the period which change geometric changes for following periods. The notations are given as follows for the mathematical model: s : nmber of nonpayment/term periods f : payment nmbers/term nmbers for an inpayment period h : nonpayment nmbers/term nmbers for a nonpayment period, p : loan (debt) amont g : relative geometric gradients in payment amont, ( G = 1 + g) r : interest rate for period/term ( R = 1 + r) b : payment amont by cstomer identified : payment nmbers/terms by cstomer identified d k : payment amont for k th inpayment period n : nmber of all terms for all loan repayments (inpayment and nonpayment periods) The nmber of all terms for all loan repayments is defined as Eq.(1). n = + s( f + h) + f (1) as The present vale for a payment amont in period with r can be calclated b (1+ r) i = br i and it is mentioned that the first nmber of payments are is eqal by cstomer identified and the rest of the payments have geometric gradients. This case can be defined as follows in Eq.(3). d = dg k ; k = 0,1,..., s (3) k The repayment of a loan taken from banks and/or credit instittions, sing the predetermined and fixed interest rate of a period is r, is based to eqality with present vale of loan and sm of the present vale of all payments. Here, we can obtain some eqations that are given as follows: (2) 98
5 b p = + Dokz Eylül Üniversitesi İktisadi ve İdari Bilimler Fakültesi Dergisi Cilt:29, Sayı:2, Yıl:2014, ss ( ) d s + k f + h + f k i (4) i i= 1 R k= 0 i= + k ( f + h) + 1 R s+ 1 f ( ) ( ) ( ) rr ( GR 1) b GR 1 ( R 1) d 1 R GR 1 + ; G R = f b( R 1) + d ( s + 1)( 1 R ) ; G = R rr f + h f + h (4a) (4b) and ( ) ( ) prr GR 1 + b(1 R ) GR 1 1 ; G R s+ f ( f h) ( 1 R + ) ( GR ) 1 d = rpr b( R 1 ) ; f G = R f ( s + 1)( 1 R ) f + h + h (5a) (5b) If we se g = 0, we can obtain as G = 1+ g = 1. In addition, if we se Eq.(4a) and Eq.(5a) in the case of g = 0, we can also obtain the first nmber of payments are is eqal. These eqations are given as follows in Eq.(6) and Eq.(7) respectively for determining payment amont. f ( s+ 1) ( ) ( ) rr ( R 1) b R 1 ( R 1) + d 1 R R 1 p = d = ( ) f ( s+ 1) ( R ) R ( ) prr R 1 + b(1 R ) R Then, if we se b = 0 in Eq.(6) and Eq.(7), we can also obtain generalized formlae for the first nmber of nonpayments periods and the following periods with eqal amont. (6) (7) 99
6 f ( s+ 1) ( ) d 1 R R 1 p = rr d = prr ( R 1) ( R 1) f ( s+ 1) ( R ) R 1 1 E. AYDEMİR R. EROĞLU In addition, if we se = 0 in Eq.(8) and Eq.(9), we can obtain generalized formlae for rhythmic skips with eqal amont of payments. f ( s+ 1) ( ) d 1 R R 1 p = r R d = ( 1) ( 1) pr R f ( s+ 1) ( R ) R 1 1 (8) (9) (10) (11) 3. QUANTITIVE ANALYSIS 3.1. Nmerical Example 1 A cstomer wants to apply for loan amont of TL (Trkish Liras) from a bank and/or credit instittion. Abot the payment plan, the amont of the first three instalments are as 650 TL that are determined by the cstomer, the rest of the instalments will inclde three payments periods with two payment months and one nonpayments month is totally six instalments. In payment periods, there is geometric gradient series with 3.5% from one payment period to following payment period. In addition, monthly interest rate is determined by 1.2%. Ths, the model parameters are defined as p = 16000, = 3, s = 2, f = 2, h = 1, b = 650, r = 0.012, R = 1.012, g = and also G = Then, if Eq.(5a) is sed, the payment amont is d = TL in a payment period. Conseqently, this payment plan is demonstrated with Table 1 by sing Eq.(5). 100
7 Dokz Eylül Üniversitesi İktisadi ve İdari Bilimler Fakültesi Dergisi Cilt:29, Sayı:2, Yıl:2014, ss Table 1. The Payment Model for Example 1 Months Payment Amont (TL) Rest of Loan Amont (TL) * = * = * = * = * = * = * = * = * = * = * = 3.2. Nmerical Example 2 According to nmerical example 2, a cstomer wants to apply for loan amont of TL from a bank and/or credit instittion. Abot the payment plan, the amont of the first two instalments are as 650 TL that are determined by the cstomer, the rest of the instalments will inclde three payments periods with two payment months and one nonpayments month is totally six instalments. In payment periods, there is geometric gradient series with % from one payment period to following payment period. In addition, monthly interest rate is determined by 1%. Ths, the model parameters are defined as p = 16000, = 2, s = 2, f = 2, h = 1, b = 650, r = 0.01, R = 1.01, g = and also G R + + f h 2 1 = = 1.01 = Then, if Eq.(5b) is sed, the payment amont is d = TL in a payment period. Conseqently, this payment plan is demonstrated with Table 2 by sing Eq.(5). 101
8 E. AYDEMİR R. EROĞLU Table 2. The Payment Model for Example 2 Months Payment Amont (TL) Rest of Loan Amont (TL) * = * = * = * = * 1.01 = * = * = * 1.01 = * = * = 3.3. Nmerical Example 3 As a nmerical example 3, a cstomer wants to apply for loan amont of TL from a bank and/or credit instittion. Abot the payment plan, the first two instalments are nonpayment periods that are determined by the cstomer, the rest of the instalments will inclde three payments periods with one payment month is totally six instalments. In addition, monthly interest rate is determined by 2%. Ths, the model parameters are defined as p = 12000, = 2, s = 1, f = 3, h = 1, r = 0.02 and also R = Then, if Eq.(9) is sed, the payment amont is d = TL in a payment period. 102
9 Dokz Eylül Üniversitesi İktisadi ve İdari Bilimler Fakültesi Dergisi Cilt:29, Sayı:2, Yıl:2014, ss Table 3. The Payment Model with NonPayments Start to Loan for Example 3 Months Payment Amont (TL) Rest of Loan Amont (TL) * 1.02 = * 1.02 = * = * = * = * 1.02 = * = * = * = 4. CONCLUSIONS and DISCUSSIONS In fact, we know the fixed repayment models is not satisfied to the cstomers in variable interest rates at the crrent economies. Today economies are designed with higher variability in the cases from globalism. So, the cstomer types and the debt volmes are also changed in this process. The cstomers are reqested the new repayment financial models for their variable debts levels in globally market conditions. As a reslt of these changes, the new repayment models mst be derived in alternatively for loan. Hereafter, the loan financial companies and banks mst be presented more alternatives repayments models to the cstomers nder market conditions which are varieties of the interest rates, crrencies, revenes, solvency levels, debt volmes etc. If the variations of the payment series are increased, the nmber of cstomer type is also increased in simltaneosly. For this reason, the derivation of new financing (payment) models both in terms of financial instittions both the cstomers is great importance. In addition, the variability of revenes from cstomers in the event of time sometimes the payment and/or nonpayments terms may be sefl to differentiate cstomers. 103
10 E. AYDEMİR R. EROĞLU In this stdy, in the beginning periods (months) a certain nmber of repayments (instalments) the amont specified by the cstomer the next eqal instalments circits formed in the period of repayments from one period to the piecewise geometric change (gradients) demonstrate that arise when solving problems sed in the new loan payment models and the general formlae are derived and crrent examples of the model by fnctioning is shown. This stdy was the determination of the amont of payment for cstomers and pay the appropriate balance is important in forming. The reslts are obtained with the nmerical samples with a repayment schedle is shown in clearly. ACKNOWLEDGEMENTS The athors thank the cited researchers and the anonymos referees whose work largely constitte this research paper. REFERENCES BLANK, L., TARQUIN, A. (2005), Engineering Economy, Sixth Edition, McGraw Hill Companies, New York, USA. DAI, T.S., CHIU, C.Y. (2013), Pricing barrier stock options with discrete dividends by approximating analytical formlae, Qantitative Finance, 14(8), EROGLU, A., AYDEMIR E., SAHIN Y., KARAGUL N., KARAGUL K. (2013), Generalized formlae for the periodic fixed and geometricgradient series payment models in a skip payment loan with rhythmic skips, Jornal of Alanya Faclty of Bsiness, 5(3), EROGLU, A., OZDEMIR, G. (2012a), A home financing model based on partnership with piecewise geometric gradient series repayments, Jornal of the Faclty of Engineering and Architectre of Gazi University, 27(1), EROGLU, A., OZDEMIR, G. (2012b), A loan payment model with rhythmic skips, 3rd International Symposim on Sstainable Development, Sarajevo, Bosnia and Herzegovina, EROGLU, A., KARAOZ, M. (2002), Generalized formla for the periodic linear gradient series payment in a skip payment loan with arbitrary skips, The Engineering Economist, 47(1),
11 Dokz Eylül Üniversitesi İktisadi ve İdari Bilimler Fakültesi Dergisi Cilt:29, Sayı:2, Yıl:2014, ss EROGLU, A. (2000), Bir borcn taksitlerle geri odenmesi problemlerine cozüm onerileri, Süleyman Demirel University Jornal of the Faclty of Economics and Administrative Sciences, 5(1), EROGLU, A. (2001), Atlamalı taksitli bir borcn parcalı geometrik ve aritmetik degisimli taksitlerle ödenmesi problemlerine cözüm önerileri, Dmlpınar University Jornal of Social Sciences, 2001, 5: FORMATO, R.A. (1992), Generalized formla for the periodic payment in a skip payment loan with arbitrary skips, The Engineering Economist, 3(4), GOEL, R. K., MEHROTRA, A. N. (2012), Financial payment instrments and corrption, Applied Financial Economics, 22(11), GRAHAM, J. R., LI, S., QIU, J. (2008), Corporate misreporting and bank loan contracting, Jornal of Financial Economics, 89, GUNDUZ, Y., UHRIGHOMBURG, M. (2011), Predicting credit defalt swap prices with financial and pre datadriven approaches, Qantitative Finance, 11(12), HANCOCK, D., HUMPHREY, D. B. (1998), Payment transactions, instrments and systems: A srvey, Jornal of Banking & Finance, 21, HERTZEL, M. G., OFFICER, M. S. (2012), Indstry contagion in loan spreads, Jornal of Financial Economics, 103, MAHAYNI, A., SCHNEIDER, J. C. (2012), Variable annities and the option to seek risk: Why shold yo diversify?, Jornal of Banking & Finance, 36, MASKARA, P. K., MULLINEAUX, D. J. (2011), Information asymmetry and selfselection bias in bank loan annoncement stdies, Jornal of Financial Economics, 101, MOON, I. (1994), Generalized formla for the periodic geometric gradient series payment in a skip payment loan with arbitrary skips, The Engineering Economist, 39(2), PARK, C. S. (1997), Contemporary Engineering Economics, Second Edition, AddisonWesley Pblishing Com. Inc. PARLOUR, C. A., RAJAN, U. (2003), Payment for order flow, Jornal of Financial Economics, 68,
12 E. AYDEMİR R. EROĞLU PARVEZ, M. (2006), Time vale of money: application and rationalityan approach sing differential eqations and definite integrals, Jornal of Mathematics and Mathematical Sciences, 21, PENG, J., LEUNG, K. S., KWOK, Y. K. (2012), Pricing garanteed minimm withdrawal benefits nder stochastic interest rates, Qantitative Finance, 12(6), SHAO, S. P., SHAO, L. P. (1998), Mathematics for Management and Finance, Eight Edition, SothWestern College Pblishing. 106
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