Mathematical Model of Banking Firm as Tool for Analysis, Management and Learning

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1 Mahemaical Model of Baning Firm as ool for Analysis Managemen and Learning Vicor Selyuin 12 and Margaria Rudeno 1 1 Research Insiue of Mechanics and Applied Mahemaics vvs1812@gmail.com riusi@mail.ru 2 Economic Faculy of Souhern Federal Universiy 22/1 av. Sachi 3449 Rosov-on-on Russia Absrac. An essenial concern for baning firms is he problem of asses and liabiliies managing ALM. Over las years a lo of model ools were offered for solving his problem. We offer he novel approach o ALM based on ranspor equaions for loan and deposi dynamics. Given he ban's iniial sae and various deposi inflow scenarios he model allows provide simulaions including sress-esing and can be used for assessmen of liquidiy ris for eamine loan issue decisions o choose reasonable soluion and in he learning purposes. Keywords. Asse- liabiliy managemen ifferenial equaions Liquidiy ris uraion Key erms. Baning Mahemaical Modelling ecision maing 1 Inroducion A baning firm is raher a comple sysem wihin he cone of managemen problem. I is caused by a considerable number of financial flows and he funds having a various origin and differing by dynamic and probabilisic characerisics and a he same ime forming he unified sysem. Sable funcioning of he sysem is provided due o hierarchy eernal prudenial supervision and inernal regulaors and resricions and feedbacs. Among he mahemaical models of baning firms i is possible o separae wo basic groups. here are models of opimizaion of asses porfolio saic single and muli-period using linear and dynamic programming mainly [1-2] and models of asses and liabiliy managemen ALM using mehodology and he echnique of he sochasic differenial equaions [3-5]. One of he problem solved by models ALM is managemen of various riss especially credi ris and ineres-rae ris including he problem of defaul probabiliy decrease.

2 42 V. Selyuin and M. Rudeno In connecion wih compuer engineering developmen from he middle of 7h years of he las cenury he compuer models of bans focused on problems of planning and decision-maing suppor sysems began o appear. However such projecs had no furher developmen [6-8]. hen we will urn our aenion o one of he possible approaches o ban modelling as a dynamic sysem which can be called hybrid. he main ass which he model developed mus solve are he analysis and managemen of liquidiy and sressesing of a ban. In addiion i can be used for opimizaion of asses profile. Aggregaion of elemens of balance shee can be varied according o he objecives of modelling and principles developing of sae variables vecor. We will use he following simplified schemaic ab. 1. Fied asses of ban we will ignore aing ino accoun only financial flows. Obviously balance shee equaion aes place: A= S+B+Q+=+C+M=L 1 where equiy capial of a ban C is a balancing variable. For deailed modelling of credi riss loans issued can be divided by caegories of he debors having various reliabiliies. ivision of deposis on ime and demand is necessary for calculaion of insan liquidiy. I is ignored in considered below version of he model for simpliciy. able 1. he aggregaed balance shee of commercial ban. Asses A Loans issued : Securiies Reserves S Business Privae cusomers buyer`s credis morgage ec. Oher bans Shares Q Bonds B Cash Res fund loan loss reserves ec. Liabiliies L eb ime deposis On-demand deposis and curren accouns Iner-ban credis M Equiy including reained profi of las periods C Formally i is possible o mar hree groups of operaions in he balance-shee able: Reallocaion of asses beween separae iems Reallocaion of liabiliies beween separae iems Idenical change of asses and liabiliies a one period hough he ban opens a posiion in liabiliies wih gran of a loan opening of a credi line a one ime from he formal poin of view his operaion is resolved ino reallocaion of asse`s iems.

3 Mahemaical Model of Baning Firm 43 Similarly if he deposi remains unclaimed in mauriy dae i eiher is prolonged or is ransferred in demand deposis wih no ineres accruing or wih he minimum percenage according o conac condiions. Acually in his case here is a reallocaion of liabiliy`s iems. A las when ineres on loans or oher ypes of income or epenses are received or repaid a one ime i is changed boh asses and liabiliies own capial of ban increases or decreases. 2 Model wih Cerain erms of Loans and Araced Funds he main difficuly in modelling of asses and liabiliies dynamics is concerned wih necessiy aing ino accoun erms of loans and deposis. ue o hese he sae variables mus depend on wo parameers - curren ime and curren "age" or he remained erm o mauriy -. ha is why dynamics of he issued loans can be described by following ranspor equaion: u 2 In addiion d - oal amoun loans issued e d - presen value of loans Т erm of loans. Movemen of ime deposis is described similarly: y y v y d - oal amoun of ime deposis y e d - presen value of ime deposis erm of deposis. Variables u and v denoe he flows of issued loans emporary ouflow of financial resources of ban and deposis emporary inflow disribued by ime aing ino accoun amorizaion ineres paymen or insallmen credis. Accordingly oal inpus of loans U and deposis V or is presen values U and V is described as: U u d U u e d V v d V v e d Soluion of he equaions 2-3 can be represened in he closed form: 3

4 44 V. Selyuin and M. Rudeno d u d v y where and - iniial disribuions of loans and deposis by "age" or may be obained by use corresponding equaions wih finie differences. ynamics of reserves S and equiy C is described by he equaions including sochasic members which consider random naure of change in value of shares and possible loans losses: M B dj QdW Qd d Z M B V U ds M B dj QdW Qd d Z M B dc where dw incremen of Wiener sochasic process dj incremen of compound Poisson process wih eponenial disribued size of jumps loan losses Z operaion epenses and paymen for dividends; repaymen of a loans in mauriy dae B M accordingly ineres on loans deposis bonds income cos of credis; - average porfolio reurn of rading securiies - volailiy of securiies porfolio. Invesmens in liquid asses - shares Q and bonds B can be considered as some parameers of managemen and o be calculaed proceeding from srucure of asses chosen or planned by ban aing ino accoun loan demand. Similarly he volume of received loans M can be selec depending on ban`s requiremen in financial resources. I is necessary o add he equaions of dynamics of duraion o he equaions of movemen of asses and liabiliies o model liquidiy ris aing ino accoun change of ineres raes If r he annual ineres rae so in his case Macaulay duraion for an asse is defined by epression: d e 1 where = ln1+r. Similarly duraion of anoher financial flows y u v are calculaed. I is possible o show ha dynamics of duraion is described by any of presened below he equaions which is chosen according o liquidiy research ass. u U d d 1 u U d d 1

5 Mahemaical Model of Baning Firm 45 d d [ where d 1 d u ] 1 u Model 2-3 has been ransformed o sysem of difference equaions and realized as compuer program [9]. he user independenly chooses one of wo operaing modes of he program: calculaion in case of predefined planning horizon or calculaion wih possible correcion of parameers seing physical speed of calculaion. he program is ineracive as he user can change values of some ey parameers in he process of calculaion wihou inerruping is wor. As ey parameers are chosen: a fracion of cash invesed in various inds of asses revenues ineres raes a duraion of demand deposis credi demand inflow of deposis crediing scenarios disribuion of loans by ime. ynamics of inflows and ouflows of cashes; diagram of change of duraions of asses and liabiliies; disribuions of loans and deposis and also inpu flow by ime are displayed on he screen of compuer. A sress-esing is provided in he program. he user can choose he period of sress-esing and such sresses-scenarios as decrease in inflow of deposis decrease in duraion of deposis he scenario of ouflow of deposis; decrease in accessible volume of araced funds on he inerban mare. 3 Model wih Fied erms of Lending and Borrowing Model 2-3 presened above is raher difficul in numerical realizaion and does no allow o consider some imporan facs for eample dependence on ineres raes from differen erms of lending or borrowing. herefore we will consider simplified modificaion of previous model under supposing ha erms of loans and deposis are fied. I is possible o fi he mos ypical erms according o he classificaion used in he ban reporing in spie of he fac ha erms of loans or deposis can be arbirary. Boh loans and ime deposis are srucured by erms as follows: ill 3 days from 31 ill 9 days from 91 ill 18 days from 181 days ill 1 year from 1 year ill 3 years over 3 years. Accordingly i is possible o esablish several ypical periods for each of hem ime ransacions are described by he parial differenial equaion of he firs order a 4 wih a boundary condiion =u and he iniial condiion =. Iniial and boundary condiion should be consisen ha is u=. Here - curren ime < - elapsed ime since he momen of selemen of ransacion "age" of an loan or deposi < a- value of "amorizaion" of an loan or deposi inflaion insallmen credi ec..

6 46 V. Selyuin and M. Rudeno Similarly 2-3 he variable is he allocaed variable characerizing some credi ools accouned in asses or in liabiliies loans for limied period ime deposis inerban lending or borrowing coupon bonds or oher asses and liabiliies wih he fied erm of repaymen. Furher i will be assumed ha a 5 i.e. repaymen of credis occurs proporionally o heir volume wih coefficien which is no dependen on age. I can be used and oher schemes when credi repaymen begins no a once and or occurs in advance esablished equal shares. I is easy o verify ha he soluion of he equaion 4 loos lie a ravelling wave u ep 6 For consisency an iniial and boundary condiions a < i is necessary o predeermine u on an inerval [-. From 4-6 follows u ep 7 and afer replacemen for - u ep under -< 8 he oal value of he considered loan or deposi are obained by inegraion on age Subsiuing 6 in 9 we have d u ep d 1 Inegraing 4 we obain he ordinal differenial equaion d u u u ep 11 d As asses wih differen erms of repaymen are in porfolio of asses or liabiliies so i is possible o replace scalar variable in 11 wih vecor. Vecor`s componens are financial ools wih differen erms of repaymen d u u u ep 12 d For simpliciy furher we will suppose = where - he erm epressed in monhs. ime ools issued loans bonds inerban credis ime deposis from he mahemaical poin of view are similar ha is why we will consider hem in he cone of one and only consrucion giving he general designaion: - o ime ools in asses and in liabiliies. hen he previous model can be presened as: d u u u ep 13 d ds Sd SdW f d 14 9

7 Mahemaical Model of Baning Firm 47 d d dq d d d d d dz d g f 15 v y v v ep 16 dz d z Z w 17 where w - inflow of on-demand deposis v - inflow of ime deposis and borrowed funds; f - purchase + or sale - rading securiies /s; g operaion coss on carrying ou of aciviies of ban; - securiies porfolio reurn; - volailiy of securiies porfolio; W - Wiener sochasic process; - ineres on he ime deposis and borrowed funds; - ineres on issued loans; z - duraion characerisic urnover ime on-demand deposis. I is easily o obain he equaion of dynamics of equiy by differeniaion of balance equaliy and corresponding subsiuions As follows dc ds f g 18 d d For simpliciy i is supposed complee wihdrawal of deposis afer erm in his version of model. However i is easy o ae ino accoun possibiliy of prolongaion of he deposi or is ransfer in caegory on-demand deposis. I is considered ha dividends are no paid. Besides credi riss defaul ris or a delay of paymens are no considered ha also i is possible o ae ino accoun by enering of corresponding adjusmens. I is considered ha ineress on he araced funds and he received credis are paid according o accrual. However i is easy o se and oher scheme in which ineress are accumulaed on deposiary accouns and are paid afer erm of deposi. Le and - - srucure of ime loans and deposis. Besides for simpliciy we will assume ha here are no invesmens in rading securiies. hen dynamics of he capials are described by he equaion: dc g 19 d I is giving eviden represenaion abou sensiiviy of dynamics of capial o changes of main parameers of asses and liabiliies. Main objecive of shareholders and ban managemen is he increase in capial: dc ma 2 d subjec o resricions on financial resources and riss credi and mare loss of liquidiy banrupcy.

8 48 V. Selyuin and M. Rudeno 4 Conclusions he approach o mahemaical modelling of cash flow moving in asse and liabiliy accouns of he commercial ban based on he parial differenial equaions is novel and has no analogues in he lieraure. A he same ime he given approach is quie logic as reflecs process of change of acives simulaneously in ime and on "age". epending on paricular heoreical or pracical problems he given approach can be realized in he various modificaions wo of which are presened in he aricle. As he preliminary esing has shown he compuer program creaed by use model 2-3 allows provide various simulaions including sress-esing and can be used in he educaional purposes o provide he bes undersanding of he dynamic processes aing place in baning firm. I is necessary he furher developmen of he offered modelling approach such as improvemen of program ool and also as required model deailed elaboraion o use hese models as par of decision suppor sysem for asse and liabiliy managemen in commercial ban. he modified model has been proposed for hese goals. References 1. Chi G. ong H. Sun.: ecision Maing Model of Ban s Asses Porfolio Based on Muli-period ynamic Opimizaion. Sysems Engineering heory & Pracice Kruger M.: A Goal Programming Approach o Sraegic Ban Balance Shee Managemen. Baning Financial Services and Insurance. In: Proc. SAS Global Forum Paper Kosmidou K. Zopounidis C.: Asse Liabiliy Managemen echniques. Handboo of Financial Engineering pp Springer Science+Business Media LLC Muuddem-Peersen J. Peersen M.A.: Ban Managemen via Sochasic Opimal Conrol. Auomaica Mulvey J.M. Shey B.: Financial Planning via Muli-sage Sochasic Opimizaion. Compuers & Operaions Research Solyanin A.A.: Compuerizaion of he Financial Analysis and Forecasing in Ban. Fin- SaInform Мoscow 1998 in Russian 7. Robinson R.S.: BANKMO: an Ineracive Simulaion Aid for Ban Financial Planning. J. Ban Res Moynihan G.P. Purushohaman P. McLeod R.W. Nichols W.G.: SSALM: a ecision Suppor Sysem for Asse and Liabiliy Managemen. ecision Suppor Sys Aleseev I.V. Selyuin V.V.: Ineracive Compuer Model of Ban`s Asse and Liabiliy ynamics. erra Economicus 94 Par in Russian