The theoretical aspects of mathematical modeling in the banking on the ex ample of compound interest

The compound interest is used in banking. It is one of the ways of establishing the interests (the compound interest), so the fee for the right to have the money at the disposal. It is associated with the multiple calculating of the same percentage from the capital whereby every time the interests are capitalized; they increase the base from which the next percentage is calculated. In this paper are discussed the matters concerning the annual, subperiod and continue capitalization and also associated with them the term of percentage: annual, subperiod, nominal, effective, continuing, and average. The reader will find here the mathematical models enabling to calculate:


Introduction
Bank is the institution which deals with the cash operations. The term interest is associated with its activity. It is the fee for the right to operate the monetary capital calculated as the percentage. The value of the interest is depended on the percentage, the capital size, the time duration on which the capital was made available for and also the rules according to which the calculations are made. The basics of one of them is the presented in this paper the model of calculations called the compound percentage.

Material and methods
The compound percentage is the type of calculating which is characterized by the capitalization of the interest-after the passage The factor ( ) 1 r + which is in all formulas is called the annual percentage factor. The equations (25) and (26) create the model of compound percentage within the annual capitalization (also known as the model of annual percentage).
From the mathematical point of view within the compound interest, the future value of F capital is the exponential function of percentage time. In practice the dynamics of the process of capital changes within the time is crucial. Its mathematical measurements are the absolute and relative increase. Using the annual percentage model, it is possible to appoint all the values. From the model of the annual compound interest the following conclusions can be made: • The values of the capital • The sequence n I , which is the reflection of the interest for the subsequent years 0,1, 2,3, n = is the geometrical sequence in which the first word is equal to Pr and the quotient 1 r + . • The compound interest is equal to simple interest within the same interest rate r at the end of the first year ( 1 n = ). In each subsequent year the compound interest is higher than the simple interest.
Within the annual interest rate r and the annual capitalization of the interest, the capital will double its value in the time.
The simple percentage as the approximation of the compound interest for the period shorter than the time of the capitalization of interest While calculating the interest for the period of time shorter than the period of their capitalization in the bank's practice, the compound interest is replaced with the simple percentage.
We assume that capital P is yield in n time. The number n is presented in the form of the sum of two components n m m =+ (7) where   mn = total n . We approve the rule that the interest for the m time is calculated as the compound interest, while for m -as the simple percentage. Where 0 m = , the future F value is calculated using the mathematical formula (2).
When 01 m  , calculating the future value of the capital is divided into two stages. First, we calculate the future value of capital F after m periods of compound interest which is ( ) Next, we determine the approximate value of F capital after m time of simple percentage which is equal to: In the case, when n is the integer number there is the equality FF = . If n is not the integer number FF  , from which stems that for the creditor the approximate percentage is more beneficial to the compound one.

The capitalization of the interest within the subperiods
In the banking the capitalization of the interest is also done after time shorter than the year. The established time after which the interest is capitalized is named the subperiod of capitalization while the percentage corresponding the subperiod -subperiod percentage. The number which represents how many times within the year the interest is capitalized is called the frequency of capitalization.
We introduce the following indications: P -the initial value of the capital F -the future (final) value of the capital n -the time of the percentage in years constructed from the total number of subperiods k -the frequency of capitalization within the year where 2 k = , the capitalization is semiannual where 4 k = , the capitalization is quarterly where 12 k = , the capitalization is monthly Equations (12) and (13) are named the model of compound percentage within the subperiod capitalization (the model of subperiod percentage).
After taking into account the equality (10) and (11) the formula (12) and (13) From the equations (14) and (15) we de duce that within the fixe nominal percentage k r , the frequency k of the capitalized interest will be increased (the length of the subperiod will be shortened), the value of the interest and the final capital will be increasing. Therefore, it can be said that the shortening of the capitalization subperiod increases the pace of growth of the final capital. (Bijak, Podgórska, Utkin, 1994).
The relationship of the value of the capital from the end of any year to the value from the beginning of the year is named the annual factor of the percentage and is indicated with k  symbol. It has the fixed value within the percentage time (Podgórska, Klimkowska, 2005), which is equal to: Especially the annual percentage factor within the annual capitalization ( 1 k = ) has the value: 1 k r  =+ (18) If the time of the percentage within the subperiod capitalization is equal to n years, the value of the capital within every year increases k  -multiple. At the end of n year its value will n k  -multiple higher from the value of the initial capital.
Therefore, the value of the final capital can be described as the following mathematical formula: Within the fixed nominal percentage, the annual percentage factor is the higher than the shorter time of the capitalization of interest is. In practice, k  factor is applied as the measure of the pace of the increase of the capital in the condition of the subperiod capitalization.

The continuing capitalization within the compound interest
The issue of the continuing capitalization concerns the problem how the frequency of the capitalization can be increased in order to, at the given nominal where e is the fixed mathematical named the number of Eulera (Fihtenholz, 1985), that is: In the situation when the frequency of the capitalization is uureservedly increasing ( k → + ) it is said about the continuing capitalization of the interest (continuing percentage). The nominal percentage c r which is the equivalent of this capitalization is named the nominal continuing percentage.
In the condition of the continuing percentage within the continuing percentage c r , the intial capital P after n years is increasing to the final value: The equations (21) and (22)  After taking it into the account, the formula (21) and (22)  The obtained equality is the condition of the equivalence of the interest rate k i and r within n time.

The effective interest rate
The annual interest rate equivalent to the compound interest rate is the effective interest rate. It means how much the percentage increases the value of the capital within the year. In the percentage given of the effective interest rate ef r , the annual percentage factor is equal 1 ef r + . In the subperiod percentage of Comparing the both factors we got: The mathematical formula for calculating of the effect interest rate within the usage of nominal interest rate is of the following:  • the effective percentage is higher than the nominal one, when the capitalization period is shorter than the year; • the effective percentage is higher the more often the interest rate is capitalized; • the effective percentage is higher within the continue capitalization.
The average interest rate The average interest rate of the capital P within n time is called the annual percentage r , within which the capital P generates in n time the interest rate equal to the interest rate within the differentiation of the percentage in the particular periods.
If the capital P yield in n years time, whereby the effective percentage in the next years is The average percentage r is equal (Podgórska, Klimkowska, 2005): The final value of the capital after the passage of m subperiods describes the following model: whereby the average percentage is equal: which means that it is the exponential function at the base e. The average simple percentage is the solving of the equation: Therefore the average continue percentage is the arithmetic average of the continue percentage changeable in time.

Results and discussion
In the condition of the compound interest the future value of the capital is the growing exponential function of percentage time. Therefore, the following conclusions can be made: 1. higher percentage generates higher interest 2. shortening the capitalization time (therefore increasing its frequency) increases the pace of the capital growth; 3. the highest increase of the value of the capital gives the continue capitalization; 4. at the same percentage the equality of simple and compound interest is for the time equal to one period of capitalization, for the longer period of time the compound interest is higher than the simple one.
According to the rule of compound interest the term deposits are yield. The client makes the contract with the bank concerning the entrusted money for the set time. The conditions of yielding the deposits are determined by two parameters-the nominal interest rate and the frequency of the capitalization of the interest.
The presented mathematical models enable the possibility: -to calculate the amount for which the paid-in capital for the deposit will gain after the certain time, -to calculate the amount which has to be paid-in on the deposit in order to generate the expected final capital after the fixed time, -to calculate the time of the percentage, in order to, the paid-in money for the deposit, bring the expected profit, -to calculate the value of the percentage on the basis of the initial and final capital after the appointed time.
The banks offer the term deposits and they give their percentage in a year. Knowing the mathematical tools enable in the case of when the time of the capitalization of the interest is different than a year, to calculate the effective percentage and to establish how much the paidin amount of money increases in reality.
Making the deposits is the form of investing the money. The banks present different offers. Thanks to the mathematical models it is possible to assess their profitabilities. The basics of the comparison of the conditions of the percentage can be the calculating of the effective interest rate or calculating of the equivalent percentage in time within different capitalization periods. Banking transactions made enable to choose the most profitable offers.

Conclusions
The client of the bank who deposits her, his savings becomes its creditor. Of this title, he or she gains the profit in the form of the interest which value is established with the usage of the mathematical instruments. The better knowledge of the mathematics enables to make better financial decisions.
Without the basic mathematical knowledge, it is not possible to understand complicated banking matters. The acquaintance of the terms and the models presented in this paper gives the chance of the efficient functioning in the condition of the compound percentage.