To an Axiomatic Model of Rate of Growth

In the paper an axiomatic approach to express rates of growth is presented. The formula is given of rate of growth at a point as the limit case of rate of growth on an interval and the inverse formula is derived to compute present and future value of capital for an integrable rate of growth. Incidentally some inconsistencies in currently used formulas are pointed out.


Introduction
The concept of an average change of an objective function plays a crucial role in financial mathematics.Reflecting the objective function f, it is called an interest rate, an inflation rate, and so on.It is given as the value of ( ) ( ) ( ) ( ) For a steady state function the same result may be obtained from the formula ( ) ( ) ( ) In macroeconomics a similar, but instantaneous measure, related to a point is needed.Baro (2003) employs the formula ( ) ( ) [1]) which is in fact an average change of the first derivative.The relation between an average change on an interval and an average change of its derivative has not been tackled in the literature.This leads to the problem of ( ) ( ) ( ) ( ) ( ) . Further, it is desirable to find a formula that gives the future value of the objective function including the case when the rate of growth is neither constant nor piecewise constant function.For a constant rate of growth function with values ξ we have the formula ( ) ( ) ( ) = does not yield ( ) ( ) ( ) . Accordingly the aims of the paper are as follows.
1) To define the concept of a rate of change by means of axioms (Section 2).
2) To formulate the notion of a steady state function to model existing interest rates and to find corresponding computation formulas (Section 3).
3) To derive a limit version of a rate of growth (Section 4).
4) To find the inverse formula that enables to calculate the values of a state function (Sections 5 and 6).
5) To point out to some impacts on currently used formulas in financial mathematics (Section 7).

Axioms
The symbol denotes the set of real numbers.Consider a quantity attaining values , for 1 respectively.A function is said to be a generalized rate of growth function (shortly rate of growth function) if the following Axioms A1-A4 are satisfied: x y x y x t y x t y for any (invariance with respect to shift of time).t ∈  Axiom A2.

)
x y x y x y k x y k κ κ = ⋅ ⋅ for any (invariance with respect to homoteties).k ∈  Axiom A3. κ is increasing with respect to the first and fourth variables and decreasing with respect to the second and third variables.

Definition
Let κ be a rate of growth function.For a function the function : For the simplicity we omit if it is clear from the context.Verbally, F f does not depend on the choice κ κ 1 x , 2 x .

Lemma
, , , , y x y x y x x y κ λ which is decreasing with respect to the first variable, increasing with respect to the second variable and it holds .

Theorem
Let be a continuous κ-steady state function.Then f is an exponential function, i.e.
be given.Then there holds x

f x x h f x h x h f x h x h f x h
, , , and with a view to (1) we get As λ is injective in any variable, it holds Further, by From here it follows that the values of f at all equidistant points form a geometric sequence.Moreover, the implication -Since this set is dense in , the proof is completed be cause of we obtained for all and coosing for instance we obt ) , where ( )  ( ) , , , y x y x y y (2) Proof: From the assumption for f it follows that there holds 4 ( ) ( ) , e , , e , e , , e , e c o n s t e for all B. Further, putting ( ) and usin 1) we get g ( ( ) ( ) ( ) as required.

Note athematics the translation In financial m
is employed and consequently the rate of growth function is of the form ( ) which is called a a compound interest (per unit of time).Besides (more or less from historical reasons) also a simple interest (per unit of time) is used, given by where y 0 is preselected constant, usually the value in a predetermined initial time.This rate does not satisfy Axiom A2, and hence there is no rational reason to use it.
Due to this rate polynomials of the first degree ( ( )

Infinitesimal Version
In macroeconomics an instantaneous measure of rate of growth is often needed.This may for a function f be naturally given by a limiting process as (see ( 2)) The number ( )( ) In macroeconomics a measure is used, denoted by ( 7) In an analogous way we may use the th which represents the relative change of the composite function with respect to the change of the argument of t ction.Notice, that the same limit has the simple in (see (5)) letting κ 1 2 x x → .

Consequence for the Interest Rate Calculations
th Using (2), e expression  ( ) It is known, that banks at the beginning of t past century (due to practical reasons stemming from the he nonexistence of computers) used to find the value 1 t κ for small 1 t the approximation by Taylor polynomial of the first degree of function (11) which gives the result ( ) ( ) interval of adding of interests" was introduced with the clause, that if the current interval was shorter than that (where O is Bachmann-Landau big-O).Consequently, supposing interest rate was known for some time interval (e.g. a month), the interest rate for shorter intervals (e.g. a day) was calculated dividing by 30 instead of as the 30th root.To legalize this inaccuracy, the notion of "an under assumption, the interest will be calculated multiplying only by a linear part of the increment of the interest rate.Hence function f representing the state of account being in a steady state was changed from exponential to piecewise linear having with the original exponential curve common only breaking points.This practice is still surviving, despite banks use software that is definitely capable to calculate the roots.The reason rests (probably) with the shortage of management theoretical competence.The difference between the exact value and its approximation, i.e. an error of approximation is an increasing function when time approaches to infinity having finite limit e 1 This limit is employed in a number of books on financial mathematics, its interpretation although is rather problematic.When we calculate compound interest and manipulate with a compound interest as with a simple interest in such a way that we divide time interval in equidistant subintervals and apply the interest tha linear part of the approximation for these subintervals, we obtain the result, whose limit for the number of subin t is the tervals approaching to infinity is given by formula (12) A magic appearance of Euler constant in this calculation gave birth the notion of continuous compounding.It may be simply verified that it is in fact a compound interest, where in formula (4) the value ( ) may be obtained as a rate of growth when putting φ = ln in (2) and then by limiting we get ν as in (8).

Inverse Problem
Let us use for the rate of growth formula ( 7) and denote 1 ν ν = with argument t in the sequel.Then we have for a fixed t 0 ( ) Supposing f is given, then (13) is the formule to find rnatively, when ν is given, then ) t or the rate of growth ν.Alte ( 13) is a differential equation to get the function f.This equation can be rearranged equivalently to h el instance if we substitute a constant interest rate in (15), we do not obtain the form or a compound interest!The following example illustrates the use Example.We assume that the inflation rate per a unit of time (e.g. a year) at time 0 and time po (15) Althoug the formula (15) is clearly simplier than (14), it has disadvantage, because it yields quantitativ y bad results.For ula f of formula (14).
1 is known.Supse that the inflation rate per unit of time at time 0 is 0.1 and 0.2 at time 1.Deliberate on the inflation rate on interval 1, 0 .It is evident that this depends on the changes of the inflation rate on 1, 0 .Consider the following four cases of the inflation rate: ( ) Notice that the first and the last cases are trivial-the rate is constant and the interval has a unit length and thus the inflation rate should be the same constant.The general formula must give the same result.By (14) we have Applying ( 16) we get consecutively (setting ) + ( ) st and the last one) which is an evid ( ) ( ) ( ) ) ) ( )

Theorem
Formula (14) is a limit case of formula (17).
Proof: First we show, that for every continuous function f defined on a closed interval, there exists a sequence of piecewise constant functions ( ) and the proof is completed.e a e derived the new formula for the rate of rowth at a point by limiting process.This formula enables to assign to state reover formula is given to find a state function on condition its rate of growth function at any point is k wn (see ( 14)).

Interest Rate of Simple Compo
Although the choice of Axioms A1-A4 seems to be n that any exponential function is a tion," e-Print Archive of Coronell University, 2003.

Conclusions
In th rticle we presented an explicit formula for all possible rates of growth possessing natural properties (described by Axioms A1-A4) (see (2)).Further w g function its rate of growth (see (7)).Mo no natural, the conditio steady state function is of crucial importance.It is an open problem of finding a simpler condition or to show that this condition may be derived from the axioms.

⋅
(see [2] among others) does not work, because the substitution of constant function with value ( ) t ξ ξ The results are surprisingly not equal (particularly the fir ent failure.Formula for the future value of the compound interest in case of constant interest rate is given by .

unding
Let f be a continuous fun tion.Due to the as-Diam U is a diameter of U. Consider a partition of As an impact of the preceding considerations let us point to the issue of simple compouding.Simple compounding i i and the p .