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Applied Mathematics, 2013, 4, 1326-1332 http://dx.doi.org/10.4236/am.2013.49179 Published Online September 2013 (http://www.scirp.org/journal/am) To an Axiomatic Model of Rate of Growth Václav Studený1, Ivan Mezník2 1Department of Applied Mathematics, Faculty of Economics, and Administration, Masaryk University, Brno, Czech Republic 2Institute of Informatics, Faculty of Business and Management, Brno University of Technology, Brno, Czech Republic Email: Vclv.St@gmail.com, meznik@fbm.vutbr.cz Received July 7, 2012; revised January 6, 2013; accepted January 13, 2013 Copyright © 2013 Václav Studený, Ivan Mezniík. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT 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. Keywords: Interest Rate; Inflation Rate; Rate of Growth 1. Introduction The concept of an average change of an objective func- tion plays a crucial role in financial mathematics. Re- flecting the objective function f, it is called an interest rate, an inflation rate, and so on. It is given as the value of ()() () () 1. f tftf+− t For a steady state function the same result may be ob- tained from the formula ()() () () 1 . ft ft ft δ δ +− In macroeconomics a similar, but instantaneous meas- ure, related to a point is needed. Baro (2003) employs the formula () () f tft ′ (see [1]) which is in fact an aver- age 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 ()() () () () 1 0 lim ftft ft δ δ δ →+− . Further, it is desir- able 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 func- tion. For a constant rate of growth function with values ξ we have the formula ()( )() 01 t ft f ξ =⋅+ d, but its generalization () () 0 0e x s s f ξ ⋅ (see [2] among others) does not work, because the substitution of constant function with value () t ξξ = does not yield ()( ) () 01 t ft f ξ =⋅+ . 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 (Sec- tion 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). 2. Axioms The symbol denotes the set of real numbers. Con- sider a quantity attaining values , for 1 1 y2 y∈ x , 2 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∈4→: κ Axiom A1. ()( 112 21122 ,,,,, , ) x yx yxtyxty κκ =++ for any (invariance with respect to shift of time). t∈ Axiom A2. ()( 112 2112 2 ,,,, ,, ) x yx yxykxyk κκ =⋅⋅ 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 C opyright © 2013 SciRes. AM V. STUDENÝ, I. MEZNÍK 1327 second and third variables. Axiom A4. () 12 ,, ,0xyx y κ = for any ( initial condition-y∈ κ has zero value for constant functions). 3. Steady State Functions 3.1. Definition Let κ be a rate of growth function. For a function the function :f→ ()() ( () 121122 :,, ,, f ) F xxxfxxfx κ is called a -rate of growth of f related to κ 12 , x x. A function f is called a -steady state function if Ff is a constant function. For the simplicity we omit if it is clear from the context. Verbally, Ff does not depend on the choice κ κ 1 x , 2 x . 3.2. Lemma 1) Every constant function is a -steady state func- tion for any rate of growth function . κ κ 2) Let be rate of growth function. Then there exists a function such that κ f2 :→ () 2 112 221 1 ,,, , y xyx yxx y κλ =− ) (1) which is decreasing with respect to the first variable, increasing with respect to the second variable and it holds . () ,0 0x λ ≡ Proof: The statement 1) follows immediately from Axiom A4. Now, by Axiom 1 ()( 112 21212 ,,,0,, , x yx yyxxy κκ =− and by Axiom 2 () 2 12 122 1 1 0,,,0,1,, y yxxyxx y κκ −= − . Putting 22 21 21 11 ,0,1,, yy xx xx yy λκ −= − we get (1). The properties of λ are obvious and hence the statemnt 2) holds true. 3.3. Theorem Let be a continuous κ-steady state function. Then f is an exponential function, i.e. :f→ :e B x fx A for some constants A a B. Proof: Let 1 x a 21 x xh=+ be given. Then there holds ) ()( ) () () ( () 222 2 111 1 ,,, ,,2,2 xfxx hfx h x hf xhxhf xh κ κ ++ =++ ++ Since f is a κ-steady state function, from Definition 3.1 it follows ()( ) () ()( ) () 222 2 111 1 ,,, ,,, x fx x hfx h x fx x hfx h κ κ ++ =+ + and with a view to (1) we get () () () () 1 21 11 ,, . f xh +fxh hh fx h fx λλ + = + As λ is injective in any variable, it holds () () () () () () 121 2 12 1 f xhfxhfx+++ == h fx hfxfx+ and hence () () () () 2 1 1 1 2. fx h fx hfx + += Further, by induction () () () () () () () 1 11 1 1 1 1 . n fx h f xnh+= fx nh fx fx h fx +⋅ +− + = From here it follows that the values of f at all equidis- tant points form a geometric sequence. Moreover, the implication () () () () () () () () 111 11 1 2 1 1 2 2 2 f xhfxhfxh fxfx hfx fx h fx +++ = + + = holds true. Therefore f attains the values of some expo- nential function at all points of the set ,, 2b ah ab ∈∈ . - Since this set is dense in , the proof is completed be cause of we obtained () () ( )() 1 1 1 1 x x h fx h fx fx − − + =fx for all and coosing for instance we obt x∈ ain 10x=, 1h= () () ()() () ln 10 0e f fx fx f⋅ =⋅ so :e B x f xA, where () 0 A f=, t solving fun () () () () 10Bf f=−. ctional equations see (For more details abou Copyright © 2013 SciRes. AM V. STUDENÝ, I. MEZNÍK 1328 [3]). . Theorem 3.4 Let an exponential function :e B x fx A be a κ-steady state function. Then there exing function φ sts an increasi with the property () 21 1 2 1122 1 ,,, y xyx yy κφ = . (2) xx− Proof: From the assumption for f it follows that there holds 4 () () 3 12 123 4 ,e ,,e,e , ,e Bx Bx BxBx xA xAxAxA κκ = for all 12 x x<, 34 x x<. Denoting 21 hx x=− we get (in a view of (1)) () () 12 2 2 Bx Bh x λ =− 1 12 1 ,e , ,e e ,,econst e Bx Bx Bx xAxA A x h A κ λ = = for all B. Further, putting () 1 ln h Bz= and usin1) we get g ( () () () 11 1 ln ln ,,e1,e1, hh hzz h hz hz λλλλ === and () 21 21 1 2 112 2 1 1 2 1 , ,,0,1,1,xx xx y xyx yy y y κκ φ − − = = (3) as required. 3.5. Note athematics the translation In financial m :1 x x φ − is employed and consequently the rate of growth function is of the form () 21 1 2 11122 1 ,,, 1xyx yy κ =− (4) which is called xx y− 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 ) () ( 21 112 2 021 ˆ,,, yy xyxyyxx κ − =⋅− (5) where y0 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 ( () 01122 ˆ :1,,, ) . f ty xyxy κ ⋅+ ⋅t 4. Infinitesimal Version In macroeconomics an instantaneous measure of rate of growth is often needed. This may for a function f be na- turally given by a limiting process as (see (2)) ()( )( )() () () () () () 0 0 00 00 lime . xx fx φφ → == 000 1 lim,,, xx fx xx fx fxxfx xfx νκ → ′ − = fx (6) The number () () 0 f x ν is called a ν-rate of growth of f at point x0. An(see (4)) we have d for κκ =1 () () () () 0 0 10 e1 fx fx fx ν ′ =−. In macroeconomics a measure is used, denoted by (7) ν obtained from (6) choosing φ = ln, ( () ) () () 0. 0 0 f x fx f x (8) In an analogous way we may use the th ν ′ = same function for e rate of growth on interval 01 , x x yielding () () () () () () () () () 0011 ,,,xfxxfx 01 1 01 0ln ln xx 10 1 ln f x fx − == which represents the relative change of the composite function fx− (9) fx x x− κ ln f he fun terest with respect to the change of the argu- ment of tction. Notice, that the same limit has the simple in (see (5)) letting ˆ κ 12 x x→. 5. Consequence for the Interest Rate Calculations thUsing (2),e expression ) () () (() () 21 1 1122 ,,, 2 1 x x f xfxxfx κφ − = x fx (10) time. For is the rate of growth of function f per unit of Copyright © 2013 SciRes. AM V. STUDENÝ, I. MEZNÍK 1329 ins ents how the state a dead account drawals) depends on time as- 1f time and the unit of time is a year, then tance, if f repres (neither deposits nor with suming x, x are moments o 2 () () () 1122 ,,, x fxx fx κ ereas if we choose in (2) is the interest rate per a year, wh () 1, t xx φ =− we get (denoting the resulting function by t κ ) () () () () () 21 1 2 1 1 , 1 xx t fx xf x κ − − which is a compound interest related t 1 2 2 ,, xx fxf = o time segment . Besides, it holds 21 tx x=− 11 11. t κκ =+− t (11) 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 1t κ for small 1t the approximation by Taylor polynomial of the first degree of function (11) which gives the result () () 1.2 1 11 11 tO t κ κκ +−=+ 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 multi- plying only by a linear part of the increment of the inter- est rate. Hence function f representing the state of ac- count being in a steady state was changed from exponen- tial to piecewise linear having with the original exponen- tial curve common only breaking points. This practice is still surviving, despite banks use software that is defi- nitely 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 e1 κ κ −− because it holds lim11lim11 e1. t t tt tt κ κ κ κκ →∞ →∞ +−=+− =− (12) This limit is employed in a number of books on finan- cial 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 sub- in 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 () 1 ln 1 κ + instead of 1 κ is applied. The number () 1 ln 1 κ + may be obtained as a rate of growth when putting φ = ln in (2) and then by limiting we get ν as in (8). 6. 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 t0 () () () () 0 0 0e1 ft ft ν =−. (13) ft ′ 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 () () ()( ln1 lntf ν ′ += () () () () ln 1 f ttft ν ′=+⋅ with the solution () () () () () 0ln1 d e0, tss ft f ν + = (14) where () s ν nt s. is the interest rate per unie me Performing the same calculation for t of time at th mo ν (see , we get 0 (8)) () () ()() 00 f tftft ν ′ = with the solution () () () 0d e tss ft f ν = 0. h el instance if we substitute a constant interest rate in (15), we do not obtain the formor a com- pound 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) Althougthe formula (15) is clearly simplier than (14), it has disadvantage, because it yields quantitativy bad results. For ula f of formula (14). 1 is known. Sup- se 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 in- terval 1,0 . It is evident that this depends on the Copyright © 2013 SciRes. AM V. STUDENÝ, I. MEZNÍK 1330 changes of the inflation rate on 1,0 . Consider the fol- lowing four cases of the inflation rate: () 1 0.1, if1u ι < 0.2, if1,u≥ u= () 2 20.1, 10 u u ι =+ () 30.1, 10 u u ι =+ () 0.1, if0u u ι ≤ = 40.2, if0.u> 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 gen- eral formula must give the same result. By (14) we have ()( ) () () () () 1 0ln1d 10eiuu ff ι 1. =⋅ − (16) Applying (16) we get consecutively (setting ) + () 01f= () () () () () () () () () 12 0 1 0 ln 1.110 d ln 1.110 d for2 :1e10.132945354 for3: 1 e10.149637533 uu uu if if + + == −= == −= () () () 1 0 1 0 ln 1.1 d ln 1.2 d for1:1e1 0.1 for4 :1e10.2. u u if if == −= == −= Now, applying (15) we obtain results 1 0 12 0 1 0 1 0 0.1d 1100.1d 1100.1d 0.2d e1 0.105170918 e1 0.142630812 e1 0.161834243 e1 0.221402758. u uu uu u + + −= −= −= −= The results are surprisingly not equal (particularly the fir ent failure. For- mula for the future value of the compound interest in case of constant interest rate is given by . st and the last one) which is an evid ()( )() 01 t ft f ξ =⋅+ (17) In case of piecewise constant interest rate, i.e. if i I me are the values of constant interest rate per year on ti intervals ( 1 , ii tt +, 0,,in=, then the interest rate per ( 1 0, n ii itt + = is given by () () 6.1. Theorem Fo n 4) for aiecewise c Proof: Ass 1 1 0 1. ii ntt i i I+ −− = + ∏ (18) rmula (17) is a special case of formula (18) for a co- stant interest rate and formula (18) is a special case of formula (1 ponstant interest rate. ume () tI ν = is constant. Then there holds () () () () ( ( 0ln1 ln 1dln 1 eee tI Iu tI ) ) () 1 tt I + +⋅+ ===+ and hence the fi let rst part of the statement holds true. Now ν be piecewii I se constant possessing values on intervals ( 1 , ii tt +, 0, ,in= and A χ be a character- istic function of set A. We have () () 1 , ii i tt tI ιχ + =⋅ and hence ( ( ( , 0ln 1 e t tt ii χ + +⋅ ) ) ) () () () ()( 1 1 1 01 d ln 1ln 11ln1 00 ee i n ii ii iii i Iu tItI ntt I ii − + =+ +− +− −⋅+ == == ∏ ) () () 6.2. Theorem Formula (14) is a limit case of formula (17). Proof: First we show, that for every continuous func- tion f defined on a closed interval, there exists a sequence of piecewise constant functions () 11 0 11 ln 1 e1 tt ii ii i i nn tt I i I − ++ = −− − + == + ∏∏ and the proof is completed. i ξ with the property i f ξ →. Let f be a continuous funtion. Due to the as-c sumption () Dom f is the c tive real number. For eve ompact set. Let be posi- ry () mDo x f at () () ∈ we find a such thneighborhood O () x () () 2 f Ox ⊂Of x . () { } () Dom x f Ox ∈ forms a covering of ( Dom ) f . Choose a finite subcovery Ω and define () () min Diam UU δ ∈Ω =, where () Diam U is a diameter of U. Consider a parti- tion of () Dom f given by n disjoint subintervals i () n i J of the length δ . In every subinterval i J we choose a pd denote () ii oint xi, an y fx=. Furer, define () i th x y ζ = for all i x J∈. Then () Dom for every x f∈ it holds () )( fx x≤ and fo ζ − r 1 2n n ξζ = the above sfied. Since the functionproperty is satial () ( 0ln 1d :e t ) s s ι ι + Φ is continue topology of uniform convergence, t () () () lim lim ii ψψ →∞ →∞ Φ=Φ =ΦΨ ous in th we ge roof is completed unding As an impact of the preceding considerations let us point to the issue of simple compouding. Simple compounding ii and the p. 7. Interest Rate of Simple Compo Copyright © 2013 SciRes. AM V. STUDENÝ, I. MEZNÍK 1331 is a situation in which dependence of a quantity on time is a polynomial of the first degree (let us call the de- pendence of the quantity on time a state funcon). In this situation, special rate of growth is used (see )) but this tual flaws. One of them rests with the mixing of different ways of measuring the rate ti (5 rate has fundamental concep of growth. From the above considerations we can conclude, that in all situations the only one rate of growth is sufficient given by (2). In what follows we compute the rate of growth of a quantity, which is simply compounded (it may be called “compound interest rate of a simple com- pounding”). It is evident if the rate of growth function is constant and positive, then the state function is increasing and convex. Further if the rate of growth function is positive and decreases sufficiently quickly, then the state function is increasing but concave. Now we are looking for the rate of growth function, which makes the state function affine, i.e. it has the form of a polynomial of the first degree. To find it, we consider the state function (see (14)) () () () 0ln1d 0e . t s s tf ξ + (19) Its derivative is given by () () () () () 0ln 1d 0ln1 e t s s tf t ξ ξ + + and second derivative by () () () () () () () () () () () () () 0ln 1d 0e tss tf ξ + 22 ln 1ln 1 1 tt t ξξ ξ ′++ ++ ⋅+ Since the state function is polynomial of the first de- gree its second derivative must be equal to zero. If and , then second derivative is equal at are solution of the differen- ( The solution of (20) is . tt ξξ () 00f≠ tial equation () 0t ξ > to zero for such () t ξ , th () () () () () () () () 22 ln 1ln 10.tt tt ξξ ξξ ′++ ++= 20) () 1 e1 tC t ξ − =− (21) onstant the given value () 0 ξ we get from (21) for any c C. For () () 1 ln 10 C ξ =− + () () () () () () () 1 1 ln 10 1 ln 101 10 1 t ξ ξ ++ =+ − e1 . t t ξ ξ + + =− If we substitute this rate into (14), we get (22) ()( ) () () () () () () () () 1 ln 101 0ln 10d 0e 0ln10 1 ts s ft f ft ξ ξ ξ ++ + =+ + (23) which is really an affine function, i.e. a stat a simple compounding. Applying (5), substituting = e function of 1 x and 2 x arbitrary and setting coresponding () 11 y fx=, () 22 y fx= fo and due to (23), while rate of growth function of (23) is () 00yf= we (23) obtain rmula for the rate of simple compounding of () () ln 10. ιξ =+ 1 e1 t ι ι ⋅+ − . e a e de- rived the new formula for the rate ofrowth 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 kwn (see (14)). 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. 8. Conclusions In thrticle 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. REFERENCES1 [1] R. J. Barro and X. Sala-i-Martin, “Economic Growth,” 2nd Edition, 2003. [2] J. Dupačová, J. Hurt and J. Štepán, “Stochastic Modeling in economics and Finance,” Kluwert Academic Publish- ers, 2002. [3] J. Aczél, “Lectures on Functional Equations and Their Applications,” Academic Press, New York, 1966. [4] V. Studený, “Functional Equation of the Rate of Infla- 1The first usage of the new way and new formulas can reader find in [4] The usual approach to the problem—problematic, as shown in this article—can be seen in [5-8]. For more details of mathematical analysis used here see [9]. and hence the partial solution is Copyright © 2013 SciRes. AM V. STUDENÝ, I. MEZNÍK Copyright © 2013 SciRes. AM 1332 http://arxiv.or Z, Praha, 2000. cs, Vol. 10-III,” Academic Press, New lassification Codes: SC (Mathematics Subject Classification) g/abs/math/0307395 [5] J. C. Van Horne, “Financial Markets, Rates and Flows,” Prentice Hall, Englewood Clifs, 1978. [6] R. C. Merton, “Continuous Time Finance,” Blackwell, Cambridge, 1992. [7] I. Karatzas and S. Shreve, “Methods of Mathematical Finance,” Springer, New York, 1998. [8] T. Cipra, “Mathematics of Securities,” H [9] J. Dieudonné, “Treatise on Analysis, Vol. III, Pure and Applied Mathemati York, London, 1972. C M 26A12, Rate of growth of functions, orders of infinity, slowly varying functions 62P20, Applications to economics 39B22, Equations for real functions 91B02, Fundamental topics (basic mathematics, methodology; applicable to economics in general) 91B28, Finance, portfolios, investment 91B24, Price theory and market structure ctions 34A05, Explicit solutions and redu JEL (Journal of Economic Literature) E4, Money and Interest Rates: General (includes measurement and data) Structure of Interest Rates C43, Index Numbers and Aggregation E43, Determination of Interest Rates; Term C02, Mathematical Methods C63, Computational Techniques C65, Miscellaneous Mathematical Tools |