Chapter 11 Language of Descriptive Statistics

Section 11.2 Frequency Distributions and Percentage Calculation

11.2.4 Continuous Compounding Interest

The expression an = (1+ r n )n with r can also be interpreted as a map depending on n

a:      ,    na(n)  =   an   =   (1+ r n )n .

A map n an is called a real sequence. The pairs (n, an ) can be interpreted as points in the Euclidean plane. In this sense, the sequence an = (1+ 0.4 n )n is shown in the figure below as a sequence of points in the Euclidean plane.

Two properties of this sequence can immediately be seen from the figure above:
  • The sequence an , n is monotonically increasing, i.e. for ij ai aj , for all i,j.
  • The sequence approaches the value a as n increases. This number a is called limit of the sequence an , and is written

    limn an   =  a.

In the lecture mathematics 1, the natural exponential function

exp:      ,    xexp(x)=   ex

will be studied in detail.

The natural exponential function

There, the following statement will be shown:
Info 11.2.17
For an arbitrary number x, we have

limn (1+ x n )n   =   ex .

For x=1, the limit of this sequence is Euler's number (named after the Swiss mathematician Leonhard Euler, 1707-1783):

limn (1+ 1 n )n   =  e    2.7182.

It can be shown (with some difficulty) that Euler's number e is an irrational number, and hence it cannot be written as a fraction.
The exponent rules apply to the natural exponential function with arbitrary real numbers as its exponents:
  • exp(x+y)= ex+y = ex · ey =exp(x)·exp(y) for x,y.
  • exp(x·y)= ex·y = ( ex )y = ( ey )x for x,y.

Information on the compound interest process can be gained if the number of times n gets very large using the exponential function and the relation to the sequence (1+ x n )n : the capital is multiplied by a factor of (1+ r n )n every year if the interest at a rate of r n is credited to the initial capital S0 at n different times in the year. After t years, t, the initial capital has increased to

S0 · (1+ r n )n·t .

If n, the limit of this sequence is

limn ( S0 · (1+ r n )n·t )  =   S0 · er·t .

For increasing n the interest is paid more and more frequently:
Info 11.2.18
The limiting case is called the continuous compounding interest. For positive real numbers t, the formula

s(t)  =   S0 · er·t

specifies to which amount an initial capital S0 has increased after t years if continuous compounding interest is applied at a rate r per year.

Example 11.2.19
An investment of 5,000 EUR is deposited for t=8 years in a bank account where continuous compounding interest is applied at a yearly interest rate of 9%. After t=8 years, this results in an investment of

5,000· e0.09·8   =  5,000· e0.72     10,272.17   EUR .