Chapter 11 Language of Descriptive Statistics

Section 11.2 Frequency Distributions and Percentage Calculation

11.2.4 Continuous Compounding Interest

The expression $a_n={(1+\frac{r}{n})}^n$ with $r\in ℝ$ can also be interpreted as a map depending on $n\in \mathbb{N}$

${\displaystyle a:\mathrm{\hspace{0.5em}\hspace{0.5em}}\mathbb{N}\hspace{0.5em}\to \hspace{0.5em}ℝ\mathrm{\hspace{0.5em}\hspace{0.5em}}\mathrm{\hspace{0.5em}\hspace{0.5em}},\mathrm{\hspace{0.5em}\hspace{0.5em}}\mathrm{\hspace{0.5em}\hspace{0.5em}}n\hspace{0.5em}⟼\hspace{0.5em}a(n)\mathrm{\hspace{0.5em}\hspace{0.5em}}=\mathrm{\hspace{0.5em}\hspace{0.5em}}{a}_n\mathrm{\hspace{0.5em}\hspace{0.5em}}=\mathrm{\hspace{0.5em}\hspace{0.5em}}{(1+\frac{r}{n})}^n\hspace{0.5em}.}$

A map $\mathbb{N}\ni n\to a_n\in ℝ$ is called a real sequence. The pairs $(n,a_n)$ can be interpreted as points in the Euclidean plane. In this sense, the sequence $a_n={(1+\frac{0.4}{n})}^n$ 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 $a_n$, $n\in \mathbb{N}$ is monotonically increasing, i.e. for $i\le j$ $a_i\le a_j$, for all $i,j\in \mathbb{N}$.
  
• The sequence approaches the value $a\in ℝ$ as $n\in \mathbb{N}$ increases. This number $a$ is called limit of the sequence $a_n$, and is written
  
  ${\displaystyle \underset{n\to \infty }{lim}{a}_n\mathrm{\hspace{0.5em}\hspace{0.5em}}=\mathrm{\hspace{0.5em}\hspace{0.5em}}a\hspace{0.5em}.}$
  
  
  

In the lecture mathematics 1, the natural exponential function

${\displaystyle \exp :\mathrm{\hspace{0.5em}\hspace{0.5em}}ℝ\hspace{0.5em}\to \hspace{0.5em}ℝ\mathrm{\hspace{0.5em}\hspace{0.5em}}\mathrm{\hspace{0.5em}\hspace{0.5em}},\mathrm{\hspace{0.5em}\hspace{0.5em}}\mathrm{\hspace{0.5em}\hspace{0.5em}}x\hspace{0.5em}⟼\hspace{0.5em}\exp (x)\hspace{0.5em}=\mathrm{\hspace{0.5em}\hspace{0.5em}}{e}^x}$

will be studied in detail.

There, the following statement will be shown:

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

${\displaystyle \underset{n\to \infty }{lim}{(1+\frac{1}{n})}^n\mathrm{\hspace{0.5em}\hspace{0.5em}}=\mathrm{\hspace{0.5em}\hspace{0.5em}}e\mathrm{\hspace{0.5em}\hspace{0.5em}}\approx \mathrm{\hspace{0.5em}\hspace{0.5em}}2.7182\dots \hspace{0.5em}.}$

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:

• $\exp (x+y)=e^{x+y}=e^x·e^y=\exp (x)·\exp (y)$ for $x,y\in ℝ$.
  
• $\exp (x·y)=e^{x·y}={\left(e^x\right)}^y={\left(e^y\right)}^x$ for $x,y\in ℝ$.
  

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 $(1+\frac{x}{n}{)}^n$: the capital is multiplied by a factor of ${(1+\frac{r}{n})}^n$ every year if the interest at a rate of $\frac{r}{n}$ is credited to the initial capital $S_0$ at $n$ different times in the year. After $t$ years, $t\in \mathbb{N}$, the initial capital has increased to

${\displaystyle S_0·{(1+\frac{r}{n})}^{n·t}\hspace{0.5em}.}$

If $n\to \infty$, the limit of this sequence is

${\displaystyle \underset{n\to \infty }{lim}(S_0·{(1+\frac{r}{n})}^{n·t})\mathrm{\hspace{0.5em}\hspace{0.5em}}=\mathrm{\hspace{0.5em}\hspace{0.5em}}{S}_0·e^{r·t}\hspace{0.5em}.}$

For increasing $n\in \mathbb{N}$ the interest is paid more and more frequently: