#### Chapter 11 Language of Descriptive Statistics

Section 11.2 Frequency Distributions and Percentage Calculation

# 11.2.4 Continuous Compounding Interest

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

$a:\mathrm{ }ℕ \to ℝ\mathrm{ }\mathrm{ },\mathrm{ }\mathrm{ }n ⟼ a\left(n\right)\mathrm{ }=\mathrm{ }{a}_{n}\mathrm{ }=\mathrm{ }{\left(1+\frac{r}{n}\right)}^{n} .$

A map $ℕ\ni n\to {a}_{n}\in ℝ$ is called a real sequence. The pairs $\left(n,{a}_{n}\right)$ can be interpreted as points in the Euclidean plane. In this sense, the sequence ${a}_{n}={\left(1+\frac{0.4}{n}\right)}^{n}$ 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 ${a}_{n}$, $n\in ℕ$ is monotonically increasing, i.e. for $i\le j$ ${a}_{i}\le {a}_{j}$, for all $i,j\in ℕ$.
• The sequence approaches the value $a\in ℝ$ as $n\in ℕ$ increases. This number $a$ is called limit of the sequence ${a}_{n}$, and is written

$\underset{n\to \infty }{lim}{a}_{n}\mathrm{ }=\mathrm{ }a .$

In the lecture mathematics 1, the natural exponential function

$\mathrm{exp}:\mathrm{ }ℝ \to ℝ\mathrm{ }\mathrm{ },\mathrm{ }\mathrm{ }x ⟼ \mathrm{exp}\left(x\right) =\mathrm{ }{e}^{x}$

will be studied in detail.

The natural exponential function

There, the following statement will be shown:
##### Info 11.2.17

For an arbitrary number $x\in ℝ$, we have

$\underset{n\to \infty }{lim}{\left(1+\frac{x}{n}\right)}^{n}\mathrm{ }=\mathrm{ }{e}^{x} .$

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

$\underset{n\to \infty }{lim}{\left(1+\frac{1}{n}\right)}^{n}\mathrm{ }=\mathrm{ }e\mathrm{ }\approx \mathrm{ }2.7182\dots .$

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:
• $\mathrm{exp}\left(x+y\right)={e}^{x+y}={e}^{x}·{e}^{y}=\mathrm{exp}\left(x\right)·\mathrm{exp}\left(y\right)$ for $x,y\in ℝ$.
• $\mathrm{exp}\left(x·y\right)={e}^{x·y}={\left({e}^{x}\right)}^{y}={\left({e}^{y}\right)}^{x}$ 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 $\left(1+\frac{x}{n}{\right)}^{n}$: the capital is multiplied by a factor of ${\left(1+\frac{r}{n}\right)}^{n}$ 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, $t\in ℕ$, the initial capital has increased to

${S}_{0}·{\left(1+\frac{r}{n}\right)}^{n·t} .$

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

$\underset{n\to \infty }{lim}\left({S}_{0}·{\left(1+\frac{r}{n}\right)}^{n·t}\right)\mathrm{ }=\mathrm{ }{S}_{0}·{e}^{r·t} .$

For increasing $n\in ℕ$ 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

$s\left(t\right)\mathrm{ }=\mathrm{ }{S}_{0}·{e}^{r·t}$

specifies to which amount an initial capital ${S}_{0}$ 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

$5,000·{e}^{0.09·8}\mathrm{ }=\mathrm{ }5,000·{e}^{0.72}\mathrm{ }\approx \mathrm{ }10,272.17\mathrm{ }\text{EUR} .$