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      <text>4.4.3 (Application - exponential growth (finance))</text>
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      <text><![CDATA[<h1>Compound interest</h1>
<p>Recall that Exponential growth is a universal principle and can be used to describe any increase that is proportional to what is already there. In other words, even though the growth rate remains constant, each successive period of time the amount of growth is greater than the previous period. These ideas have great implications in finance, perhaps the most straightforward is that of compound interest.</p>
<p>Suppose that you invest £100 at a rate of \(5\%\) per year. If we followed a <em>simple interest</em> formula this means that you receive an interest of \(£5\) per year. However, in reality this doesn't occur in real world finance. The reason for this is because the interest is frequently <em>compounded</em> which means that the \(5\%\) interest is paid on the <em>full balance</em> rather than the initial investment. This is called <strong>compound interest</strong>.We can see the difference if we calculate the total money you receive after the first few years. Let \(A_n\) be the total money you receive after year \(n\). Therefore if you follow the simple interest plan you get: \[ 
\begin{align*} 
A_0 = £100 + £5 = £105 \\ A_1 = £105 + £5 = £110 \\ A_2 = £110 + £5 = £115. \end{align*} 
\] 
However, if you follow the compound interest plan you will receive: 
\[
\begin{align*} A_0 = £100 + 0.05 \cdot £100 = £105 \\ A_1 = £105 + 0.05 \cdot £105 = £110.25 \\ A_2 = £110 + 0.05 \cdot £110.25 = £115.76. 
\end{align*} 
\] So, we can see that you will get more money if you compound the interest. The difference doesn't look large here however if we plot the balance over a greater number of years we can see the difference.</p>
<p> </p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Compound%20vs%20simple%20interest.png" alt="plot showing compound vs simple interest" width="656" height="286" /></p>
<br/>
<p>In this case, the red line is the compound interest and the blue is simple interest. We can see that the red line is following an exponential curve. This is why compound interest is linked with exponential functions. The aim we have now, is to find a function which governs simple growth and compound growth. The equation of simple growth is straightforward as it is  a straight line. In this case the simple growth is given by 
\[
\mbox{Accumulated balance} = \mbox{Principle (initial) investment}(1 + \mbox{number of periods}\cdot \mbox{periodic interest rate}).
\]
We often denote this as 
\[
A =P (1 + rt).
\]
To find a compound interest formula is slightly different. Recall that in the example above, for compound interest we have a growth factor of 0.05 at each stage. Looking at our values for \(A_0, A_1, A_2\) for the compound interest scheme note that we could have written 
\[
A_1 = £105 + 0.05^2 \cdot £100 = £105.25 \qquad \mbox{ and } \qquad A_2 = £110 + 0.05^3 \cdot £100 = £115.76.
\]
Therefore the exponent appears in the interest rate term and we can see that it corresponds to the number of time periods we require to pass (in this case the number of years). The fact we can express these relationships in terms of these exponents is the reason why the growth is governed by an exponential equation.</p>

<p>We can perform a similar analysis to get the general compound interest rate formula which is
\[
\mbox{Accumulated balance} = \mbox{Principle investment}(1 + \mbox{periodic interest rate})^{\mbox{number of periods}}.
\]
We often denote this as 
\[
A = P(1 + r)^t
\]</p>]]></text>
<file name="Compound vs simple interest.png" path="/" encoding="base64">[encoded data, 72336 characters]</file>
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