Interest
Interest is the amount of money paid or earned for the use of capital over time. In finance, it is common to distinguish between simple interest and compound interest.Another important concept is capitalization, which describes how interest is added to the capital.
Interactive chart
In the following GeoGebra activity, sequences of points represent the growth of simple interest (red), compound interest (green), and capitalized interest (blue). By checking the “cap” box, the buttons for selecting the capitalization period will appear along with the corresponding equation for each case. By activating the “exp” button, the graph of the associated continuous exponential function is displayed (purple), together with its equation. You can adjust the values using the sliders for the initial capital (C), the interest rate (r), and the number of points (i). Additionally, you can select the capitalization period using the buttons on the left: “1 y” (annual), “6 m” (semiannual), “4 m” (four-month period), “3 m” (quarterly), “1 m” (monthly), and “1 d” (daily). You can also drag the pink cross on the x-axis to compare the values corresponding to each period.
Simple Interest
Under simple interest, the interest is calculated only on the initial principal. If the initial capital is \(C\), the annual interest rate is \(r\), and the time is \(t\), then the accumulated amount is given by:
$$ M = C(1 + rt) $$
In this case, the growth is linear because the interest generated in each period is always the same.
Example: if \(C = 1000\), \(r = 0.08\), and \(t = 3\), then:
$$ M = 1000(1 + 0.08 \cdot 3) = 1240 $$
Compound Interest
Under compound interest, the interest generated in each period is added to the capital, so the next period's interest is calculated on a larger amount. If the capital is compounded once per period, the accumulated amount is:
$$ M = C(1 + r)^t $$
This produces exponential growth, since the interest is earned on both the initial capital and the previously accumulated interest.
Example: if \(C = 1000\), \(r = 0.08\), and \(t = 3\), then:
$$ M = 1000(1.08)^3 \approx 1259.71 $$
Capitalization
Capitalization refers to the frequency with which interest is added to the principal. If the nominal annual rate is \(r\) and capitalization occurs \(n\) times per year, then the accumulated amount after \(t\) years is:
$$ M = C\left(1 + \frac{r}{n}\right)^{nt} $$
For example, if interest is capitalized monthly, then \(n = 12\). If it is capitalized daily, then \(n = 365\) is often used.
The more frequent the capitalization, the greater the final amount, assuming the same nominal annual rate.
Example: if \(C = 1000\), \(r = 0.12\), \(n = 12\), and \(t = 2\), then:
$$ M = 1000\left(1 + \frac{0.12}{12}\right)^{12 \cdot 2} \approx 1268.25 $$
Comparison
Simple interest follows a linear model, while compound interest follows an exponential model. Because of this, compound interest usually produces a larger amount over long periods of time.
In summary:
$$ \text{Simple interest: } M = C(1 + rt) $$
$$ \text{Compound interest: } M = C(1 + r)^t $$
$$ \text{Capitalized } n \text{ times per year: } M = C\left(1 + \frac{r}{n}\right)^{nt} $$