Compound Interest

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Let's say you are given a task of calculating how much money you have after 53 months when your annual interest rate is 30%. Your initial investment is $1,000.

"per centum" is Latin and means "by a hundred". In other words the number of shares in a hundred. So 30% is equal to 0.3. You originally had 100%. So if you get 30% extra you now have 130 shares or 130%. This is equal to 1.3.

${\mathrm{1.3}}^{\frac{1}{12}}$ means 1.3 raised to the power of $\frac{1}{12}$ (one twelfth). If you don't understand this, read this article first: Exponentiation

In order to do this you need a calculator. Let's first calculate the rate of our interest each month. We get 30% per year, which means we get: (using the multiplication sign $\cdot$) $$({\mathrm{1.3}}^{\frac{1}{12}}-1)\cdot \mathrm{100}\approx 2.21\%per\; month.$$

Now let's calculate how much money we have after 53 months $${\left({\mathrm{1.3}}^{\frac{1}{12}}\right)}^{\mathrm{53}}\cdot \$1,000={\mathrm{1.3}}^{\frac{53}{12}}\cdot \$1,000\approx \$3,186.03$$

Elaboration:
Compund interest is interest which is compounded over and over again until you after a certain number of iterations reach your final result. You apply the interest if your next iteration to the product of the previous iteration. It is also known as the term interest-on-interest. Your total amount of savings after five years with 20 percent of interest each year is calculated as (1.2 * (1.2 * (1.2 * (1.2 * (1.2 * initial balance))))) = final balance. The final value method is used in investment calculation to assess the profitability of an investment. The method is closely linked to the present value method, but instead of calculating the value of the investment at the time of investment, the value is obtained at the end of the economic life. The method generates a final value, the total value of an investment's cash flows, discounted forward in time to the end of its useful life. It is obtained by interest-on-interest calculation of all payment streams, with the calculation interest as the interest rate. The final value method is less common in the course literature than the present value method, and it is not used as much by larger companies. However, the method is more intuitive than the present value method, and is often used by individuals and small businesses, even if they don't know it. The final value is what is calculated when you save for something; a future nominal value as a result of an investment. If the present value, or net present value, has already been calculated, it is easy to calculate the final value and the profit, respectively. The formulas below use the same notation as under present value, with the addition of SV, the final value, and V, future "profit" (the difference between the final value and discounted investment costs). Otherwise, the final value is calculated directly from the investment's cash flows. If the payment surpluses are equal, the calculation can be simplified. If we reverse the last formula, to calculate the annuity instead, we get the answer to the question "how much should we save per year to be able to afford a new car in two years?". This can also be developed to calculate a real amount with a given growth factor for inflation. "If inflation is g, how much do I have to save per year to have a real value of SV when I retire in n years?.