Financial Maths, Part 2

So I am talking about simple interest. In my last post, I explained how to calculate the interest after the money has been invested for one period – one year in our example. It turns out that by investing $1000 at a simple interest rate of 3%, you earn $30 after one year. This means you have a total of $1000 + $30 = $1030 after one year. What if you want to know what you have after 5 years?

There are two ways of doing this: sequentially or directly. The sequential method has the advantage of showing how your money is growing each period, and the formula is very useful for entering in spreadsheet applications like MS Excel. Let’s first discuss this sequential method.

So after the first year, you have $1030. Each year, the interest rate of 3% is applied to the initial $1000 investment, and you get an additional $30. Below is a table of how the investment grows each year. I will explain the headings and the calculations afterwards:


So according to this table, you will have $1150 after 5 years. So what are the column headings?

The first 4 were defined in my last post, but I’ll repeat them here. n is the period number. It starts at 0 since this indicates when time starts. You only get interest after the money has been invested for 1 period (a year in this case). P is the principal which in this scenario, is the amount originally invested. r is the interest rate. I is the amount of interest earned. From my last post, this is calculated as I = Pr or I = Pr/100, depending if you use the decimal equivalent of r or not (see my last post). An is the total amount you have after n periods. Using a subscript like this is very common in maths. A0 is the initial amount after 0 periods. A1 is the amount after 1 period. A5 is the amount after 5 periods.

I know the table is a bit repeating with the $1000 and the $30 repeated throughout the table, but I did this so you can see the difference between simple interest and the eventual compound interest that I will talk about later.

Notice that the difference between An for each adjacent period is $30, that is, $30 is added to the previous An to get the next period total amount, An+1. So the sequential formula to calculate the next period’s amount is:

An+1 = An + 30

This is called a recursion formula as you recursively calculate the next period’s total amount by knowing the previous period’s amount. So starting with A0:

A1 = A0 + 30 = $1000 + $30 = $1030
A2 = A1 + 30 = $1030 + $30 = $1060
and so on until
A5 = A4 + 30 = $1120 + $30 = $1150

Now let’s generalise this formula for any interest rate. The $30 in the above example is the interest I from the formula I = Pr or I = Pr/100. So the general recurring formula for the total amount of interest in a simple interest investment is:

An+1 = An + Pr (decimal equivalent r) or An+1 = An + Pr/100

That’s all well and good for a spreadsheet formula, but what if you only want to know how much money will you have after 10 years? Do you need to apply this formula 10 times to get the answer? The answer is “no” because we can get a formula that directly calculates an answer.

If you are adding the same amount each year, after 10 years, the total amount added is 10 × the amount after 10 years. So in our example, after 10 years, the total amount is $1000 + 10×30 = 1000 + 300 = $1300. Notice that I will get the same result as in the table above after 5 years: 1000 + 30×5 = 1000 + 150 = $1150. So you just need to multiply the same amount of interest each year by the number of periods desired. In general,

An = P + Prn (decimal equivalent r) or An = P + Prn/100

Next time I will introduce compound interest. But to prepare you for this a bit, notice that I can factor out a P from the above formula to get an equivalent one ( please see my posts on the Distributed Property if you need a review):

An = P(1 + rn) (decimal equivalent r) or An = P(1 +rn/100)

Knowing how to use this form of the equation will help you understand the compound interest formulas.