Financial Maths, Part 4

In my last post, I developed a recursion formula for compound interest where the interest is compounded annually:

An+1 = An + An × r/100 where you use the percentage value of r or
An+1 = An + An × r where the decimal equivalent of r is used.

These are good formulas if you want to know how much money you will have next year, but will take a while if you want to know how much you will have in 10 years. However, we can use the recursion formula to help develop the direct formula.

I will use the decimal equivalent equation so I don’t have to keep writing “/100”. I will add that at the end of this development.

Using the recursion formula, at the end of year 1 you will have

A1 = A0 + A0 × r = A0(1+r)

Please review my post on factoring, specifically the distributive property, if you do not understand why A0 + A0 × r = A0(1+r).

At the end of year 2, you will have A2 = A1 + A1 × r = A1(1+r). But we have A1 in terms of our initial investment A0 above. So substituting A1 = A0(1+r) into A2 = A1(1+r) results in

A2 = A1(1+r) = A0(1+r)(1+r) = A0(1+r)²

Wait! The exponent 2 of (1+r) is the same as the current period (the subscript of A2. Wouldn’t it be nice if this pattern continued? Let’s do one more iteration, substituting A2 with the above equation:

A3 = A2 + A2 × r = A2(1+r) = A0(1+r)²(1+r) = A0(1+r)³

Wow! It looks like this is true. In fact, it is. The exponent of the (1 + r) part is the same as the period of interest. If you want to find how much you will have after 10 years, you can use the direct formula:

An = A0(1 + r)n

So in our example of investing $1000 at 3% interest per annum (that is each year), compounded annually, you will have after 10 years

A10 = A0(1 + r)10 = 1000(1 + 0.03)10 = $1343.92

Of course, you need a calculator that can do exponents, or you can use the all-knowing internet. The percentage form of the direct equation is

An = A0(1 + r/100)n

where you would substitute the “3” in for r instead of “0.03”.

I said last time that interest can be applied (compounded) more frequently than annually. I will continue this example next time where we do just that.