Compound interest theory is at the center of financial analysis.

What is compound interest? It’s the interest earned on money invested over time. Invested at some fixed interest rate.

Let’s assume $1 is invested at 6% interest (.06 mathematically) for exactly one year. At the end of the year, the $1 has grown to $1 + $1(.06), or $1.06.

Now, let’s assume the $1.06 is left invested at 6% interest for a second year. At the end of the second year, the $1.06 has grown to $1.06 + $1.06(.06).

This sum can be re-written as: ($1.06)(1 + .06). Which can be re-written as: ($1.06)(1.06). Which equals $(1.06)^{2}.

So now we see the pattern. If $1 is invested at 6% interest for __n years__, the $1 will grow to $(1.06)^{n}. More generally, we can say that if $1 is invested at some interest rate __i__ for n years, the $1 will grow to $(1 + i)^{n}.

If i = 4% (.04) per year, and n = 50 years, $1 will grow to $(1.04)^{50}, which equals $7.11, rounded to the nearest whole cent. I calculated the $7.11 amount using Microsoft Excel.

We’ve just gotten the key to solving __half__ of compound interest problems. What is the __other half__? The __other half__ are __present value__ problems.

A present value (PV) problem is stated this way mathematically, for example: What is __the value today__ (PV) of $7.11 to be received in 50 years assuming 4% interest?

The answer to the question is: PV = $7.11/(1.04)^{50}.

What is $7.11/(1.04)^{50}? It’s $7.11/7.11, which equals $1.

Chew on all this for the next week, and then we’ll dig a little deeper and gain a deeper understanding of compound interest.

by Jon Tidd, Esq