Deriving the Future Value Formula

I had a difficult time understanding how the future value formula actually compounded the interest. It wasn't till I used the numbers in expanded form did I understand what was happening. 

1. First Attempt 

Example 

What is the future value of 1$ compounded annually at a interest rate of 5% in two years.

FV=PV(1.00+i)ⁿ

FV=1$(1.00+.05)²

FV=1$(1.05)²

FV=$1(1.1025)

FV=$1.1025 ≅ $1.10

But what is happening? I wasn't understanding how the interest was being calculated on the interest. Here's how it works

1.05² = 1.05 x 1.05 

= (1 + .05) x (1 + .05)

Distribute:

1 x 1 = 1

1 x .05 = .05 (these two equal the first period interest and principal calculation)

.05 x 1 = .05

.05 x .05 = .0025 (this is the interest on the interest part of the calculation)

2. Second Attempt

Here's a much better way of understanding the formula.

Example

What is the future value of $4000 compounded semiannually at a interest rate of 12% in one year.

Periodic interest rate = 12% / 2 = 6%

At the end of the first six months (one compounding period) of the loan, the value of the loan is:

FV = PV + i x PV

      =  $4000 + 0.06 x $4000

      = $4240

In the above calculation FV =  PV + i x PV can be rewritten as FV = PV(1 + i)

How do find the value after the second compounding period?

FV = PV(1 + i) x (1 + i)

Working through this step by step you can see why this works

FV = $4000(1 + 0.06) x (1 + 0.06)

      = ($4000 + 240) x (1 + 0.06)

      = $4240 x (1 + 0.06)   

      = $4240 + $254.40

      = $4494.40

You can see on the third step of the calculation the value of the loan at the end of the first compounding period of $4240. The fourth line gives us the interest calculated for the second compounding period. Finally we add them together to get the future value at the end of the year. 

The equation above of FV = PV (1 + i) x (1 + i) can be simplified to FV = PV (1 + i)²

And now we have derived are compound interest formula.