# BM4.5 – Future Value of Money Using Compound Interest

Chapter 4, Lesson 5

In this lesson students will:

• understand where the formula to calculate the future value of compound interest comes from
• determine the future value of an investment/debt given information about the compound interest rate, principle amount and length of term

## Future Value Formula for Compound Interest

Recall in Lesson 4.4 we determined the future value of money after one compounding period using the formula $FV = \displaystyle P \left( 1 + \frac{r}{f} \right)$. What if we would like to calculate the future value after several compound periods? The key is to think of our first future value as the principle amount during the next step. This means the future value after two compounding periods would be

\begin{aligned} FV &= P \left( 1 + \frac{r}{f} \right) \left( 1 + \frac{r}{f} \right) \\ &= P \left( 1 + \frac{r}{f} \right)^{2} . \end{aligned}

Similarly, if we wanted to find the future value after three compounding periods, we have

\begin{aligned} FV &= P \left( 1 + \frac{r}{f} \right)^{2} \left( 1 + \frac{r}{f} \right) \\ &= P \left( 1 + \frac{r}{f} \right)^{3} . \end{aligned}

So, in other words, the power of the term $\displaystyle \left( 1 + \frac{r}{f} \right)$ tells us how many compounding periods our money has accrued interest over. This means a more general future value formula that would tells us the future value of our money after $n$ compounding periods is

\begin{aligned} FV &= P \left( 1 + \frac{r}{f} \right)^{n}. \end{aligned}

So the question now becomes, given a time period and a rate of compound interest, how do we determine the numerical value of $n$? This is relatively straight-forward to do: take the number of years $t$ we would like to accrue interest over and multiply by the frequency number $f$ given. For instance, if we want to invest money for three years at a rate that is compounded monthly, how many compounding periods will our money earn interest? Well, it would earn interest exactly $n = 3 \times 12 = 36$ times. The relationship between the number of compound periods $n$, the time in years $t$, and the freqeuncy number $f$ is summarized below.

\begin{aligned} n &= t \times f \end{aligned}

Substituting this relationship into the formula above gives the future value formula for compound interest.

\begin{aligned} FV &= P \left( 1 + \frac{r}{f} \right)^{t \times f} \end{aligned}

The following video will work through a few examples where we utilize the future value formula for compound interest problems.