# BM5.2 – The Future Value Formula

Chapter 5, Lesson 2

In this lesson students will:

• recognize the form of a geometric series
• determine the sum of a geometric series using the closed-form formula
• use a time diagram to develop the future value formula
• calculate the future value of an ordinary simple annuity

## Future Value – Ordinary Simple Annuities

#### Geometric Series

In this lesson, our goal is to develop the future value formula for a series of equivalent payments that all earn compound interest over time. To fully understand how the formula is developed, we must first briefly look at geometric series.

If a sum of numbers has a certain format it is known as a geometric series. The format of the numbers being added together begins with the number $1$ and then all numbers following are multiplied by the same number, called the common ratio $r$. Let’s refer to the sum of the geometric series as $S$. The format can then be summarized as follows, where we take $n-1$ as the power of the last term for convenience:

\begin{aligned} S &= 1 + r + r^{2} + r^{3} + ... + r^{n-1} \end{aligned}

Next, we would like to develop a formula for calculating the sum $S$ easily. To do this, we can multiply all the terms in our formula above by the common ratio $r$ and then subtract the original sum $S$.

\begin{aligned} rS &= r + r^{2} + r^{3} + r^{4} + ... + r^{n}\\ rS - S &= (r + r^{2} + r^{3} + r^{4} + ... + r^{n}) - (1 + r + r^{2} + r^{3} + ... + r^{n-1})\\ rS - S &= r^{n} - 1 \end{aligned}

Finally, by factoring the left side of our equation and dividing by $r-1$ we obtain a closed-form formula for finding the sum of a geometric series.

\begin{aligned} rS - S &= r^{n} - 1 \\ S(r-1) &= r^{n}-1\\ S &= \frac{r^{n}-1}{r-1} \end{aligned}

The video below will guide you through two examples of how to use the closed-form formula.

#### The Time Diagram Approach

Next, let’s turn our attention to setting up a time diagram to develop the future value formula. We would like to consider an ordinary simple annuity. This means that equivalent payments, which we label as $PMT$, occur at the end of each compound period; and that compound frequency is the same as the payment frequency. To denote the time periods in which compounding and payments occur, we use $t_{n}$. The video below will guide you through the formulation of the future value formula for an ordinary simple annuity.

#### Future Value Formula Examples

As we saw in our previous video, the future value formula for an ordinary simple annuity is:

\begin{aligned} FV &= PMT \left[ \frac{(1+i)^{n}-1}{i} \right] \end{aligned}

Let’s work through a couple examples to help solidify our use of the formula.