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.

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