# BM5.5 – Calculating the Term of the Investment

Chapter 5, Lesson 5

In this lesson students will:

• calculate the term of an ordinary simple annuity given information about the future value
• calculate the term of an ordinary simple annuity given information about the principle amount

## Calculating the Total Number of Payments

In practice we are often interested in knowing the length, or term, of an annuity. For example, perhaps we would like to know how long it will take us to pay off a loan that receives equal-sized payments every month. Or perhaps we would like to save up a certain amount of money by making equal-sized payments into an account, and we are curious about how long this will take. Since there are two formulas for ordinary simple annuities, one for $FV$ and one for $P$, we will divide this lesson into two parts.

#### Given the Future Value

Recall the future value formula for ordinary simple annuities.

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

Using logarithm properties from Lesson 4.3, we solve for the number of equal-sized payments $n$ in the following video. Using the formula $t = nf$, where $f$ represents the frequency of the equal-sized payments, we would be able to determine the term of the annuity in years, if needed.

So we see that it is possible to solve for the number of equivalent payments, given the future value, using the formula

\begin{aligned} n &= \frac{\ln\left( \frac{i \; FV}{PMT} + 1 \right)}{\ln(1+i)}. \end{aligned}

Now, let’s consider the following example: If you make equal payments of $\250$ at the end of every month into an account that earns $3\%$ compounded monthly, how long will it take to save $\15,000$? We will solve this question below.

#### Given the Principle Amount

Recall the present value formula for ordinary simple annuities.

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

Again, using logarithm properties from Lesson 4.3, we can solve for the number of equal-sized payments $n$. This is done in the following video. Using the formula $t = nf$, where $f$ represents the frequency of the equal-sized payments, we would be able to determine the term of the annuity in years, if needed.

So we see that it is possible to solve for the number of equivalent payments, given the future value, using the formula

\begin{aligned} n &= \frac{-\ln\left( 1 - \frac{i \; P}{PMT} \right)}{\ln(1+i)}. \end{aligned}

Next, let’s consider the following example: A loan of $\200,000$ will be paid off by making year-end payments of $\10,000$. How long will it take to pay off the loan if the interest rate is $4.5\%$ compounded annually? Let’s look at this example below.