# BM8.3 – Calculating the Term of the Investment

Chapter 8, Lesson 3

In this lesson students will:

• calculate the term of a general annuity due given information about the future value
• calculate the term of a general annuity due given information about the principle amount

## Calculating the Total Number of Payments

#### Given the Future Value

In Lesson 6.4 we developed the formula for solving for the number of equal sized payments for an annuity due. Recall that the procedure to solve a problem involving a general annuity is to first calculate an equivalent interest rate $i_{2}$. Once we know this periodic rate, we can substitute it into one of our ordinary annuity formulas. If we know the future value of the annuity we are working with then the formula below will apply.

\begin{aligned} n &= \frac{\ln \left( \frac{i_{2} \; FV}{PMT(1+i_{2})} + 1 \right)}{\ln(1+i_{2})} \end{aligned}

Let’s consider the following example: If you make equal payments of $\250$ at the beginning of every week into an account that earns $3\%$ compounded monthly, how long will it take to save $\15,000$? We will solve this question using both the formula and the TMV buttons on our calculator.

#### Given the Principle Amount

Suppose we are given the principle amount, or present value, of an annuity and would like to determine the term. In this case, we can first calculate the equivalent interest rate $i_{2}$ and then substitute this into our formula.

\begin{aligned} n = \frac{-\ln \left( 1-\frac{i_{2} \; P}{PMT(1+i_{2})} \right)}{\ln(1+i_{2})} \end{aligned}

Let’s consider the following example: A loan of $\200,000$ will be paid off by making payments of $\10,000$ at the beginning of each year. How long will it take to pay off the loan if the interest rate is $4.5\%$ compounded bi-weekly? We will solve this example using both the formula, as well as the TMV buttons on our calculator.