# BM9.1 – Perpetuities

Chapter 9, Lesson 1

In this lesson students will:

• understand where the formulas for perpetuities come from
• calculate the present value of an ordinary perpetuity in both the simple and general case
• calculate the present value of a perpetuity due in both the simple and general case

## Perpetuities

Some lottery companies in Canada offer a scratch ticket in which a player might win $\1000$ every week for the rest of their lives. Depending on when you receive this payment, we could think about this as an annuity. To make this lottery win more exciting, let’s assume that this payment stream actually lasts forever! These are the kind of annuities we would like to discuss in this lesson. A perpetuity is an annuity in which the equal-sized payments continue forever.

#### Ordinary Perpetuities

To start off this lesson, let’s take the equal-sized payments to occur at the end of the payment period. This means that we will be working with a ordinary annuity. If the payments occur for the rest of time, then we call this an ordinary perpetuity. Be mindful that if $f_{1} \neq f_{2}$ (the compounding frequency is not the same as the payment frequency), you are working with a general perpetuity, and you will need to apply the equivalent interest formula $i_{2} = \left( 1 + i_{1} \right)^{f1/f2} - 1$ before applying the formula.

We have seen in the previous video that the formula for the present value of an ordinary perpetuity is

\begin{aligned} P &= \frac{PMT}{i} \end{aligned}

Let’s apply the ordinary perpetuity formula in a couple of examples.

#### Perpetuities Due

Next, consider equal-sized payments that occur at the beginning of the payment period. This means that we will be working with an annuity due. To start off this lesson, let’s take the equal-sized payments to occur at the end of the payment period. This means that we will be working with a ordinary annuity. If the payments occur for the rest of time, then we call this a perpetuity due. Of course, if $f_{1} = f_{2}$ (the compounding frequency is equal to the payment frequency), you are working with a simple perpetuity, you do not need to calculate $i_{2}$.

We have seen in the previous video that the formula for the present value of a perpetuity due is

\begin{aligned} P &= PMT + \frac{PMT}{i} \end{aligned}

Let’s apply the perpetuity due formula in a couple of examples.