## Annuities

After reading this section, you will know how to identify, define, and calculate an annuity's present and future value. An annuity is the structure of a financial instrument that is a finite series of level payments that have a definite end. When you are finished, you will be able to recognize the two types of annuities: an ordinary annuity and an annuity due, and explain how they are different. You will also be able to calculate each of these types of annuities and contrast them to their opposites: perpetuities. Annuities are key to understanding because they mimic the payment structure of a bond's coupon payment. This section is foundational for being able to calculate bond prices.

### Calculating Annuities

Annuities can be calculated by knowing four of the five variables: PV, FV, interest rate, payment size, and number of periods.

#### LEARNING OBJECTIVE

• Calculate a perpetuity

#### KEY TAKEAWAYS

##### Key Points
• There are five total variables that go into annuity calculations: PV, FV, interest rate (i or r), payment amount (A, m, pmt, or p), and the number of periods (n).
• The calculations for ordinary annuities and annuities-due differ due to the different times when the first and last payments occur.
• Perpetuities don't have a FV formula because they continue forever. To find the FV at a point, treat it as an ordinary annuity or annuity-due up to that point.

##### Key Terms
• perpetuity: An annuity in which the periodic payments begin on a fixed date and continue indefinitely.
• ordinary annuity: An annuity where the payments occur at the end of each period.
• annuity-due: An annuity where the payments occur at the beginning of each period.

There are five total variables that go into annuity calculations:

• Present value ($PV$)
• Future value ($FV$)
• Interest rate ($i$ or $r$)
• Payment amount ($A$, $m$, $pmt$, or $p$)
• Number of periods ($n$)

So far, we have addressed ways to find the $PV$ and $FV$ of three different types of annuities:

• Ordinary annuities: payments occur at the end of the period ( and )
• Annuities-due: payments occur at the beginning of the period ( and )
• Perpetuities: payments continue forever . Perpetuities don't have a FV because they don't have an end date. To find the FV of a perpetuity at a certain point, treat the annuity up to that point as one of the other two types.

$P V=\frac{A}{r}$

PV of a Perpetuity: The PV of a perpetuity is the payment size divided by the interest rate.

The Future Value of an Annuity due = $\frac{m\left[(1+r / n)^{k+1}-1\right]}{r / n}-m$

FV Annuity-Due: The $FV$ of an annuity with payments at the beginning of each period: m is the amount amount, r is the interest, n is the number of periods per year, and $t$ is the number of years.

$P_{n}=P \frac{(1+i)^{n}-1}{i}(1+i)$

PV Annuity-Due: The $PV$ of an annuity with the payments at the beginning of each period

$F V(A)=A \cdot \frac{(1+i)^{n}-1}{i}$

FV Ordinary Annuity: The $FV$ of an annuity with the payments at the end of each period

$P_{0}=\frac{P_{n}}{(1+i)^{n}}=P \cdot \sum_{k=1}^{n} \frac{1}{(1+i)^{n+1-k}}=P \frac{1-(1+i)^{-1}}{i}$

PV Ordinary Annuity: The $PV$ of an annuity with the payments at the end of each period

Each formula can be rearranged within a few steps to solve for the payment amount. Solving for the interest rate or number of periods is a bit more complicated, so it is better to use Excel or a financial calculator to solve for them.

This may seem like a lot to commit to memory, but there are some tricks to help. For example, note that the $PV$ of an annuity-due is simply 1+i times the $PV$ of an ordinary annuity.

As for the $FV$ equations, the $FV$ of an annuity-due is the same as the $FV$ of an ordinary annuity plus one period and minus one payment. This logically makes sense because all payments in an ordinary annuity occur one period later than in an annuity-due.

Unfortunately, there are a lot of different ways to write each variable, which may make the equations seem more complex if you are not used to the notation. Fundamentally, each formula is similar, however. It is just a matter of when the first and last payments occur (or the size of the payments for perpetuities). Go carefully through each formula and the differences should eventually become apparent, which will make the formulas much easier to understand, regardless of the notation.