## 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.

### Future Value of Annuity

The future value of an annuity is the sum of the future values of all of the payments in the annuity.

#### LEARNING OBJECTIVE

• Calculate the future value of different types of annuities

#### KEY TAKEAWAYS

##### Key Points
• To find the $FV$, you need to know the payment amount, the interest rate of the account the payments are deposited in, the number of periods per year, and the time frame in years.
• The first and last payments of an annuity due both occur one period before they would in an ordinary annuity, so they have different values in the future.
• There are different formulas for annuities due and ordinary annuities because of when the first and last payments occur.

##### Key Terms
• annuity-due: An investment with fixed-payments that occur at regular intervals, paid at the beginning of each period.
• ordinary repair: expense accrued in normal maintenance of an asset
• annuity-due: a stream of fixed payments where payments are made at the beginning of each period
• ordinary annuity: An investment with fixed-payments that occur at regular intervals, paid at the end of each period.

The future value of an annuity is the sum of the future values of all of the payments in the annuity. It is possible to take the $FV$ of all cash flows and add them together, but this isn't really pragmatic if there are more than a couple of payments.

If you were to manually find the $FV$ of all the payments, it would be important to be explicit about when the inception and termination of the annuity is. For an annuity-due, the payments occur at the beginning of each period, so the first payment is at the inception of the annuity, and the last one occurs one period before the termination.

For an ordinary annuity, however, the payments occur at the end of the period. This means the first payment is one period after the start of the annuity, and the last one occurs right at the end. There are different $FV$ calculations for annuities due and ordinary annuities because of when the first and last payments occur.

There are some formulas to make calculating the $FV$ of an annuity easier. For both of the formulas we will discuss, you need to know the payment amount ($m$, though often written as $pmt$ or $p$), the interest rate of the account the payments are deposited in ($r$ ,though sometimes $i$), the number of periods per year ($n$), and the time frame in years ($t$).

The formula for an ordinary annuity is as follows:

$\mathrm{A}=\frac{\mathrm{m}\left[(1+\mathrm{r} / \mathrm{n})^{\mathrm{nt}}-1\right]}{\mathrm{r} / \mathrm{n}}$

where $m$ is the payment amount, $r$ is the interest rate, $n$ is the number of periods per year, and $t$ is the length of time in years.

In contrast, the formula for an annuity-due is as follows:

$\mathrm{A}=\frac{\mathrm{m}\left[(1+\mathrm{r} / \mathrm{n})^{\mathrm{nt}+1}-1\right]}{\mathrm{r} / \mathrm{n}}-\mathrm{m}$

Provided you know $m$, $r$, $n$, and $t$, therefore, you can find the future value ($FV$) of an annuity.