Annuities

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.

The $PV$ of an annuity can be found by calculating the $PV$ of each individual payment and then summing them up.

LEARNING OBJECTIVE

• Calculate the present value of annuities

KEY TAKEAWAYS

Key Points

• The $PV$ for both annuities-due and ordinary annuities can be calculated using the size of the payments, the interest rate, and number of periods.
• The $PV$ of a perpetuity can be found by dividing the size of the payments by the interest rate.
• Payment size is represented as $p$, $pmt$, or $A$; interest rate by $i$ or $r$; and number of periods by $n$ or $t$.

Key Terms

• perpetuity: An annuity in which the periodic payments begin on a fixed date and continue indefinitely.

The Present Value ($PV$) of an annuity can be found by calculating the $PV$ of each individual payment and then summing them up . As in the case of finding the Future Value ($FV$) of an annuity, it is important to note when each payment occurs. Annuities-due have payments at the beginning of each period, and ordinary annuities have them at the end.

Recall that the first payment of an annuity-due occurs at the start of the annuity, and the final payment occurs one period before the end. The $PV$ of an annuity-due can be calculated as follows:

$\mathrm{P}_{0}=\frac{\mathrm{P}_{\mathrm{n}}}{(1+\mathrm{i})^{\mathrm{n}}}=\mathrm{P} \frac{1-(1+\mathrm{i})^{-\mathrm{n}}}{\mathrm{i}} \cdot(1+\mathrm{i})$

where $P$ is the size of the payment (sometimes $A$ or $pmt$), i is the interest rate, and $n$ is the number of periods.

An ordinary annuity has annuity payments at the end of each period, so the formula is slightly different than for an annuity-due. An ordinary annuity has one full period before the first payment (so it must be discounted) and the last payment occurs at the termination of the annuity (so it must be discounted for one period more than the last period in an annuity-due). The formula is:

$\mathrm{P}_{0}=\frac{\mathrm{P}_{\mathrm{n}}}{(1+\mathrm{i})^{\mathrm{n}}}=\mathrm{P} \cdot \sum_{\mathrm{k}=1}^{\mathrm{n}} \frac{1}{(1+\mathrm{i})^{\mathrm{n}+\mathrm{k}-1}}=\mathrm{P} \cdot \frac{1-\left[\frac{1}{(1+\mathrm{i})^{\mathrm{n}}}\right]}{\mathrm{i}}$

where, again, $P$, $i$, and $n$ are the size of the payment, the interest rate, and the number of periods, respectively.

Both annuities-due and ordinary annuities have a finite number of payments, so it is possible, though cumbersome, to find the $PV$ for each period. For perpetuities, however, there are an infinite number of periods, so we need a formula to find the $PV$. The formula for calculating the $PV$ is the size of each payment divided by the interest rate.

Example 1

Suppose you have won a lottery that pays $1,000 per month for the next 20 years. But, you prefer to have the entire amount now. If the interest rate is 8%, how much will you accept?

Consider for argument purposes that two people, Mr. Cash, and Mr. Credit, have won the same lottery of $1,000 per month for the next 20 years. Now, Mr. Credit is happy with his $1,000 monthly payment, but Mr. Cash wants to have the entire amount now. Our job is to determine how much Mr. Cash should get. We reason as follows: If Mr. Cash accepts x dollars, then the x dollars deposited at 8% for 20 years should yield the same amount as the $1,000 monthly payments for 20 years. In other words, we are comparing the future values for both Mr. Cash and Mr. Credit, and we would like the future values to be equal.

Since Mr. Cash is receiving a lump sum of x dollars, its future value is given by the lump sum formula:

$x\left(\frac{1+.08}{12}\right)^{240}$

Since Mr. Credit is receiving a sequence of payments, or an annuity, of $1,000 per month, its future value is given by the annuity formula:

$\frac{1000\left[\left(\frac{1+0.08}{12}\right)^{240}-1\right]}{\frac{0.08}{12}}$

The only way Mr. Cash will agree to the amount he receives is if these two future values are equal. So we set them equal and solve for the unknown:

$\begin{aligned}&\mathrm{x}\left(\frac{1+.08}{12}\right)^{240}=\frac{1000\left[\left(\frac{1+0.08}{12}\right)^{240}-1\right]}{\frac{0.08}{12}} \\&\mathrm{x} \cdot(4.9268)=1,000 \cdot(589.02041) \\&\mathrm{x} \cdot 4.9268=589,020.41 \\&\mathrm{x}=119,554.36\end{aligned}$

The reader should also note that if Mr. Cash takes his lump sum of $119,554.36 and invests it at 8% compounded monthly, he will have $589,020.41 in 20 years.

Example 2

Find the monthly payment for a car costing $15,000 if the loan is amortized over five years at an interest rate of 9%.

Again, consider the following scenario: Two people, Mr. Cash and Mr. Credit, go to buy the same car that costs $15,000. Mr. Cash pays cash and drives away, but Mr. Credit wants to make monthly payments for five years. Our job is to determine the amount of the monthly payment.

We reason as follows: If Mr. Credit pays x dollars per month, then the x dollar payment deposited each month at 9% for 5 years should yield the same amount as the $15,000 lump sum deposited for 5 years. Again, we are comparing the future values for both Mr. Cash and Mr. Credit, and we would like them to be the same.

Since Mr. Cash is paying a lump sum of $15,000, its future value is given by the lump sum formula:

$\$ 15,000 \cdot\left(\frac{1+.09}{12}\right)^{60}$

Mr. Credit wishes to make a sequence of payments, or an annuity, of x dollars per month, and its future value is given by the annuity formula:

$\frac{x\left[\left(\frac{1+0.09}{12}\right)^{60}-1\right]}{\frac{0.09}{12}}$

We set the two future amounts equal and solve for the unknown:

$\begin{aligned}&15,000 \cdot\left(\frac{1+.09}{12}\right)^{60}=\frac{\mathrm{x}\left[\left(\frac{1+0.09}{12}\right)^{60}-1\right]}{\frac{0.09}{12}} \\&15,000 \cdot 1.5657=\mathrm{x} \cdot 75.4241 \\&311.38=\mathrm{x}\end{aligned}$

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.