## 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.

### Present Value of 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 of119,554.36 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}