6.5 Annuities

Robert Foth

6.5 Annuities

6.5: Annuities

Learning Objectives

Upon completion of this section, you should be able to

Determine the future value in applications of an annuity.
Determine payment values needed to achieve a future value in applications of an annuity.
Determine the principal amount needed for a given payment in applications of a payout annuity
Determine the payment needed for a given principal amount in applications of a payout annuity

Ordinary Annuity

Imagine you’re planning for an important goal in your life. Maybe you’re saving up for nursing school tuition, setting aside money for a car, or building a fund for a career change. Whatever your aim, you probably can’t afford to put aside a large sum at once to let it grow to your financial goal. Instead, you might choose to save a small amount from each paycheck. This common financial strategy closely mirrors a concept called an annuity – a series of equal payments made at fixed intervals over time into an account that is compounding interest.

In this section we will examine a situation that is more precisely called an ordinary annuity, where deposits are made at the end of each period rather than the beginning. When deposits are made at the beginning it is called an annuity due. We’ll learn how to calculate:

The future value of an annuity (how much your savings will grow to)
The payment amount needed to reach a specific financial goal

Annuity Formula

The future value of an annuity can be calculated using the following formula:

$F V = \frac{p y m t ({(1 + \frac{r}{n})}^{(n \cdot t)} - 1)}{(\frac{r}{n})}$

The payment needed to reach a future value can be calculated using the following formula:

$p y m t = \frac{F V (\frac{r}{n})}{({(1 + \frac{r}{n})}^{(n \cdot t)} - 1)}$

$F V$ is the balance in the account after $t$ years.
$p y m t$ is the regular deposit (the amount you deposit each year, each month, etc.)
$r$ is the annual interest rate in decimal form.
$n$ is the number of compounding periods in one year.
$t$ is the amount of time deposits are being made (in years).

If the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year. Without this assumption the formula would yield incorrect results. For example,

If you make your deposits every month, use monthly compounding, $n = 12$ .
If you make your deposits every year, use yearly compounding, $n = 1$ .
If you make your deposits every quarter, use quarterly compounding, $n = 4$ .

When do you use this

Annuities assume that you put money in the account on a regular schedule (every month, year, quarter, etc.) and let it sit there earning interest.

Compound interest discretely or continuously formulas assume that you put money in the account once and let it sit there earning interest.

Compound interest: One deposit
Annuity: Many deposits.

Example 1

A traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month into an IRA earning 6% interest, how much will you have in the account after 20 years?

Solution

In this example,

$\begin{array}{l} p y m t & = $ 100 & the monthly deposit \\ r & = 0.06 & 6% annual rate \\ n & = 12 & since we’re doing monthly deposits, we’ll compound monthly \\ t & = 20 & we want the amount after 20 years \end{array}$

Putting this into the annuity formula we can find the future value:

$\begin{array}{l} F V = \frac{100 ({(1 + \frac{0.06}{12})}^{20 (12)} - 1)}{(\frac{0.06}{12})} \\ F V = \frac{100 ({(1.005)}^{240} - 1)}{(0.005)} \\ F V = \frac{100 (3.310 - 1)}{(0.005)} \\ F V = \frac{100 (2.310)}{(0.005)} = $ 46200 \end{array}$

The account will grow to $46,200 after 20 years.

Notice that you deposited into the account a total of $24,000 ($100 a month for 240 months). The difference between what you end up with and how much you put in is the interest earned. In this case it is $$ 46, 200 - $ 24, 000 = $ 22, 200$ .

A word on rounding. The work above rounded the different steps while doing the computation. It is always better to do the entire computation at once. If you had done that the total $F V$ would have been closer to $46,204.09.

Try it Now 1

For the last month Madeline orders a Mocha Frappuccino (Grande) from StarBucks® every day. The cost with tax is about $5 when she goes. If instead she invested the $5 a day for one year in an account that pays 3% interest compounded daily, then how much will she have? How much is from interest?

Hint 1 (click to Show/Hide)

We have to be careful with these investment problems that we are able to identify the difference between a single deposit problem and multiple equal deposits into an account spread out evenly over a fixed time period.

The key words to focus on here is that we have deposits happening each day over a time period. This means we are dealing with an annuity problem and need to determine what we need to find with the annuity formula.

You are given the fixed deposit amounts of $5 which represents $p y m t$ .
You are told the rate $r$ is 3%.
You are told the frequency of the deposits $n$ being daily ( $n = 365$ ).
Lastly we are told that the money is being deposited for one year which is represented by $t$ .

Answer (click to Show/Hide)

Start by listing out what we know:

$\begin{array}{l} r & = 0.03 & 3% annual rate \\ n & = 365 & since we’re doing daily deposits \\ t & = 1 & 1 year \\ p y m t & = $ 5 & the daily deposit \end{array}$

Use the Annuity Formula to find the future value, FV, in the account:

$F V = \frac{5 ({(1 + \frac{0.03}{365})}^{365 \times 1} - 1)}{\frac{0.03}{365}} = $ 1, 852.57$

We found that in the account there would be $1,852.57 after one year.

For the interest we know she would have deposited a total of $$ 5 \cdot 365 \cdot 1 = $ 1, 825$ , so the extra money in the account is from interest. Take the difference of the money in the account and the deposits:

$$ 1, 852.57 - $ 1825.00 = $ 27.57$

She earned $27.57 in interest in that account over the one year.

Next let us look at a situation where we are saving for some future goal.

Example 2

You want to have $200,000 in your account when you retire in 30 years. Your retirement account earns 8% interest compounded monthly. How much do you need to deposit each month to meet your retirement goal?

Solution

In this example, we’re looking for the monthly deposits into the account to reach a future goal. Let us start with writing out the given information:

$\begin{array}{l} r & = 0.08 & 8% annual rate \\ n & = 12 & since we’re depositing monthly \\ t & = 30 & 30 years \\ F V & = $ 200, 000 & the amount we want to have in 30 years \end{array}$

In this case, we’re solving for $p y m t$ .

$\begin{array}{l} p y m t = \frac{F V (\frac{r}{n})}{({(1 + \frac{r}{n})}^{(n \cdot t)} - 1)} \\ p y m t = \frac{200, 000 (\frac{0.08}{12})}{({(1 + \frac{0.08}{12})}^{(12 \cdot 30)} - 1)} \\ p y m t \approx $ 134.20 \end{array}$

So you would need to deposit $134.20 each month to have $200,000 in 30 years if your account earns 8% interest.

As you look back at this answer if we add up all the deposits made each month for 30 years we have a total of $$ 134.20 \cdot (12) \cdot 30 = $ 48, 312.00$ .

The difference between the future value and the sum of the deposits is the amount of interest that was earned over that time in the account: $$ 200, 000 - $ 48, 312.00 = $ 151, 688$ .

The interest earned was over three times the amount that was deposited over the 30 years.

Try it Now 2

The owners of a local pizza restaurant wants to replace the pizza oven in two years. Instead of financing the purchase they will make monthly payments into a savings account that earns 2.78% interest compounded monthly. How much do those deposits need to be in order to have $19800 in two years?

Hint 1 (click to Show/Hide)

We are looking for the $p y m t$ amount to reach a future value for an annuity. The future value is the
cost of the pizza oven, $19800.

Answer (click to Show/Hide)

Start with listing out the given information about the future value, rate, time, and compounding

$\begin{array}{l} F V = $ 19, 800 \\ r = 0.0278 \\ t = 2 \\ n = 12 \end{array}$

Next enter the values into the annuity formula and solve for the unknown $p y m t$ amount (the monthly deposits).

$\begin{array}{l} p y m t = \frac{F V (\frac{r}{n})}{({(1 + \frac{r}{n})}^{(n \cdot t)} - 1)} \\ p y m t = \frac{19, 800 (\frac{0.0278}{12})}{({(1 + \frac{0.0278}{12})}^{(12 \cdot 2)} - 1)} \\ p y m t \approx $ 803.23 \end{array}$

They would need to deposit approximately $803.23 each month for the next two years to reach the goal of $19,800.

When companies are using this type of savings strategy to pay off a future debt it is called a sinking fund.

Payout Annuities

In the above section you learned about annuities. In an annuity, you start with nothing, put money into an account on a regular basis, and end up with money in your account.

In this section, we will learn about a variation called a Payout Annuity. With a payout annuity, you start with money in the account, and pull money out of the account on a regular basis. Any remaining money in the account earns interest. After a fixed amount of time, the account will end up empty if the amount pulled out is high enough.

Payout annuities are typically used after retirement. Perhaps you have saved $500,000 for retirement, and want to take money out of the account each month to live on. If you want the money to last you 20 years and take payments (withdrawals) out to yourself over that time this is a setup for a payout annuity (as long as that money was sitting in an account that was compounding interest). The formula is the same as what you saw for amortized loans and should look familiar.

Payout Annuity Formula

The payments (withdrawals) for a payout annuity with a known starting balance can be found using the following formula:

$p y m t = \frac{P (\frac{r}{n})}{(1 - {(1 + \frac{r}{n})}^{(- n \cdot t)})}$

The starting balance in the account for a payout annuity with fixed payments can be found using the following formula:

$P = \frac{p y m t (1 - {(1 + \frac{r}{n})}^{(- n \cdot t)})}{(\frac{r}{n})}$

$P$ is the balance in the account at the beginning (starting amount, or principal).
$p y m t$ is the regular withdrawal (the amount you take out each year, each month, etc.)
$r$ is the annual interest rate (in decimal form. Example: 5% = 0.05)
$n$ is the number of compounding periods in one year.
$t$ is the number of years we plan to take withdrawals

Like with annuities, the compounding frequency is not always explicitly given, so we will assume it is how often you take the the withdrawals.

When do you use this

Payout annuities assume that you take money from the account on a regular schedule (every month, year, quarter, etc.) and let the rest sit there earning interest.

Compound interest formulas: One deposit
Annuity: Many deposits
Payout Annuity: Many withdrawals

Example 3

After retiring, you want to be able to take $1000 every month for a total of 20 years from your retirement account. The account earns 6% interest. How much will you need in your account when you retire?

Solution

Start with writing out the given information:

$\begin{array}{l} p y m t & = $ 1000 & the monthly withdrawal \\ r & = 0.06 & 6% annual rate \\ n & = 12 & since we’re doing monthly withdrawals, we’ll compound monthly \\ t & = 20 & since were taking withdrawals for 20 years \end{array}$

We’re looking for $P$ ; how much money needs to be in the account at the beginning. Putting this into the formula:

$\begin{array}{l} P = \frac{1000 (1 - {(1 + \frac{0.06}{12})}^{- 20 (12)})}{(\frac{0.06}{12})} \\ P = \frac{1000 \times (1 - {(1.005)}^{- 240})}{(0.005)} \\ P = \frac{1000 \times (1 - 0.302)}{(0.005)} = $ 139, 600 \end{array}$

You will need to have $139,600 in your account when you retire.

Notice that you withdrew a total of $240,000 ($1000 a month for 240 months). The difference between what you pulled out and what you started with is the interest earned during that time from when you started taking withdrawal till the last withdrawal. In this case it is:

$$ 240, 000 - $ 139, 600 = $ 100, 400$

With the account starting with $139,600 and taking the withdraws it still earned $100,400 in interest over the 20 years.

A word on rounding. The work above rounded the different steps while doing the computation. It is always better to do the entire computation at once. If you had done that the value for $P$ would have been closer to $139,580.77 (about a $1000 difference).

Example 4

You know you will have $500,000 in your account when you retire. You want to be able to take monthly withdrawals from the account for a total of 30 years. Your retirement account earns 8% interest. How much will you be able to withdraw each month?

Solution

In this example, we’re looking for the amount you can withdraw each month, $p y m t$ . Start with listing out what we know first:

$\begin{array}{l} P & = $ 500, 000 & we are beginning with $500,000 \\ r & = 0.08 & 8% annual rate \\ n & = 12 & since we’re doing monthly withdrawals \\ t & = 30 & taking withdrawals for 30 years \end{array}$

In this case, we’re going to use the other formula to solve for $p y m t$ :

$\begin{array}{l} p y m t = \frac{P (\frac{r}{n})}{(1 - {(1 + \frac{r}{n})}^{(- n \cdot t)})} \\ p y m t = \frac{500, 000 (\frac{0.08}{12})}{(1 - {(1 + \frac{0.08}{12})}^{(- 12 \cdot 30)})} \\ p y m t \approx $ 3, 368.82 \end{array}$

You would be able to withdraw $3,368.82 each month for 30 years.

If we had ignored the interest being earned the $500,000 payments over 30 years would have been $\frac{$ 500, 000}{12 \cdot 30} = $ 1, 388.89$ , which is less than half of the payments actually received.

The ability for the savings to continue to earn interest (grow) allows that $500,000 to be stretched into a much higher payment. The total of all the payments would actually give you $$ 3, 368.82 \cdot 12 \cdot 30 = $ 1, 212, 775.20$ over the 30 years.

Try it Now 3

A donor gives $100,000 to a university, and specifies that it is to be used to give annual scholarships for the next 20 years. If the university can earn 4% interest, how much can they give in scholarships each year?

Hint 1 (click to Show/Hide)

A fixed amount is used to distribute money from an account over a fixed time period. In this scenario we are dealing with a payout annuity. If we are depositing a fixed amount into an account that is a general annuity problem. So the key thing here is we are told money is coming out of an account that earns interest. Take a look at the payout annuity formula where we are solving for $p y m t$ and try to determine what values represent what variables in the formula:

$p y m t = \frac{P (\frac{r}{n})}{(1 - {(1 + \frac{r}{n})}^{(- n \cdot t)})}$

Hint 2 (click to Show/Hide)

Below is the given information listed out:

$\begin{array}{l} P & = $ 100, 000 & we are beginning with $100,000 \\ r & = 0.04 & 4% annual rate \\ n & = 1 & since we’re doing annual scholarships \\ t & = 20 & taking withdrawals for 20 years \end{array}$

Answer (click to Show/Hide)

The withdrawal amount, $p y m t$ is unknown, but we are given the following

$\begin{array}{l} P & = $ 100, 000 & we are beginning with $100,000 \\ r & = 0.04 & 4% annual rate \\ n & = 1 & since we’re doing annual scholarships \\ t & = 20 & taking withdrawals for 20 years \end{array}$

Use the given information into the payout annuity formula where $p y m t$ is solved for:

$\begin{array}{l} p y m t = \frac{P (\frac{r}{n})}{(1 - {(1 + \frac{r}{n})}^{(- n \cdot t)})} \\ p y m t = \frac{100, 000 (\frac{0.04}{1})}{(1 - {(1 + \frac{0.04}{1})}^{(- 1 \cdot 20)})} \\ p y m t \approx $ 7, 358.18 \end{array}$

Solving for $p y m t$ gives $7,358.18 each year that they can give in scholarships.

It is worth noting that sometimes donors instead specify that only interest is to be used for scholarship, which makes the original donation last indefinitely. If this donor had specified that $$ 100, 000 (0.04) = $ 4, 000$ a year would have been available.

Exercises

You deposit $200 each month into an account with an annual rate of 3% compounded monthly.
1. How much will you have in the account in 30 years?
2. How much total money will you put into the account?
3. How much total interest will you earn?
You deposit $1000 each year into an account with an annual rate of 8% compounded annually.
1. How much will you have in the account in 10 years?
2. How much total money will you put into the account?
3. How much total interest will you earn?
Jose has determined he needs to have $800,000 for retirement in 30 years. His account has an annual rate of 6% compounded monthly.
1. How much would you need to deposit in the account each month?
2. How much total money will you put into the account?
3. How much total interest will you earn?
You plan to save $600 a month for the next 30 years for retirement. What annual interest rate compounded monthly would you need to have $1,000,000 at retirement? Approximate the solution for “r ” to the nearest percent. Hint: start with a guess for what “r” is by trying something like 5%, if you don’t reach a million after 30 years you can then increase/decrease the value of r by a few percentage points and try again.
You once heard that if you were to save just a little each month that it will add up to a nice cushion for retirement. If you were to save just $25 a month over the next 40 years and deposited the money into an account with an annual rate of 4.5% compounded monthly how much would you have at the start of retirement in that account?
Compare the following scenarios:
1. Deposit $100 a month for twenty years into an account with an annual rate of 5% compounded monthly.
2. Wait ten years – deposit $200 a month for ten years into an account with an annual rate of 5% compounded monthly.
3. In the second scenario what amount needs to be deposited in order to have the same amount after the ten years as you did in part “a”.
You want to be able to withdraw $30,000 each year for 25 years. Your account has an annual rate of 8% compounded annually.
1. How much do you need in your account at the beginning
2. How much total money will you pull out of the account?
3. How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20 years from an account with an annual rate of 8% compounded annually?
1. How much do you need in your account at the beginning
2. How much total money will you pull out of the account?
3. How much of that money is interest?
You have $500,000 saved for retirement. Your account has an annual rate of 6% compounded monthly. How much will you be able to pull out each month, if you want to be able to take withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout annuity for 30 years in an account paying an annual rate of 10% compounded monthly, how much could the annuity provide each month?

Attributions

This page contains modified content from David Lippman, “Math In Society, 2nd Edition.” Licensed under CC BY-SA 4.0.
This page contains content by Robert Foth, Math Faculty, Pima Community College, 2021. Licensed under CC BY 4.0.

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Ordinary Annuity

When do you use this

Solution

Solution

Payout Annuities

When do you use this

Solution

Solution

Exercises

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