6.5 Annuities
5.6
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
Annuities
For most of us, we aren’t able to put a large sum of money in the bank today and leave that money sit for
many years as our only savings strategy. Instead, we save for the future by depositing a smaller amount of money
from each paycheck into the bank. This idea is called a savings annuity. Most retirement plans
like 401k plans and IRA plans are examples of something like a savings annuities as you are making smaller
frequent contributions to those accounts.
In this section we will examine a situation that is more precisely called an ordinary annuity.
Deposits made at the end of each period rather than the beginning. When deposits are made at the beginning it is
called an annuity due and changes the formula you would use slightly. The purpose of this
section is to introduce students to a way to calculate the future value and payment amount for an annuity
scenario and we will only focus on applications of ordinary annuities to simplify the number of formulas needed
to get the point across.
Annuity Formula
FV is the balance in the account after t years.
pymt 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.
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 the compounding frequency isn’t stated:
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.
Etc.
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,
Putting this into the annuity equation we can find the future value:
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
.
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 FV 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
Answer
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 or pymt.
In this case, we’re going to have to set up the equation, and solve for pymt.
So you would need to deposit $134.20 each month to have $200,000 in 30 years if your account earns 8%
interest
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
Answer
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. You want the money to last you 20
years. This is a payout annuity. The formula is similar to that of savings annuities, but the same
as what you saw for amortized loans.
Payout Annuity Formula
P is the balance in the account at the beginning (starting amount, or principal).
pymt 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, but is determined by how often
you take 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
In this example,
We’re looking for P; how much money needs to be in the account at the beginning. Putting this
into the equation:
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
in interest.
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?
In this example, we’re looking for the amount you can withdraw each month, pymt.
In this case, we’re going to have to set up the equation, and solve for d.
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 be
, 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
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
Hint 2
Answer
Exercises
- You deposit $200 each month into an account with an annual rate of 3% compounded monthly.
- How much will you have in the account in 30 years?
- How much total money will you put into the account?
- How much total interest will you earn?
- You deposit $1000 each year into an account with an annual rate of 8% compounded annually.
- How much will you have in the account in 10 years?
- How much total money will you put into the account?
- 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.- How much would you need to deposit in the account each month?
- How much total money will you put into the account?
- 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:
- Deposit $100 a month for twenty years into an account with an annual rate of 5% compounded monthly.
- Wait ten years – deposit $200 a month for ten years into an account with an annual rate of 5%
compounded monthly. - 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.- How much do you need in your account at the beginning
- How much total money will you pull out of the account?
- 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?- How much do you need in your account at the beginning
- How much total money will you pull out of the account?
- 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?