8.2 Annuites
For most of us, we aren’t able to put a large sum of money in the bank today. 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 or IRA plans are examples of savings annuities.
Suppose we will deposit $100 each month into an account paying 6% interest. How much will we have after a year? We assume that the account is compounded with the same frequency as we make deposits unless stated otherwise. In this example:
or 6%
(12 compounds/deposits per year)
(our deposit per month)
year
With ordinary annuities we assume the payment is made at the end of the period. The $100 we deposit at the end of the first month will earn interest for 11 months and at the end of the year will be worth
The $100 deposited at the end of the second month will have 10 months to grow, and will be worth at the end of the year. This pattern continues down to the last deposit, which has no time to compound, and will be worth .
In total, we will have accumulated:
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any easier! To simplify things, multiply both sides of the equation by 1.005:
Distributing on the right side of the equation gives:
Now we’ll line this up with like terms from our original equation, and subtract each side:
Almost all the terms cancel on the right-hand side when we subtract, leaving:
Now we can solve this equation for :
Recall 0.005 was r/k, 100 was the deposit d, and 12 was the number of months, kt. Generalizing this result, we get the saving annuity formula.
Annuity Formula
- is the balance in the account after years.
- is the regular deposit (the amount you deposit each year, each month, etc.)
- is the annual interest rate in decimal form.
- 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.
For example, if the compounding frequency isn’t stated:
- If you make your deposits every month, use monthly compounding, .
- If you make your deposits every year, use yearly compounding, .
- If you make your deposits every quarter, use quarterly compounding, .
Etc.
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 assumes that you put money in the account once and let it sit there earning interest.
Compound interest: One deposit
Annuity: Many deposits
Example of an Annuity
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?
In this example,
the monthly deposit
6% annual rate
since we’