Annuities
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. An annuity can be described recursively in a fairly simple way. Recall that basic compound interest follows from the relationship For a savings annuity, we simply need to add a deposit, d, to the account with each compounding period: Taking this equation from recursive form to explicit form is a bit trickier than with compound interest. It will be easiest to see by working with an example rather than working in general. Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the account is compounded with the same frequency as we make deposits unless stated otherwise. In this example: r = 0.06 (6%) k = 12 (12 compounds/deposits per year) d = $100 (our deposit per month) Writing out the recursive equation gives Assuming we start with an empty account, we can begin using this relationship: Continuing this pattern, after m deposits, we’d have saved: In other words, after m months, the first deposit will have earned compound interest for m-1 months. The second deposit will have earned interest for m-2 months. Last months deposit would have earned only one month worth of interest. The most recent deposit will have earned no interest yet. 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 Solving for Pm Replacing m months with 12N, where N is measured in years, gives Recall 0.005 was r/k and 100 was the deposit d. 12 was k, the number of deposit each year. Generalizing this result, we get the saving annuity formula.
Annuity Formula
PN is the balance in the account after N years.
d is the regular deposit (the amount you deposit each year, each month, etc.)
r is the annual interest rate in decimal form.
k 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, k = 12.
If you make your deposits every year, use yearly compounding, k = 1.
If you make your deposits every quarter, use quarterly compounding, k = 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 assumes that you put money in the account once and let it sit there earning interest.
Compound interest: One deposit
Annuity: Many deposits.