Financial Mathematics: Annuities

An annuity is any stream of payments.  Examples include savings accounts where regular deposits are made, loans in which regular payments are made, annuities (the financial product offered by many insurance companies) and so on.

There are a variety of types of annuities and it would be difficult to cover them all. We'll look at a few common ones to get a flavor for how annuities work.

A distinction is made between annuity-immediate and annuity-due. It's important to note which type we're dealing with so we can use the correct mathematical formula.

In annuity-immediate, the first payment is made after the first period. For example, you take out a loan today and you make your first monthly payment one month from today.

In annuity-due, the first payment is made ("due") right now. As an example, if you were to win the Mega Millions jackpot and took the annuity option, the first payment would be made ("due") when you collected your winnings.

There also can be differences in how the payments are made. In some cases, the payments are the same for each period. Many loans (such as traditional 30 year mortgage) work this way. You pay the same amount each period (e.g. $500 per month). Sometimes, payments may increase or decrease. And the way in which they increase can differ (e.g. arithmetically or geometrically). As an example, if you play the Powerball lottery and win the annuity jackpot, you'll receive fixed payments that increase (geometrically) by 4% per year.

And in some cases, annuities do not have a finite number of periods. These are typically referred to as perpetuities.

We'll be looking at a few different types of annuities and how the mathematics work for them all related to the time value of money. 

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