Annuity is similar concept to perpetuity, but it is finite. So, payments are either paid or received for a set amount of time. Think about your parents’ mortgage or car loans. You make same payments every single month for a specified number of years.
In order to conceptually understand annuity, we have to play with the perpetuity formula. Check out the graph.
Here, we calculated perpetuity for year 0, and year 5. They are both $10,000. However, in order to compare apples to apples, we have to discount $10,000 from year 5 to year 0. The difference will be present value of annuity.
So, pretty much what we did is we took payments of $500 forever from year 0, and payments of $500 forever from year 5. If you subtract all payments of $500 forever from year 5, than you’re left over only with payments from years 1 through 5. Therefore, the formula for PV of annuity is: