What Is an Annuity?
An annuity is a series of equal payments made at equal intervals. Annuity payments don’t have to be made annually but can be made monthly, weekly, or even daily.
The critical factors are:
- The payments equal each other and;
- The interval between the payments is the same.
Annuity Due is the one in which payments are made at the beginning of each period.
Formula
More generally, for any size of payment and number of time periods, the future value of an annuity due is equal to
\[\begin{matrix}F{{V}_{\text{annuity due}}}=PMT\times \frac{{{\left( 1+i \right)}^{n}}-1}{i}\times \left( 1+i \right) & \left( 1 \right) \\\end{matrix}\]
Where
PMT=the periodic payment in the annuity
i=the interest rate
n= the number of payments
Example 1 FV of an Ordinary Annuity and an Annuity Due
You have been provided an investment chance that will pay you $5,000 per year for next 6 years. What is the value of this stream of cash flows at the end of year 6 if you get payments at the end of each year and you expect a return of 7%? What if you obtain payments at the beginning of each year?
Solution
In this example,
\[\begin{align}& PMT=5,000 \\& i=0.07 \\& n=6 \\\end{align}\]
For the ordinary annuity timing, the solution is
\[\begin{align}& F{{V}_{\text{ordinary annuity}}}=PMT\times \frac{{{\left( 1+i \right)}^{n}}-1}{i} \\& =\$5,000\times\frac{{{\left(1.07\right)}^{6}}-1}{0.07}=\$35,766.45\\\end{align}\]
For the annuity due timing, the solution is
\[\begin{align}& F{{V}_{\text{annuity due}}}=PMT\times \frac{{{\left( 1+i \right)}^{n}}-1}{i}\times \left( 1+i \right) \\& =\$5,000\times\frac{{{\left(1.07\right)}^{6}}-1}{0.07}\times\left(1.07\right)=\$38,270.11\\\end{align}\]