The article explains the concept of annuities, focusing on the future value of an annuity due, where payments are made at the beginning of each period. It provides a formula for calculating the future value and compares the outcomes for ordinary annuities and annuities due using an example.
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
Annuity Due 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
\[\begin{align}& F{{V}_{\text{annuity due}}}=PMT\times \frac{{{\left( 1+i \right)}^{n}}-1}{i}\times \left( 1+i \right) \\& =\$5,000\times\frac{{{\left(1.07\right)}^{6}}-1}{0.07}\times\left(1.07\right)=\$38,270.11\\\end{align}\]
Future Value of an Annuity Due Key Takeaways
The future value of an annuity due is essential in financial planning and investment strategies, as it helps in calculating the value of a series of payments over time. By comparing ordinary annuities and annuities due, individuals and businesses can make informed decisions regarding investment choices, retirement planning, and loan management, ultimately maximizing the returns on their investments or minimizing the costs of financing. The ability to calculate the future value accurately ensures better financial forecasting and strategic decision-making.