# find the value of a three-year \$45 annuity if the interest rate is 5% per year

lfryerda | Certified Educator

An annuity is an investment where the an amount is paid into the investment regularly until the end of the investment.  In this case, there is no additional information given and the interest rate is per year, so the annuity amount of \$45 is invested each year.  We need to calculate the amount of the investment at the end of the third year.

For more complicated annuities (longer investment periods), you should use the annuity formula for it, but in this case, it is probably easier to calculate it directly.

At the beginning of the investment, \$45 is paid, which accumulates interest at the end of the 1st, 2nd and 3rd year.

The beginning of the second year, another \$45 is invested, which accumulates interest at the end of the 2nd and 3rd year.

Finally, at the beginning of the 3rd year, the last \$45 is invested, which accumulates interest at the end of the 3rd year.

The total investment is the sum of all those amounts.

The first amount is `A=45(1+0.05)^3=52.09`.

The second amount is `A=45(1+0.05)^2=49.61` .

The third amount is `A=45(1+0.05)=47.25` .

The total value (also called future value) is the sum of those three amounts `A=52.09+49.61+47.25=148.95` .

This can also be calculated using the future-value annuity formula:

`A={(1.05)((1.05)^3-1)}/0.05=148.96`   where the penny discrepancy from earalier was due to rounding errors in the intermediate calculations.

The total value of the annuity is \$148.96.