Determine the future value of an annuity due into which quarterly deposits of $450 are made for nine years if the annuity pays 10% compounded quarterly.
To determine the future value of an annuity due, you can use the formula for the future value of an ordinary annuity and then adjust for the annuity due.
The formula for the future value of an ordinary annuity is:
FV = P * [(1 + r)^n - 1] / r
Where:
- FV is the future value of the annuity
- P is the periodic payment (also known as the deposit amount)
- r is the interest rate per period
- n is the number of periods
In this case, the deposit amount (P) is $450, the interest rate (r) is 10% (or 0.10) compounded quarterly, and the number of periods (n) is 9 years, which is equivalent to 36 quarters.
First, let's calculate the future value of the ordinary annuity:
FV_ordinary = $450 * [(1 + 0.10/4)^(4*9) - 1] / (0.10/4)
Simplifying this equation:
FV_ordinary = $450 * [(1 + 0.025)^36 - 1] / 0.025
Next, we need to adjust for the annuity due. To do that, we multiply the future value of the ordinary annuity by (1 + r), which in this case is (1 + 0.10/4).
FV_annuity_due = FV_ordinary * (1 + 0.10/4)
Substituting the FV_ordinary value:
FV_annuity_due = [$450 * [(1 + 0.025)^36 - 1] / 0.025] * (1 + 0.10/4)
Now, you can calculate the future value of the annuity due by plugging in the values and performing the calculations.