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. a. $26,430.20 b. $29,073.31, c. $26,430.25 or d. $27,751.79
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, we can use the formula:
\[FV = P \times \left(\frac{(1 + r)^n - 1}{r}\right) \]
Where:
FV = Future Value
P = Payment (Quarterly Deposit)
r = Interest Rate per period
n = Number of periods
In this case, the payment (P) is $450, the interest rate (r) is 10% or 0.10 per quarter, and the number of periods (n) is 9 years, which is equivalent to 36 quarters since there are 4 quarters in a year.
Substituting these values into the formula, we get:
\[FV = 450 \times \left(\frac{(1 + 0.10)^{36} - 1}{0.10}\right) \]
Calculating the expression within the brackets first:
\[((1 + 0.10)^{36} - 1) = (1.10^{36} - 1) \]
Using a calculator or spreadsheet, we find that (1.10)^36 = 5.744 and therefore:
\[((1 + 0.10)^{36} - 1) = (5.744 - 1) = 4.744 \]
Now, substituting this value back into the original formula:
\[FV = 450 \times \left(\frac{4.744}{0.10}\right) \]
Calculating this expression, we find:
\[FV = 450 \times 47.44 = $21,348\]
Therefore, none of the answer choices given (a, b, c, d) match the calculated value of $21,348. It is possible that there is an error or typo in the provided answer choices.