At the beginning of each period for 12 years, Merl Agnes invests $580 semiannually at 6%. What is the cash value of this annuity due at the end of year 12? (Do not round intermediate calculations. Round your answer to the nearest cent.)
Question
At the beginning of each period for 12 years, Merl Agnes invests $580 semiannually at 6%. What is the cash value of this annuity due at the end of year 12? (Do not round intermediate calculations. Round your answer to the nearest cent.)
Solution
To calculate the cash value of this annuity due at the end of year 12, we need to use the formula for the future value of an annuity due. The formula is:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n)
Where: FV = future value of the annuity P = amount invested each period ($580) r = annual interest rate (6% or 0.06) n = number of times the interest is compounded per period (2, because it's semiannually) t = number of periods (12 years)
First, we need to convert the annual interest rate to the semiannual rate and the number of years to the number of periods:
r = 0.06 / 2 = 0.03 t = 12 * 2 = 24
Then, we can substitute these values into the formula:
FV = 580 * [(1 + 0.03)^24 - 1] / 0.03
Now, calculate the expression in the brackets:
= 580 * [2.03^24 - 1] / 0.03
= 580 * [20.48 - 1] / 0.03
= 580 * 19.48 / 0.03
Finally, calculate the future value:
FV = $37,000.16
So, the cash value of this annuity due at the end of year 12 is $37,000.16.
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