A young couple wants to have a college fund that will pay $25,000 at the end of each half-year for 8 years. If they can invest at annual rate of 8%, compounded semiannually, how much do they need to invest at the end of each 6-month period for the next 18 years in order to begin making their college withdrawals 6 months after their last investment? Round your answer to the nearest cent.

Question

A young couple wants to have a college fund that will pay $25,000 at the end of each half-year for 8 years. If they can invest at annual rate of 8%, compounded semiannually, how much do they need to invest at the end of each 6-month period for the next 18 years in order to begin making their college withdrawals 6 months after their last investment? Round your answer to the nearest cent. ​

Quizplus · Accepted Answer

To solve this, we use the formula for the future value of an annuity to calculate the amount needed for the college fund, and then use the present value of an annuity formula to determine how much they need to invest each half-year. The future value of the college fund can be calculated using the formula $FV = P \times \frac{(1 + r)^n - 1}{r}$, where $P$ is the payment amount, $r$ is the interest rate per period, and $n$ is the total number of payments. Given an 8% annual interest rate compounded semiannually, $r = 0.04$ (since 8% annual rate divided by 2), and with $25,000 needed at the end of each half-year for 8 years, $n = 16$ (since 8 years times 2). Plugging these values into the formula gives the future value needed for the college fund. Then, to find out how much they need to invest at the end of each 6-month period for the next 18 years to reach this future value, we use the present value of an annuity formula, considering the same rate and doubling the number of periods (since 18 years times 2). The correct calculation will yield the amount they need to invest each period, which matches one of the given options.

A Young Couple Wants to Have a College Fund That