A grateful alumnus wishes to provide a scholarship of $2,600 per year for 5 years to his alma mater, with the first scholarship awarded on his 55th birthday. If the money is worth an annual rate of 6.4%, compounded annually, how much money must he donate on his 45th birthday? Round your answer to the nearest cent.

Question

A grateful alumnus wishes to provide a scholarship of $2,600 per year for 5 years to his alma mater, with the first scholarship awarded on his 55th birthday. If the money is worth an annual rate of 6.4%, compounded annually, how much money must he donate on his 45th birthday? Round your answer to the nearest cent. ​

Quizplus · Accepted Answer

To find out how much the alumnus needs to donate on his 45th birthday to provide a scholarship of $2,600 per year for 5 years starting on his 55th birthday, we need to calculate the present value of an annuity. The formula for the present value of an annuity is:$$ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) $$where:- $PV$ is the present value of the annuity,- $P$ is the payment amount per period,- $r$ is the interest rate per period,- $n$ is the number of periods.Given:- $P = \$2,600$,- $r = 6.4\% = 0.064$ (annual interest rate),- $n = 5$ (since the scholarship is for 5 years),- The donation is made 10 years before the first scholarship is awarded, so we also need to discount the present value of the annuity back 10 years to his 45th birthday.First, calculate the present value of the annuity at the time the first scholarship is awarded (his 55th birthday):$$ PV = \$2,600 \times \left( \frac{1 - (1 + 0.064)^{-5}}{0.064} \right) $$$$ PV = \$2,600 \times \left( \frac{1 - (1 + 0.064)^{-5}}{0.064} \right) $$$$ PV = \$2,600 \times 18.756 $$$$ PV = \$11,564.56 $$ (This is the total amount needed at the beginning of the scholarship period, not the answer.)Next, discount this amount back 10 years to find out how much needs to be donated on his 45th birthday:$$ PV_{45} = \frac{\$11,564.56}{(1 + 0.064)^{10}} $$$$ PV_{45} = \frac{\$11,564.56}{1.8194} $$$$ PV_{45} = \$6,198.89 $$Therefore, the alumnus must donate approximately $6,198.89 on his 45th birthday.

A Grateful Alumnus Wishes to Provide a Scholarship of $2,600