Vincent would like to have $50,000 per year to live off in his retirement. He expects to retire on his 65^th birthday, make his first retirement withdrawal on his 66^th birthday, and to live until his 85^th birthday. He is celebrating his 30^th birthday today and will make his first savings deposit one year from today. If he can earn 4% per year on his savings, the amount he must save each year to meet his goal, with the last savings deposit on his 65^th birthday, is closest to: A) $7,257.59. B) $7,547.90. C) $7,849.81.

Question

Quizplus · Accepted Answer

To solve this, we use the formula for the future value of an annuity to calculate the amount needed at retirement to withdraw $50,000 annually for 20 years (from age 66 to 85) at a 4% interest rate. Then, we use the formula for the present value of an annuity to determine how much Vincent needs to save annually from age 31 to 65 (35 years) to reach that amount.1. Calculate the future value needed at retirement:$$PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)$$$$50,000 \times \left( \frac{1 - (1 + 0.04)^{-20}}{0.04} \right) = 50,000 \times 14.5949 = 729,745$$2. Calculate the annual savings required:$$PMT = FV \times \left( \frac{r}{1 - (1 + r)^{-n}} \right)$$$$729,745 \times \left( \frac{0.04}{1 - (1 + 0.04)^{-35}} \right) \approx 7,547.90$$Therefore, Vincent needs to save approximately $7,547.90 annually to meet his retirement goal.

Vincent Would Like to Have $50,000 Per Year to Live