Jasmine has decided that she wants to build enough retirement wealth that, if invested at 6 percent per year, will provide her with $3,000 of monthly income for 30 years. To date, she has saved nothing but she still has 25 years until she retires. Jasmine believes that she can earn 9 percent on her investments until she retires. How much money does she need to contribute per month to reach her goal?
A) $446.32
B) $521.84
C) $667.13
D) $722.05

Question

Quizplus · Accepted Answer

To solve this, we first need to calculate the future value of the annuity that will provide Jasmine with $3,000 monthly for 30 years at a 6% annual interest rate. Then, we calculate how much she needs to save monthly for 25 years at a 9% annual interest rate to reach that future value.1. Calculate the present value (PV) of the retirement annuity:$$ PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) $$Where:- $PMT = \$3,000$ (monthly payment)- $r = \frac{0.06}{12} = 0.005$ (monthly interest rate)- $n = 30 \times 12 = 360$ (total number of payments)$$ PV = \$3,000 \times \left( \frac{1 - (1 + 0.005)^{-360}}{0.005} \right) \approx \$545,372.98 $$2. Calculate the monthly savings needed to reach the PV of the retirement annuity:$$ PMT = PV \times \left( \frac{r}{1 - (1 + r)^{-n}} \right) $$Where:- $PV = \$545,372.98$- $r = \frac{0.09}{12} = 0.0075$ (monthly interest rate)- $n = 25 \times 12 = 300$ (total number of payments)$$ PMT = \$545,372.98 \times \left( \frac{0.0075}{1 - (1 + 0.0075)^{-300}} \right) \approx \$446.32 $$Therefore, Jasmine needs to contribute approximately $446.32 per month to reach her goal.

Jasmine Has Decided That She Wants to Build Enough Retirement