Question
If Ali wants to save for retirement and plans to withdraw $2,000 every month for the next 20 years, how much should he save today if the interest rate is 5 percent per year? Give the answer rounded to the nearest hundredths.(1 point)
Answers
GPT-4o mini
To determine how much Ali should save today to withdraw $2,000 every month for the next 20 years at an interest rate of 5% per year, we'll use the present value of an annuity formula.
First, we need to calculate the monthly interest rate and the number of total withdrawals:
- Annual interest rate = 5% or 0.05 per year.
- Monthly interest rate (r) = 0.05 / 12 = 0.00416667
- Total number of withdrawals (n) = 20 years × 12 months/year = 240 months
The present value of an annuity formula is given by:
\[
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
\]
Where:
- \(PV\) = present value (the amount he should save today)
- \(P\) = payment amount per period ($2,000)
- \(r\) = monthly interest rate (0.00416667)
- \(n\) = total number of payments (240)
Now we can plug in the values:
\[
PV = 2000 \times \frac{1 - (1 + 0.00416667)^{-240}}{0.00416667}
\]
Calculating the term \((1 + 0.00416667)^{-240}\):
1. \(1 + 0.00416667 = 1.00416667\)
2. \((1.00416667)^{-240} \approx 0.3678794412\)
Now substituting back into the formula:
\[
PV = 2000 \times \frac{1 - 0.3678794412}{0.00416667}
\]
\[
PV = 2000 \times \frac{0.6321205588}{0.00416667}
\]
\[
PV = 2000 \times 151.2841207 \approx 302568.2414
\]
So, rounding to the nearest hundredths, the present value is:
\[
PV \approx 302568.24
\]
Therefore, Ali should save approximately **$302,568.24** today to ensure he can withdraw $2,000 every month for the next 20 years at an interest rate of 5%.
First, we need to calculate the monthly interest rate and the number of total withdrawals:
- Annual interest rate = 5% or 0.05 per year.
- Monthly interest rate (r) = 0.05 / 12 = 0.00416667
- Total number of withdrawals (n) = 20 years × 12 months/year = 240 months
The present value of an annuity formula is given by:
\[
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
\]
Where:
- \(PV\) = present value (the amount he should save today)
- \(P\) = payment amount per period ($2,000)
- \(r\) = monthly interest rate (0.00416667)
- \(n\) = total number of payments (240)
Now we can plug in the values:
\[
PV = 2000 \times \frac{1 - (1 + 0.00416667)^{-240}}{0.00416667}
\]
Calculating the term \((1 + 0.00416667)^{-240}\):
1. \(1 + 0.00416667 = 1.00416667\)
2. \((1.00416667)^{-240} \approx 0.3678794412\)
Now substituting back into the formula:
\[
PV = 2000 \times \frac{1 - 0.3678794412}{0.00416667}
\]
\[
PV = 2000 \times \frac{0.6321205588}{0.00416667}
\]
\[
PV = 2000 \times 151.2841207 \approx 302568.2414
\]
So, rounding to the nearest hundredths, the present value is:
\[
PV \approx 302568.24
\]
Therefore, Ali should save approximately **$302,568.24** today to ensure he can withdraw $2,000 every month for the next 20 years at an interest rate of 5%.