Question
How much should you deposit at the end of each month in an IRA that pays 5% compounded monthly to earn
$110,000 per year from interest alone, while leaving the principal untouched, to be withdrawn at the end of
each year after you retire in 50 years?
$732
$787
$829
None of these
$110,000 per year from interest alone, while leaving the principal untouched, to be withdrawn at the end of
each year after you retire in 50 years?
$732
$787
$829
None of these
Answers
GPT-4o mini
To determine how much you should deposit at the end of each month in an IRA to reach a goal of earning $110,000 per year from interest alone, we need to calculate the required principal amount needed at retirement and then work backwards to find the monthly deposits.
1. **Calculate the required principal amount (P)**:
To earn $110,000 per year from interest alone at a 5% annual interest rate, we can use the formula for interest earnings:
\[
\text{Interest} = P \times r
\]
Where \( r \) is the annual interest rate (0.05 for 5%).
Rearranging the formula to find \( P \):
\[
P = \frac{\text{Interest}}{r} = \frac{110,000}{0.05} = 2,200,000
\]
Thus, you need a principal (P) of $2,200,000 at retirement.
2. **Calculate the future value of the monthly deposits**:
You will be making monthly deposits \( PMT \) into your IRA for 50 years (600 months). The formula for the future value of a series of cash flows (ordinary annuity) compounded monthly is:
\[
FV = PMT \times \frac{(1 + r/n)^{nt} - 1}{(r/n)}
\]
Where:
- \( FV \) is the future value, which we calculated as $2,200,000.
- \( PMT \) is the monthly deposit.
- \( r \) is the annual interest rate (0.05).
- \( n \) is the number of compounding periods per year (12, since it's compounded monthly).
- \( t \) is the number of years (50).
Plugging in the values:
\[
2,200,000 = PMT \times \frac{(1 + 0.05/12)^{12 \times 50} - 1}{(0.05/12)}
\]
First, we need to calculate \( (1 + 0.05/12)^{600} \):
\[
(1 + 0.0041667)^{600} \approx (1.0041667)^{600} \approx 18.679
\]
Now substitute that back into the formula:
\[
2,200,000 = PMT \times \frac{18.679 - 1}{0.0041667}
\]
\[
\frac{18.679 - 1}{0.0041667} \approx \frac{17.679}{0.0041667} \approx 4245.4
\]
Thus, the equation becomes:
\[
2,200,000 = PMT \times 4245.4
\]
Now solve for \( PMT \):
\[
PMT = \frac{2,200,000}{4245.4} \approx 518.41
\]
3. **Conclusion**:
The calculated monthly deposit is approximately $518.41, which does not match any of the options given. Therefore, the answer is:
**None of these.**
1. **Calculate the required principal amount (P)**:
To earn $110,000 per year from interest alone at a 5% annual interest rate, we can use the formula for interest earnings:
\[
\text{Interest} = P \times r
\]
Where \( r \) is the annual interest rate (0.05 for 5%).
Rearranging the formula to find \( P \):
\[
P = \frac{\text{Interest}}{r} = \frac{110,000}{0.05} = 2,200,000
\]
Thus, you need a principal (P) of $2,200,000 at retirement.
2. **Calculate the future value of the monthly deposits**:
You will be making monthly deposits \( PMT \) into your IRA for 50 years (600 months). The formula for the future value of a series of cash flows (ordinary annuity) compounded monthly is:
\[
FV = PMT \times \frac{(1 + r/n)^{nt} - 1}{(r/n)}
\]
Where:
- \( FV \) is the future value, which we calculated as $2,200,000.
- \( PMT \) is the monthly deposit.
- \( r \) is the annual interest rate (0.05).
- \( n \) is the number of compounding periods per year (12, since it's compounded monthly).
- \( t \) is the number of years (50).
Plugging in the values:
\[
2,200,000 = PMT \times \frac{(1 + 0.05/12)^{12 \times 50} - 1}{(0.05/12)}
\]
First, we need to calculate \( (1 + 0.05/12)^{600} \):
\[
(1 + 0.0041667)^{600} \approx (1.0041667)^{600} \approx 18.679
\]
Now substitute that back into the formula:
\[
2,200,000 = PMT \times \frac{18.679 - 1}{0.0041667}
\]
\[
\frac{18.679 - 1}{0.0041667} \approx \frac{17.679}{0.0041667} \approx 4245.4
\]
Thus, the equation becomes:
\[
2,200,000 = PMT \times 4245.4
\]
Now solve for \( PMT \):
\[
PMT = \frac{2,200,000}{4245.4} \approx 518.41
\]
3. **Conclusion**:
The calculated monthly deposit is approximately $518.41, which does not match any of the options given. Therefore, the answer is:
**None of these.**