To calculate the monthly deposit needed in an IRA paying 11% compounded monthly to earn $90,000 a year from interest, we use the future value of an annuity formula. A financial calculator or software is needed to find the exact deposit amount, considering an 11% annual interest rate, compounded monthly over 30 years.
To determine how much should be deposited at the end of each month into an IRA that pays 11% interest compounded monthly, we need to find the monthly deposit that will yield $90,000 from interest alone each year after retirement. To do this, we use the formula for the future value of an annuity, which is a series of equal payments made at regular intervals. The formula we will use takes into account the periodic deposit (PMT), the interest rate per period (i), and the number of periods (n):
FV = PMT × `(({(1+i)^n} - 1) / i)`
In this case, the future value (FV) we desire is the amount that, when multiplied by the monthly interest rate, will yield $90,000 a year. Since the interest is compounded monthly, there will be 12 periods in a year, hence:
$90,000 / 12 months = $7,500 needed from interest each month
Since the annual interest rate is 11%, the monthly interest rate is 11% / 12 months = 0.009167. We use this to find the future value necessary to yield $7,500 per month from interest:
$7,500 / 0.009167 = $818,558.66
This future value needs to be reached by the time of retirement in 30 years. Since n is the number of monthly deposits for 30 years:
n = 30 years × 12 months/year = 360 months
Now that we have n and i, we can plug these values back into the formula for the future value of an annuity to solve for PMT, the monthly deposit we're seeking:
`PMT = FV / (({(1+i)^n} - 1) / i)`
Due to the complexity of the calculation and to provide an accurate answer, a financial calculator or software (such as Excel) should be used to compute PMT. Remember to round up to the nearest dollar as required in the question.