In the following ordinary annuity, the interest is compounded with each payment, and the payment is made at the end of the compounding period. How much must you invest each month in a mutual fund yielding 13.5% compounded monthly to become a millionaire in 10 years? (Round your answer to the nearest cent.)
To find out how much you must invest each month to become a millionaire in 10 years with an interest rate of 13.5% compounded monthly, we can use the formula for the future value of an ordinary annuity:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = future value (the target amount of one million dollars)
P = payment amount (the amount you need to invest each month)
r = interest rate per compounding period (13.5% divided by 12 months)
n = number of compounding periods (10 years multiplied by 12 months per year)
Let's plug in the values and solve for P:
FV = 1000000
r = 0.135 / 12
n = 10 * 12
1000000 = P * [(1 + 0.135/12)^(10*12) - 1] / (0.135/12)
Now, we can solve for P by rearranging the equation to isolate P:
P = 1000000 * (0.135/12) / [(1 + 0.135/12)^(10*12) - 1]
Calculating this equation will give us the monthly payment needed to become a millionaire in 10 years. Round the answer to the nearest cent.
Please note: Since the interest rate is compounded monthly and the payment is made at the end of the compounding period, this calculation assumes that your first investment is made at the end of the first month.
Calculating this equation may require a calculator or a spreadsheet program.
To calculate the monthly investment needed to become a millionaire in 10 years with a mutual fund yielding 13.5% compounded monthly, we can use the formula for the future value of an ordinary annuity.
The formula for the future value of an ordinary annuity is:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future Value
P = Monthly Payment
r = Interest Rate per compounding period (monthly rate in this case)
n = Number of compounding periods (number of months in this case)
In this scenario, we want to find the monthly payment (P) needed to reach a future value (FV) of one million dollars.
FV = $1,000,000
r = 13.5% / 12 = 0.135 / 12 = 0.01125 (monthly rate)
n = 10 years * 12 months/year = 120 months
Substituting these values into the formula:
$1,000,000 = P * [(1 + 0.01125)^120 - 1] / 0.01125
Now we can solve for P:
$1,000,000 * 0.01125 = P * [(1 + 0.01125)^120 - 1]
$11,250 = P * [(1.01125)^120 - 1]
$11,250 = P * (1.489036 - 1)
$11,250 = P * 0.489036
P = $11,250 / 0.489036
P ≈ $23,023.26
Therefore, you would need to invest approximately $23,023.26 each month in the mutual fund yielding 13.5% compounded monthly to become a millionaire in 10 years.