At the age of 34
,
to save for retirement, you decide to deposit $80
at the end of each month in an IRA that pays 6
%
compounded monthly.
a.
Use the following formula to determine how much you will have in the IRA when you retire at age 65.
Upper A equals StartFraction Upper P left bracket left parenthesis 1 plus r right parenthesis Superscript t Baseline minus 1 right bracket Over r EndFraction
or
Upper A equals StartStartFraction Upper P left bracket left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript nt Baseline minus 1 right bracket OverOver left parenthesis StartFraction r Over n EndFraction right parenthesis EndEndFraction
b.
Find the interest.
a. In this problem, P = $80, n = 12 (since there are 12 months in a year), r = 6%/100 = 0.06 (as r should be in decimal form), and t = 65 - 34 = 31 years.
Let's use the second formula:
A = P * [ ((1 + (r/n))^(nt) - 1 ) / (r/n) ]
Substituting the given values, we get:
A = 80 * [ ((1 + (0.06/12))^(12*31) - 1 ) / (0.06/12) ]
Calculate the expression inside the brackets first.
A = 80 * 97.9233
Multiply the two:
A = $7,833.86
So, when you retire at age 65, you will have approximately $7,833.86 in your IRA.
b. The interest earned on the account is the final account balance minus the total contributions. You made monthly contributions over 31 years, which is 31*12 = 372 months.
Compute the total of your contributions by multiplying the number of months by the amount of contribution per month.
So, total contributions = $80 * 372 = $29,760
Subtract this from the final account balance to find the interest.
Interest = $78,338.66 - $29,760 = $48,578.66
a. To determine how much you will have in the IRA when you retire at age 65, you can use the formula:
Upper A = (Upper P * [(1 + r)^t - 1]) / r
Where:
- Upper A represents the amount in the IRA when you retire
- Upper P represents the monthly deposit ($80 in this case)
- r represents the interest rate per compounding period (6% or 0.06 as a decimal)
- t represents the number of compounding periods (12 compounding periods per year from age 34 to 65, so t = 31 years * 12 months/year = 372 months)
Substituting the given values into the formula:
Upper A = (80 * [(1 + 0.06)^372 - 1]) / 0.06
b. The interest earned can be calculated by subtracting the total amount deposited from the total amount in the IRA when you retire:
Interest = Upper A - (Upper P * t)
Substituting the given values into the formula:
Interest = [(80 * [(1 + 0.06)^372 - 1]) / 0.06] - (80 * 372)