Use the savings plan formula to answer the following question.
A friend has an IRA with an APR of [tex]$7.25 \%$[/tex]. She started the IRA at age 23 and deposits [tex]$35[/tex] per month. How much will her IRA contain when she retires at age 65? Compare that amount to the total deposits made over the time period.
To calculate how much money the IRA will contain after retirement, use the formula shown below, where PMT is the regular payment (deposit) amount, APR is the annual percentage rate (as a decimal), [tex]$n$[/tex] is the number of payment periods per year, and [tex]$Y$[/tex] is the number of years money is saved.
[tex]$A=P M T \times \frac{\left[\left(1+\frac{A P R}{n}\right)^{(n Y)}-1\right]}{\left(\frac{A P R}{n}\right)}$[/tex]
Answer (1)
Calculate the future value of the IRA using the savings plan formula: A = PMT × ( n A PR ) [ ( 1 + n A PR ) ( nY ) − 1 ] .
Substitute the given values: PMT = $35 , A PR = 0.0725 , n = 12 , and Y = 42 to find A = $114 , 810.19 .
Calculate the total deposits made: Total Deposits = PMT × n × Y = $35 × 12 × 42 = $17 , 640 .
Compare the future value of the IRA to the total deposits made. The IRA contains $114 , 810.19 compared to the total deposits of $17 , 640 .
$114 , 810.19
Explanation
Understanding the Problem We are asked to calculate the future value of an IRA using the savings plan formula and then compare that value to the total deposits made over the investment period. The formula is:
A = PMT × ( n A PR ) [ ( 1 + n A PR ) ( nY ) − 1 ]
Where:
A is the amount in the IRA after Y years.
PMT is the regular payment (deposit) amount.
A PR is the annual percentage rate (as a decimal).
n is the number of payment periods per year.
Y is the number of years the money is saved.
Identifying Given Information We are given the following information:
PMT = $35 per month
A PR = 7.25% = 0.0725
n = 12 (monthly deposits)
Y = 65 − 23 = 42 years
Calculating the Future Value of the IRA First, we calculate the amount A in the IRA after 42 years using the formula:
A = 35 × ( 12 0.0725 ) [ ( 1 + 12 0.0725 ) ( 12 × 42 ) − 1 ]
A = 35 × 0.00604167 [ ( 1 + 0.00604167 ) 504 − 1 ]
A = 35 × 0.00604167 [ ( 1.00604167 ) 504 − 1 ]
A = 35 × 0.00604167 [ 1.648898 − 1 ]
A = 35 × 0.00604167 0.648898
A = 35 × 107.402
A = 114810.19
So, the IRA will contain $114 , 810.19 when she retires.
Calculating Total Deposits Next, we calculate the total deposits made over the time period:
Total Deposits = PMT × n × Y
Total Deposits = 35 × 12 × 42
Total Deposits = 35 × 504
Total Deposits = $17 , 640
Comparing the Future Value to Total Deposits Finally, we compare the amount in the IRA at retirement to the total deposits made:
Amount in IRA: $114 , 810.19 Total Deposits: $17 , 640
The IRA contains $114 , 810.19 , while the total deposits made were $17 , 640 . The interest earned over the 42 years is $114 , 810.19 − $17 , 640 = $97 , 170.19 .
Final Answer The IRA will contain $114 , 810.19 when she retires at age 65. The total deposits made over the 42-year period will be $17 , 640 .
Examples
Saving for retirement is a crucial aspect of financial planning. By understanding the savings plan formula, individuals can estimate the future value of their investments and make informed decisions about their savings strategy. For instance, consider two friends, Alice and Bob, who both start saving for retirement at age 25. Alice invests $200 per month with an APR of 6%, while Bob invests $300 per month with an APR of 5%. Using the savings plan formula, they can project their savings at age 65 and adjust their contributions accordingly to meet their retirement goals. This proactive approach ensures a financially secure future.
