Catherine Dohanyos plans to retire in 20 years. She will make 20 years of monthly contributions to her retirement account. One month after her last contribution, she will begin the first of 10 years of withdrawals. She wants to withdraw $2800 per month. How large must her monthly contributions be in order to accomplish her goal if the account earns interest of 7.1% compounded monthly for the duration of her contributions and the 120 months of withdrawals?
We can use the formula for the future value of an annuity to solve this problem. Let's break it down into steps:
1. Calculate the future value of Catherine's monthly contributions over 20 years, using the formula:
FV = PMT × ((1 + r/n)^(n*t) - 1) / (r/n)
where:
- PMT = the monthly contribution
- r = the annual interest rate (7.1%)
- n = the number of compounding periods per year (12, since it's compounded monthly)
- t = the number of years (20)
Plugging in the numbers, we get:
FV = PMT × ((1 + 0.071/12)^(12*20) - 1) / (0.071/12)
FV = PMT × 603.833
2. Calculate the present value of Catherine's withdrawals over 10 years, using the formula:
PV = PMT × (1 - (1 + r/n)^(-n*t)) / (r/n)
where:
- PMT = the monthly withdrawal
- r = the annual interest rate (7.1%)
- n = the number of compounding periods per year (12, since it's compounded monthly)
- t = the number of years (10)
Plugging in the numbers, we get:
PV = 2800 × (1 - (1 + 0.071/12)^(-12*10)) / (0.071/12)
PV = 2800 × 219.613
3. Use the formula for the present value of an annuity to find the required monthly contributions:
PV = PMT × ((1 - (1 + r/n)^(-n*t)) / (r/n))
where:
- PV = the present value we just calculated
- PMT = the monthly contribution (what we're trying to find)
- r = the annual interest rate (7.1%)
- n = the number of compounding periods per year (12, since it's compounded monthly)
- t = the number of years (30, since it's 20 years of contributions plus 10 years of withdrawals)
Plugging in the numbers, we get:
2800 × 219.613 = PMT × ((1 - (1 + 0.071/12)^(-12*30)) / (0.071/12))
PMT = 2800 × 219.613 / ((1 - (1 + 0.071/12)^(-12*30)) / (0.071/12))
PMT = 400.43
Therefore, Catherine needs to contribute $400.43 per month in order to withdraw $2800 per month for 10 years, assuming an interest rate of 7.1% compounded monthly.