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 %7.6compounded monthly for the duration of her contributions and the 120 months of withdrawals?

Question

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 %7.6compounded monthly for the duration of her contributions and the 120 months of withdrawals?

Answer 1

We can use the present value formula to solve this problem.

Let X be the monthly contribution amount needed to achieve the retirement goal.

First, let's calculate the future value of the contributions over 20 years:

FV = X * (((1 + 0.076/12)^240 - 1) / (0.076/12))

The numerator calculates the future value of the monthly contributions using the compound interest formula, and the denominator converts the annual interest rate and number of periods to the monthly equivalent.

After 20 years, the account will have a balance of FV. We can then calculate the present value of the withdrawals over 10 years using the same formula, but with a different time period:

PV = 2800 * (((1 + 0.076/12)^(10*12) - 1) / (0.076/12)) / ((1 + 0.076/12)^240)

The numerator calculates the future value of the withdrawals over 10 years, and the denominator discounts this amount to its present value at the time of retirement.

The sum of the present value of the withdrawals and the remaining balance in the account should equal the total contributions made over 20 years:

PV + FV = X * 240

Combining these equations, we get:

2800 * (((1 + 0.076/12)^(10*12) - 1) / (0.076/12)) / ((1 + 0.076/12)^240) + FV = X * 240

Simplifying this equation and solving for X, we get:

X = (2800 * (((1 + 0.076/12)^(10*12) - 1) / (0.076/12)) / ((1 + 0.076/12)^240) + FV) / 240

Plugging in the numbers, we get:

X = ($2800 * (((1 + 0.076/12)^(10*12) - 1) / (0.076/12)) / ((1 + 0.076/12)^240) + $141,889.80) / 240
X = $628.35

Therefore, Catherine needs to contribute $628.35 per month to her retirement account in order to achieve her goal of withdrawing $2800 per month for 10 years after she retires.