Solving for the Monthly Contribution

Oct 13, 2025 by sorusgentrification.com 128 views

Introduction

Mike is planning for his retirement by making contributions to his account for 20 years. After the last contribution, he will start withdrawing $3,000 a month for 15 years. To determine how large his monthly contributions must be, we need to consider the future value of his account and the present value of his future withdrawals. In this article, we will use the concept of compound interest to calculate the required monthly contributions.

Compound Interest Formula

The formula for compound interest is given by:

A = P(1 + r/n)^(nt)

Where:

A is the future value of the investment
P is the principal amount (initial investment)
r is the annual interest rate (in decimal form)
n is the number of times the interest is compounded per year
t is the time the money is invested for (in years)

In this case, we want to find the present value of Mike's future withdrawals, so we will use the formula for present value:

PV = FV / (1 + r/n)^(nt)

Where:

PV is the present value of the investment
FV is the future value of the investment
r is the annual interest rate (in decimal form)
n is the number of times the interest is compounded per year
t is the time the money is invested for (in years)

Calculating the Future Value of Mike's Account

To calculate the future value of Mike's account, we need to know the number of years he will contribute to the account, the monthly interest rate, and the monthly contribution. Let's assume Mike contributes $x$ dollars per month for 20 years. The future value of his account can be calculated using the formula:

FV = x * (((1 + r/n)^(n*t) - 1) / (r/n))

Where:

FV is the future value of the investment
x is the monthly contribution
r is the annual interest rate (in decimal form)
n is the number of times the interest is compounded per year
t is the time the money is invested for (in years)

Calculating the Present Value of Mike's Future Withdrawals

To calculate the present value of Mike's future withdrawals, we need to know the number of years he will withdraw from the account, the monthly interest rate, and the monthly withdrawal. Let's assume Mike withdraws $3,000 per month for 15 years. The present value of his future withdrawals can be calculated using the formula:

PV = 3000 * (((1 + r/n)^(n*t) - 1) / (r/n))

Where:

PV is the present value of the investment
3000 is the monthly withdrawal
r is the annual interest rate (in decimal form)
n is the number of times the interest is compounded per year
t is the time the money is invested for (in years)

Solving for the Monthly Contribution

Now that we have the formulas for the future value of Mike's account and the present value of his future withdrawals, we can solve for the monthly contribution. We want to find the value of x that makes the present value of the future withdrawals equal to the future value of the account.

PV = FV

3000 * (((1 + 0.08/12)^(12*15) - 1) / (0.08/12)) = x * (((1 + 0.08/12)^(12*20) - 1) / (0.08/12))

Simplifying the equation, we get:

3000 * 134.93 = x * 134.93

x = 3000

So, Mike must contribute $3,000 per month for 20 years to have enough money to withdraw $3,000 per month for 15 years, assuming an annual interest rate of 8% compounded monthly.

Conclusion

In this article, we used the concept of compound interest to calculate the required monthly contributions for Mike's retirement account. We found that Mike must contribute $3,000 per month for 20 years to have enough money to withdraw $3,000 per month for 15 years, assuming an annual interest rate of 8% compounded monthly. This calculation assumes that Mike will not make any withdrawals during the 20-year contribution period and that the interest rate remains constant at 8% per annum.

Limitations

There are several limitations to this calculation. First, it assumes that the interest rate remains constant at 8% per annum, which may not be the case in reality. Second, it assumes that Mike will not make any withdrawals during the 20-year contribution period, which may not be the case in reality. Finally, it assumes that the monthly contribution remains constant at $3,000, which may not be the case in reality.

Future Research

There are several areas for future research. First, we could investigate the impact of different interest rates on the required monthly contributions. Second, we could investigate the impact of different withdrawal rates on the required monthly contributions. Finally, we could investigate the impact of different contribution periods on the required monthly contributions.

References

[1] Investopedia. (2022). Compound Interest Formula.
[2] Investopedia. (2022). Present Value Formula.
[3] Khan Academy. (2022). Compound Interest.
[4] Khan Academy. (2022). Present Value.