Suppose that you are 34 years old now, and that you would like to retire at the age of 75. Furthermore, you would like to have a retirement fund from which you can draw an income of $70,000 annually. You plan to reach this goal by making monthly deposits into an investment plan until you retire. How much do you need to deposit each month? Assume an APR of 8% compounded monthly, both as you pay into the retirement fund and when you collect from it later.

This problem involves two stages:

1. **Accumulating a retirement fund**: You will make monthly deposits into an investment account until you retire.
2. **Drawing from the retirement fund**: Once retired, you will withdraw $70,000 annually from the accumulated fund.

We'll go step by step to solve this problem.

### 1. Amount Needed at Retirement

First, calculate the amount you need in your retirement fund when you retire at 75 years old to sustain $70,000 annually. Since you will draw $70,000 per year from the fund, let's assume that you will draw this amount for 20 years (from age 75 to 95).

The retirement fund will continue to earn interest while you are withdrawing from it. The annual income requirement ($70,000) is withdrawn while earning an 8% annual percentage rate (APR), compounded monthly. The future value of these withdrawals is determined by the formula for the present value of an annuity:

\[
PV = \frac{PMT \times \left(1 - (1 + r)^{-n}\right)}{r}
\]

Where:
- $ PV $ is the present value (the amount needed at the start of retirement),
- $ PMT = 70,000 \div 12 = 5833.33 $ is the monthly withdrawal,
- $ r = 0.08 / 12 = 0.00667 $ is the monthly interest rate,
- $ n = 20 \times 12 = 240 $ is the number of months over 20 years.

Let's calculate the amount needed at the start of retirement.

\[
PV = \frac{5833.33 \times \left(1 - (1 + 0.00667)^{-240}\right)}{0.00667}
\]

### 2. Monthly Deposits to Accumulate the Retirement Fund

Once we know how much is needed in the fund when you retire, we will calculate the monthly deposits required to reach that amount.

For this, we use the formula for the future value of an annuity (regular monthly deposits):

\[
FV = \frac{PMT \times \left((1 + r)^n - 1\right)}{r}
\]

Where:
- $ FV $ is the future value (the amount needed at retirement),
- $ PMT $ is the monthly deposit,
- $ r = 0.08 / 12 = 0.00667 $ is the monthly interest rate,
- $ n = (75 - 34) \times 12 = 492 $ is the number of months from now until retirement.

We solve for $ PMT $ once we know $ FV $ from the first calculation. Let's proceed with the calculations.The amount you need in your retirement fund at the age of 75 to support a $70,000 annual withdrawal for 20 years is approximately **$697,400**.

### 2. Monthly Deposits to Accumulate the Retirement Fund

Next, we calculate the monthly deposit required to reach this amount. Using the future value formula for monthly deposits, let's proceed with the calculation.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to proceed manually, or should I try again? Feel free to ask any further questions.

Calculate the monthly deposits needed to accumulate a retirement fund that supports a $70,000 annual income. This problem involves annuity formulas, compound interest, and future value calculations with an 8% APR compounded monthly. Suitable for advanced high school or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation