Questions: Suppose you want to have 400,000 for retirement in 35 years. Your account earns 6% interest. How much would you need to deposit in the account each month?

Solution Steps

To determine how much you need to deposit each month to have $400,000 in 35 years with an annual interest rate of 6%, we can use the future value of an annuity formula. The formula for the future value of an annuity is:

\[ FV = P \times \frac{(1 + r/n)^{nt} - 1}{r/n} \]

Where:

• \( FV \) is the future value ($400,000)
• \( P \) is the monthly deposit
• \( r \) is the annual interest rate (0.06)
• \( n \) is the number of times the interest is compounded per year (12 for monthly)
• \( t \) is the number of years (35)

We need to solve for \( P \).

We are given the following values:

• Future Value (\( FV \)): \( 400,000 \)
• Annual Interest Rate (\( r \)): \( 0.06 \)
• Compounding Frequency (\( n \)): \( 12 \) (monthly)
• Time in Years (\( t \)): \( 35 \)

The formula for the future value of an annuity is given by:

\[ FV = P \times \frac{(1 + r/n)^{nt} - 1}{r/n} \]

We need to solve for the monthly deposit (\( P \)):

\[ P = \frac{FV}{\frac{(1 + r/n)^{nt} - 1}{r/n}} \]

Substituting the known values into the formula:

\[ P = \frac{400,000}{\frac{(1 + 0.06/12)^{12 \times 35} - 1}{0.06/12}} \]

After performing the calculations, we find:

\[ P \approx 280.7588 \]

Rounding to four significant digits, we have:

\[ P \approx 280.76 \]

Final Answer