To determine how much you need to deposit each month to accumulate [tex]$300,000 for retirement in 35 years, given that your account earns 7% annual interest, we use the formula for the future value of an annuity.
Here’s a detailed, step-by-step solution for the question:
1. Establish the initial variables:
- Desired retirement amount (\(FV\)): $[/tex]300,000
- Number of years until retirement ([tex]\(n\)[/tex]): 35 years
- Annual interest rate ([tex]\(r\)[/tex]): 7%
2. Convert the parameters into a monthly context:
- Number of months until retirement:
[tex]\[
\text{months} = \text{years} \times 12 = 35 \times 12 = 420 \text{ months}
\][/tex]
- Monthly interest rate:
[tex]\[
\text{monthly interest rate} = \frac{\text{annual interest rate}}{12} = \frac{0.07}{12} \approx 0.005833
\][/tex]
3. Use the formula for the future value of an annuity to determine the future value factor:
[tex]\[
FV = P \times \left[ \frac{(1 + r)^{nt} - 1}{r} \right]
\][/tex]
Here, [tex]\(FV\)[/tex] is the future value, [tex]\(P\)[/tex] is the monthly deposit, [tex]\(r\)[/tex] is the monthly interest rate, and [tex]\(nt\)[/tex] is the number of months.
In our case, we need to rearrange this formula to solve for [tex]\(P\)[/tex] (the monthly deposit):
[tex]\[
P = \frac{FV}{ \left[ \frac{(1 + r)^{nt} - 1}{r} \right]}
\][/tex]
Calculate the future value factor:
[tex]\[
\text{Future Value Factor} = (1 + r)^{nt} - 1 \approx (1 + 0.005833)^{420} - 1 \approx 10.506
\][/tex]
4. Substitute all known values into the formula:
[tex]\[
P = \frac{300,000}{\left[ \frac{10.506}{0.005833} \right]}
\][/tex]
5. Perform the final division:
[tex]\[
P = \frac{300,000}{1800.357} \approx 166.57
\][/tex]
So, you would need to deposit approximately [tex]$166.57 in the account each month to accumulate $[/tex]300,000 for retirement in 35 years, considering the account earns 7% interest annually.
