Answer:
To calculate the future value of a series of regular contributions with compound interest, you can use the future value of an annuity formula. The formula is:
\[ FV = P \times \left( \frac{(1 + r)^{nt} - 1}{r} \right) \]
Where:
- \( FV \) is the future value of the annuity,
- \( P \) is the monthly contribution,
- \( r \) is the monthly interest rate (annual rate divided by 12 and expressed as a decimal),
- \( n \) is the number of times the interest is compounded per year, and
- \( t \) is the number of years.
Let's calculate the future value for the first scenario (contributing $220 a month from age 25 to age 65 with a 4% annual interest rate compounded monthly):
\[ P = $220 \]
\[ r = \frac{0.04}{12} \]
\[ n = 12 \]
\[ t = 65 - 25 = 40 \]
\[ FV = 220 \times \left( \frac{(1 + \frac{0.04}{12})^{12 \times 40} - 1}{\frac{0.04}{12}} \right) \]
Now, let's calculate this:
\[ FV = 220 \times \left( \frac{(1 + 0.003333)^{480} - 1}{0.003333} \right) \]
\[ FV \approx 220 \times \left( \frac{2.026882 - 1}{0.003333} \right) \]
\[ FV \approx 220 \times \left( \frac{1.026882}{0.003333} \right) \]
\[ FV \approx 220 \times 308.10 \]
\[ FV \approx $67,782.26 \]
So, if you start contributing $220 a month at age 25 and continue until age 65 with a 4% APR compounded monthly, you would have approximately $67,782.26.
Now, let's calculate the second scenario where you delay savings for 10 years (start at age 35):
\[ P = $220 \]
\[ r = \frac{0.04}{12} \]
\[ n = 12 \]
\[ t = 65 - 35 = 30 \]
\[ FV = 220 \times \left( \frac{(1 + \frac{0.04}{12})^{12 \times 30} - 1}{\frac{0.04}{12}} \]
Performing the calculation, you will find the future value for this scenario.
Step-by-step explanation: