Answer:
To determine how much more Cole will need to invest each month than Dylan to have the same amount of money by the time they are both 55, we need to calculate the future values of their investments.
**For Dylan:**
- Monthly investment: $100
- Annual interest rate: 5.4%
- Compounding frequency: Monthly
- Investment duration: 35 years (from age 20 to age 55)
The future value \( FV \) of an annuity (regular monthly investments) is given by the formula:
\[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \]
where:
- \( P \) is the monthly investment
- \( r \) is the monthly interest rate
- \( n \) is the total number of investments (months)
For Dylan:
- \( P = 100 \)
- \( r = \frac{5.4\%}{12} = 0.0045 \)
- \( n = 35 \times 12 = 420 \)
So, the future value \( FV_D \) is:
\[ FV_D = 100 \times \left( \frac{(1 + 0.0045)^{420} - 1}{0.0045} \right) \]
**For Cole:**
- Annual interest rate: 7.2%
- Compounding frequency: Monthly
- Investment duration: 18 years (from age 37 to age 55)
For Cole:
- \( r = \frac{7.2\%}{12} = 0.006 \)
- \( n = 18 \times 12 = 216 \)
We need to determine the monthly investment \( P_C \) that will give Cole the same future value as Dylan. So, we set the future value formulas equal and solve for \( P_C \):
\[ FV_C = P_C \times \left( \frac{(1 + 0.006)^{216} - 1}{0.006} \right) = FV_D \]
First, we calculate \( FV_D \):
\[ FV_D = 100 \times \left( \frac{(1 + 0.0045)^{420} - 1}{0.0045} \right) \]
Let's calculate this using Python to get an accurate value.
Dylan's investment will be worth approximately $124,252.52 by the time he is 55 years old.
Now, we need to determine the monthly investment \( P_C \) for Cole to reach the same future value in 18 years (216 months) with an interest rate of 7.2% per annum compounded monthly.
The future value formula for Cole is:
\[ FV_C = P_C \times \left( \frac{(1 + 0.006)^{216} - 1}{0.006} \right) \]
We set \( FV_C \) equal to \( FV_D \) and solve for \( P_C \):
\[ 124252.52 = P_C \times \left( \frac{(1 + 0.006)^{216} - 1}{0.006} \right) \]
Let's calculate \( P_C \) using Python.
Cole will need to invest approximately $282.34 per month to match the future value of Dylan's investment by the time they are both 55 years old.
Therefore, Cole will need to invest about $182.34 more per month than Dylan to achieve the same investment value.