To determine which retirement plan will yield the largest contribution to the cafe start-up costs, let's evaluate the future value of each plan based on the given payments, interest rates, and compounding periods over 30 years. The future value (FV) of an annuity can be calculated using the formula:
[tex]\[ FV = P \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right) \][/tex]
where:
- [tex]\( P \)[/tex] is the payment amount per period
- [tex]\( r \)[/tex] is the annual interest rate
- [tex]\( n \)[/tex] is the number of compounding periods per year
- [tex]\( t \)[/tex] is the number of years
Given values:
- Plan A:
- Payments: \[tex]$450 per month
- Annual Rate: 2.3\%
- Compounding Period: Monthly
- Plan B:
- Payments: \$[/tex]150 per week
- Annual Rate: 0.5\%
- Compounding Period: Weekly
- Plan C:
- Payments: \[tex]$250 every two weeks
- Annual Rate: 1.196\%
- Compounding Period: Bi-Weekly
Let's calculate the future values:
Future Value for Plan A:
\( P = 450 \)
\( r = 0.023 \)
\( n = 12 \)
\( t = 30 \)
Using the future value formula:
\[ FV_A = 450 \left( \frac{(1 + 0.023/12)^{12 \times 30} - 1}{0.023/12} \right) \]
After evaluating:
\[ FV_A ≈ 232,998.10 \]
Future Value for Plan B:
\( P = 150 \)
\( r = 0.005 \)
\( n = 52 \)
\( t = 30 \)
Using the future value formula:
\[ FV_B = 150 \left( \frac{(1 + 0.005/52)^{52 \times 30} - 1}{0.005/52} \right) \]
After evaluating:
\[ FV_B ≈ 252,448.35 \]
Future Value for Plan C:
\( P = 250 \)
\( r = 0.01196 \)
\( n = 26 \)
\( t = 30 \)
Using the future value formula:
\[ FV_C = 250 \left( \frac{(1 + 0.01196/26)^{26 \times 30} - 1}{0.01196/26} \right) \]
After evaluating:
\[ FV_C ≈ 234,506.71 \]
Comparing the future values:
- Plan A: \$[/tex]232,998.10
- Plan B: \[tex]$252,448.35
- Plan C: \$[/tex]234,506.71
The plan which will yield the largest contribution to the cafe start-up costs is Plan B with a future value of approximately \$252,448.35. Therefore, select Plan B.
