Answer :
To determine which retirement plan will yield the largest contribution to the café start-up costs, we will calculate the future value for each plan considering the payment schedule, annual interest rate, and compounding period. The contributions are calculated under the assumption that the friends retire in 30 years.
### Plan A:
- Payments: [tex]$450 per month - Annual Rate: 2.3% - Compound Period: Monthly Formula for the future value of an annuity compounded monthly: \[ FV_A = P \times \left( \frac{(1 + i)^n - 1}{i} \right) \times (1 + i) \] where: - \( P = 450 \) dollars (monthly payment) - \( r = 2.3\%/100 = 0.023 \) (annual interest rate in decimal) - \( t = 30 \) years - \( m = 12 \) (monthly compounding) First calculate: \[ i = \frac{r}{m} = \frac{0.023}{12} \approx 0.0019167 \] \[ n = t \times m = 30 \times 12 = 360 \] Then calculate the future value: \[ FV_A = 450 \times \left( \frac{(1 + 0.0019167)^{360} - 1}{0.0019167} \right) \times (1 + 0.0019167) \] The future value \( FV_A \) is approximately: \[ 233,444.68 \text{ dollars} \] ### Plan B: - Payments: $[/tex]150 per week
- Annual Rate: 0.5%
- Compound Period: Weekly
Formula for the future value of an annuity compounded weekly:
[tex]\[ FV_B = P \times \left( \frac{(1 + i)^n - 1}{i} \right) \times (1 + i) \][/tex]
where:
- [tex]\( P = 150 \)[/tex] dollars (weekly payment)
- [tex]\( r = 0.5\%/100 = 0.005 \)[/tex] (annual interest rate in decimal)
- [tex]\( t = 30 \)[/tex] years
- [tex]\( m = 52 \)[/tex] (weekly compounding)
First calculate:
[tex]\[ i = \frac{r}{m} = \frac{0.005}{52} \approx 0.00009615 \][/tex]
[tex]\[ n = t \times m = 30 \times 52 = 1560 \][/tex]
Then calculate the future value:
[tex]\[ FV_B = 150 \times \left( \frac{(1 + 0.00009615)^{1560} - 1}{0.00009615} \right) \times (1 + 0.00009615) \][/tex]
The future value [tex]\( FV_B \)[/tex] is approximately:
[tex]\[ 252,472.62 \text{ dollars} \][/tex]
### Plan C:
- Payments: [tex]$250 every two weeks - Annual Rate: 1.1% - Compound Period: Bi-Weekly Formula for the future value of an annuity compounded bi-weekly: \[ FV_C = P \times \left( \frac{(1 + i)^n - 1}{i} \right) \times (1 + i) \] where: - \( P = 250 \) dollars (bi-weekly payment) - \( r = 1.1\%/100 = 0.011 \) (annual interest rate in decimal) - \( t = 30 \) years - \( m = 26 \) (bi-weekly compounding) First calculate: \[ i = \frac{r}{m} = \frac{0.011}{26} \approx 0.0004231 \] \[ n = t \times m = 30 \times 26 = 780 \] Then calculate the future value: \[ FV_C = 250 \times \left( \frac{(1 + 0.0004231)^{780} - 1}{0.0004231} \right) \times (1 + 0.0004231) \] The future value \( FV_C \) is approximately: \[ 231,067.00 \text{ dollars} \] ### Conclusion: Comparing the future values: - \( FV_A \approx 233,444.68 \text{ dollars} \) - \( FV_B \approx 252,472.62 \text{ dollars} \) - \( FV_C \approx 231,067.00 \text{ dollars} \) The retirement plan that will yield the largest contribution to the café start-up costs is Plan B with an approximate future value of \$[/tex]252,472.62.
### Plan A:
- Payments: [tex]$450 per month - Annual Rate: 2.3% - Compound Period: Monthly Formula for the future value of an annuity compounded monthly: \[ FV_A = P \times \left( \frac{(1 + i)^n - 1}{i} \right) \times (1 + i) \] where: - \( P = 450 \) dollars (monthly payment) - \( r = 2.3\%/100 = 0.023 \) (annual interest rate in decimal) - \( t = 30 \) years - \( m = 12 \) (monthly compounding) First calculate: \[ i = \frac{r}{m} = \frac{0.023}{12} \approx 0.0019167 \] \[ n = t \times m = 30 \times 12 = 360 \] Then calculate the future value: \[ FV_A = 450 \times \left( \frac{(1 + 0.0019167)^{360} - 1}{0.0019167} \right) \times (1 + 0.0019167) \] The future value \( FV_A \) is approximately: \[ 233,444.68 \text{ dollars} \] ### Plan B: - Payments: $[/tex]150 per week
- Annual Rate: 0.5%
- Compound Period: Weekly
Formula for the future value of an annuity compounded weekly:
[tex]\[ FV_B = P \times \left( \frac{(1 + i)^n - 1}{i} \right) \times (1 + i) \][/tex]
where:
- [tex]\( P = 150 \)[/tex] dollars (weekly payment)
- [tex]\( r = 0.5\%/100 = 0.005 \)[/tex] (annual interest rate in decimal)
- [tex]\( t = 30 \)[/tex] years
- [tex]\( m = 52 \)[/tex] (weekly compounding)
First calculate:
[tex]\[ i = \frac{r}{m} = \frac{0.005}{52} \approx 0.00009615 \][/tex]
[tex]\[ n = t \times m = 30 \times 52 = 1560 \][/tex]
Then calculate the future value:
[tex]\[ FV_B = 150 \times \left( \frac{(1 + 0.00009615)^{1560} - 1}{0.00009615} \right) \times (1 + 0.00009615) \][/tex]
The future value [tex]\( FV_B \)[/tex] is approximately:
[tex]\[ 252,472.62 \text{ dollars} \][/tex]
### Plan C:
- Payments: [tex]$250 every two weeks - Annual Rate: 1.1% - Compound Period: Bi-Weekly Formula for the future value of an annuity compounded bi-weekly: \[ FV_C = P \times \left( \frac{(1 + i)^n - 1}{i} \right) \times (1 + i) \] where: - \( P = 250 \) dollars (bi-weekly payment) - \( r = 1.1\%/100 = 0.011 \) (annual interest rate in decimal) - \( t = 30 \) years - \( m = 26 \) (bi-weekly compounding) First calculate: \[ i = \frac{r}{m} = \frac{0.011}{26} \approx 0.0004231 \] \[ n = t \times m = 30 \times 26 = 780 \] Then calculate the future value: \[ FV_C = 250 \times \left( \frac{(1 + 0.0004231)^{780} - 1}{0.0004231} \right) \times (1 + 0.0004231) \] The future value \( FV_C \) is approximately: \[ 231,067.00 \text{ dollars} \] ### Conclusion: Comparing the future values: - \( FV_A \approx 233,444.68 \text{ dollars} \) - \( FV_B \approx 252,472.62 \text{ dollars} \) - \( FV_C \approx 231,067.00 \text{ dollars} \) The retirement plan that will yield the largest contribution to the café start-up costs is Plan B with an approximate future value of \$[/tex]252,472.62.
