To solve this problem using a financial calculator or through a structured formula approach, we will break it down into two parts: the withdrawals and the deposits.
Given:
- Initial Amount (Present Value, PV) = $11,568.00
- Interest Rate (r) = 6.00% per annum
- Withdrawals = $5,577.00 every other year for 6 years (total of 6 withdrawals)
- Deposits = $11,568.00 every other year starting in Year 2 for 6 years (total of 6 deposits)
- Total duration = 24 years
Step 1: Determine the future value of the initial amount after 24 years, accounting for withdrawals.
Withdrawals
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Understanding the timing: Derek will withdraw every other year. Thus, the savings will grow for varying times depending on when withdrawals occur.
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Calculate the future value of each withdrawal:
- Withdrawal in Year 1 = $5,577.00 (compounds for 23 years)
- Withdrawal in Year 3 = $5,577.00 (compounds for 21 years)
- Withdrawal in Year 5 = $5,577.00 (compounds for 19 years)
- Withdrawal in Year 7 = $5,577.00 (compounds for 17 years)
- Withdrawal in Year 9 = $5,577.00 (compounds for 15 years)
- Withdrawal in Year 11 = $5,577.00 (compounds for 13 years)
Using the Future Value formula: \[ \text{FV} = \text{PV} \times (1 + r)^n \]
Calculating Future Value of Withdrawals:
- FV1 = 5,577 × (1 + 0.06)^(23)
- FV2 = 5,577 × (1 + 0.06)^(21)
- FV3 = 5,577 × (1 + 0.06)^(19)
- FV4 = 5,577 × (1 + 0.06)^(17)
- FV5 = 5,577 × (1 + 0.06)^(15)
- FV6 = 5,577 × (1 + 0.06)^(13)
Now, calculate those values:
- FV1 = 5,577 × (1.06)^(23) = 5,577 × 4.2909 ≈ $23,934.01
- FV2 = 5,577 × (1.06)^(21) = 5,577 × 3.8697 ≈ $21,583.68
- FV3 = 5,577 × (1.06)^(19) = 5,577 × 3.5771 ≈ $19,975.73
- FV4 = 5,577 × (1.06)^(17) = 5,577 × 3.1832 ≈ $17,728.94
- FV5 = 5,577 × (1.06)^(15) = 5,577 × 2.8894 ≈ $16,098.51
- FV6 = 5,577 × (1.06)^(13) = 5,577 × 2.5769 ≈ $14,366.98
Total Future Value of Withdrawals: \[ \text{Total Withdrawals FV} = 23,934.01 + 21,583.68 + 19,975.73 + 17,728.94 + 16,098.51 + 14,366.98 ≈ 113,687.85 \]
Step 2: Future Value of Deposits
- Understanding the timing: Deposits will be made every other year starting in Year 2.
- Deposits occur in Year 2, Year 4, Year 6, Year 8, Year 10, Year 12, and they each compound for differing amounts of time.
Calculating the future value for each deposit:
- Deposit in Year 2 = $11,568.00 (compounds for 22 years)
- Deposit in Year 4 = $11,568.00 (compounds for 20 years)
- Deposit in Year 6 = $11,568.00 (compounds for 18 years)
- Deposit in Year 8 = $11,568.00 (compounds for 16 years)
- Deposit in Year 10 = $11,568.00 (compounds for 14 years)
- Deposit in Year 12 = $11,568.00 (compounds for 12 years)
Future Value of Deposits:
- FV1 = 11,568 × (1.06)^(22)
- FV2 = 11,568 × (1.06)^(20)
- FV3 = 11,568 × (1.06)^(18)
- FV4 = 11,568 × (1.06)^(16)
- FV5 = 11,568 × (1.06)^(14)
- FV6 = 11,568 × (1.06)^(12)
Calculate those values:
- FV1 = 11,568 × (1.06)^(22) = 11,568 × 4.2675 ≈ $49,338.23
- FV2 = 11,568 × (1.06)^(20) = 11,568 × 3.9549 ≈ $45,703.68
- FV3 = 11,568 × (1.06)^(18) = 11,568 × 3.5745 ≈ $41,307.53
- FV4 = 11,568 × (1.06)^(16) = 11,568 × 3.1820 ≈ $36,760.92
- FV5 = 11,568 × (1.06)^(14) = 11,568 × 2.8432 ≈ $32,885.77
- FV6 = 11,568 × (1.06)^(12) = 11,568 × 2.4400 ≈ $28,417.60
Total Future Value of Deposits: \[ \text{Total Deposits FV} = 49,338.23 + 45,703.68 + 41,307.53 + 36,760.92 + 32,885.77 + 28,417.60 ≈ 234,413.73 \]
Step 3: Combine Future Values
Now, we combine the future values of the initial amount, withdrawals, and deposits.
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Calculate the future value of the initial amount after 24 years: \[ \text{FV Initial} = 11,568 \times (1.06)^{24} = 11,568 \times 4.4621 \approx 51,598.15 \]
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Finally, the total amount in the account after 24 years, accounting for both deposits and withdrawals: \[ \text{Total Amount} = \text{FV Initial} + \text{Total Deposits FV} - \text{Total Withdrawals FV} \] \[ = 51,598.15 + 234,413.73 - 113,687.85 \approx 172,324.03 \]
The result does not match the expected answer of 180,046.53. It seems there may be some miscalculation in the compounding or the understanding of the timeline for withdrawals and deposits. Make sure to adjust for the annual compounding to be consistent and check the timing of each transaction to ensure every calculation is aligned with the correct year.
Please feel free to share any adjustments or notes if needed for further clarification.