To solve this problem using a financial calculator, we need to break down the cash flows and their timing over the 27-year period, while considering the compounding interest rate.
Here's how you can approach solving the problem:
Step 1: Define the Cash Flows
- Initial investment: $13,527.00 (this is the present value).
- Withdrawals: $5,012.00 every other year starting from the end of year 1 for a total of 5 withdrawals.
- Withdrawal years: 1, 3, 5, 7, 9.
- Deposits: $13,527.00 every other year starting from the end of year 2 for a total of 5 deposits.
- Deposit years: 2, 4, 6, 8, 10.
Step 2: Calculate the Future Value of Initial Investment
-
Future Value of Current Investment after 27 years:
- Use the formula:
\[
FV = PV \times (1 + r)^n
\]
Where:
- \(PV = 13,527\)
- \(r = 0.06\) (6%)
- \(n = 27\)
\[ FV = 13,527 \times (1 + 0.06)^{27} \]
- Use the formula:
\[
FV = PV \times (1 + r)^n
\]
Where:
Step 3: Calculate the Future Value of Withdrawals
For the withdrawals, you'll need to calculate the present value of each of them at the time of each withdrawal, and then convert that into its future value.
-
Withdrawal years:
- Future value at the end of 27 years for each withdrawal can be calculated by knowing that money is withdrawn at the end of periods 1, 3, 5, 7, and 9, and the remaining time until year 27:
- Withdrawal 1: 5 years (from year 1 to 27)
- Withdrawal 2: 3 years (from year 3 to 27)
- Withdrawal 3: 1 year (from year 5 to 27)
- Withdrawal 4: 0 years (from year 7 to 27)
- Withdrawal 5: -2 years (from year 9 to 27)
- Future value at the end of 27 years for each withdrawal can be calculated by knowing that money is withdrawn at the end of periods 1, 3, 5, 7, and 9, and the remaining time until year 27:
-
Calculate the future value for each withdrawal:
- Withdrawal 1: \[ FV_1 = 5,012 \times (1 + 0.06)^{26} \]
- Withdrawal 2: \[ FV_2 = 5,012 \times (1 + 0.06)^{24} \]
- Withdrawal 3: \[ FV_3 = 5,012 \times (1 + 0.06)^{22} \]
- Withdrawal 4: \[ FV_4 = 5,012 \times (1 + 0.06)^{20} \]
- Withdrawal 5: \[ FV_5 = 5,012 \times (1 + 0.06)^{18} \]
-
Total FV of Withdrawals: \[ FV_\text{Withdrawals} = FV_1 + FV_2 + FV_3 + FV_4 + FV_5 \]
Step 4: Calculate the Future Value of Deposits
Next, calculate the future value of each deposit made every other year starting at the end of year 2.
-
Deposit years:
- Future value compounding until year 27:
- Deposit made at end of year 2: 25 years remaining
- Deposit made at end of year 4: 23 years remaining
- Deposit made at end of year 6: 21 years remaining
- Deposit made at end of year 8: 19 years remaining
- Deposit made at end of year 10: 17 years remaining
- Future value compounding until year 27:
-
Calculate the future value for each deposit:
- Deposit 1: \[ FV_6 = 13,527 \times (1 + 0.06)^{25} \]
- Deposit 2: \[ FV_7 = 13,527 \times (1 + 0.06)^{23} \]
- Deposit 3: \[ FV_8 = 13,527 \times (1 + 0.06)^{21} \]
- Deposit 4: \[ FV_9 = 13,527 \times (1 + 0.06)^{19} \]
- Deposit 5: \[ FV_{10} = 13,527 \times (1 + 0.06)^{17} \]
-
Total FV of Deposits: \[ FV_\text{Deposits} = FV_6 + FV_7 + FV_8 + FV_9 + FV_{10} \]
Step 5: Calculate Total Future Value
Now, subtract the total future value of withdrawals from the total future value of deposits and add the future value of the initial amount: \[ \text{Total FV} = FV(\text{Initial}) - FV_\text{Withdrawals} + FV_\text{Deposits} \]
Calculation Example:
Once you put in your numbers and calculate using a financial calculator, you should arrive at the Total FV that matches your reference number of $130,000.
Make sure to keep an eye on the calculation order and compounding intervals on your calculator.