Question 1200986
**a) Calculating the Required Initial Deposit in Account A**

**i) Account A starts to pay interest one year after retirement (Ordinary Annuity)**

* **Formula:** Present Value of Ordinary Annuity = Annual Withdrawal * [(1 - (1 + Interest Rate)^(-Number of Years)) / Interest Rate]

* **Calculation:** 
    * Present Value = $120,000 * [(1 - (1 + 0.05)^(-30)) / 0.05]
    * Present Value = $120,000 * [0.768623 / 0.05]
    * Present Value = $1,844,694.12

* **Required Deposit:** $1,844,694.12

**ii) Account A starts to pay interest on the day of retirement (Annuity Due)**

* **Formula:** Present Value of Annuity Due = Annual Withdrawal * [(1 - (1 + Interest Rate)^(-Number of Years)) / Interest Rate] * (1 + Interest Rate)

* **Calculation:** 
    * Present Value = $120,000 * [(1 - (1 + 0.05)^(-30)) / 0.05] * (1 + 0.05)
    * Present Value = $120,000 * [0.768623 / 0.05] * 1.05
    * Present Value = $1,926,943.33

* **Required Deposit:** $1,926,943.33

**b) Calculating Monthly Deposits into Account B**

* **Future Value of Annuity Due (from part a(ii)):** $1,926,943.33 (This is the target amount to accumulate in Account B)

* **Monthly Interest Rate:** 12% / 12 = 1% per month

* **Number of Periods:** 25 years * 12 months/year = 300 months

* **Formula for Future Value of Ordinary Annuity Due:**
    * Future Value = Monthly Deposit * [(1 + Monthly Interest Rate)^(Number of Periods) - 1] * (1 + Monthly Interest Rate) / Monthly Interest Rate

* **Rearrange to solve for Monthly Deposit:**
    * Monthly Deposit = Future Value / [(1 + Monthly Interest Rate)^(Number of Periods) - 1] * (1 + Monthly Interest Rate)

* **Substitute values:**
    * Monthly Deposit = $1,926,943.33 / [(1 + 0.01)^(300) - 1] * (1 + 0.01)
    * Monthly Deposit = $1,926,943.33 / [20.0855 - 1] * 1.01
    * Monthly Deposit = $1,926,943.33 / 19.0855 * 1.01
    * Monthly Deposit = $101.45

* **Required Monthly Deposit:** $101.45

**b) (ii) Single Lump-Sum Deposit into Account B**

* **Formula for Future Value of a Single Sum:**
    * Future Value = Present Value * (1 + Interest Rate)^Number of Periods

* **Rearrange to solve for Present Value:**
    * Present Value = Future Value / (1 + Interest Rate)^Number of Periods

* **Number of Periods:** (60 years - 40 years) * 12 months/year = 240 months

* **Substitute values:**
    * Present Value = $1,926,943.33 / (1 + 0.01)^240
    * Present Value = $1,926,943.33 / 10.8926
    * Present Value = $176,547.85

* **Required Single Lump-Sum Deposit:** $176,547.85

**Note:** These calculations assume consistent interest rates and no changes in withdrawal or deposit amounts.