SOLUTION: Continued with part (aii). Suppose Carter has just had his 35th birthday today and decided to begin his retirement (exactly) 25 years from now, at his age of 60. To ensure having s

Question 1200972: Continued with part (aii). Suppose Carter has just had his 35th birthday today and decided to begin his retirement (exactly) 25 years from now, at his age of 60. To ensure having sufficient funds to meet his goal, Carter plans to start depositing a fixed amount at the end of every month to a retirement savings account (Account B) that pays an interest of 12%, compounded monthly. The first deposit will be made today (on his 35th birthday) and the last on his 58th birthday.
(i) Compute the size of the monthly deposit into Account B that will allow Carter to meet the financial goal of his retirement. (8 marks)
(ii) If Carter is going to make one single (lump-sum) deposit into Account B on his 40th birthday instead, how much will that be for him to achieve the goal? (2 marks)
Answer by asinus(45) (Show Source): You can put this solution on YOUR website!
**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.

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