An S&L pays 5% per annum compounded quarterly. Bert and Bertha are now 50 years old. They will deposit $200 per quarter at the end of each

 quarter until they are 65 years old.

a.       How much is in their retirement account at the end of this period i.e. at 65?

b.       3 months after their last deposit they start withdrawing equal amounts each quarter until they are 80 i.e. for the next 15 years Find the size of the withdrawals.

c.       How much cash did they deposit? How much cash did they withdraw? What was the total interest?

### Part A: Amount in the Retirement Account at Age 65

We are given:
- Quarterly deposit = $200
- Annual interest rate = 5%, compounded quarterly → Quarterly interest rate $ i = \frac{5\%}{4} = 0.0125 $
- Number of quarters from age 50 to 65 = $ 15 \text{ years} \times 4 \text{ quarters/year} = 60 \text{ quarters} $

The formula for the future value of an annuity (regular deposits) compounded periodically is:

\[
FV = P \times \frac{(1 + i)^n - 1}{i}
\]

Where:
- $ P $ = Quarterly deposit = $200
- $ i $ = Quarterly interest rate = 0.0125
- $ n $ = Number of periods = 60 quarters

Substituting the values:

\[
FV = 200 \times \frac{(1 + 0.0125)^{60} - 1}{0.0125}
\]

Let's calculate that first.At the end of 65 years old, Bert and Bertha will have approximately **$17,714.90** in their retirement account.

### Part B: Size of Withdrawals from Age 65 to 80

We are given:
- The account balance at age 65 = $17,714.90
- Quarterly withdrawals for 15 years (from 65 to 80) → Number of quarters $ = 15 \times 4 = 60 $
- Quarterly interest rate $ i = 0.0125 $

The formula for calculating the periodic withdrawals from an account (annuity) is:

\[
W = \frac{FV \times i}{1 - (1 + i)^{-n}}
\]

Where:
- $ W $ = Quarterly withdrawal amount
- $ FV $ = Future value of the account = $17,714.90
- $ i $ = Quarterly interest rate = 0.0125
- $ n $ = Number of periods = 60 quarters

Now, let's calculate the withdrawal size.Bert and Bertha can withdraw approximately **$421.44** per quarter from their retirement account for 15 years, starting 3 months after their last deposit at age 65.

### Part C: Total Deposits, Total Withdrawals, and Total Interest

1. **Total Deposits:**
   - Quarterly deposit = $200
   - Number of quarters = 60
   \[
   \text{Total Deposits} = 200 \times 60 = 12,000
   \]

2. **Total Withdrawals:**
   - Quarterly withdrawal = $421.44
   - Number of quarters = 60
   \[
   \text{Total Withdrawals} = 421.44 \times 60
   \]

3. **Total Interest:**
   \[
   \text{Total Interest} = \text{Total Withdrawals} - \text{Total Deposits}
   \]

Let's calculate the total withdrawals and the total interest.### Summary of Results for Part C:

1. **Total Deposits**: $12,000
2. **Total Withdrawals**: $25,286.18
3. **Total Interest Earned**: $13,286.18

This means that Bert and Bertha deposited $12,000, withdrew $25,286.18, and earned a total of $13,286.18 in interest.

---

Do you need further details or clarification? 

Here are 5 related questions:
1. How would the results change if the interest was compounded monthly instead of quarterly?
2. What happens if they decide to make monthly instead of quarterly deposits?
3. How does a change in the interest rate (say to 6%) affect their withdrawals?
4. Can the withdrawal amount be adjusted if they plan for a longer retirement period (e.g., until age 85)?
5. How does delaying their retirement by 5 years affect their future balance?

**Tip**: Regularly review and adjust your retirement contributions to align with your future goals and inflation.

An S&L pays 5% per annum compounded quarterly. Bert and Bertha are now 50 years old. They will deposit $200 per quarter at the end of each quarter until they are 65 years old. a. How much is in their retirement account at the end of this period i.e. at 65? b. 3 months after their last deposit they start withdrawing equal amounts each quarter until they are 80 i.e. for the next 15 years. Find the size of the withdrawals. c. How much cash did they deposit? How much cash did they withdraw? What was the total interest?

Discover how Bert and Bertha save for retirement with quarterly deposits of $200 at 5% annual interest compounded quarterly. Learn to calculate the future value at age 65, determine equal quarterly withdrawals for 15 years from age 65 to 80, and evaluate total deposits, withdrawals, and interest earned. Suitable for advanced high school and college students.

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